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Algorithmic aspects of throughput-delay performance for fast data collection in wireless sensor networks

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Algorithmic aspects of throughput-delay performance for fast data collection in wireless sensor networks

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Content ALGORITHMIC ASPECTS OF THROUGHPUT-DELAY PERFORMANCE FOR FAST DATA COLLECTION IN WIRELESS SENSOR NETWORKS by Amitabha Ghosh A Dissertation Presented to the FACULTY OF THE USC GRADUATE SCHOOL UNIVERSITY OF SOUTHERN CALIFORNIA In Partial Fulfillment of the Requirements for the Degree DOCTOR OF PHILOSOPHY (ELECTRICAL ENGINEERING) August 2010 Copyright 2010 Amitabha Ghosh Dedication To my loving parents, whose sacrifice and blessings have always guided me through life. ii Acknowledgments I have been very fortunate to interact and work with some of the brightest minds during my graduate studies in the last 5 years. First and foremost, is my academic advisor Prof. Bhaskar Krishnamachari, without whose support, enthusiasm, and meaningful guidance this thesis would not have been possible. I have learned things from Bhaskar that go well beyond the regular, traditional Ph.D. advice, which have helped me to grow both in my professional and personal life. Besides my academic advisor, I am grateful to have interacted with my collaborators Prof. Anil Vullikanti from the Department of Computer Science and Bioinformatics Institute at Virginia Tech, Dr. Ozlem Durmaz Incel from the Networking Laboratory at the Bogazici University, Turkey, and my colleague/lab-mate Yi Wang. I would also like to thank my committee members Prof. Michael Neely, Prof. Cauligi Raghavendra, Prof. John Silvester, Prof. Murali Annavaram, and Prof. Gaurav Sukhatme whose useful feedback has helped to improve the quality of this thesis. Specifically, all the materials in Chapter 3 and parts of Chapter 5 were joint ef- forts with Bhaskar, Anil, and Ozlem. Some of the results presented in Chapter 3 were published in the 6th IEEE International Conference on Mobile Adhoc and Sensor Sys- tems (MASS), 2009 [50], and are under revision along with some additional materials iii contributed by Ozlem on empirical evaluations of raw-data convergecast in the IEEE Transactions on Mobile Computing (TMC), 2009 [69]. The approximation algorithms described in Chapter 3 on multi-channel scheduling, and those in Chapter 5 on bicrite- ria formulations are the result of many brainstorming sessions with Anil and Bhaskar. Some of these results got published as a poster in the 29th Annual IEEE Conference on Computer Communications (INFOCOM), 2010 [52], and also under submission in the IEEE/ACM Transactions on Networking, 2010 [51]. Chapter 6 on topology control is a joint work with Yi that we did as a class project guided by Bhaskar. This work was pub- lished in the 4th Annual IEEE Communications Society Conference on Sensor, Mesh and Ad Hoc Communications and Networks (SECON), 2007 [53]. Lastly, the materi- als in Chapter 4 on SINR multi-channel scheduling are again joint work with Anil and Bhaskar. Apart from the visible contributions cited above, I would like to take this opportunity to thank all the past and present members of the Autonomous Networks Research Group (ANRG) [6] of which I have been a member throughout my graduate studies. In partic- ular, I thank Marco, Shyam, Kiran, Dongjin, Joon, Sundeep, Avinash, Hua, Ying, and Mahesh for not only being my lab mates but also great friends who have helped me on numerous occasions and have shared many happy moments that I will always cherish. In the last five years, I spent a lot of time playing table tennis as part of the USC table tennis team Ping Pong Posse [142]. I made some great friends there. In particular, I would like to thank Adam, Imran, Kedar, Jordon, Jason, Jimmy, Subal, Misha, Jack, iv Jeremy, Akhilesh, Lei, Cathy, Sarah, and Alice who accompanied me at various local tournaments as well as in the collegiate nationals organized by NCTTA [114]. Table ten- nis has become an integral part of my life, and, through the support and encouragement of my friends has often given me the strength and a sense of accomplishment when I had felt down. I would also like to thank my house mates, Jeff, Sophie, Sherry, Annabelle, Min, Shuyan, and Tao Ran, with whom I had a wonderful time living under the same roof during the final year of my Ph.D. Through our traditional weekly family dinners, the barbecue parties in the backyard, the birthday trips to Yogurt Land, the hikes to the beautiful trails of Santa Monica and Malibu mountains, and the skiing adventures to the Big Bear Lake [127], we shared numerous moments that have made my life richer. I have also been blessed to have some wonderful friends in my life who have seen me through times of joy and sorrow. In particular, I feel a deep sense of gratitude to- ward my long time friends Prateek, Shraya, Ayush, Subbu, and Janhavi who have helped me during times of crisis as well as celebrated with me during times of happiness; my loving sister Pinky; my friends from undergrad, Anup, Amit, and Nihar; and my friend Neelanjana with whom I have shared an invisible loving bond. v Table of Contents Dedication ii Acknowledgments iii List Of Figures ix Abstract xiv Chapter 1: Introduction 1 1.1 Wireless Sensor Networks . . . . . . . . . . . . . . . . . . . . . . . . 3 1.2 Data Communication Patterns in Sensor Networks . . . . . . . . . . . . 8 1.3 Data Acquisition Models in Sensor Networks . . . . . . . . . . . . . . 11 1.4 Link Scheduling in Wireless Networks . . . . . . . . . . . . . . . . . . 12 1.4.1 Medium Access Control Mechanisms . . . . . . . . . . . . . . 14 1.4.2 Communication Using Multiple Frequency Channels . . . . . . 15 1.5 Routing Topologies in Sensor Networks . . . . . . . . . . . . . . . . . 20 1.6 Power Control Mechanisms in Sensor Networks . . . . . . . . . . . . . 22 1.7 Contributions and Organization . . . . . . . . . . . . . . . . . . . . . . 24 Chapter 2: Background 27 2.1 Link Scheduling for Fast Data Collection . . . . . . . . . . . . . . . . 27 2.1.1 Single-Channel Scheduling . . . . . . . . . . . . . . . . . . . . 29 2.1.1.1 Maximizing Raw-Data Convergecast Throughput . . 29 2.1.1.2 Maximizing Aggregated Convergecast Throughput . . 36 2.1.2 Multi-Channel Scheduling . . . . . . . . . . . . . . . . . . . . 40 2.1.2.1 Maximizing Raw-Data Convergecast Throughput . . 43 2.2 Spanning Trees for Fast Data Collection . . . . . . . . . . . . . . . . . 47 2.2.1 Bicriteria Network Design Problems . . . . . . . . . . . . . . . 49 2.3 Topology Control in Wireless Networks . . . . . . . . . . . . . . . . . 55 2.3.1 Why Topology Control? . . . . . . . . . . . . . . . . . . . . . 55 vi Chapter 3: Maximizing Convergecast Throughput in Tree-Based Sensor Net- works 60 3.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 3.2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 3.2.1 Model and Assumptions . . . . . . . . . . . . . . . . . . . . . 62 3.2.2 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . 66 3.3 Multi-Channel Scheduling on General Graphs . . . . . . . . . . . . . . 67 3.3.1 Complexity of Multi-Channel Scheduling and Frequency Assignment Problems . . . . . . . . . . . . . . . . . . . . . . 68 3.3.2 Scheduling Under Sufficient Frequencies . . . . . . . . . . . . 75 3.4 Multi-Channel Scheduling on Unit Disk Graphs . . . . . . . . . . . . . 80 3.4.1 Known Routing Topologies . . . . . . . . . . . . . . . . . . . 80 3.4.1.1 Frequency Assignment . . . . . . . . . . . . . . . . 81 3.4.1.2 Time Slot Assignment . . . . . . . . . . . . . . . . . 85 3.4.2 Unknown Routing Topologies . . . . . . . . . . . . . . . . . . 88 3.5 Multi-Channel Scheduling on General Disk Graphs . . . . . . . . . . . 91 3.5.1 An Integer Linear Programming Formulation . . . . . . . . . . 92 3.6 Evaluation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96 3.6.1 Frequency Bounds . . . . . . . . . . . . . . . . . . . . . . . . 97 3.6.2 Multiple Frequencies on Schedule Length . . . . . . . . . . . . 98 3.6.3 Scheduling Under SINR Model . . . . . . . . . . . . . . . . . 99 3.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 Chapter 4: Multi-Channel SINR Scheduling for Fast Convergecast 102 4.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 4.2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104 4.2.1 The SINR Model . . . . . . . . . . . . . . . . . . . . . . . . . 104 4.2.2 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . 106 4.3 Multi-Channel Scheduling Under SINR . . . . . . . . . . . . . . . . . 107 4.3.1 Overall Approach . . . . . . . . . . . . . . . . . . . . . . . . . 107 4.3.1.1 Link Diversity . . . . . . . . . . . . . . . . . . . . . 107 4.3.1.2 Reusing Time Slots Under SINR . . . . . . . . . . . 108 4.3.2 O(g(E T )) Approximation Algorithm . . . . . . . . . . . . . . 113 4.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116 Chapter 5: Optimal Spanning Trees for Maximizing Throughput and Minimiz- ing Packet Delays 118 5.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118 5.2 Bicriteria Problem Formulation . . . . . . . . . . . . . . . . . . . . . . 124 5.3 Routing Tree Construction . . . . . . . . . . . . . . . . . . . . . . . . 125 5.3.1 An Approximation Algorithm . . . . . . . . . . . . . . . . . . 126 5.3.2 Algorithm Analysis . . . . . . . . . . . . . . . . . . . . . . . . 130 5.4 Evaluation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134 vii 5.4.1 Schedule Length and Maximum Delay . . . . . . . . . . . . . . 135 5.4.2 Multiple Frequencies on Schedule Length . . . . . . . . . . . . 137 5.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140 Chapter 6: Efficient Distributed Topology Control in 3-Dimensional Wireless Networks 143 6.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143 6.2 Preliminaries and Approach . . . . . . . . . . . . . . . . . . . . . . . 147 6.3 Phase 1: Multi-Dimensional Scaling in 3-D . . . . . . . . . . . . . . . 149 6.4 Phase 2: Orthographic Projections . . . . . . . . . . . . . . . . . . . . 150 6.5 Phase 2: Spherical Delaunay Triangulation . . . . . . . . . . . . . . . . 156 6.6 Evaluation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163 6.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171 Chapter 7: Conclusions and Future Work 173 7.1 Multi-Channel TDMA Scheduling . . . . . . . . . . . . . . . . . . . . 174 7.2 Optimal Routing Topologies . . . . . . . . . . . . . . . . . . . . . . . 176 7.3 Transmission Power Control . . . . . . . . . . . . . . . . . . . . . . . 177 7.4 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178 7.4.1 Real Testbed Implementation . . . . . . . . . . . . . . . . . . 178 7.4.2 Other Joint Objectives . . . . . . . . . . . . . . . . . . . . . . 179 7.4.3 Traffic Patterns . . . . . . . . . . . . . . . . . . . . . . . . . . 179 7.4.4 Cross Layer Solutions . . . . . . . . . . . . . . . . . . . . . . 180 7.4.5 Realistic Interference Models . . . . . . . . . . . . . . . . . . 181 References 183 viii List Of Figures 1.1 (a) Components of a sensor node. (b) A Micaz node. . . . . . . . . . . 4 1.2 (a) Multicast - data flows from a single nodes to a set of nodesa,b, and c. (b) Convergecast - data flows from nodesa,b, andc to a single nodes. 10 1.3 Multi-channel hidden terminal problem . . . . . . . . . . . . . . . . . 16 2.1 Optimal time scheduling for a10-node line network with minimum sched- ule length 11. Upper part shows the schedule for packet distribution, bottom part shows the schedule for convergecast, which is obtained by symmetry, i.e., by reflecting the upper schedule with respect to the ficti- tious horizontal line. Note that nodes that are closer than or at 2 hops do not transmit concurrently to respect interference issue. . . . . . . . . . 31 2.2 (a) A linear network with initial state assignment (R, I, T) depending on the hop distance from the sinks. (b) State transition of the nodes. (c) A multi-line network as a composition of multiple linear networks. . . . . 34 2.3 Raw-data convergecast using algorithm LOCAL-TIMESLOTASSIGNMENT: (a) Schedule length 7 when secondary conflicts are eliminated. (a) Sched- ule length 10 when secondary conflicts are present. . . . . . . . . . . . 45 2.4 Benefits of topology control: (a) Network without topology control in which nodes are configured at maximum transmission range leading to high node degree. (b) Network without topology control in which nodes are configured at minimum transmission range leading to network parti- tion. (c) Network with topology control in effect in which nodes adjust their transmission range leading to low average node degree and yet a connected network. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 ix 3.1 (a) Concurrent transmissions on adjacent edgese 1 ande 2 cause primary conflict. (b) Concurrent transmissions on edges e 1 and e 2 cause sec- ondary conflict if their receivers are on the same frequency, sayf 1 , and if either of the receivers is within the range of the other transmitter. . . . 64 3.2 Aggregated convergecast: (a) Schedule length of 6 time slots using one frequency. Dotted lines represent secondary conflicts. (b) Schedule length of 3 time slots using two frequencies. Note that, all the secondary conflicts are eliminated. (c), (d) Nodes from which aggregated data is received by their corresponding parents in each time slot over 2 consec- utive frames for (a) and (b), respectively. . . . . . . . . . . . . . . . . . 65 3.3 Reduction for the Multi-Channel Scheduling Problem: (a) Gadget for eachv i inG ′ when the number of frequenciesq is2. (b) InstanceG ′ of the vertex color problem. (c) Instance G of the Multi-Channel Scheduling Problem as constructed fromG ′ forq = 2. . . . . . . . . . . . . . . . . 70 3.4 Reduction from the Vertex Color problem to the Frequency Assignment Problem. The left part of the figure shows as instance of the Vertex Color problem, which is colored using 3 colors. The right part of the figure shows the corresponding instance of the Frequency Assignment Problem, which also requires 3 frequencies to remove all the secondary conflicts. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 3.5 (a) Original graphG; receiver nodes are shaded. (b) Constraint graphG C and a frequency assignment to the receivers according to Largest Degree First. Here, 4 frequencies are sufficient to remove all the secondary con- flicts, i.e., frequencies on adjacent nodes are different. . . . . . . . . . . 77 3.6 An optimal time slot assignment according to Algorithm BFS-TIMESLOT- ASSIGNMENT yielding a schedule length 3 after all the secondary con- flicts are removed using4 frequencies. . . . . . . . . . . . . . . . . . . 78 3.7 (a) Frequency assignment according to Algorithm FREQUENCY-GREEDY. Load on frequencies: ℓ (f 1 ) = 5,ℓ (f 2 ) = 5. White colored nodes trans- mit on frequency f 1 ; gray colored nodes transmit on frequency f 2 . (b) Four pair-wise disjoint sets of time slotsγ 1 ,γ 2 ,γ 3 , andγ 4 schedule the whole network. Each set γ i maps to a unique color. Edges whose re- ceivers lie in non-adjacent cells can be scheduled simultaneously, e.g., edges(u,v) and(u ′ ,v ′ ). . . . . . . . . . . . . . . . . . . . . . . . . . 84 x 3.8 Number of frequencies required to remove all the secondary conflicts as a function of network density on shortest path trees. . . . . . . . . . . . 97 3.9 Effect of multiple frequencies: Schedule Lengths for SPT with network size forK = 1,3, and5 frequencies. . . . . . . . . . . . . . . . . . . . 99 3.10 Percentage of nodes whose schedules conflict in the SINR model for different network sizes and three different number of frequencies (K = 1,3,5) on an SPT. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 4.1 Coloring the grid cells inC k corresponding to the length classE k using four distinct colorsγ 1 ,γ 2 ,γ 3 , andγ 4 . Size of each grid cell isη k =δ·2 k . 109 4.2 For an edgee i = (s i ,r i )∈ E k , the distance between its receiverr i and the senders j of another edgee j = (s j ,r j )∈ E k whose corresponding receiverr j lies in one of the non-adjacent layer 1 cells of the same color, is at leastδ·2 k −2 k+1 . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 5.1 Shortest path tree (SPT): High node degrees and minimum hop distances to the sink. Dark lines represent tree edges and dotted lines represent interfering links. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119 5.2 (a) Minimum interference tree (MIT): Low node degrees but large num- ber of hops to the sink. Dark lines represent tree edges, dotted lines rep- resent interfering links. (b) Cost of an edge(u,v) is equal to the number of nodes lying within the union of the two disks centered at nodesu and v, each of radiusd(u,v); here cost is11. . . . . . . . . . . . . . . . . . 120 5.3 Schedule length on minimum interference trees with different network sizes forK = 1,3, and5 frequencies. . . . . . . . . . . . . . . . . . . 121 5.4 Backbone tree construction: Filled black circles represent local roots (chosen arbitrarily from each non-empty cell), and shaded cells are non- empty adjacent cells ofs. Iteration 1: Local rootsr 1 ,r 2 ,r 3 ,r 4 ,r 5 , andr 6 from non-empty adjacent cells ofs are connected. Nodesr 3 andr 5 are connected to sinks via helper nodew 1 , and noder 4 via helper nodew 2 . Noder 7 is out of range of any helper node, and is not connected in the first iteration. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128 5.5 Local tree construction on an induced complete graph within each cell for maximum node degree Δ ∗ = 4; filled black circle represents the local root. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129 xi 5.6 Traversing the edge (u k ,u k+1 ) along the shortest path P G in graph G. Local rootsr k andr k+1 are at most distance3R away from each other. . 131 5.7 (a) Schedule Length, and (b) Maximum Delay (tree radius) with increas- ing network size on three different types of trees (BDMRST, SPT, and MIT) for single frequency scheduling. . . . . . . . . . . . . . . . . . . 136 5.8 Maximum Node Degree with increasing network size on three different types of trees (BDMRST, SPT, and MIT) for single frequency scheduling. 137 5.9 Effect of multiple frequencies on schedule lengths for BDMRST with different network sizes forK = 1,3, and5 frequencies. . . . . . . . . . 138 5.10 Node Degree Distribution of BDMRST, SPT, and MIT for two different network sizes with (a)N = 150, and (b)N = 800 nodes. . . . . . . . . 139 5.11 A BDMRST constructed on the same deployment of 800 nodes of Fig- ure 5.1 and 5.2(a). The node degrees are more uniform compared to those on an SPT and MIT. . . . . . . . . . . . . . . . . . . . . . . . . 140 6.1 Three non collinear points p i , p j , and p k on the surface of a sphere uniquely determine a spherical cap. The radius of the base of the cap isr, andh is its height.~ n denotes normal to the cap. . . . . . . . . . . . 148 6.2 An empty sector of angleθ aroundu i ’s projected location on thexy plane. 151 6.3 Projected locations on xy and zx planes of node u i and its neighbors N i (P max ) whenu i transmits at maximum power. . . . . . . . . . . . . 153 6.4 Spherical Delaunay Triangulation illustrating empty circle property: Spher- ical capCap(a,b,c) is empty in its interior. . . . . . . . . . . . . . . . 157 6.5 The 3-D cone withCap(p,q,r) as its base andu i as its apex is empty. Black dots show the actual locations of the neighbors, blue dots show their projected locations on the surface of the spherical ball.△ pqr is a spherical Delaunay triangle. . . . . . . . . . . . . . . . . . . . . . . . 158 6.6 Spherical Delaunay triangulation using the Quickhull algorithm of a set of100 points randomly distributed on the surface of a sphere of radius50. 160 xii 6.7 (a) Network topology of the original maximum power graph. (b) Net- work topology after running the SDT-based algorithm. Here n = 200 andP max = 40. Node degrees have drastically reduced. . . . . . . . . . 164 6.8 (a) Node degrees of the maximum power graph and that of the final topol- ogy forn = 200,P max = 40. (b) Final assigned minimal transmission power levels of nodes, forn = 200,P max = 40. . . . . . . . . . . . . . 165 6.9 (a) Dependency of average node degree with network size. (b) Depen- dency of the average node degree with maximum power, for n = 200; node degree remains almost flat in the final topology based on the SDT algorithm. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166 6.10 Dependency of average transmission power with network size. . . . . . 168 6.11 CPU execution time of the SDT based algorithm and the one [12] based on the proceduregap−3D α () for different random topologies withP max = 40. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169 6.12 Probability of network connectivity as theθ constraint is satisfied on 1, 2, or all3 orthogonal planes. . . . . . . . . . . . . . . . . . . . . . . . 170 xiii Abstract Convergecast, namely the many-to-one flow of data from a set of sources to a common sink over a tree-based routing topology, is a fundamental communication primitive in wireless sensor networks. For real-time, mission-critical, and high data-rate applications, it is often critical to maximize the aggregated data collection rate (throughput) at the sink node, as well as minimize the time (delay) required for packets to get there. In this thesis, we look into the algorithmic aspects of jointly optimizing both throughput and delay for aggregated data collection in sensor networks. Our contributions are in designing efficient algorithms with provably good, worst-case performance bounds for arbitrarily deployed networks. To the best of our knowledge, we are the first ones to address these two mutually conflicting performance objectives – throughput and delay – under the same optimization framework and develop techniques to meet the stringent requirements for fast data collection. Our approach in addressing the throughput-delay performance trade-off comprises three techniques: (i) multi-channel scheduling, (ii) routing over optimal topologies, and (iii) transmission power control. We exploit the benefits of multiple frequency channels to design efficient TDMA scheduling algorithms, both under the graph-based and the xiv SINR-based interference models. In particular, by decoupling the joint frequency and time slot assignment problem into two separate subproblems of frequency assignment and time slot assignment, we show that our scheduling algorithms have constant factor and logarithmic approximation ratios on the optimal throughput for random geometric graphs as well as for SINR-based models. In order to further enhance the data collection rate and bound the maximum delay, we study the degree-radius trade-off in spanning trees and propose algorithms under the bi- criteria optimization framework. In particular, we construct bounded-degree-minimum- radius spanning trees that have constant factor approximations on the maximum node degree as well as the tree radius. We also show that our multi-channel scheduling al- gorithms perform much better on such trees in maximizing the aggregated throughput and minimizing the maximum delay, thus achieving the best of both worlds. Lastly, we design efficient, distributed power control schemes for sensor networks deployed in 3- D, where very high density of nodes causes high interference resulting in low network throughput. Our proposed algorithms have low computational overhead compared to the state-of-the-art, and by using local geometric information and tools from computational geometry produce sparse yet connected topologies in 3-D, thus reducing interference and allowing for high throughput. xv Chapter 1 Introduction We live in an era of immense technological revolution where computers and digital com- munications continue to transform the ways we live. From Richard Feynman’s famous 1959 lecture titled There’s Plenty of Room at the Bottom [39], to the fabrication of mod- ern Micro-Electro-Mechanical Systems (MEMS) have led the widespread development and use of tiny computing machines that have permeated our lives. Moving away from Freeman Dyson’s view on technology, that the ones “which have had the most profound effects on human life are usually simple” [34], we are entering into a time of miniatur- ization and disappearing of technologies [49], [139] leading to the world of embedded computing and nanodevices. While not necessarily being simpler, MEMS-enabled em- bedded devices [36] continue to become smaller, cheaper, and yet more powerful and capable of high storage, making them useful to a wide range of devices. Over the last decade, Wireless Sensor Networks (WSN) [3] have appeared as one of the most prominent enabling technologies of MEMS, which combines automated sens- ing, embedded computing, and wireless capabilities into tiny devices, bringing promises 1 of understanding and instrumenting nature at scales that were unimaginable before. Just like the invention of microscope has let us see things that were previously invisible to the naked eye, wireless sensor networks have enabled us not only to detect and measure a physical phenomenon with accuracy even at the microscopic level, but also to commu- nicate the measured information across distances using the wireless medium. The earliest research efforts on WSN date back to the late 1990’s, when the United States Defense Advanced Research Project Agency (DARPA) focused on developing low-power sensing devices to enable large-scale, distributed, networked sensor systems through the SensIT project [32]. Since then, numerous research and commercial ef- forts, such as the WINS [121] and Sensorsim [118] from UCLA, Smart Dust [77] and PicoRadio [19] from UC Berkeley have advanced the field from traditional simple low data-rate environmental monitoring applications, to more complex ones ranging from smart-homes and factory automation, to high data-rate mission-critical applications, such as security-surveillance, structural health monitoring, and health-care. This thesis is mo- tivated by the need of such complex applications which require fast and timely collection of large amounts of data. Our aim is to provide mechanisms that address the challenges of fast data collection and develop algorithmic solutions that fulfill their requirements. In this chapter, we first provide an overview of sensor networks in the context of this thesis by describing the general properties of sensor devices and enlisting potential applications that pertain to fast data collection. Next, we discuss the key challenges for designing efficient solutions, and present different data collection models that lead us to 2 the research questions addressed in this thesis. Finally, we describe our contributions and the organization for the rest of the chapters. 1.1 Wireless Sensor Networks A sensor network typically comprises a large number of low-power, low-cost, tiny em- bedded devices with sensing capabilities, which are networked together to collect, pro- cess, and deliver information about a physical phenomenon of interest. The position of the nodes could be engineered or predetermined, such as in structural health monitor- ing, where nodes are placed at optimal locations to maximize the fidelity of measured vibrations for accurate and reliable diagnosis about the health of the structure. On other occasions, nodes could be placed randomly, allowing deployment of networks over in- accessible terrains or in disaster recovery operations. An individual sensor node consists of the following four basic components, as illustrated in Figure 1.1(a). • Sensing Unit: It includes a variety of sensors, such as temperature, pressure, hu- midity, acoustic, light, vibration, etc; and optionally a few actuators, such as speak- ers, buzzers, and LEDs. • Processing Unit: Typically, the processing unit is a 8-bit or 16-bit microprocessor that is responsible for processing raw measurements, such as carrying out simple 3 Power Unit Battery Processing Unit Sensing Unit Transceiver Unit GPS (a) (b) Figure 1.1: (a) Components of a sensor node. (b) A Micaz node. computations, or fusing data from neighboring nodes into more meaningful quan- tities desired by the underlying application. Some examples of microprocessors used in current hardware are the following: – 8 MHz Texas Instruments MSP430 microcontroller [73] with 10 KBytes RAM and 48 KBytes Flash memory used on Tmote Sky [131] platforms. – MPR400 and MPR2400, which are based on Atmel ATmega128L and run MoteWorks from its internal flash memory. These are used on Mica2 and Mi- caz [30] platforms, and can be configured to run the sensor application/processing and the network/radio communication stack simultaneously. The size of the EEPROM and measurement flash memory on both these microcontrollers are 4 KBytes and 512 KBytes, respectively. – Atmel AT91RM9200 [7] system on a chip (SOC) integrated circuit compris- ing ARM920T ARM Thumb processor used on the more advanced SunSPOT platforms [106]. These usually have a larger memory size consisting of 4 4 MBytes NOR Flash and 512 KBytes pseudo-static random access memory (pSRAM). • Transceiver Unit: It comprises a low data-rate, low-power, short-range radio that usually operates on unlicensed bands, such as the 868–915 MHz band, or the 2.4 GHz industrial, scientific, and medical (ISM) band. The transmit power could vary from -24 dBm to 10 dBm with receiver sensitivity in the range of -95 dBm to -100 dBm. Typical power consumption varies in the range of 10 mA to 40 mA, and data rates in a range of 50 Kbps to 250 Kbps. Some of the state-of-the- art transceivers can also support multiple frequency channels. Common examples include the IEEE 802.15.4 compliant TI CC1000 [71] and CC2400 [72] (formerly Chipcon) radios used on Tmote Sky and SunSPOTs, and the Nordic NrF905 [130] radios. • Power Unit: The sensor nodes typically run on rechargeable AA batteries supply- ing a voltage of 2.7 V – 3.3 V , such as on Mica2, Micaz, and Tmote Sky platforms, or could be charged using USB power like in SunSPOTs. Since these batteries are not replenishable once the network is deployed, energy efficiency is a big concern with protocols that run on these devices. In addition to the above four basic components, a sensor node can optionally have a satellite-based Global Positioning System (GPS) attached to it. Since in many cases the measurements need to be location stamped, the easiest way to obtain such positioning information for applications deployed in the outdoors is via a GPS device. However, for 5 networks deployed in the indoors, nodes must obtain their locations indirectly through network localization algorithms. Being equipped with a variety of sensors and wireless communication capabilities, the possible applications of sensor networks are numerous. We list a few areas in the following, acknowledging that it can certainly be extended with the growing interest in both academia and industry. • Environmental: Some of the environmental applications of sensor networks in- clude tracking the movements of birds (Great Duck Island [100]), small animals, and insects; monitoring temperature and soil conditions that affect crops and live- stock [123]; precision agriculture [10]; marine networks [43] for monitoring coral reefs; pollution study; monitoring Alpine permafrost [63]; etc. • Military and Security Surveillance: As with many other information technolo- gies, sensor networks originated primarily in military-related research, where unat- tended networks are envisioned as the key ingredient toward network-centric war- fare systems. They can be an integral part of military command, control, surveil- lance, reconnaissance, target enemy classification, and tracking [104]. • Disaster Early Warning Systems: Sensor networks can be useful in providing early warnings about imminent flood [33], wildfire [159], volcano eruption [151], as well as hazardous substance detection [120], such as chemical contamination, etc. 6 • Structural and Seismic Monitoring: A growing class of application of sensor networks pertains to monitoring the condition of civil infrastructures [156], [26], such as buildings, bridges, roads, aircrafts, etc. Traditional structural safety as- sessment methods are often dependent on visual inspection or using technologies such as X-rays and ultrasound, which are manual, expensive, and time consuming. Unattended networked sensing can examine the structural integrity in an automatic and efficient manner, thus proactively detecting and preventing future damages. A particularly compelling futuristic application is the development of controllable structures, which would contain actuators to react to real-time sensor information to perform “echo-cancellation” on seismic waves in order to protect it from exter- nal disturbances, such as earthquakes. • Industrial and Building Automation: In industrial manufacturing facilities, such as multi-stage chemical processing plants, sensors and actuators can be used for process monitoring and control [113] [122]. The key advantage of using wireless networks in these environments is the reduced cost and improved flexibility asso- ciated with installing, maintaining, and upgrading wired systems. In the context building automation, sensor networks can be used for controlling heating, ventila- tion, and air conditioning (HV AC), lighting, refrigeration, etc. • Health Care: Continuous and remote monitoring of physiological patient data and tracking doctors and patients inside hospitals [60] are some of the typical health 7 care applications of sensor networks. In addition, recent development of intelli- gent physiological sensors can be integrated into a wearable wireless body sensor network (BSN) [13], which can be used for computer assisted rehabilitation and even early detection of medical conditions. 1.2 Data Communication Patterns in Sensor Networks Sensor networks are characterized by ad hoc multi-hop networks that are capable of self-organizing without the help of any external infrastructure. Once a network is de- ployed, the nodes collect data about a physical phenomenon of interest, process it lo- cally, and send it toward a common sink node, which can perhaps fuse all the received data and make intelligent decisions. The sink node is typically high-powered, such as a laptop, with larger memory and processing power. Although in small-scale, single-hop networks, direct communication between the nodes and the sink is possible, the most common form of communication in large-scale, multi-hop networks is peer-to-peer, i.e., among neighboring nodes. This peer-to-peer communication over short distances is ideal for low-power, short-range radios, and allows nodes to cooperate and collectively work toward a common goal. There are three basic data communication patterns in sensor networks: (i) converge- cast, (ii) unicast or local broadcast, and (iii) multicast. We describe each of them in details below. 8 • Convergecast: It is a many-to-one communication pattern [46, 47], where data flows from a set of nodes toward a common sink over a tree-based routing topol- ogy. This is the most common form of data communication and constitutes a fun- damental operational primitive in sensor networks. When the sensor readings are correlated due to spatial/temporal proximity of the nodes, or when the application requires summarized information, data is often combined or aggregated at each hop en route to the sink. Data aggregation [98] has been put forward as an essen- tial paradigm for wireless routing in sensor networks, where the idea is to combine the data coming from different sources before transmitting to the upstream node toward the sink. It has been shown that aggregation can eliminate redundancy and minimize the number of transmissions, thus saving energy. This paradigm shifts the focus from traditional address-centric approaches for networking (find- ing short routes between pairs of addressable end-nodes) to a more data-centric approach, i.e., finding routes from multiple sources to a single destination that al- lows in-network consolidation of redundant data. We refer to the convergecast process under aggregation as aggregated convergecast [50, 51], and distinguish it from raw-data convergecast [69, 70] when there is no aggregation. • Unicast: It is a form of local broadcast where a node exchanges data with its local neighbors, for instance, to perform collaborative data processing and fusion instead of transmitting raw sensor readings. 9 s a b c (a) s a b c (b) Figure 1.2: (a) Multicast - data flows from a single nodes to a set of nodesa,b, andc. (b) Convergecast - data flows from nodesa,b, andc to a single nodes. • Multicast: It can be considered as opposite to convergecast, i.e., one-to-many communication pattern in which data is disseminated from the sink to a set of nodes. Multicast could be used, for instance, in reprogrammable sensor networks where the sink supplies automatic network-wide updates of system software or reconfiguration information to all the nodes. Figure 1.2 shows a simple example that illustrates the characteristics of a typical multi- cast and convergecast. In a multi-cast, as shown in Figure 1.2(a), nodes is the message source and nodesa,b, andc are its expected recipients. Nodea hears the message directly froms and forwards a copy to nodesb andc using a single local broadcast. In case of a convergecast, as shown in Figure 1.2(b), nodesa,b, andc each has a message destined to sinks, anda serves as a relay forb andc. In this case, we assume there is no aggregation, and so nodea transmits three messages as indicated by the three arrows. 10 1.3 Data Acquisition Models in Sensor Networks Convergecast or many-to-one communication being the most fundamental form of data collection in sensor networks, it is natural to ask what triggers the data collection process; in other words, what are the data acquisition models in sensor networks. Here, we present a classification of the most common types of data acquisition models [143] and give examples of applications to which they are relevant. • Continuous/Periodic Delivery: This form of data collection is most relevant for real-time, mission-critical applications where sensing and collection are performed synchronously, or for applications where periodic notification of events is required. Such data collection usually takes place over long durations of time ranging from low data-rate scenarios, such as in surveillance and habitat monitoring, to high data-rate scenarios, such as in structural health and permafrost monitoring. Since the nodes need to continuously sense and transmit data, energy efficiency is a big concern for continuous and periodic data collection. • Query-Driven Delivery: In this model [21], the nodes send data only when trig- gered by an external query fed into the network by the sink node. Since the nodes could sleep most of the time and wake up to collect data only when triggered by the queries, this delivery model is efficient in terms of energy consumption. Typi- cal application scenarios for query-driven data delivery includes getting a snapshot 11 view of the network, or sending back data/acknowledgments in response to soft- ware reconfigure/upgrade messages sent by the sink. Such data delivery usually spans over short intervals. • Event-Driven Delivery: The sensor nodes could be programmed to deliver data whenever an event of interest occurs within the network. This mode of data acqui- sition is useful when the events are rare but critical. However, such events could also trigger huge bursts of data that require immediate delivery. Since the nodes need to sense the environment continuously for possible occurrence of the events, but transmit only when an actual event occurs, energy consumption is a lesser se- vere concern compared to the case of continuous/periodic delivery. In this thesis, we focus on continuous/periodic data collection over long durations of time for high data-rate applications. 1.4 Link Scheduling in Wireless Networks The nodes in a terrestrial wireless network communicate using radio frequencies (RF), commonly known as the spectrum. It is well understood that the strength of a radio sig- nal attenuates according to the inverse square of distance between the transmitter and the receiver under a free-space propagation model when a clear unobstructed line-of- sight exists. However, in most cases, due to the presence of obstacles, such as buildings, mountains, etc, and other environmental parameters, a radio signal undergoes reflection, 12 absorption, scattering, and diffraction, which makes the signal strength attenuate and fade nonuniformly over distance. This makes wireless communications unreliable and error-prone. Moreover, multiple transmissions within close proximity of each other tak- ing place at the same time and on the same frequency might interfere and may not be correctly decoded by their intended receivers. This calls for efficient mechanisms for scheduling links appropriately in the frequency, time, or code domain, so as to avoid mutual conflicts. We refer to any combination of time/frequency/code as a channel, and the problem of assigning channels to the links as the link scheduling problem. With the rapid growth of wireless networking and embedded computing, leading to the proliferation of ubiquitous wireless devices into our everyday lives, it is likely that extremely high premiums are placed on the communications spectrum. The scarcity of spectrum necessitates efficient channel assignment mechanisms. Whether the channel sharing is based on Time Division Multiple Access (TDMA), Frequency Division Mul- tiple Access (FDMA), Code Division Multiple Access (CDMA), or any combination thereof, there is a fundamental limit on the number of nodes or links sharing the same channel simultaneously. This has motivated the need for spatial reuse of the channels by having links that are sufficiently far apart to use the same frequency, time slot, or code. Scheduling link transmissions so as to optimize one or more of the performance ob- jective, such as throughput, delay, or energy, has spurred a lot of research interest in the last few decades. Unlike wireline networks, where all the links have fixed band- width, the effective bandwidth of a wireless link is influenced by channel variations due 13 to fading, transmission power, routing, changes in network topology, etc. Moreover, in resource constrained wireless networks, such as in sensor networks where the nodes are low-powered and equipped with a single half-duplex radio capable of either transmitting or receiving at any given time, designing efficient scheduling protocols in the face of varying channel conditions and network topology brings in additional challenges. 1.4.1 Medium Access Control Mechanisms For ad hoc networks in general, and sensor networks in particular, there are two primary medium access control (MAC) mechanisms: (i) contention-based, and (ii) contention- free. In contention-based medium access, such as time-slotted Carrier Sense Multi- ple Access with Collision Avoidance (CSMA/CA), each node needs to contend for the medium before being able to transmit. These protocols are known to have lower delay and promising throughput at lower traffic loads, such as in query-driven data collec- tion. However, when the network load is high, there is a higher chance of collisions and back-offs, thus wasting a lot of bandwidth. On the other hand, in contention-free medium access, such as Time Division Multiple Access (TDMA), the nodes do not need to contend for the medium; instead, time is slotted (fixed or variable durations), and each node is assigned a particular time slot to transmit. This eliminates collisions and mini- mizes retransmissions. Such contention-free protocols are better suited for higher traffic load that sustains over long durations, such as in the continuous/periodic data collection 14 model [102]. However, they might incur higher delay and lower throughput for low net- work traffic, because some time slots might go wasted if nodes do not have any packets to transmit. In wireless sensor networks, since radio communication drains the maximum amount of energy compared to other computational processing required by the sensor nodes, re- ducing the number of retransmissions required to successfully send a packet by avoiding collisions saves a lot of energy. Due to low-cost, small form factor, and energy efficiency requirements, the radio capabilities on the sensor nodes are limited compared to other wireless devices. In addition, the nodes transmit at a low power making the effective bandwidth go down even further. Considering all these limitations, and the fact that we focus on continuous/periodic data collection for high data-rate applications, we propose to use contention-free TDMA scheduling protocols that will help in achieving improved throughput and delay. 1.4.2 Communication Using Multiple Frequency Channels Until recently, most of the protocols developed for wireless sensor networks used only a single frequency channel for communication. However, this turns out to be not good enough to provide reliable and timely delivery for large amounts of data generated by high data-rate applications. This calls for the need of using multiple frequency channels. The basic idea behind communication using multiple frequencies is to tune neighboring 15 a b c d RTS RTS CTS CTS DATA ACK DATA Control Channel Data Channel 1 Data Channel 2 Collision Time Figure 1.3: Multi-channel hidden terminal problem transmitters and receivers on different orthogonal frequencies, such that concurrent trans- missions do not interfere with each other, thus leading to increased network throughput. Newer generations of commercially available radios, such as the CC2420 and Nordic NrF905 radios, support multiple frequencies in order to comply with the IEEE 802.15.4 standard. The IEEE 802.15.4 standard is used as the basis for ZigBee [116], Wire- lessHART [137], and MiWi specifications, and provides a framework for low data-rate communication systems, such as wireless personal area networks (WPAN). Along with its advantages, multiple frequencies also bring in two additional chal- lenges in terms of synchronization that need special care. These are known as the multi- channel hidden terminal problem [135] and the deafness problem [99], which occur due to the fact that the nodes may be tuned to listen to different frequencies at different times. We briefly explain these two problems in the following. 16 • Multi-Channel Hidden Terminal Problem: Suppose there areK frequency chan- nels available; one channel is used for exchanging control messages, known as the control channel, and all others are used for exchanging data and acknowledgments, known as the data channels. When a node is neither transmitting or receiving, it listens to the control channel. Suppose that the underlying MAC protocol is CSMA/CA. When nodea wants to send a packet to nodeb, it sends out a ready- to-send (RTS) message to node b on the control channel, possibly with a list of data channels it is willing to use (cf. Figure 1.4.2). On receiving the RTS, nodeb chooses a data channel from the list, say channel 1, and sends out a clear-to-send (CTS) message on the control channel to nodea. At this point, the handshake is complete and if there is no other node trying to transmit in the neighborhood, no collision occurs. However, suppose when nodeb sent the CTS to nodea, there was another node c in the neighborhood that was busy receiving on another channel, say channel 2, from some other noded, and so it did not hear the CTS. Thus, not knowing that nodeb is receiving on channel 2, nodec might initiate a communica- tion with noded also on channel 1, resulting in a collision at nodeb. • Deafness Problem: The deafness problem is also associated with CSMA/CA RTS/CTS based protocols. It occurs when a node sends an RTS message on the control channel to initiate a transmission when the destination is tuned to a differ- ent data channel. After trying multiple times, the transmitter might give up and conclude that the destination is not reachable anymore. 17 There are three primary channel assignment methods in multi-channel communica- tion. • Fixed Channel Assignment: In this method, the network is partitioned into differ- ent clusters, and nodes within each cluster are assigned the same channel, which remain unchanged throughput the operation of the network. An example of this type of channel assignment is presented in [158]. The clusters are assigned chan- nels such that the interference among neighboring clusters is minimized. This static assignment strategy has the advantage of ease of implementation and incur- ring almost no overhead as any sender-receiver pair is always tuned to the right channel. However, it is not adaptive to network dynamics and can partition the network when certain links go down. For example, consider a network topology where a cluster of nodes is connected to the rest of the network using only a single link that is tuned a fixed channel. If that link goes down due to excessive neigh- borhood transmissions by other devices on the same channel, the network will get partitioned. Different from the idea of clustering nodes into different frequencies, Vedantham et al. [147] introduce the concept of component-based channel assign- ment. In this approach, all links in a connected component, induced by a flow graph between sources and destinations, operate on a single channel. • Semi-Dynamic Channel Assignment: In this approach, the nodes are assigned fixed channels to start with, but can change channels during communication if they 18 experience interference, thus avoiding the network partitioning problem. Semi- dynamic assignment strategy, however, needs a coordinator node to make sure that a sender-receiver pair is tuned to the same frequency while transmitting in order to avoid the deafness problem. Moreover, in case of CSMA/CA RTS/CTS based protocols, such semi-dynamic channel assignment might run into multi-channel hidden terminal problem. Examples of semi-dynamic assignment are presented in [99] for wireless mesh networks, and in [132] for multi-hop packet radio net- works where receivers are assigned fixed channels and the transmitters switch to those channels for communication. • Dynamic Channel Assignment: In this method, every sender-receiver pair tunes on a particular channel on-the-fly before each packet transmission. Although such dynamic channel assignment strategy does not require time synchronization, it might incur overhead giving rise to unacceptable delays, especially for real-time data collection. In addition, when a dedicated control channel is used for data-ack channel negotiation, such as in the protocols presented in [74, 92], it runs into the control channel bottleneck problem due to frequent use. There also exist other methods of dynamic channel assignment, such as a split phase-based approach, used in [24, 135, 162], which eliminates the hidden terminal and deafness prob- lems, and a frequency hopping-based approach used in [11, 109, 140, 145]. 19 In this thesis, since we consider continuous/periodic data collection in high data-rate applications, we exploit the benefits of using multiple frequency channels in reducing in- terference and increasing the number of concurrent transmissions. This helps in achiev- ing better network throughput. However, since the sensor nodes are resource constrained in terms of having a single half-duplex radio and limited energy source, frequent per- packet channel negotiation under a dynamic channel assignment strategy might cause unacceptable overheads, and so special care needs to be taken for optimal channel as- signments. 1.5 Routing Topologies in Sensor Networks Routing topologies play an important role in the performance of sensor networks in data collection [96] and data dissemination [66]. The Collection Tree Protocol (CTP) [55] is the defacto routing protocol standard for data collection. Since remote monitoring is one of the key drivers in sensor networks, data from individual nodes must be sent to a sink, often located far from the network, through the use of specific routing paths. Typically, these routing paths comprise a spanning tree rooted at the sink node [146]. During data collection, a node can send its data to its unique parent node, which in turn can send it to its parent and so on, until the data reaches the sink. The structure of the routing tree plays an important role in data collection. While one hand budgeting and technological constraints may require the placement of nodes at specific locations, thus sometimes 20 limiting network performance in terms of throughput and delay, one might, on the other hand, construct specific routing topologies to optimize performance metrics. A routing topology can either be static or dynamic. A static routing topology has the advantage of having fixed routing paths that are a priori known to the network designer and can be leveraged for easy maintenance and optimizing certain performance metrics, such as energy consumption, throughput, delay, etc. However, since sensor nodes are subject to failure due to battery power depletion or environmental causes, static routing paths can at times be fatal in disconnecting the network. This might hamper network operations and requires additional resources for network maintenance. In addition, when new nodes join the network, recomputing some of the routing paths might be expensive. On the other hand, dynamic routing topologies, such as those used in Dynamic Source Routing (DSR) [76], routing paths are discovered only when a packet needs to be sent to a given destination. Computing such on the fly routing paths requires control messages to flow from the source to the destination, that could possibly include link quality esti- mation, congestion indicators, etc, which could help in establishing robust routing paths. Clearly, dynamic routing thus incurs overhead as compared to static routing. In the context of data collection, which is the primary focus of this thesis, we consider routing topologies that are given a priori, as well as construct specific topologies that are optimized for enhancing the throughput-delay trade-off. In particular, given a deploy- ment of nodes, our scheduling algorithms run on routing topologies that are optimized 21 for maximizing aggregated convergecast throughput and minimizing data collection de- lay. We propose algorithms to construct routing topologies that have low node degree as well as low depths. Having a low degree helps in reducing bottlenecks in the presence of multiple frequencies, and low depth helps in reducing delays. 1.6 Power Control Mechanisms in Sensor Networks In recent years, power control has received intense attention in the domain of cellular networks [160] as well as in ad hoc networks [31]. The design of efficient power control mechanisms is crucial to the successful operation of a sensor network [136]. Power control not only saves energy consumption for these resource constrained devices, thus enhancing network lifetime, but also helps in reducing packet collision probability and wireless interference [35, 87, 136], thus allowing for higher throughput and lower delay. When combined with link reliability assessment algorithms, power control techniques can also be used to improve the reliability of links. For instance, upon detecting link reliability below a certain threshold, the MAC protocol can increase the transmission power, thus lowering the probability of receiving corrupted data. Power consumed in transmission and reception is responsible for up to70% of the total energy consumption for a typical sensor node such as Tmote Sky and Micaz, and thus designing efficient power control protocols is a key research area. There have been several studies in designing efficient power control mechanisms and analyzing their benefits. Instead of globally defining a transmission range that keeps a 22 network connected, wireless networks should adjust transmission ranges on each link. The average traffic capacity per node is constant even when more nodes are added to a fixed area network so long the network employs transmission power control. This is not true for fixed power levels, because more nodes will interfere with each other and lower the capacity. A sensor node can perform calculations based on several readings and the network topology in order to identify the ideal transmission power. Some of these readings are: (i) Received Signal Strength Indicator (RSSI), (ii) Sensitivity, and (iii) Battery voltage. RSSI is the received signal strength measured by the transceiver and it depends on ex- ternal factors, such as environmental conditions and presence of obstacles. Sensitivity is an indicator of the least power level at which a transceiver is able to detect and decode data correctly. RSSI values read from the radio are calculated with battery voltage as a reference. Thus, in order to convert any RSSI reading into an actual reception power, the battery voltage must be known. In the context of this thesis, we use transmission power control for 3-dimensional networks to construct sparser topologies with the goal to maintain a connected network. Since the density of nodes is very high in 3-D for a network to be connected, adjusting the transmission power can provide significant benefits in reducing interference. 23 1.7 Contributions and Organization The rest of the thesis is organized as follows. In Chapter 2, we provide a background on relevant topics comprising this thesis. In particular, we describe existing works on link scheduling for fast data collection in sensor networks, both with single channel and multi-channel scheduling. We also describe spanning trees and bicriteria network design problems that are suitable for fast data collection. Lastly, we describe existing works on the need for topology control and a few important topology control algorithms. In Chapter 3, we propose a multi-channel scheduling algorithm for maximizing the aggregated convergecast throughput under a graph-based interference model. We show that by decoupling the joint frequency and time slot assignment problem into two sub- problems of frequency assignment and time slot assignment can achieve provably good worst-case performance bounds for two classes of random geometric graphs. For unit disk graphs, where nodes have a uniform transmission range, the scheduling algorithm gives a constant factor approximation, while for general disk graphs, where nodes can have different transmission ranges, it gives a logarithmic approximation. The proposed algorithms are combinatorial and use tools from approximation algorithms, graph the- ory, and linear programming. Along the way, we also prove several NP-hardness results on the the scheduling complexity for arbitrary networks. We evaluate the algorithms through simulations and show various trends of the scheduling performance with respect to network size and the number of frequencies. 24 In Chapter 5, we combine two metrics – throughput and delay – and construct effi- cient routing topologies that help in optimizing both in the context of aggregated con- vergecast. In particular, we formulate our joint throughput and delay optimization prob- lem as a bicriteria optimization problem, and design an algorithm that minimizes the the depth (radius) of a spanning tree given a certain bound on the node degree. We call such a tree a bounded-degree minimum-radius spanning tree, and show that our proposed al- gorithm gives a constant factor bicriteria approximation on both degree and depth of the tree. Once the routing tree is constructed, we ran the scheduling algorithms designed in Chapter 3 to evaluate the scheduling and delay performance. We show that schedul- ing on such a tree indeed achieves the best of both worlds in terms of maximizing the aggregated convergecast throughput as well as minimizing the maximum delay. In Chapter 6, we consider 3-dimensional networks and designed distributed topol- ogy control algorithms from local neighborhood information with the goal to construct connected sparse topologies. We extended previous results on 2-dimensional topology control and proposed a heuristic based on orthographic projections that achieves very good performance in practice, although it does not theoretically guarantees network con- nectivity at all times. In addition, we also used tools from computational geometry, such as spherical Delaunay triangulation, to design a robust algorithm that achieves global network connectivity at all times, while incuring very low computational overhead as compared to existing techniques in 3-D. We evaluated our topology control algorithms 25 through simulations and showed various trends with average node degree, transmission power, and CPU execution time. In Chapter 4, we consider multi-channel scheduling in the context of aggregated con- vergecast under a more realistic interference model, the so called signal-to-interference- plus-noise-ratio (SINR) model, and extended the algorithms designed in Chapter 3 to work under SINR. By using the notion of link diversity and appropriately dividing the network into grid cells of a certain size, we show that the worst-case scheduling com- plexity can be bounded by the number of non-empty length classes which, in practice, is a small constant. In Chapter 7, we provide a summary of the contributions in this thesis and discuss various open problems and directions for future research. 26 Chapter 2 Background As described in Chapter 1, this thesis focuses on throughput-delay performance for fast data collection in tree-based wireless sensor networks. In particular, we address the prob- lem of jointly optimizing two mutually conflicting performance objectives – converge- cast sink throughput and packet delays – by utilizing three techniques: (i) multi-channel scheduling, (ii) routing over optimal topologies, and (iii) transmission power control. This chapter introduces the existing literature in each of these categories, and describe their differences with our work. 2.1 Link Scheduling for Fast Data Collection Wireless interference and bandwidth limitation are perhaps the key limiting factors to achieving high convergecast throughput in large-scale sensor networks that require peri- odic, real-time, and high-rate data collection. Interference restricts the number of con- current transmissions that can take place within a given neighborhood, thus reducing 27 the effective bandwidth of the wireless channel and, in turn, network throughput. In addition, typical sensor nodes are equipped with a single radio that can transmit only in the order of few tens of Kbps to a few hundreds of Kbps, such as the older gen- eration CC1000 [71] transceivers operating at 38.4 Kbps, to the newer state-of-the-art CC2420 [72] transceivers operating at 250 Kbps using spread spectrum capabilities and larger frequency bands of 5 MHz per channel. Furthermore, these radios operate on the 868/915 MHz and 2.4 GHz unlicensed ISM frequency bands, which limit their transmis- sion power to a maximum of 10 dBm. This in turn results in a maximum transmission range of about 10 to 30 meters depending on environmental conditions. Given these lim- itations due to interference and bandwidth, techniques that are able to schedule a large number of spatially well separated links are required in order to achieve high through- put. To this end, utilizing multiple frequency channels gives a promising approach to overcome some of the interference-related limitations. In fact, most of the commer- cial radios available today support multiple frequency channels, such as the 16 orthogo- nal frequencies supported by CC2420 radios, or the 512 channels supported by Nordic NrF905 radios. Thus, designing scheduling protocols by exploiting the benefits of mul- tiple frequencies is imperative to the successful operation of large-scale sensor networks deployed for fast data collection. In this section, we first describe the relevant scheduling literature that aim to max- imize the data collection rate using a single frequency channel, and then describe the works that consider multiple frequency channels. We note that although multi-channel 28 communication is a well-studied research topic in wireless ad hoc networks, particu- larly in packet radio networks [78] and wireless mesh networks [29], such research is relatively new in the domain of sensor networks, caused partially due to the unique chal- lenges brought forth by severe resource constraints and bandwidth limitations. 2.1.1 Single-Channel Scheduling 2.1.1.1 Maximizing Raw-Data Convergecast Throughput We first consider the case of raw-data convergecast, i.e., when every packet generated by a source node needs to be delivered to a common sink without being aggregated at intermediate hops. This type of data collection is relevant for applications that require delivery or raw sensor readings, or when the redundancy in data is minimal. The prob- lem of maximizing the convergecast sink throughput can be formulated as minimizing the number of time slots required per frame (referred to as the schedule length) under TDMA scheduling. Several variants of the problem exist depending on network topol- ogy, interference model, packet generation scheme, buffer constraints, antenna models, etc. One of the early works in this category is by Florens et al. [40–42], who address the problem of scheduling for packet distribution in sensor networks, and show that it can be considered as an inverse operation of convergecast. Assuming protocol interference model, they propose optimal centralized algorithms for special network topologies, such as line, multi-line, and tree networks, for both omnidirectional and directional antennas. 29 For line networks where the sink sendsp(i)≥ 0 packets to nodei that isi hops away, the basic idea is to send packets destined to the furthest node first, then to the second furthest node, and so on, as quickly as possible respecting channel reuse constraints. Nodes between the sink and a packet’s destination are required to forward a packet as soon as it arrives (i.e., in the next time slot following its arrival). This basic idea can also be extended to a multi-line and tree networks. The upper part of Figure 2.1 shows an example for packet distribution on a 10-node line network for directional antennas with p(1) = 2, p(2) = 1, p(8) = 1, and p(9) = 1. Once an optimal schedule is found for the distribution problem, a schedule for the convergecast problem, where nodei sends p(i) packets to the sink, is constructed by symmetry as shown in the bottom part of Figure 2.1. In particular, a transmission from nodei toi+1 occurring at time slotj for the distribution problem corresponds to a transmission from nodei+1 toi in time slot T−j +1 for the convergecast problem, whereN is the total number of nodes andT is the minimal schedule length. Ergen et al. [37] prove that the problem of minimizing the schedule length is NP- hard by reducing it from the Graph Coloring problem. Under a graph-based interference model, they show that a conflict-free schedule can be found by coloring a conflict graph. A conflict graph is defined as one in which every node represents an edge in the original graph, and two nodes are connected if their corresponding edges interfere in the original graph, i.e., give rise to primary or secondary conflicts. Using such a graph coloring strat- egy, they propose a node-based and a level-based scheduling heuristic, and show that one 30 1 s 2 4 6 5 3 9 7 8 1 4 7 9 11 Time slot Time slot 3 2 11 8 5 distribution convergecast reflect 2 1 1 1 Figure 2.1: Optimal time scheduling for a10-node line network with minimum schedule length 11. Upper part shows the schedule for packet distribution, bottom part shows the schedule for convergecast, which is obtained by symmetry, i.e., by reflecting the upper schedule with respect to the fictitious horizontal line. Note that nodes that are closer than or at 2 hops do not transmit concurrently to respect interference issue. 31 outperforms the other depending on the distribution of packet generation. In particular, node-based scheduling is better for topologies that have equal density of packets across the network or higher density of packets at low levels of the tree, whereas level-based scheduling is better for topologies when the packet density is higher at upper levels. A Virtual Node Expansion-based approach that also uses graph coloring to find a minimum-length, conflict-free schedule, where every node generates a single packet in each frame is proposed by Lai et al. [90]. They first construct a conflict graph from the original graph and then expands it by creating, for each parent node, a number of virtual nodes equal to the size of the subtree rooted at that node in the original tree. This graph expansion is done to accommodate multiple transmissions by intermediate parent nodes which relay packets from nodes in its subtree. Since the virtual nodes also conflict with any node that has an edge to its original node, edges are added between the virtual nodes and the conflicting node. Similarly, an edge is added between each virtual node and its original node. Once the expanded conflict graph is constructed, an approximate coloring algorithm, originally due to Li et al. [94], is used to find a time slot assignment. The coloring algorithm works by finding a vertex with the least degree and removing it from all its adjacent edges. This is repeated until all the vertices are removed, after which it greedily assigns colors in the reverse order of removing the vertices. This results in a conflict-free schedule for each of the edges in the original graph. The authors of [90] also discuss the effect of different routing structures on the sched- ule length and propose a disjoint-strip based approach to construct an efficient routing 32 topology. This results in uniform flow of data along different paths in the network and prevents certain nodes from being overloaded. The basic idea is to construct several disjoint, equally spaced node strips, all with the same number of nodes. Two distanced strips are likely to relay data simultaneously without interfering with each other. It is shown that this disjoint-strip routing, although increases the total number of transmis- sions, yields a shorter schedule length for unbalanced node deployments as compared to shortest-path routing, which is suitable for balanced deployments minimizing the total number of transmissions but not necessarily the schedule length. Choi et al. in [28] formulate the scheduling problem as a Minimum Information Gathering Time Problem, where every node sends a single packet, and the goal is to find routing paths from the nodes to the sink as well as an optimal time slot assignment. By reducing it from the Partition Problem [105], they prove that the problem is NP-complete on general graphs and propose algorithms for line and tree topologies that take at most 3N−3 time slots to deliver all the packets to the sink. For general networks a heuristic is proposed, which starts with a minimum spanning tree and trims the edges such that transmissions on different branches of the tree do not interfere with each other and can be scheduled in parallel. This results in a backbone forest whose segments are then scheduled independently respecting adjacency and two-hop interfering constraints. Following a strategy similar to [42], Gandham et al. in [46, 47] propose distributed scheduling algorithms for raw-data convergecast where every node generates a single packet in one data collection cycle. They give an Integer Linear Programming (ILP) 33 a b c d e f s R R I T I T (a) R I T t t+1 t+2 (b) a b c s e f d (c) Figure 2.2: (a) A linear network with initial state assignment (R, I, T) depending on the hop distance from the sinks. (b) State transition of the nodes. (c) A multi-line network as a composition of multiple linear networks. formulation of the problem and propose a distributed time slot assignment scheme that takes (i) at most 3N− 3 time slots for linear networks, which is optimal, (ii) at most max(3n k − 1,N) time slots for multi-line and tree networks, where the lower bound for multi-line networks is max(3n k −3,N), and (iii) at most 3N time slots for general networks. Heren k represents the maximum number of nodes in any subtree of the routing structure. Similar results are also obtained by Tsai et al. [144]. In addition to minimizing the schedule length, the proposed algorithm in [47] also considers memory constraints on the sensor nodes and requires storage for at most two packets in each node buffer. Links are assumed to be symmetric and the interference model is assumed to be graph-based, with the interference range of a node equal to its transmission range. Their results also extend to the case where nodes generate multiple packets and when channel propagation characteristics are not ideal. Starting from linear networks, the algorithm proposed by Gandham et al. [47] is generalized for multi-line networks, which are multiple linear networks intersecting with 34 the sink, and also to tree networks. The basic idea for linear networks is that each node is assigned an initial state depending on its hop count from the sink. As illustrated in Figure 2.2(a), a node at hop distanceh is assigned a state of: transmitting (T) ifh mod 3 is 1, idle (I) ifh mod 3 is 2, and receiving (R) ifh mod 3 is 0. A node comes back to this initial state after every 3 time slots and follows the state transition diagram as shown in Figure 2.2(b). Effectively, this implies that for every node in state R, there is only one node in the neighborhood which is in stateT , and so packet transmission is always successful resulting in exactly one packet reception by the sink in every 3 time slots. The above basic idea extends to more complex topologies, such as multi-line net- works, as shown in Figure 2.2(c); tree networks where transmissions are scheduled in parallel along multiple branches; and general networks where packets are routed over a Breadth First Search (BFS) tree. However, for the distributed algorithm to work, each node must know the branch ID and the number of nodes in all other branches, but need not be aware of the entire network topology. The scheduling rule for multi-line networks is that the branch with the largest number of remaining packets and whose root has at least one packet gets priority to transmit (ties are broken based on the lowest branch ID). This results in a schedule length ofmax(3n k −1,N). For general networks, since there are interfering edges that are not part of the spanning tree, the goal is to first eliminate interference by constructing a BFS tree and then scheduling as before. However, in ad- dition to knowing the number of nodes in all the branches and branch IDs, for general 35 networks, a node also has to know a conflict map at the initialization phase. This gives a schedule length of3N, although the simulations presented require only1.5N time slots. The use of orthogonal codes, such as Direct Sequence Spread Spectrum (DSSS) and Frequency Hopping Spread Spectrum (FHSS), to eliminate interference is studied by Annamalai et al. [5]. They propose a greedy, top-down, tree construction scheme that chooses the children of a node based on the nearest neighbor criterion starting from the sink and traversing the graph in BFS order. To reduce interference, nodes that fall within the transmission range of a parent other than their own, are assigned different codes if available; otherwise, the code that is least used by the interfering neighbors is used. Once the channel allocation is done, time slots are assigned in a greedy fashion such that a parent does not transmit before its children. Simulation results indicate that the schedule length on such a tree constructed specifically for convergecast is shorter than on a tree constructed for broadcast [27]. However, one limitation of this approach is that the miniature hardware design of sensor nodes may not permit employing complex radio transceivers required for spread spectrum codes or frequency bands systems. 2.1.1.2 Maximizing Aggregated Convergecast Throughput Unlike raw-data convergecast where every packet generated by a node needs to be de- livered to the sink, periodic data collection often requires delivery of only summarized information in the form of aggregated packets. In general, such aggregated convergecast requires less number of time slots than raw-data convergecast because of the reduced 36 traffic volume. Under this setting, it is assumed that every node generates a single packet at the beginning of every frame and perfect data aggregation is possible, i.e., each node is capable of aggregating all the packets received from its children as well as its own into a single packet before transmitting to its parent. It is also assumed that the size of an aggregated packet is constant and is independent of the size of the raw sensor readings. Since the goal is to minimize the schedule length, each parent node ideally waits to receive all packets from its children and then aggregate those with its own before transmitting. Thus, in aggregated convergecast, a node transmits only once per frame and maintains an intrinsic order of transmission with respect to its children. When the routing tree is not specified as part of the problem, the algorithms also construct routing trees suitable for aggregation prior to scheduling. One of the early works is by Chen et al. [25], where the problem is slightly general- ized by considering only a subsetS⊆ V of nodes generating data. Assuming uniform transmission range and a Unit Disk Graph (UDG) model, they formulate it as a Minimum Data Aggregation Time (MDAT) problem with the goal to find a collision-free schedule that routes data from the subset of nodes to the sink in minimum possible time. They prove that MDAT is NP-complete, even when restricted to UDGs, by reducing it from the restricted planar 3-SAT problem [48]. They design a centralized(Δ−1)-approximation algorithm, whereΔ+1 is the maximum number of nodes within the transmission range of any node. Their proposed approach does not assume that the routing tree is known a priori, instead the algorithm finds the data aggregation tree after the schedule is made. If 37 the height of the routing tree ish, then a trivial lower bound on the schedule length is max{h,log 2 |S|}. The basic idea of the algorithm, called the Shortest Data Aggregation (SDA), is to incrementally construct smaller and smaller shortest path trees (SPT) rooted at the sink that span nodes possessing all data, i.e., in the current iteration, the SPT rooted at the sink spans a set of nodes that possess all data aggregated fromS till the previous iteration. The current iteration produces a collision-free schedule that comprises a set of simultaneously transmitting senders, which are selected from the leaves of the SPT based on the number of non-leaf neighbors in the graph, and a set of corresponding receivers. Malhotra et al. [101] consider the joint routing and scheduling problem for aggre- gated convergecast with the goal to construct an optimal routing tree that will help mini- mizing the schedule length. The basic idea of the tree construction algorithm is to create an SPT and balance the number of children per node so that more parallel transmis- sions can take place without any single node causing a bottleneck. They show that for given a tree, a lower bound on the schedule length is max i∈V (ξ i +h i ), whereξ i andh i are the number of children and hop distance from the sink, respectively, for nodei. To balance the number of children per node, an optimal semi-matching formulation on bi- partite graphs, originally due to Harvey et al. [62], is used with the goal to assign nodes from level h + 1 to the parents at level h, such that every parent has an equal number of children. Once the balanced tree is constructed, a ranking-based heuristic is used for scheduling, which ranks all eligible nodes in decreasing order of their weights, taken as 38 the number of non-leaf neighbors. A higher weight gives a higher relative priority to a node to be scheduled in the current slot over other eligible nodes. A variation of the aggregated convergecast problem where nodes can adjust their transmission ranges is studied by Shang et al. [133]. They propose an approximation algorithm that gives a constant factor guarantee on the optimal schedule length for UDG. It first constructs a BFS tree rooted at the sink, and then constructs a maximal independent set using the greedy First-Fit algorithm by choosing nodes in order of their increasing hop distances from the sink. This results in a dominating set, which contains the sink but is not necessarily connected. Then, a minimal number of connector nodes is added to construct a connected dominating set,V CDS , and the transmission ranges of all the nodes in this set are set to one. Next, the scheduling phase runs in two stages. In the first stage, nodes inV\V CDS are scheduled first so that all their data reach the nodes inV CDS . In the second stage, data are sent from the nodes inV CDS to the sink. It is shown that the first stage takes15log 2 |V\V CDS | time slots, whereas the second stage takes16d(T BFS )−12 time slots, whered(T BFS ) is the depth of the BFS tree. Combining the two, it is shown that the schedule length is at most 31 times the optimal. Zhang et al. extend their prior work on minimal time convergecast scheduling for raw data convergecast [46] to aggregated convergecast scheduling in [163]. When the size of data is much smaller than the size of the data frame, nodes aggregate the received pack- ets instead of sending single packets. They use the same scheduling algorithm proposed in [46,47], and show that packet aggregation requires at mostN+2 time slots in a linear 39 network. According to the algorithm, the sink receives the first packet in the first time slot; then in every 3 time slots it receives an aggregated packet which contains data from 3 original unaggregated packets due to 2-hop scheduling to prevent interference. Hence, the total number of required time slots is1+3 N−1 3 ≤N+2. For multi-line networks, the algorithm achieves a schedule length ofmax n k +3 N+2k 3 , wherek is the number of branches andn k is the maximum number of nodes in a branch. Finally, for tree net- works aggregation-enabled convergecast requires max ˆ n, l N+2L+2(L−k) 3 m , whereL is the number of leaf nodes,k is the number of one-hop subtrees, ˆ n = max i (n i +4l i −2), n i is the number of nodes, andl i is the number of leaf nodes in thei th one-hop-subtree. 2.1.2 Multi-Channel Scheduling The use of multi-channel communication is a well-studied research topic in wireless ad hoc networks and cellular networks. In cellular networks [79], base stations use differ- ent frequency domains within a cell, while clients share the time domain to access the wireless medium. However, this approach is either single-hop or infrastructure based, and thus is not suitable for multi-hop networks that are deployed over large geographical regions. For multi-hop ad hoc networks, there exist several works that aim to increase the sys- tem throughput [74, 128, 135]. Most of these approaches are based on the IEEE 802.11 protocols. For instance, IEEE 802.11b allows 11 channels that are spaced 5 MHz apart. 40 However, the IEEE 802.11 protocols are very expensive in terms of energy consump- tion and do not meet the requirements of WSN. Kyasanur et al. [89] study the capac- ity of multi-channel wireless ad hoc networks by extending the analysis of Gupta and Kumar [59], which shows that the asymptotic achievable throughput per node for a ran- domly distributed set of N source-destination pairs with one-to-one communication is bounded by Θ W √ N logN , whereW is the transmission capacity. They investigate the impact of multiple channels and the number of radio interfaces on the network capacity, and show that, even with smaller number of interfaces than that of available channels, multi-channel communication can enhance the networks capacity. In the domain of sensor networks, research using multiple channels is relatively new due to their several differences with ad hoc networks, such as simpler radios, bandwidth and energy limitations, scalability, etc. Although the focus of this thesis is to utilize mul- tiple channels to alleviate the impact of interference and improve network performance in terms of throughput and delay, we first briefly discuss a few important studies that use multiple channels to achieve other objectives. The use of multi-channel communication against jamming is discussed in [4, 152, 157], where channel surfing mechanisms have been introduced such that the jammed nodes dynamically change their operating frequency. Multi-channel clustering is dis- cussed in [58] where nodes that hold correlated data are clustered together and commu- nicate on the same frequency, which is different from the communication frequency of other clusters. Cluster heads are assumed to be the aggregation points to process raw 41 data before relaying toward the sink node. The key objective in this work is to min- imize energy consumption by reducing the effects of collisions that may occur if the clusters operate on the same frequency. Multi-channel communication is also used in reliable data dissemination [95,155]. The Typhoon protocol [95] uses channel switching to reduce contention in the broadcast medium which, in turn, reduces the completion time of data dissemination. Multi-channel communication can also be used to overcome the congestion that can occur due to contention and interference in the network. Ex- amples of joint channel assignment and congestion control do exist in wireless ad hoc networks [54, 110]. However, in WSN, congestion avoidance with multiple channels is very briefly addressed in [154]. Just as there exist different channel assignment strategies for ad hoc networks, such as fixed, semi-dynamic, and dynamic, as discussed in Chapter 1, similar assignments also exist in sensor networks. Fixed channel assignment, where nodes are clustered into dif- ferent frequencies, are presented in [18, 58, 154]. The IEEE 802.15.4 standard also uses fixed channel assignment, but it is possible for the beacon node to change the operating frequency if other nodes report excessive interference. In [154], it is argued that frequent channel switching may cause potential packet losses, however, once channel switching is done synchronously, this overhead can be eliminated. Examples of semi-dynamic assignment, where nodes are assigned fixed channels but can switch in order to commu- nicate with other nodes, are presented in [99, 132]. Y-mac [81] is the first example that uses dynamic channel assignment in WSN, where a combination of a dedicated control 42 channel and a frequency hopping method is used. In [148], V oigt extends the D-MAC protocol [97] and proposes to use multi-channel communication to reduce interference. 2.1.2.1 Maximizing Raw-Data Convergecast Throughput Multiple channels to eliminate interference and increase network throughput has been studied by Incel et al. in [69]. They explore and evaluate a number of different tech- niques using realistic simulation models to study the data collection rate for raw-data convergecast. First, a simple spatial-reuse TDMA scheme is employed to minimize the schedule length, which is then combined with multiple frequency channels and transmis- sion power control to achieve further improvement. A receiver-based channel assign- ment (RBCA) scheme is proposed where the receivers of the tree are statically assigned a channel, and the children of a common receiver transmit on that channel. This avoids pair-wise, per-packet channel negotiation overheads. Once multiple frequencies are used to completely eliminate interference (i.e., secondary conflicts), it is shown that the lower bound on the schedule length ismax(2n k −1,N), and a time slot assignment scheme is proposed that achieves this bound with no nodes requiring to buffer more than one packet at any time. Here, as in [47],n k is the maximum number of nodes in any branch of the tree. Next, the authors also show that once interference is eliminated, the data collection rate often becomes limited by the routing topology. To overcome this, trees with spe- cific properties are constructed, which help in further enhancing the data collection rate. In particular, capacitated minimum spanning trees [117], which aim to have an equal 43 Algorithm 1 LOCAL-TIMESLOTASSIGNMENT 1. node.buffer = full 2. if node is sink then 3. Among all eligible top-subtrees, choose one with the largest number of remaining packets, say top-subtreei 4. Schedule link(root(i),s) respecting interfering constraint 5. else 6. if node.buffer == empty then 7. Choose a random childc of node whose buffer is full 8. Schedule link(c,node) respecting interfering constraint 9. c.buffer = empty 10. node.buffer = full 11. end if 12. end if number of nodes on each branch, are shown to achieve a factor of two improvement as compared to single-channel TDMA scheduling on minimum-hop SPTs. The key idea behind the algorithm in [69], which is formally presented in Algorithm 1 as LOCAL-TIMESLOTASSIGNMENT, is to: (i) schedule transmissions in parallel along multiple branches of the tree, and (ii) keep the sink busy in receiving packets for as many time slots as possible. Each node maintains a buffer and its associated state, which can either be full or empty depending on whether it contains a packet or not. Initially, all the buffers are full because every node has a packet to send. The first block of the algorithm in lines 2-4 gives the scheduling rules between the sink and the roots of the top-subtrees. A top-subtreeTS(r) is defined as one whose root r is a child of the sink, and it is said to be eligible ifr has at least one packet to send. For instance, in Figure 2.3(a), the top-subtrees are{1,4},{2,5,6}, and{3,7}. For a given time slot, the root of an eligible top-subtree which has the largest number of total remaining packets is scheduled. If none of the top-subtrees are eligible, the sink does not 44 (a) (b) Figure 2.3: Raw-data convergecast using algorithm LOCAL-TIMESLOTASSIGNMENT: (a) Schedule length 7 when secondary conflicts are eliminated. (a) Schedule length 10 when secondary conflicts are present. receive any packet during that time slot. Inside each top-subtree, nodes are scheduled according to the rules in lines 5-12. A subtree is defined to be active if there are still packets left in it (excluding its root) to be relayed. If a node’s buffer is empty and the subtree rooted at this node is active, one of its children is scheduled at random whose buffer is not empty. The algorithm guarantees that in an active subtree there will always be at least one child whose buffer is not empty, and so whenever a node empties its buffer, it will receive a packet in the next time slot, thus emptying buffers from the bottom of the subtree to the top. Figure 2.3(a) shows an illustration of the working of the algorithm. In slot 1, since the eligible top-subtree containing the largest number of remaining packets is{2,5,6}, link(2,s) is scheduled and the sink receives a packet from node 2. In slot 2, the eligible top-subtrees are{1,4} and{3,7}, both of which have 2 remaining packets. We choose 45 one of them at random, say{1,4}, and schedule the link (1,s). Also, in the same time slot since node 2’s buffer is empty, it chooses one of its children at random, say node 5, and schedule the link (5,2). In slot 3, the eligible top-subtrees are{2,5,6} and{3,7}, both of which have 2 remaining packets. We choose the first one at random and schedule the link (2,s), and so the sink receives a packet from node 5 (relayed by node 2). We also schedule the link(4,1) in slot 3 because node 1’s buffer is empty at this point. This process continues until all the packets are delivered to the sink, yielding an assignment that requires 7 time slots. Note that, in this example, 2n k −1 = 5, and so max(2n k − 1,N) = 7. In Figure 2.3(b), an assignment is shown when all the interfering links are present, yielding a schedule length of10. A similar result ofmax(2n k −1,N) is obtained by Song et al. [138] where they also extended it to the case when nodes have different number of packets to send. Multiple channels that minimize the schedule length for raw-data convergecast in WirelessHART networks [137] is studied by Zhang et al. [161]. One significant difference between Wire- lessHART networks and sensor networks is that the former performs channel hopping on a per-packet basis, while most existing TDMA convergecast schemes do not support this feature. Thus, parallel transmissions scheduled in the same time slot must use differ- ent channels, whereas most of the existing TDMA-based multi-channel protocols first statically assign channels to eliminate potential interference and then perform time slot scheduling. Like in [69] and [138], they also consider buffer requirements at each node, and show that when nodes can store at most one packet, the minimum schedule length 46 for line topologies is 2N−1 using at most⌈ N/2⌉ channels. However, when the nodes can buffer multiple packets, the optimal convergecast time remains the same while the number of channels required can be reduced to l N− p N(N−1)/2 m . The basic idea of their proposed approach, which is similar to [69] and [138], is to schedule as many transmissions as possible in each time slot in order to maximize the use of available channels, and to make sure that a node which does not have a packet at the beginning of a time slot receives one packet at the beginning of the next time slot. 2.2 Spanning Trees for Fast Data Collection Several optimization problems arising in the design of communication networks can be modeled as constructing optimal network topologies [164], in particular, spanning trees. Spanning trees are widely used in communication networks as a means to disseminate information from one node to all other nodes, as in broadcast or multi-cast, and/or to collect information from a set of sources to a single designated node, as in converge- cast. Depending on the constraints, the problem of computing spanning trees has been characterized by different complexity classes. Perhaps the simplest among all these is the problem of constructing a minimum cost spanning tree in an edge weighted graph, which is efficiently solvable in polynomial time for both sequential and parallel cases. However, when simple constraints are imposed on the structure of the resulting tree, such as bounded node degree or diameter, the problem frequently becomes NP-hard. 47 In this thesis, we focus on computing spanning trees that help in improving the throughput-delay trade-off for fast data collection in large-scale sensor networks. In [70], Incel et al. show that, in addition to spatial-reuse TDMA scheduling using multiple frequency channels, which helps to eliminate interference and enable more concurrent transmissions, thereby enhancing the data collection rate, the structure of the routing tree also plays an important role in the performance of convergecast throughput. In particular, routing trees with high node degree are likely to create bottlenecks within the network, because the children of a common parent need to be scheduled at different time slots due to half-duplex transceivers. On the other hand, trees with low node degree might allow for more concurrent transmissions in the presence of multiple frequencies. Thus, if we aim to maximize the convergecast sink throughput and have the flexibility to construct a routing tree, computing one with minimum degree might be a good choice. We note that, sometimes, however, the routing tree on which a given network should operate might be fixed and specified a priori due to socio-economic constraints, in which case we do not have the flexibility to construct one that is best suited for the objectives we want to optimize. It is also true that for a given deployment of nodes, a spanning tree with low node degree also has a large number of hops to the sink. Thus, if packet delays are measured purely in terms of hop counts, a tree with low node degree is likely to incur high de- lays as opposed to one with high node degree. These two opposing tree properties – node degree and hop distance – therefore, underscore the role of the routing topology in 48 maximizing the data collection rate and minimizing packet delays. In the following, we describe some of the existing works that aim to construct spanning trees with different structural constraints so as to optimize multiple conflicting objective functions. These works fall under the general theme of multi-criteria network design problems [125], and in particular, bicriteria problems when the number objectives is two. 2.2.1 Bicriteria Network Design Problems A good example of a network design problem that calls for a multi-criteria optimization framework is multicast. Designing networks that are capable of accommodating mul- timedia (both audio and voice) traffic in a multicast (simultaneous transmission of data to multiple destinations) environment has gained considerable interest in recent years. One of the popular solutions to multicast routing, as discussed by Kompella et al. [83], involves tree construction. Two optimization criteria – (i) worst-case transmission delay, and (ii) total cost – are typically sought to be minimized in constructing these trees. In addition, it is often the case that these multiple optimization criteria also have different cost functions and budget associated with each measure. For example, as pointed out in [83], in the problem of finding good multicast trees, each edge has two costs asso- ciated with it: a construction cost and a delay cost. The construction cost is typically a measure of the amount of buffer or channel bandwidth used, and the delay cost is a combination of propagation, transmission, and queuing delays. Clearly, these costs have associated budgets under which they should operate. Such multi-criteria network design 49 problems with separate cost functions for each optimization criterion also occur naturally in information retrieval and VLSI design where one aims to find minimum cost spanning or Steiner trees given delay bound constraints on source-sink connections. Network design problems where even one cost measure needs to be minimized are often NP-hard, with the exception of a few, such as the Minimum Diameter Spanning Tree problem. Here the goal is to construct a spanning tree such that the tree diameter, defined as the longest hop distance between any pair of nodes, is minimized. On Euclidean graphs, this problem is solved in polynomial time Θ(n 3 ), and the result extends to any complete graph whose edge weights satisfy a distance metric [67]. The most recent result on general graphs is obtained by Hassin and Tamir [65] by showing its equivalence with the Absolute 1-Center (A1CP) problem that runs in O(mn +n 2 logn) time, where m is the number of edges. Note that, this reduces to the Θ(n 3 ) bound for complete graphs becausem =n 2 for complete graphs. Most notably among the NP-hard unicriterion optimization problems is the Mini- mum Degree Spanning Tree (MDST) problem, where the goal is to construct a spanning tree of a graph G = (V,E) with n vertices, such that the maximal degree is smallest among all spanning trees ofG. This problem is a generalization of the Hamiltonian Path Problem and is NP-hard, thus leaving us hope only in the space of approximation algo- rithms. SupposeT ∗ be an optimal spanning tree whose maximal degree isΔ ∗ . Furer and Raghavachari [44] propose an iterative polynomial time approximation algorithm that computes a spanning tree of maximal degree at mostO(Δ ∗ +logn). This result is then 50 refined by the same authors to produce a spanning tree of degree at mostΔ ∗ +1, which is the best bound achievable in polynomial time, unlessP =NP . In the Steiner version of the problem, along with the input graph, we are also given a distinguished set of vertices D⊆ V , and the goal is to compute a tree of minimum degree spanning all the nodes inD. The Minimum Degree Steiner Tree problem [45] is more general and the MDST problem is its special case whenD =V . Agrawal et al. [82] have shown that the Steiner version can be approximated to within alogn factor using multicommodity flow. Taking a step further from unicriterion optimization problems, such as the MDST, we now move on to describe some of the existing works that consider bicriteria network design problems. As one could expect, such problems are also NP-hard and render us in the domain of approximation algorithms. Definition 1. A generic bicriteria optimization problem is defined asP = (X,Y,H), whereX andY are the two objectives, andH represents a membership requirement in the class of subgraphs. The problemP specifies a given budget on the first objectiveX under a cost functionc X , and the goal is to find a network topology that minimizes the second objectiveY under possibly a different cost functionc Y , such that the network is within the budget of the first objective. An example of a bicriteria optimization problem is that of finding a spanning tree that yields low-cost and low-transmission-delay in multimedia networks for multicast, which can be modeled as (Diameter, Total Cost, Spanning Tree)-bicriteria problem as follows. Given an undirected graphG = (V,E) with two cost functionsc andd for each edgee∈ 51 E modeling construction and delay costs, respectively, and a boundD on the total delay, find a minimumc-cost spanning tree such that the diameter of the tree under thed-cost is at mostD. It is easy to see that the notion of bicriteria problems can easily be extended to the more general multi-criteria optimization problems. We note that such optimization problems that involve minimizing two cost measures have also been addressed in the literature by attempting to minimize a functional combination of the two, thus converting them into unicriterion problems, however, this approach fails when the two criteria are very disparate. It is shown by Marathe et al. [103] that bicriteria formulation is quite generic and robust because the quality of approximation is independent of which of the two criteria the budget is imposed on, and it subsumes the case where one wishes to optimize a functional combination of the two objectives. Recall that an approximation algorithm for a unicriterion optimization problem pro- vides a performance guaranteeρ if for every instance, the value of the solution returned by the algorithm is within a factorρ of the optimal value for that instance. This notion of performance guarantee can be extended to bicriteria optimization problems as follows. An (α,β) approximation algorithm for a bicriteria minimization problem (X,Y,H) is defined as a polynomial time algorithm that produces a solution in which the value of the first objectiveX is at mostα times the budget, and the value of the second objectiveY is at mostβ times the minimum for any solution that is within the budget ofX . The so- lution produced must also belong to the subgraph-classH. Analogous definitions can be given whenX and/orY are maximization objectives. In [103], Marathe et al. show that 52 an(α,β) approximation algorithm for a problem(X,Y,H) can be easily transformed in polynomial time into a (β,α) approximation algorithm for problem (Y,X,H), and so the definition of a bicriteria approximation is independent of the choice of the criterion that is budgeted in the formulation. The (Diameter, Total Cost, Spanning Tree) problem, also known as the Bounded Diameter Minimum Spanning Tree problem is NP-hard even in the special case when the two cost functions are identical [67]. Awerbuch et al. [9] gave an approximation algorithm with (O(1),O(1)) performance guarantee for finding a spanning tree that has small diameter as well as small cost, both under the same cost function. Khuller et al. [80] studied an extension called Light, Approximate Shortest-path Tree (LAST) and gave an approximation algorithm with(O(1),O(1)) performance guarantee. Kadaba and Jaffe in [16] and Kompella et al. in [83] considered the (Diameter, Total Cost, Steiner Tree) problem with two edge costs and presented heuristics without any guarantees. A closely related problem is the (Diameter, Total Cost, s-t Path) problem, which requires finding a diameter-constrained shortest path between two pre-specified vertices s and t. This problem, termed in the literature as the Multi-Objective Shortest Path (MOSP) problem, is NP-complete, and Warburton [149] presented the first fully polynomial time approximation scheme (FPTAS) for it. Hassin [64] also provided a strongly polynomial FPTAS for this problem which improved the running time of Warburton. More general than the MDST problem is the Bounded Degree Minimum Spanning Tree problem (Degree, Total Cost, Spanning Tree) where, in addition to a degree boundΔ 53 on each node, the total cost of the tree also needs to be minimized. Ravi et al. [125] pro- pose an approximation algorithm using a matching-based augmentation technique that gives a(O(Δlog n Δ ),O(log n Δ )) guarantee. Konemann and Ravi in [85] use a Lagrangian- relaxation based approach to propose an improved(O(Δ+logn),O(1)) approximation algorithm. Chaudhuri et al. presented an improved(1,O(Δ+logn)) approximation al- gorithm in [23], and a further improved version that gives (1,O(Δ)) performance guar- antee in [22] based on push relabel framework. Their algorithm is the first one that approximates both degree and cost within a constant factor. The best possible result is proposed recently by Singh and Lau in [134] where the approximation guarantee is (1,Δ + 1). In addition to the upper bound Δ on the node degree, when there is also a lower bound Δ ′ , their algorithm returns a spanning tree where the degree of each node v satisfies the constraint Δ ′ −1 < deg(v) < Δ+1 with at most the optimal total cost. This result is also true for general cost functions (even negative). Most related to our work is the Bounded Degree Minimum Diameter Spanning Tree problem (Degree, Diameter, Spanning Tree), where the goal is to find a spanning tree such that the diameter of the resulting tree is minimized subject to the constraint that the degree of each node is no more than a given integer Δ. This problem is NP-hard for general graphs. Ravi first proposed a(O(log 2 n),O(logn)) approximation algorithm in [124] that runs inO(nmlogn) time and finds a spanning tree of degreeO(Δ ∗ logn+ log 2 n) and diameterO(Dlogn), where Δ ∗ is the optimal degree of any spanning tree of diameter at mostD, whose value is greater than the diameter of the graph. They use 54 the notion of poise of a tree, defined as the sum of the maximum degree and diameter, and use multicommodity flow results, in particular, a result on the L-bounded minimum congestion problem, to prove the approximation guarantee. A further improvement is obtained by Konemann et al. [84] who propose aO( p log Δ ∗n) approximation algorithm for complete graphs under a metric cost function. Their algorithm uses a combination of filtering and divide and conquer techniques to find a spanning tree of maximum node degree Δ and diameter O( p log Δ n·D ∗ ), whereD ∗ is the minimum diameter of all spanning trees with degree at mostΔ. 2.3 Topology Control in Wireless Networks 2.3.1 Why Topology Control? The topology of a packet radio network (PRN), and in particular, wireless multihop net- work is a set of communication links between node pairs that describe the connectivity information of the network, and is used explicitly or implicitly by a routing mechanism. Unlike the links in a wired network, which are well defined by modems and hard wires, links in a wireless network are the result of some link determination protocol subject to a transmission power limitation. A basic requirement for a link to exist between any two nodes in a topology is that the nodes are within each other’s transmission range. In most of the early works, the topologies of a PRN is defined by connecting all neighbors within a fixed transmission range, which is uniform for all nodes. 55 In order to maintain good connectivity and reliability of the network, this fixed, uni- form transmission range must be adjusted to be large enough, especially in sparse re- gions. However, setting a large, uniform transmission range will result in high node degrees in denser regions of a network, which will prevent spatial reuse of the wire- less channel, thus causing congestion and reducing network throughput. Additionally, it might also cause long packet delays. Thus, reliability and throughput are conflicting requirements. For instance, the most reliable topology is a complete graph where every node is connected to every other node, but it is the worst case for a shared radio channel in terms of throughput because no reuse is possible. On the other hand, setting a low transmission range might partition the network. One solution to the above problem associated with dense networks is topology con- trol, in which every node can carefully select a set of neighbors to establish logical data links, and then dynamically adjust its transmission range for different links by adjusting its transmission power. In establishing these logical links, it is desirable to choose only those local connections that will guarantee overall global network connectivity while sat- isfying different and often conflicting performance metrics, such as overall throughput, network utilization, and power dissipation. In a shared radio channel, a higher trans- mit power results in more forward progress (i.e., the distance by which a packet has moved in the direction toward its destination) per transmission, while a lower trans- mit power causes less interference to distant nodes. Theoretically, selecting a proper transmit power per node can result in optimal performance. In addition, controlling the 56 (a) (b) (c) Figure 2.4: Benefits of topology control: (a) Network without topology control in which nodes are configured at maximum transmission range leading to high node degree. (b) Network without topology control in which nodes are configured at minimum transmis- sion range leading to network partition. (c) Network with topology control in effect in which nodes adjust their transmission range leading to low average node degree and yet a connected network. 57 transmission power helps in extending network lifetime, which is a crucial requirement for wireless multihop networks, and in particular, resource constrained sensor networks. Figure 2.4(a) and 2.4(b) show examples of two networks without topology control in which nodes are configured at maximum and minimum transmission range, respectively, leading to high node degree and network partition. Figure 2.4(c) shows the same network with topology control in effect where nodes could adjust their transmission range lead- ing to low average node degree and yet a connected network. A list of good properties for topology control in order to maximize network throughput is the following: (i) each node should have a similar number of neighbors, and the average node degree should be small, (ii) nodes that are geographically closer should have higher priority of being log- ical neighbors to a given node, (iii) regular and uniform structures are usually preferred, and (iv) it should be difficult to partition the network by removing only a few links. In this thesis, we consider topology control for arbitrarily deployed networks with the goal to establish global connectivity by choosing only a minimal set of neighbors using local information. In particular, we consider 3-dimensional (3-D) networks where the density of nodes required to maintain connectivity in typical deployments is very high compared to 2-dimensional (2-D) networks. Although topology control in the domain of 2-D networks is a well researched topic, the number of such works in 3-D networks is only a few and they require high computational overhead. To this end, our contribu- tion in this thesis is to develop an efficient distributed topology control protocol for 3-D 58 networks that has low computational complexity. In the following, we describe some relevant literature for 2-D as well as 3-D networks. One of the prominent works that consider 2-D networks is by Wattenhofer et al. [93, 150], in which the authors propose a novel distributed Cone-Based Topology Control (CBTC) algorithm that increases network lifetime while maintaining global connectivity with reasonable throughput in multihop wireless ad hoc networks. In contrast to earlier approaches that rely on knowing and sharing global positioning information of all nodes, CBTC is a distributed algorithm that relies solely on local directional information of incoming signals from neighboring nodes. 59 Chapter 3 Maximizing Convergecast Throughput in Tree-Based Sensor Networks 3.1 Motivation Convergecast, namely the many-to-one flow of data from a set of sources to a common sink over a tree-based routing topology, is a fundamental communication primitive in sensor networks . Depending on the specific application, such data collection can be trig- gered by external events, such as user queries, to periodically get a snapshot view of the network, or can be automated internally sustaining over long durations. For real-time, mission-critical, and high data-rate applications, such as security-surveillance, health- care, structural health monitoring [156], as well as some environmental networks, such as wildfire [141] and permafrost monitoring [63] [15], it is often critical to maximize the data collection rate at the sink node. In addition, when the sensor readings are correlated Parts of this chapter are based on [50], [69], and [68]. 60 due to spatial/temporal proximity of the nodes, or when summarized information is re- quired, it is beneficial to aggregate data en route to the sink. Data aggregation helps in reducing redundancy and the number of transmissions, thus saving energy [98]. In such cases, we refer to the data collection process as aggregated convergecast [70]. In this chapter, we consider the problem of maximizing the aggregated converge- cast throughput at the sink node by utilizing multiple frequencies using TDMA schedul- ing [102]. We consider two cases with respect to our knowledge about the routing topol- ogy: (i) when the topology is known a priori, and (ii) when the topology is unknown and can be arbitrary. We show that multiple frequencies can help in eliminating interfer- ing links and enable more concurrent transmissions, thereby increasing the aggregated data collection rate. We prove NP-completeness results on the multi-channel scheduling problem, and design efficient link scheduling algorithms that have provably good, worst- case performance guarantee for arbitrarily deployed networks in 2-D. In particular, when the routing topology is known a priori, we design a constant factor approximation algo- rithm for networks modeled as Unit Disk Graphs (UDG) where every node has a uniform transmission range, and aO(Δ(T)logN)-approximation for general disk graphs where nodes could have different transmission ranges. HereN is the total number of nodes in the network, and Δ(T) is the maximum node degree on a given routing treeT . We also prove that a constant factor approximation is achievable on UDG even when the rout- ing topology is unknown, so long as the maximum in-degree of any node in the tree is 61 bounded by a constant. We evaluate our proposed algorithms using MATLAB simulation and show trends in the scheduling performance for different network parameters. The rest of the chapter is organized as follows. In Section 3.2, we describe our models, assumptions, and problem formulation. In Section 3.3, we consider the multi- channel scheduling problem for general graphs (where links could exist between any pair of nodes), and prove two NP-completeness results. In the same section, we also design a polynomial-time algorithm to schedule the whole network in minimum time when there are enough frequencies available to remove all the secondary conflicts. In Section 3.4, we design approximation algorithms for the scheduling problem on a UDG model under a given limited number of frequencies for the two cases of known a priori and unknown routing topologies. In Section 3.5, we present an Integer Linear Programming (ILP) formulation of the scheduling problem for the general disk graph model, and propose an approximation algorithm. In Section 3.6, we present our evaluation results, and finally in Section 3.7, we summarize the chapter. 3.2 Preliminaries 3.2.1 Model and Assumptions We model the network as an undirected graphG = (V,E), whereV is the set of nodes arbitrarily deployed in the 2-D plane, andE is the set of edges. An edgee = (u,v)∈E exists between any two nodesu andv if their Euclidean distanced(u,v) is no more than 62 the transmission rangeR. We assumeG to be connected, i.e., there is no isolated node . We are also given a distinguished node s∈ V that represents the sink. We denote a spanning tree onG rooted ats byT = (V,E T ), whereE T ⊆ E is the set of tree edges that needs to be scheduled. Each node is equipped with a half-duplex transceiver using which it can either trans- mit or receive a single packet at any given time slot. Consecutive slots are grouped into equal sized frames that are repeated for periodic scheduling. We assume that every node generates a single packet in the beginning of every frame, and has the ability to aggregate all the packets from its children as well as its own into a single packet before transmitting to its parent. The class of aggregation functions that falls in this category include dis- tributive and algebraic functions [98], where the size of an aggregated packet is constant regardless of the size of the raw sensor readings. Examples of such functions are MIN, MAX, MEDIAN, COUNT, SUM, A VERAGE, etc. We assume that transmissions on different frequencies are orthogonal and non-interfering with each other. This assumption may sometimes fail in practice depending on transceiver- specific adjacent channel rejection values, however, experimental results presented by Incel et al. [70] show that the scheduling performance remains similar for CC2420 and Nordic nrf905 radios. We consider a receiver-based frequency assignment strategy, in which we assign a frequency statically to each of the receivers (parents) in the tree and If there are isolated nodes, or more than one connected components in the network, then we consider only those nodes that have a path to the sink. 63 e 1 e 2 e 1 e 2 (a) f 1 e 1 f 1 e 2 (b) Figure 3.1: (a) Concurrent transmissions on adjacent edges e 1 and e 2 cause primary conflict. (b) Concurrent transmissions on edges e 1 and e 2 cause secondary conflict if their receivers are on the same frequency, sayf 1 , and if either of the receivers is within the range of the other transmitter. have the children transmit on the same frequency assigned to their parent. Due to this par- ticular static assignment scheme, each node operates on at most two frequencies, which results in less overhead compared to dynamic frequency assignment schemes, such as pair-wise negotiation of frequencies on a per-packet basis. We consider a graph-based interference model (also known as the protocol model), in which (a) the interference range of a node is equal to its transmission rangeR, and (b) concurrent transmissions on two tree edgese 1 ,e 2 ∈ E T interfere with each other if and only if either (i) they are adjacent, or (ii) both their receivers are on the same frequency and at least one of the receivers is within the range of the other transmitter. The first type of interference is known as primary conflict, and the second type as secondary conflict, as illustrated in Figure 3.1(a) and 3.1(b), respectively. 64 s 1 3 2 6 5 4 1 2 3 4 5 6 f 1 f 1 f 1 (a) s 1 3 2 6 5 4 1 2 3 1 3 2 f 1 f 1 f 2 (b) Frame 1 Frame 2 1 2 3 4 5 6 1 2 3 4 5 6 s 1 2 3 - - - (1,4) (2,5,6) 3 - - - 1 - - - 4 - - - - - 4 - - 2 - - - - 5 6 - - - - 5 6 (c) Frame 1 Frame 2 1 2 3 1 2 3 s 1 (2,5) 3 (1,4) (2,5,6) 3 1 - 4 - - 4 - 2 5 - 6 5 - 6 (d) Figure 3.2: Aggregated convergecast: (a) Schedule length of 6 time slots using one frequency. Dotted lines represent secondary conflicts. (b) Schedule length of 3 time slots using two frequencies. Note that, all the secondary conflicts are eliminated. (c), (d) Nodes from which aggregated data is received by their corresponding parents in each time slot over 2 consecutive frames for (a) and (b), respectively. 65 3.2.2 Problem Formulation We first explain the process of aggregated convergecast, and then define the multi-channel scheduling problem. Figure 3.2(a) shows a network of6 source nodes and a routing tree whose edges are marked by solid lines; dotted lines represent interfering links causing secondary conflicts. We also show a possible time slot assignment (optimal in this case) represented by the numbers beside each of the edges in which they are scheduled to transmit. The frequencies assigned to the receiver nodess, 1, and 2, are shown within the boxes. The left-most column in the table of Figure 3.2(c) lists the receivers, and the entries in each row list the nodes from which packets are received by their corresponding receivers in each time slot, represented by the columns. We note that at the end of frame 1, the sink has not yet received packets from nodes 4, 5, and 6; however, as the same schedule is repeated, aggregated packets from nodes 1 and4, and nodes2,5, and6 reach the sink starting from slot1 and slot2, respectively, in frame 2. The entries (1,4) and (2,5,6) represent single packets comprising aggregated data. Therefore, starting from frame 2 the sink continues to receive aggregated data from all the nodes in the network at the rate of once in every 6 time slots. This is when the network reaches a steady state and a pipeline gets established. We measure the data collection rate by the number of time slots required to schedule all the tree edges exactly once per frame, and call it the schedule length. Thus, maximizing the aggregated convergecast throughput under this scenario is equivalent to minimizing the schedule length. In Figure 3.2(b), we illustrate the advantage of using multiple frequencies on 66 the same network. Here, the frequency assigned to node 2 is different from that of node 1 and sink s. As a result, the secondary conflicts are eliminated and more concurrent transmissions are possible, which reduce the schedule length to only 3 time slots. The corresponding table is shown in Figure 3.2(d). We note that multiple frequencies cannot eliminate primary conflicts, which is due the inherent property of the transceivers being half-duplex. We now formally define the multi-channel scheduling problem. Multi-Channel Scheduling Problem: Given a spanning T on a general graph G, and K orthogonal frequencies, we want to assign a frequency to each of the receivers and a time slot to each of the edges inT , such that the schedule length is minimized. 3.3 Multi-Channel Scheduling on General Graphs In Figure 3.2(b), we observed that multiple frequencies, when assigned appropriately to the receivers, can help in reducing the schedule length by eliminating the secondary con- flicts. In this section, we first prove the hardness results of the Multi-Channel Schedul- ing Problem and the Frequency Assignment Problem (defined below) on general graphs. Then we design a polynomial time algorithm when sufficient number of frequencies is available that can remove all the secondary conflicts. 67 3.3.1 Complexity of Multi-Channel Scheduling and Frequency Assignment Problems Multi-Channel Scheduling Problem (decision version): Given a routing tree T on a general graphG, and two positive integersp andq, is there an assignment of time slots to the edges ofT using at mostq frequencies to the receivers, such that the schedule length is no more thanp? Theorem 1. The Multi-Channel Scheduling Problem is NP-complete. We first define the concept of distance-2-edge-coloring on trees, also known as strong edge coloring, and state a known result in Lemma 1 before proving Theorem 1. Definition 2. Two edgese,e ′ ∈ E in a graphG = (V,E) are within distance 2 of each other if either they are adjacent, or both are incident on a common edge. A distance-2-edge-coloring of G requires that every two edges that are within dis- tance 2 of each other have distinct colors. The minimum number of such colors needed is called the strong chromatic index, denoted bysχ ′ (G). Although, findingsχ ′ (G) for general graphs is known to be NP-hard [48], the following result of Lemma 1 due to is true for trees. We use this in the proof of Theorem 1. It is also easy to see that even when all the receivers inG are assigned the same frequency, the minimum schedule length is no more than the strong chromatic index. 68 Lemma 1. The strong chromatic indexsχ ′ (T) of a treeT = (V,E T ) is given by [38]: sχ ′ (T) = max (u,v)∈E T {deg(u)+deg(v)−1}. (3.1) Proof. (of Theorem 1) It is easy to show that the Multi-Channel Scheduling Problem is in NP. Given a particular assignment, one can verify in polynomial time that: (i) at most q frequencies andp time slots are used, (ii) either the receivers of every edge-pair that form secondary conflict are assigned different frequencies, or their edges are on different time slots, and (iii) all adjacent edges are on different time slots. To show NP-hardness, we reduce an instanceG ′ = (V ′ ,E ′ ) of the well known Vertex Color problem to an instanceG = (V,E) of the Multi-Channel Scheduling Problem, as illustrated with an example in Figure 3.3. Our construction is as follows. Let|V ′| = n. For every vertexv i ∈ V ′ , create a setS i ofq pairs of nodes{(u is ,v is ) : s = 1,...,q} inG, and join each pair with an edgee is , treatingu is as the parent ofv is . Then, create q 2 =q(q−1)/2 interfering links between all such pairs in eachS i as follows. Consider eachu is in turn, fors = 1,...,q−1, and create an interfering link fromu is tov il , for all l>s. Thus, every two edges inS i form an interfering edge structure. Next, for every edgee ij = (v i ,v j )∈ E ′ , createq 2 interfering links (and hence, q 2 interfering edge structures) in G by considering the two sets: S i ={(u is ,v is ) : s = 1,...,q} andS j ={(u js ,v js ) :s = 1,...,q}, and creating an interfering link from each 69 1 2 v 1 v 2 v 3 (a) v i u i1 u i2 v i1 v i2 (b) v 11 v 31 v 21 f 1 f 2 f 1 f 2 f 2 f 1 1 1 2 2 2 2 1 1 3 4 5 f 1 f 1 f 1 f 1 f 1 2 v 12 u 11 u 12 v 22 u 21 u 22 v 32 u 31 u 32 3 3 4 6 T b 1 T b 2 T b 3 s (c) Figure 3.3: Reduction for the Multi-Channel Scheduling Problem: (a) Gadget for each v i in G ′ when the number of frequencies q is 2. (b) Instance G ′ of the vertex color problem. (c) InstanceG of the Multi-Channel Scheduling Problem as constructed from G ′ forq = 2. 70 u is to eachv js . Then, for eachS i , construct a binary treeT i b creating additional nodes and edges, and treating the{u is } nodes as leaves, fors = 1,...,q. Finally, treating the roots of T i b ’s as leaves create a binary tree on top of it, and designate the root of it as the sinks. The reduction clearly runs in polynomial time and creates an instance of the Multi-Frequency Scheduling Problem. Next, we show that there exists a solution to the Vertex Color problem using at mostp colors if and only if there exists an assignment inT using at mostq frequencies and at mostp plus a constant number of time slots. SupposeG ′ is vertex colorable using at mostp colors, andv i is assigned colori. First, assign frequencyf s tou is , fors = 1,...,q, in eachS i , and any one of theq frequencies, say f 1 , to all the parents in the rest of tree. Then, assign time slot i to all the q edges connecting the pairs(u is ,v is ), fors = 1,...,q, in eachS i . Because all the receivers inS i are assigned different frequencies, assigning the same time slot to all the edges inS i does not violate the interfering link constraint within eachS i . Also, since only non-adjacent vertices inG ′ may have the same color, two sets of edgesS i andS j that are on the same time slot cannot have interfering links between each other, because interfering links exist betweenS i andS j wheneverv i andv j are adjacent inG ′ . Next, the lowest level edges, which connect to the{u is } nodes, of all the binary treesT i b ,∀i, can be scheduled using at most 2 time slots, because all the edges in each S i are assigned the same slot. Finally, all the remaining edges in the binary tree can be scheduled in polynomial time because a distance-2-edge-coloring on trees can be 71 computed in polynomial time [129], and within number of time slots no more than its strong chromatic index which, from Lemma 1, equals at most5. Conversely, suppose there exists a valid assignment inG that uses at mostq frequen- cies and at most p plus a constant number of time slots. Assign colors to the vertices in G ′ as follows. For each frequency f s , consider the set of edges E ts ={(u ts ,v ts )}, which are assigned time slott, fort = 1,...,p, in order. Since the edges inE ts are on the same time slot and their receivers are on the same frequency, they cannot be part of an interfering edge structure, and so each one of them must lie in a differentS i . Therefore, each edge inE ts has a corresponding vertex inG ′ , no two of which are adjacent. Select those edges inE ts whose corresponding vertices are unassigned, and assign colort to all of them. Repeat the above assignment for all the frequenciesf s , fors = 1,...,q. Clearly this uses at mostp colors and assigns different colors to adjacent vertices. Also, because we run the above procedure over all frequencies and over all time slots, and select an edge fromE ts only when its corresponding vertex is unassigned, exactly one edge gets picked from eachS i . Therefore, every node inG ′ gets a proper color, and the theorem follows. The NP-hardness of the Multi-Channel Scheduling Problem is due to the presence of interfering links that cause secondary conflicts, making scheduling inherently difficult. This is because many subsets of non-conflicting nodes are candidates for transmission in each time slot, and the subset chosen in one time slot affects the number of transmissions in the next time slot. A natural question to ask, therefore, is to find the minimum number 72 1 2 3 v’ 2 v’ 3 v’ 4 u 1 v 1 v 2 v 3 v 4 u 2 u 3 u 4 f 1 f 2 f 3 f 3 f 1 v’ 1 s Figure 3.4: Reduction from the Vertex Color problem to the Frequency Assignment Prob- lem. The left part of the figure shows as instance of the Vertex Color problem, which is colored using 3 colors. The right part of the figure shows the corresponding instance of the Frequency Assignment Problem, which also requires3 frequencies to remove all the secondary conflicts. of frequencies that is sufficient to eliminate all the secondary conflicts, which will then reduce the problem from being on a graph to being on a tree. We say that a secondary conflict is eliminated if the two receivers of an edge-pair are assigned different frequen- cies. We define this as the Frequency Assignment Problem, and prove its hardness result in the following. Frequency Assignment Problem (decision version): Given a tree T on a general graphG and an integerq, is there a frequency assignment to the receivers inT using at mostq frequencies such that all the secondary conflicts are removed? Theorem 2. The Frequency Assignment Problem is NP-complete. Proof. It is easy to show that the Frequency Assignment Problem is in NP. Given a particular assignment, one can verify in polynomial time (i) if at most q frequencies 73 are used, and (ii) if the receivers of every edge-pair that cause secondary conflicts are assigned different frequencies. To show NP-hardness, we reduce an instance G ′ = (V ′ ,E ′ ) of the Vertex Color problem to an instanceG = (V,E) of the Frequency Assignment Problem, as illustrated in Figure 3.4. For every vertex v ′ i ∈ V ′ , create two nodes u i and v i in G, and join them with an edge e i = (u i ,v i ), treating u i as the parent of v i . For every edge e ′ ij = (v ′ i ,v ′ j )∈ E ′ , create an interfering link in G between u i and v j . Finally, create a root node s, and add an edge e is = (u i ,s) from each u i to s, treating s as the parent of u i . This is an instance of the Frequency Assignment Problem, where the tree is given by T = (V ={u i }∪{v i }∪{s},E T ={e i }∪{e is }). Clearly, the reduction runs in polynomial time. Next, we show that there exists a solution to the Vertex Color problem using at mostq colors if and only if there exists a solution to the Frequency Assignment Problem using at mostq frequencies. Suppose G ′ is vertex colorable using at most q colors, and suppose v ′ i is assigned color j. Assign frequency f j to u i in G, and any one of the frequencies, say f 1 , to s. Clearly, this needs at most q frequencies. Since no two adjacent vertices v ′ i and v ′ j in G ′ are assigned the same color, no two nodesu i andu j inG, which are the receivers of an interfering edge structure, are assigned the same frequency, because by construction a secondary conflict exists between u i and v j whenever v ′ i and v ′ j are adjacent in G ′ . Therefore, this frequency assignment removes all the secondary conflicts. 74 Conversely, suppose there exists a solution to the Frequency Assignment Problem using at most q frequencies. If u i is assigned frequency f j , assign color j to v ′ i in G ′ . Clearly, this requires at mostq colors, because the number of receivers inG is one more than the number of vertices inG ′ . Since all the interfering link constraints are removed by such a frequency assignment, every two nodesu i andu j , which are receivers of an edge-pair that cause secondary conflict, are assigned different frequencies. And since their corresponding verticesv ′ i andv ′ j are adjacent inG ′ , they will be assigned different colors, thus yielding a proper coloring ofG ′ . Therefore, the theorem follows. 3.3.2 Scheduling Under Sufficient Frequencies Since finding the minimum number of frequencies required to remove all the secondary conflicts in a general graph in NP-hard, in this section we give an upper bound and show that when sufficient number of frequencies is available, the scheduling problem can be solved optimally in polynomial time. Lemma 2. Create a constraint graph G C = (V C ,E C ) from the original graph G as follows (cf. Figure 3.5(a) and 3.5(b) for an illustration). For each receiver (parent) in G, create a node inG C . Connect any two nodes inG C if their corresponding receivers inG are incident on two edges that form a secondary conflict. Then, the numberK max of frequencies that will be sufficient to remove all the secondary conflicts is bounded by: K max ≤ Δ(G C )+1, whereΔ(G C ) is the maximum node degree inG C . 75 Algorithm 2 BFS-TIMESLOT-ASSIGNMENT 1. Input:T = (V,E T ) 2. whileE T 6=φ do 3. e← next edge fromE T in BFS order; 4. Assign minimum time slot toe respecting adjacency constraint; 5. E T ←E T \{e}; 6. end while Proof. Since we create an edge between every two nodes in G C whenever their cor- responding receivers inG form a secondary conflict, assigning different frequencies to every such receiver-pair inG is equivalent to assigning different colors to the adjacent nodes inG C . Thus, K max is equal to the minimum number of colors needed to vertex colorG C , called its chromatic number, denoted byχ(G C ). Sinceχ(G)≤ Δ(G)+1, for any arbitrary graphG, the lemma follows. As illustrated in Figure 3.5(b), the frequencies assigned to the receivers inG C are as follows: frequency f 1 to nodes 1 and 2; f 2 to nodes 3, 4, and 8; f 3 to nodes s, 5, and 6; andf 4 to node 7. This particular frequency assignment is according to the heuristic called Largest Degree First [91], in which we consider the nodes inG C in non-increasing order of their degrees and assign the first available frequency such that no two adjacent nodes have the same frequency. Once all the secondary conflicts are eliminated by an appropriate frequency assign- ment to the receivers, the following time slot assignment scheme, called BFS-TIMESLOT- ASSIGNMENT (running in O(|E T | 2 ) time), presented in Algorithm 2 minimizes the schedule length. In each iteration (lines 2-6) of the algorithm, an edge e is chosen in 76 1 2 3 4 5 6 7 8 9 10 11 12 13 15 16 14 s (a) f 1 1 2 3 4 5 6 7 8 f 2 f 4 f 1 f 2 f 2 f 3 f 3 f 3 s (b) Figure 3.5: (a) Original graph G; receiver nodes are shaded. (b) Constraint graph G C and a frequency assignment to the receivers according to Largest Degree First. Here, 4 frequencies are sufficient to remove all the secondary conflicts, i.e., frequencies on adjacent nodes are different. 77 1 2 3 4 5 6 7 8 9 10 11 12 13 15 16 14 s 1 1 1 1 1 2 3 2 2 2 2 2 2 3 3 3 f 1 f 2 f 4 f 1 f 2 f 2 f 3 f 3 f 3 Figure 3.6: An optimal time slot assignment according to Algorithm BFS-TIMESLOT- ASSIGNMENT yielding a schedule length3 after all the secondary conflicts are removed using4 frequencies. the Breadth-First-Search (BFS) order, starting from any node, and is assigned the min- imum time slot that is different from all its adjacent edges. We prove in Theorem 3 that such an assignment gives a minimum schedule length equal to the maximum degree Δ(T) of T . An illustration of this time slot assignment scheme for the same network topology as in Figure 3.5(a) is shown in Figure 3.6. Theorem 3. After all the secondary conflicts are removed, Algorithm BFS-TIMESLOT- ASSIGNMENT gives a minimum schedule length equal toΔ(T). Proof. The proof is by induction oni. LetT i = (V i ,E i T ) denote the subtree ofT in the i th iteration constructed in the BFS order, whereE i T comprises all edges that are already 78 assigned a slot, andV i comprises the set of nodes on which the edges inE i T are incident. Note that,|E i T | = i, because at every iteration exactly one edge is assigned a slot. For i = 1, clearly the number of slots used is1, equal toΔ(T 1 ). Suppose the number of time slotsN(i) needed to schedule the edges inT i isΔ(T i ). In the(i+1) th iteration, after assigning a slot to the next edge in BFS order, the number of slots needed inT i+1 can either remain the same as before, or increase by one. Thus, N(i+1) = max{N(i),N(i)+1}. (3.2) If it remains the same,N(i+1) is still the maximum degree ofT i+1 at end of(i+1) th iteration. Otherwise, if it increases by one, the new edge must be incident on a nodev ∗ , common to both T i and T i+1 , such that the number of incident edges on v ∗ that were already assigned a time slot at the end ofi th iteration was Δ(T i ). This is so because in the BFS traversal, all the edges incident on a node are assigned a slot first before moving on to the next node, and because the slot assigned to the new edge is the minimum possible that is different from all that already assigned to the edges incident onv ∗ until thei th iteration. Thus, at the end of(i+1) th iteration, the number of slots usedN(i)+1 is equal to the number of assigned edges incident onv ∗ which, in turn, equals Δ(T i+1 ). This proves the inductive step. Therefore, it holds at every iteration of the algorithm until the end wheni =|V|−2, yielding a schedule length equal to the maximum degree Δ(T) = Δ(T |V|−1 ). Since assigning different time slots to the adjacent edges in T is 79 equivalent to edge coloringT , which requires at least Δ(T) colors, the schedule length is minimum. 3.4 Multi-Channel Scheduling on Unit Disk Graphs 3.4.1 Known Routing Topologies In the previous section, we showed that with sufficient number of frequencies available, all the secondary conflicts can be removed and a minimum length schedule can be found in polynomial time. However, typically there is a limitation on the number of frequencies over which a given transceiver can operate, and as shown in Theorem 1, the scheduling problem is NP-hard for a given (constant) number of frequencies. In this section, we take into account the limited number of frequencies available on current WSN hardware, and design an algorithm for the Multi-Channel Scheduling Problem that achieves a con- stant factor approximation on the optimal schedule length. We assume that the routing topology is known a priori. We divide the 2-D deployment region into a set of square grid cells{c i }, each of side lengthα. We define two cells to be adjacent to each other if they share a common edge or a common grid point. Thus, a cell can have either 3, 5, or 8 adjacent cells depending on whether it is a corner cell, an edge cell, or an interior cell, respectively. In our approach to design an approximation algorithm for minimizing the schedule length, we decouple the frequency assignment and the time slot assignment phases. We first 80 assign the frequencies to the receivers in T , such that the maximum number of nodes transmitting on the same frequency is minimized. Then, we employ a greedy time slot assignment scheme. We describe the two phases in detail below. 3.4.1.1 Frequency Assignment LetR i ={v 1 ,...,v i } denote the set of receivers on a given routing tree T that lie in cellc i , and letm :R i →{f 1 ,...,f K } be a mapping that assigns a frequency to each of these receivers. Note that,m(v j ) = f k implies that all the children ofv j transmit on frequencyf k due to our receiver-based frequency assignment strategy. Definition 3. We define a load-balanced frequency assignment in cellc i as an assignment of the K frequencies to the receivers inR i , such that the maximum number of nodes transmitting on the same frequency is minimized. To express this formally, we define the load on frequencyf k in cellc i under mapping m as the total number of children of all the receivers inR i that are assigned frequency f k , and denote it byℓ m i (f k ). We call the number of children of nodev j its in-degree, and denote it bydeg in (v j ). Thus, ℓ m i (f k ) = X v j ∈R i :m(v j )=f k deg in (v j ). (3.3) Then, a load-balanced frequency assignmentm ∗ inc i is defined as: m ∗ = argmin m max k {ℓ m i (f k )}. (3.4) 81 Algorithm 3 FREQUENCY-GREEDY 1. for all non-empty cellc i do 2. Sort receivers inR i in non-increasing order of in-degrees; 3. Suppose:deg in (v 1 )≥deg in (v 2 )≥...≥deg in (v i ); 4. forj = 1 toi do 5. Find frequencyf k that is least loaded (breaking ties arbitrarily); 6. Assignf k tov j ; 7. end for 8. end for We denote the load on the maximally loaded frequency under mapping m ∗ in cell c i by L m ∗ i . We show that finding a load-balanced frequency assignment is equivalent to scheduling jobs on identical machines with the goal to minimize the makespan (i.e., the last finishing time of the given jobs), and prove this equivalence in Lemma 3. Since minimizing the makespan is known to be NP-hard [48], so is a load-balanced frequency assignment. Therefore, we resort to designing an approximation algorithm. In Algorithm 3, we describe a greedy assignment scheme called FREQUENCY-GREEDY that achieves a constant factor approximation on the optimal load. The basic idea of the algorithm is as follows: For each cellc i , we sort the receivers inR i in non-increasing order of their in-degrees; let this order be: v 1 ,...,v i . Then, starting fromv 1 , we assign to each subsequent nodev j a frequency that has the least load on it so far, breaking ties arbitrarily. In Figure 3.7(a), we illustrate this scheme for two frequenciesf 1 andf 2 , and three receivers v 1 , v 2 , and v 3 , sorted in non-increasing order of their in-degrees. First, nodev 1 is assigned frequencyf 1 , incurring a load of 5, as it has 5 children. Then nodev 2 is assigned frequencyf 2 , giving a load of 3. Finally, nodev 3 is also assigned frequency 82 f 2 , becausef 2 is the least loaded so far. In this particular case, the assignment achieves an optimal load of 5 on both frequencies. In general, the following approximation holds. Lemma 3. Algorithm FREQUENCY-GREEDY in each cellc i gives a 4 3 − 1 3K approx- imation on the maximum load L m ∗ i achieved by an optimal load-balanced frequency assignment schemem ∗ . Proof. Consider a job scheduling problem withK identical machinesm 1 ,...,m K , and i jobs 1,...,i. Suppose, executing a jobj on any machine takes timet j > 0. Thus, if Ψ(k) denotes the set of jobs assigned to machinem k , then the total timem k takes is given by P j∈Ψ(k) t j , and the makespan is defined as the maximum of this quantity over all the machines, i.e., max 1≤k≤K { P j∈Ψ(k) t j }. The objective of the job scheduling problem is to find an assignment of the jobs to the machines such that the makespan is minimized. In the case of a load-balanced frequency assignment, we map each receiverv j ∈ R i to jobj, and its in-degreedeg in (v j ) to timet j . We also map each frequencyf k to ma- chinem k . The load on frequencyf k is therefore equal to the total timem k takes. Thus, minimizing the maximum load over all the frequencies is equivalent to minimizing the makespan over all the machines. Under this mapping, Algorithm FREQUENCY-GREEDY is identical to Graham’s list scheduling algorithm according to the Longest Process- ing Time (LPT) [57] first, which achieves a 4 3 − 1 3K approximation on the minimum makespan. Therefore, the lemma follows. 83 v 3 v 2 v 1 f 1 f 2 f 2 (a) e e’ f f g 1 g 1 g 1 g 1 g 2 g 2 g 2 g 2 g 3 g 3 g 3 g 3 g 4 g 4 g 4 g 4 a>2R u u’ v v’ e e’ f f u u’ v v’ (b) Figure 3.7: (a) Frequency assignment according to Algorithm FREQUENCY-GREEDY. Load on frequencies: ℓ (f 1 ) = 5,ℓ (f 2 ) = 5. White colored nodes transmit on frequency f 1 ; gray colored nodes transmit on frequencyf 2 . (b) Four pair-wise disjoint sets of time slotsγ 1 ,γ 2 ,γ 3 , andγ 4 schedule the whole network. Each setγ i maps to a unique color. Edges whose receivers lie in non-adjacent cells can be scheduled simultaneously, e.g., edges(u,v) and(u ′ ,v ′ ). 84 3.4.1.2 Time Slot Assignment Once the receivers in each cellc i are assigned frequencies according to Algorithm FREQUENCY- GREEDY, we employ a greedy time slot assignment scheme for the whole network. Lemma 4. Let γ i denote the set of time slots needed to schedule all the edges in cell c i . Then, the minimum schedule length Γ for the whole network is bounded by: Γ≤ 4·max i |γ i |, for allα≥ 2R. Proof. Consider the grid cells shown in Figure 3.7(b). In our network model, there exists a link between any two nodes if and only if they are at most a distanceR from each other. This implies that interference is spatially restricted and time slots can be reused across cells that are spatially well separated. In particular, as illustrated in Figure 3.7(b), for all α≥ 2R, two edgese = (u,v) ande ′ = (u ′ ,v ′ ), whose respective receiversv andv ′ lie in non-adjacent cells must have their transmittersu andu ′ more than distanceR away from the other receiver, and so can be scheduled on the same time slot regardless of the frequency assigned to their receivers. If each set of time slotsγ i represents a unique color, then the whole network can be scheduled using at most four distinct colors such that no two adjacent cells have the same color, i.e., four pair-wise disjoint sets of time slots, as shown in Figure 3.7(b). Therefore, the total number of time slots required is 4 times the maximum number of slots in any setγ i . Lemma 5. If L φ i denote the load on the maximally loaded frequency in cell c i under mapping φ :R i →{f 1 ,...,f K } achieved by Algorithm FREQUENCY-GREEDY, then 85 any greedy time slot assignment scheme can schedule all the edges in cellc i within2·L φ i time slots. Proof. Consider a multi-graphH = ({f 1 ,...,f K },E ′ ), where for each edgee = (v i ,v i ′ ), v i ,v i ′ ∈R i withφ(v i )6= φ(v i ′ ), we have an edge (φ(v i ),φ(v i ′ ))∈ E ′ . Note that these will be multi-edges; letn(f k ,f k ′ ) denote the number of edges betweenf k andf k ′ inH. Then,deg(f k )≤l φ i (f k ), wherel φ i (f k ) is the load onf k underφ in cellc i . By Ore’s theo- rem [115], which generalizes Vizing’s theorem [107] for edge coloring on multi-graphs, it follows that the edges inH can be colored using max k {l φ i (f k )} colors. Therefore, all edges of the forme = (v i ,v i ′ ) between two nodes inR i with different frequencies can be colored inmax k {l φ i (f k )} =L φ i colors. All the remaining edges either have only one end-point inR i , or have both end-points inR i , with the same frequency on their receivers; letS(f k ) denote the set of such edges with their end-point inR i that are assigned frequencyf k . Note that|S(f k )|≤ l φ i (f k ), and edges e∈ S(f k ),e ′ ∈ S(f k ′ ) can be assigned the same time slot if f k 6= f k ′ . So all the remaining edges can be scheduled in max k |S(f k )|≤ max k {l φ i (f k )} time slots. Therefore, all edges inc i can be scheduled within 2·max k {l φ i (f k )} = 2·L φ i time slots, and the lemma follows. We now prove the key approximation result on the optimal schedule length. Theorem 4. Given a routing treeT on an arbitrarily deployed network in 2-D, andK orthogonal frequencies, there exists a greedy algorithmA that achieves a constant factor 86 8μ α · 4 3 − 1 3K approximation on the optimal schedule length, whereμ α > 0 is a constant for anyα≥ 2R. Proof. AlgorithmA consists of two phases. In Phase 1, we run algorithm FREQUENCY- GREEDY to assign theK frequencies to the receivers in each cell. In Phase 2, we greedily schedule a maximal number of edges in each time slot. Let the schedule length of algo- rithmA beΓ A , and that of an optimal algorithm beOPT . Due to the presence of interfering links, there exists a constantμ α > 0 depending on cell sizeα and deployment distribution, such that at mostμ α edges in any cell, whose receivers are on the same frequency, can be scheduled in the same time slot by OPT . Now, regardless of the assignment chosen by an optimal strategy, it will take at least max i {L m ∗ i /μ α } time slots to schedule all the edges, becauseL m ∗ i is the minimum of the maximum number of edges that are on the same frequency in cellc i . Thus, OPT≥ 1 μ α ·max i L m ∗ i . (3.5) By running FREQUENCY-GREEDY in cellc i , Lemma 3 implies L φ i ≤ 4 3 − 1 3K ·L m ∗ i , (3.6) 87 and by scheduling a maximal number of edges in each time slot, Lemma 5 implies|γ i |≤ 2·L φ i . Then, from Lemma 4 Γ A ≤ 4·max i {|γ i |} ≤ 8·max i n L φ i o ≤ 8·max i 4 3 − 1 3K ·L m ∗ i ≤ 8μ α · 4 3 − 1 3K ·OPT. (3.7) Therefore, the theorem follows. 3.4.2 Unknown Routing Topologies In this subsection, we consider the scenario when the routing topology is not known to the algorithm designer a priori. The significance of this situation is that an optimal algorithm can choose the best routing topology given a deployment of nodes, and as a designer we would want to find properties of a routing topology that could still guarantee a constant factor approximation on the optimal schedule length. Theorem 5. Given a network modeled as a UDG andK orthogonal frequencies, there exists an algorithmH that achieves a constant factor 8μ α · Δ C approximation on the optimal schedule length, so long as the maximum in-degree of any node in the routing tree is bounded by a constant Δ C > 0, whereμ α > 0 is a constant for a given cell size α≥ 2R. 88 Proof. Let V i denote the set of nodes in cell c i . We note that the set of receivers on the tree depends on the routing topology, but the total number of nodesV i (in each cell) depends only on the graph. Because an optimal algorithm can concurrently schedule at most a constant number μ α > 0 of nodes (edges) in any cell c i whose parents are on the same frequency, the best it could do with K frequencies is distribute the nodes in V i evenly among all the frequencies, so that⌈| V i |/K⌉ is the minimum of the maximum number of nodes transmitting on the same frequency. Thus, OPT ≥ max i 1 μ α |V i | K ⇒ max i |V i | K ≤ μ α ·OPT (3.8) SupposeR i (T) ={v 1 ,...,v n } denote the set of receivers inc i on an arbitrary routing treeT , and supposeΔ in (T) be the maximum in-degree of any node inT . Define a cyclic frequency assignment under mappingψ :R i (T)→{f 1 ,...,f K } as follows: ψ(v i ) =        i modK, if i6=qK K, if i =qK (3.9) whereq∈N + is a positive integer. It is easy to see that the maximum number of receivers that are on the same frequency is⌈|R i (T)|/K⌉ . Therefore, the loadL ψ i on the maximally loaded frequency in cellc i is bounded by: 89 L ψ i ≤ |R i (T)| K · max v j ∈R i (T) deg in (v j ) ≤ |V i | K ·Δ in (T) (3.10) The load L φ i on the maximally loaded frequency produced by FREQUENCY-GREEDY cannot be more thanL ψ i ; thus L φ i ≤L ψ i ≤ |V i | K ·Δ in (T) (3.11) Then, scheduling a maximal number of edges in each time slot and using Lemma 4, Lemma 5 as before, and (3.11) it follows that: Γ H ≤ 8·max i |V i | K ·Δ in (T) (3.12) Since|V i | and Δ in (T) are independent of each other, we can take the maximum sepa- rately on the two terms as: Γ H ≤ 8·max i |V i | K ·max i Δ in (T) = 8·max i |V i | K ·Δ in (T) ≤ 8μ α ·Δ in (T)·OPT. (3.13) 90 Thus, (3.13) implies that so long as the maximum in-degree of a node inT is bounded by a constant Δ C > 0, the theorem holds. Although finding a degree-bounded spanning tree on a general graph is known to be NP-hard [48], for any UDG it is always possible to find a spanning tree of degree at most5 [153]. Therefore, the theorem follows. 3.5 Multi-Channel Scheduling on General Disk Graphs In this section, we relax our assumption of the nodes having a uniform transmission range, and consider networks where nodes could transmit at different power levels re- sulting in different transmission ranges. Such networks could be modeled as general disk graphs, where a directed edge from node u to node v exists if d(u,v)≤ R(u), where R(u) is the transmission range ofu. We consider only those edges that are bidirectional, i.e., bothR(u) andR(v) are greater than or equal to their Euclidean distance. For an edgee = (u,v), we use the convention thatu is the transmitter andv is the receiver. We denote the Euclidean length ofe byℓ (e). DefineI(e) as the set of edges that are either adjacent toe, or form a secondary conflict withe. Also, defineI ≥ (e)⊆ I(e) as the subset of edges ofI(e) whose end points have larger disks than those ofe, i.e., I ≥ (e) = {e ′ = (u ′ ,v ′ ) :e ′ ∈I(e) and max{R(u ′ ),R(v ′ )}≥ max{R(u),R(v)}}. As before, we separate the two phases of frequency assignment and time slot assign- ment. Our goal in the frequency assignment phase is to assign the frequencies in such a 91 way that minimizes the maximum number of edges that interfere with any given edge. Once this is done, we devise a time slot assignment strategy that optimizes the schedule length. 3.5.1 An Integer Linear Programming Formulation We formulate the frequency assignment subproblem as a 0-1 Integer Linear Program (ILP). Define indicator variablesX vk for edgee = (u,v) as follows: X vk =        1, if v is assigned frequencyf k 0, otherwise (3.14) A frequency assignment is therefore a 0-1 assignment to the variables X vk , for all e = (u,v)∈ E T , and for allf k . Furthermore, for an edgee = (u,v) on frequencyf k , defineZ ek as the total number of edges inI ≥ (e) that are also on frequencyf k . Thus, Z ek = X e ′ =(u ′ ,v ′ )∈I ≥ (e) X v ′ k = X v ′ n(e,v ′ )X v ′ k , (3.15) where n(e,v ′ ) =|{e ′ = (u ′ ,v ′ )∈I ≥ (e) :ℓ (e ′ )≥ℓ (e)}|. If we are given a frequency assignment ~ X, the following lemma (based on [88], [61]) shows how the schedule length is related to theZ ek ’s. 92 Lemma 6. ( [88], [61]) LetX vk andZ ek be as defined above. Then,Ω(max e,k {Z ek }) is a lower bound on the length of any schedule for the edges. Also, it is possible to schedule all the edges usingmax e,k {Z ek } time slots. Proof. (sketch) The lower bound directly follows from [88], [61]. We briefly sketch a greedy scheduling algorithm, which implies the upper bound. LetE T ={e 1 ,e 2 ,...,e n−1 }, with the edges numbered so thatℓ (e 1 )≥ℓ (e 2 ).... Our scheduling algorithm assigns a time slott(e i ) for each edgee i in the following manner: Consider the edges in the ordere 1 ,...,e n−1 , and for edgee i , assign the smallest available time slott =t(e i ) so that that following two conditions satisfy: • for each edgee j withj <i having the same receiver ase i , we havet(e i )6=t(e j ), • for eache j such thate i ,e j ∈ E k T (i.e., they are assigned the same frequency), we havet(e i )6=t(e j ). It can now be shown that the number of slots needed is at mostmax e,k {Z ek }. 93 The above lemma implies that we should find a frequency assignment that minimizes max e,k {Z ek }. We formulate this by the following ILP. Minimize λ subject to : ∀e = (u,v), ∀f k : X v ′ n(e,v ′ )X vk ≤λ (3.16) ∀e = (u,v) : X k X vk = 1, (3.17) ∀e = (u,v), ∀f k : X vk ∈{0,1} (3.18) The second constraint guarantees that each of the receivers is assigned a single fre- quency. Since solving this ILP is NP-hard, we first find the solution to the linear pro- gramming (LP) relaxation of it, which is obtained by modifying the third constraint to include fractional values for the indicator variables as X vk ∈ [0,1]. Let the optimum (fractional) values for the indicator variablesX vk obtained by solving the LP relaxation beX ∗ vk , and letλ ∗ be the corresponding objective value. We now construct integral ran- dom variablesY vk by rounding the fractional valuesX ∗ vk in the following manner: For eachv andf k , we chooseY vk = 1 with probabilityX ∗ ik . This rounding is done in a mutu- ally exclusive manner, so that for eachv, there is exactly one frequencyf k withY vk = 1; this can be done since P k X ∗ vk = 1. 94 Lemma 7. LetY vk be the rounded solution, as described above. Then, max e=(u,v),k    X e ′ =(u ′ ,v ′ )∈I ≥ (e) Y v ′ k    =O(Δ(T)logn·λ ∗ ), (3.19) with probability at least1− 1 n . Proof. Because of our randomized rounding strategy E[Y ik ] =Pr[Y ik = 1] =X ∗ ik (3.20) Let b Z ek = X e ′ =(u ′ ,v ′ )∈I ≥ (e) Y v ′ k = X v ′ n(e,v ′ )Y v ′ k . (3.21) Therefore, by linearity of expectation, E h b Z ek i = X v ′ n(e,v ′ )E[Y v ′ k ] = X v ′ n(e,v ′ )X ∗ vk ≤ λ ∗ . (3.22) Next, note thatn(e,v ′ )≤ Δ(T) for anye,v ′ . Therefore, by the weighted version of the Chernoff bound [108], it follows that for any edgee and frequencyf k , we have Pr h b Z ek ≥ Δ(T)logn·λ ∗ i ≤ 1 n 3 . (3.23) 95 Since the number of edges inT and the number of frequencies are bothO(n), we have Pr max e,k n b Z ek o ≥ Δ(T)logn·λ ∗ ≤ X e,k Pr h b Z ek ≥ Δ(T)logn·λ ∗ i ≤ 1 n , (3.24) where the first inequality follows from the union bound. The above rounded solution, along with the scheduling algorithm from Lemma 6, leads to the following approximation guarantee. Theorem 6. The schedule constructed by Lemma 6, along with the frequency assign- ment using the above randomized rounding procedure results in a schedule of length O(Δ(T)logn) times the optimum. 3.6 Evaluation In this section, we evaluate the performance of our scheduling algorithm using MATLAB simulations on networks modeled as random geometric graphs. We generate connected networks by uniformly and randomly placing nodes in a square region of maximum size 200×200, and connecting any two nodes that are at most distanceR = 25 apart. 96 0 0.1 0.2 0.3 0.4 0.5 0 5 10 15 20 25 30 Network Density Number of Frequencies on SPT Largest Degree First Upper Bound Δ(G C ) + 1 Figure 3.8: Number of frequencies required to remove all the secondary conflicts as a function of network density on shortest path trees. 3.6.1 Frequency Bounds In Figure 3.8, we compare the number of frequencies needed as a function of network density to remove all the secondary conflicts on shortest path trees, as calculated from the upper bound Δ(G C )+1, and that from Largest Degree First (LDF) assignment. Here, the number of nodes N is fixed at 200, and the length l of the square region is varied from200 to20; so the density,d =N/l 2 , varies from0.005 to0.5. The plot shows that the number of frequencies initially increases with density, reach- ing a peak at around 0.025, and then steadily going down to one. This happens because of two opposing factors. As the density goes up, the parents link up with more and more new nodes, increasing the number of secondary conflicts; however, at the same time, the number of parents on the SPT gradually decreases because the deployment region gets 97 smaller in size. As we keep on increasing the density further, the latter effect starts dom- inating, and since the number of frequencies required depends on the number of parents in the constraint graph G C , this number goes down as well, until the network finally turns into a single hop network with the sink as the only parent. We also observe that for sparser networks there is a significant gap between the upper bound and the LDF scheme, as opposed to that in denser networks. This is because in sparser settings there are many parents, resulting in higher a Δ(G C ) value, and assigning a distinct frequency to the largest degree parent according to the LDF scheme removes more secondary con- flicts at every step than it does for denser settings when the parents are fewer and have comparable degrees. 3.6.2 Multiple Frequencies on Schedule Length Since multiple frequencies can eliminate interfering links and reduce the schedule length, we now evaluate their effects on SPT for our proposed multi-channel scheduling algo- rithm. Figure 3.9 shows the schedule length with increasing network size for one, three, and five frequencies. We observe that the gains of utilizing multiple frequencies increase as the network gets larger in size. However, the schedule lengths with three and five fre- quencies are almost the same. This is due to very high node degrees on an SPT resulting in many primary conflicts in the network than secondary conflicts. Recall that primary conflicts are not removable using multiple frequencies. 98 100 200 300 400 500 600 700 800 5 10 15 20 25 30 35 40 45 50 Number of Nodes Schedule Length SPT, K=1 SPT, K=3 SPT, K=5 Figure 3.9: Effect of multiple frequencies: Schedule Lengths for SPT with network size forK = 1,3, and5 frequencies. 3.6.3 Scheduling Under SINR Model In our evaluation so far, we have considered the graph-based interference model and evaluated the proposed scheduling algorithm. However, since the protocol model de- viates from the more realistic Signal-to-Interference-plus-Noise-Ratio (SINR) model in capturing cumulative interference from distant transmitters, thus sometimes under/over estimating interference, the schedules generated from the protocol model might conflict under the SINR model. To measure the amount of conflict, we plot in Figure 3.10 the percentage of nodes on an SPT whose schedules calculated according to the protocol model conflict under the SINR model. The parameters for the SINR model are chosen according to the CC2420 radio parameters with receiver sensitivity−95 dB, path-loss 99 100 200 300 400 500 600 700 800 0 5 10 15 20 25 30 35 40 Number of Nodes Conflict Links (%) SPT, K=1 SPT, K=3 SPT, K=5 Figure 3.10: Percentage of nodes whose schedules conflict in the SINR model for differ- ent network sizes and three different number of frequencies (K = 1,3,5) on an SPT. exponent 3.5, and transmit power−6 dB. We note that for a given number of frequen- cies, as the network gets denser the amount of conflict increases, reaching almost 40% for the densest deployment with one frequency. However, with multiple frequencies, the amount of conflict is much lesser. This indicates that although the protocol model performs reasonably well under multiple frequencies, more sophisticated SINR based scheduling algorithms are needed when there is only one frequency. 3.7 Summary In this chapter, we considered the aggregated convergecast problem in tree-based sen- sor networks, and proposed multi-channel scheduling algorithms to maximize the sink throughput. We proved two NP-completeness results related to scheduling for general 100 graphs – one on finding an optimal time slot assignment that would minimize the sched- ule length, and the other on finding the minimum number of frequencies that would remove all the secondary conflicts. We showed that once all the secondary conflicts are removed using multiple frequencies, the scheduling problem reduces to on a tree from being on a graph, and can be solved optimally in polynomial time. When the number of frequencies is limited, as in most current sensor network hard- ware, we considered the scheduling problem on random geometric graphs for a given routing tree, and designed approximation algorithms that have provably good, worst-case performance bounds. In particular, for UDG our proposed algorithm gives a constant fac- tor approximation, and for general disks graph it gives anO(Δ(T)logn) approximation. When the routing tree is not known a priori, we showed that a constant factor approxima- tion on the optimal schedule length is still achievable so long as the maximum in-degree of any node in the tree is bounded by a constant. We evaluated our algorithms through simulations and showed various trends in performance for different network parameters. We observed that multiple frequencies can reduce the schedule length up to a certain extent, beyond which the structure of the routing tree plays an important role in the scheduling performance. Lastly, evaluating our algorithms on the SINR model revealed some of the weaknesses of the graph-based interference model, and motivated the need for further SINR-based scheduling algorithms. 101 Chapter 4 Multi-Channel SINR Scheduling for Fast Convergecast 4.1 Motivation In Chapter 3, we considered the aggregated convergecast problem assuming a simple, graph-based interference model and designed efficient algorithms that have provably good worst-case performance bounds. However, in our evaluation, we observed in Fig- ure 3.10 that the schedule computed by such a graph-based model might be conflicting with respect to a more realistic fading channel model, such as the SINR model, i.e., not all concurrent transmissions might be successful under SINR. Although the percentage of conflicts reduces with multiple frequencies, such a conflicting schedule might need for more retransmissions and undesirable control overhead. In this chapter, we consider the multi-channel scheduling problem under the SINR model and extend our previous algorithms. The scheduling conflict occurs because of the following reason. In a graph-based model, the network consists of two graphs: a connectivity graph and an interference 102 graph, where the vertices represent nodes in both the graphs. An edge between any two nodes exists in the connectivity graph if they are within the transmission range R of each other. Likewise, an edge in the interference graphs exists if two nodes are within a certain distance from each other, for example, within twice the transmission range, as in the 2-hop interference model. The set of edges in the interference graph is therefore typically a superset of the set of edges in the connectivity graph. Under such a setting, a transmission from a senders i to a receiverr i is always successful so long as there is no concurrent transmission from any other node s j that has an edge to node r i in the interference graph, i.e., there is no secondary conflict (assuming primary conflicts have already been taken care of). As a result, solving the scheduling problem often boils down to finding independent sets and coloring in the graphs. Although the concept of an edge in the interference graph, which is defined based on a fixed distance between the nodes, lends to elegant algorithmic analysis by assuming interference to be a binary and local phenomenon, it is an oversimplification of the phys- ical laws underlying wireless communications. An electromagnetic signal while propa- gating in the wireless medium does not stop suddenly at a particular distance boundary; it continues to propagate infinitely in space until its intensity gradually fades away with distance and due to the presence of obstacles. Thus, while a single transmission origi- nated outside the interference range of a particular receiverr i may not disrupt the signal, the effect of many, possibly far-away located, concurrent transmitters might accumulate 103 and prevent successful reception at noder i . This illustrates that a set of concurrent trans- missions forming an independent set in the interference graph might be conflicting in the underlying SINR model, and might therefore represent an unfeasible schedule. The rest of the chapter is organized as follows. In Section 4.2, we describe the SINR model and our problem formulation. In Section 4.3, we present an approximation algo- rithm for multi-channel scheduling under the SINR model... 4.2 Preliminaries 4.2.1 The SINR Model The SINR model, often known as the physical interference model, offers a more realis- tic representation of wireless communications. A successful transmission in this model accounts for the cumulative interference generated by all concurrent transmissions in the whole network, as opposed to the binary and local interference dictated by the presence of interfering edges in a graph-based model. Since the SINR depends on which nodes are being scheduled simultaneously, it is not possible to build an interference graph a priori to finding a scheduling solution. In the SINR model, the received signal strength decays proportionally to the inverse of the sender-receiver distance raised to the power of the so called path-loss exponent α, whose value is a constant and depends on external conditions (such as obstacles, humidity, temperature), as well as on the exact sender-receiver distance. Typically, it is 104 assumed thatα lies between 2 and 4. Thus, if all the nodes transmit at a uniform powerP on frequencyf, the received signal powerP r i (s i ) from senders i to its intended receiver r i is given by P r i (s i ) = P d(s i ,r i ) α . (4.1) Similarly, the received signal power at r i from any other node s j ,j 6= i transmitting simultaneously withs i on the same frequencyf , referred to as interference and denoted byI r i (s j ), is given by I r i (s j ) = P d(s j ,r i ) α , j6=i (4.2) The cumulative interference atr i is defined as the sum of the interferences caused by all the nodes that are transmitting on the same frequencyf simultaneously withs i , and is denoted byI r i = P j6=i I r i (s j ). In the SINR model, a transmission from senders i to its intended receiverr i is defined to be successful if and only if the ratio, SINR r i , of the received signal power to the cumulative interference plus an ambient noise powerN at receiverr i is more than a certain hardware-dependent thresholdβ, i.e., if and only if SINR r i = P r i (s i ) I r i +N = P r i (s i ) P j6=i I r i (s j )+N = P d(s i ,r i ) α P j6=i P d(s j ,r i ) α +N ≥β (4.3) We refer to the above condition as the SINR constraint, and assume thatβ≥ 1. Due to orthogonality, transmissions on different frequencies do not cause interference 105 4.2.2 Problem Formulation Our problem formulation is similar to the one for maximizing aggregated convergecast throughput described in Chapter 3. As before, we are given the following: (i) a set V of nodes arbitrarily distributed in the 2-D plane, (ii) a distinguished node s∈ V that represents the sink, (iii) a routing treeT = (V,E T ) rooted at the sink, whereE T is the set of edges, which define a parent-child relationship among the nodes inV and need to be scheduled, and (iv) a set ofK orthogonal non-interfering frequencies. We define the lengthℓ i of an edgee i = (s i ,r i )∈E T between a senders i and its intended receiverr i in the tree as the Euclidean distance between them, i.e.,ℓ i =d(s i ,r i ). Note that the parents in the tree (except the sink) are both senders and receivers. Each node is equipped with a single half-duplex transceiver using which it can either transmit or receive at any given time slot. We assume that every node generates only one packet in the beginning of every TDMA frame, and that it can aggregate packets from its children before transmitting to its parent. Our goal is to find a minimum length schedule using multiple frequencies under the SINR model so that aggregated packets can reach the sink once per frame after a pipeline is established. In other words, we want to assign a frequency to each of the receivers inT following a receiver-based frequency assignment strategy, and a time slot to each of the edges in E T satisfying the SINR constraint of ( 4.3). The concept of pipeline is defined in Chapter 3 106 4.3 Multi-Channel Scheduling Under SINR 4.3.1 Overall Approach In this section, we extend the multi-channel scheduling algorithm presented in Chapter 3 so that it computes a non-conflicting schedule under the SINR model. As before, we decouple the joint frequency and time slot assignment problem into two separate phases of frequency assignment and time slot assignment, and show that the resulting algorithm still guarantees a provably good worst-case performance bound. 4.3.1.1 Link Diversity We first classify the edges in E T based on their lengths, and use the notion of link di- versity g(E T ), originally proposed in [56], to represent the number of magnitudes of different lengths. Formally, g(E T ) =|{m|∃e i ,e j ∈E T :⌊ log(ℓ i /ℓ j )⌋ =m}| (4.4) In realistic scenarios, g(E T ) is usually a small constant, although in theory it could be as large as the number of nodes. Without loss of generality, we normalize the minimum edge length to one, and letE = E 0 ,...,E log(emax) denote the set of non-empty length classes, where E k is the set of edges of lengths lying in the interval 2 k ,2 k+1 , and e max is the longest edge. First, the problem instance is partitioned into disjoint length 107 classes; then the edges in each class are processed separately for frequency and time slot assignment. For length classE k , we divide the 2-D deployment region into a setC k of square grid cells of side lengthη k =δ·2 k (we will chooseδ later). Our greedy frequency assignment strategy is exactly the same as before, i.e., for each cell inC k , we consider the receivers of those edges in length classE k that lie within that cell, and assign theK frequencies in a load-balanced manner such that the maximum number of nodes transmitting on the same frequency is minimized in that cell. Recall from Lemma 3 in Chapter 3, that such a frequency assignment will give a constant factor approximation on the maximum load on any frequency for each cell inC k . 4.3.1.2 Reusing Time Slots Under SINR To assign time slots for each of the edges in length classE k , we first 4-color the grid cells inC k such that no two adjacent cells have the same color, as illustrated in Figure 4.1 (cf. Figure 3.7(b)). Note that, unlike in the previous case of a graph-based interference model, now it is no longer clear whether it is possible to reuse time slots in non-adjacent cells in order to schedule edges that are on the same frequency under the SINR model. This is because we do not have the concept of an interfering edge anymore whose effect is limited upto only a certain distance, and therefore we need to consider the cumulative interference caused by concurrent transmissions from all non-adjacent cells. In the fol- lowing theorem (similar to Theorem 5.1 in [56]), we prove that such a reuse of time slots 108 g 1 g 1 g 1 g 1 g 2 g 2 g 2 g 2 g 3 g 3 g 3 g 3 g 4 g 4 g 4 g 4 h k =d.2 k g 1 g 1 g 2 g 2 g 1 g 1 g 3 g 3 g 1 Figure 4.1: Coloring the grid cells inC k corresponding to the length classE k using four distinct colorsγ 1 ,γ 2 ,γ 3 , andγ 4 . Size of each grid cell isη k =δ·2 k . is still possible so long as the sizeη k of the grid cells for each length classE k is chosen appropriately. Theorem 7. For cells of the same color inC k corresponding to a given length classE k , we can simultaneously schedule at most one edge inE k from every non-adjacent cell that lies on the same frequency, so long as the cell sizeη k satisfies the following: η k =δ·2 k , forδ = 4 8β· α−1 α−2 1 α , (4.5) whereβ≥ 1 is the SINR threshold, andα> 2 is the path-loss exponent. 109 Proof. For any given frequency, we prove that all the transmissions (at most one per cell of the same color) scheduled on the same time slot satisfy the SINR constraint of (4.3). In particular, without loss of generality, we consider one specific edgee i = (s i ,r i )∈E k in a given cellc∈ C k , and prove that the cumulative interference caused by concurrent transmissions from all non-adjacent cells is still within the SINR thresholdβ. Since the length of edgee i satisfiesℓ i < 2 k+1 , the received power atr i is P r i (s i ) = P ℓ α i ≥ P 2 α(k+1) . (4.6) The number of non-adjacent cells of the same color that are closest (we call this layer 1) to cellc is 8. For instance, consider the center cell marked asγ 1 and its 8 non-adjacent cells of the same color also marked asγ 1 in Figure. 4.1. Every senders j of an edgee j whose corresponding receiverr j lies in one of those layer 1 non-adjacent cells must be at least a distanced(r i ,s j )≥ δ· 2 k − 2 k+1 = 2 k (δ− 2) away from the receiverr i , as illustrated in Figure. 4.2. Therefore, the interference caused atr i by such a concurrent senders j is I r i (s j )≤ P [2 k (δ−2)] α , (4.7) and consequently, the cumulative interference caused by all the 8 senders from layer 1 non-adjacent cells is 8 X j=1 I r i (s j )≤ 8P [2 k (δ−2)] α . (4.8) 110 g 1 g 1 g 2 d.2 k -2 k+1 r i r j s i s j d.2 k Figure 4.2: For an edgee i = (s i ,r i )∈ E k , the distance between its receiverr i and the senders j of another edgee j = (s j ,r j )∈E k whose corresponding receiverr j lies in one of the non-adjacent layer 1 cells of the same color, is at leastδ·2 k −2 k+1 . Next, we consider at most 16 senders whose corresponding receivers lie in layer 2 non-adjacent cells of the same color to c. Each of these senders is at least a distance 3δ· 2 k − 2 k+1 = 2 k (3δ− 2) away from the receiverr i , and thus contributes to a total interference of 25 X j=9 I r i (s j )≤ 16P [2 k (3δ−2)] α . (4.9) We continue to aggregate interferences from all non-adjacent cells of the same color in every layer. In general, the number of such cells in layer q is 8q, and a sender whose receiver lies in one of those cells is at least a distance (2q− 1)δ· 2 k − 2 k+1 = 111 2 k {(2q−1)δ−2} away from the receiver r i . Therefore, the cumulative interference caused by all concurrent senders is bounded by I r i ≤ ∞ X q=1 8qP [2 k {(2q−1)δ−2}] α = 8P 2 kα ∞ X q=1 q [(2q−1)δ−2] α ≤ 8P 2 kα ∞ X q=1 q [(2q−1)δ/2] α , for allδ≥ 4 = 8P 2 (k−1)α δ α ∞ X q=1 q (2q−1) α (4.10) Substituting2q−1 =z, the infinite sum in (4.10) can be written in the following form: ∞ X q=1 q (2q−1) α = ∞ X z=1 z +1 2z α ≤ ∞ X z=1 1 z α−1 ≤ Bound on Zeta function ≤ α−1 α−2 . (4.11) 112 Thus, using (4.6), (4.11), and (4.10), the SINR at receiverr i is (ignoring noiseN ) SINR r i = P r i (s i ) I r i > P 2 α(k+1) 8P 2 (k−1)α δ α · α−1 α−2 = δ α 4 α ·8· α−1 α−2 = β, substitutingδ = 4 8β· α−1 α−2 1 α . Therefore the theorem follows. 4.3.2 O(g(E T )) Approximation Algorithm The overall process of frequency and time slot assignment is presented in Algorithm 4. We start by forming sets of edgesE k whose lengths lie in 2 k ,2 k+1 , and consider each length class in turn. For a given length class, we partition the 2-D deployment region into a setC k of square grid cells of lengthsδ·2 k . Then, for each cell inC k , we assign theK frequencies to the receivers according to algorithm FREQUENCY-GREEDY. In the time slot assignment phase, we first 4-color all the cells inC k , and consider cells of the same color in turn. For cells of the same color, we simultaneously schedule at most one edge inE k (respecting adjacency constraint) from every non-adjacent cell with receivers on the same frequency. Note that, Lemma 5 from Chapter 3 holds even in the SINR case, because no two edges on the same frequency from any cell is scheduled simultaneously. We next prove the complexity of the algorithm in Theorem 8. 113 Algorithm 4 APPROXIMATION ALGORITHM UNDER SINR 1. Input: Routing treeT = (V,E T );K orthogonal frequencies 2. Let E = {E 0 ,...,E log(emax) }, where E k is the set of edges with lengths in 2 k ,2 k+1 ; 3. t = 1; 4. for allE k 6=φ do 5. Partition the 2-D region into a setC k of square grid cells of lengthη k =δ·2 k ; 6. For each cellc∈ C k , consider the edges inE k whose receivers lie inc, and run FREQUENCY-GREEDY; 7. 4-color the cells inC k such that no two adjacent cells have the same color; 8. for colorj = 1 to4 do 9. repeat 10. for frequencyf = 1 toK do 11. Pick at most one edge inE k (respecting adjacency constraint) from every cell of colorj whose receiver is on frequencyf; 12. Assign time slott to all those edges; 13. end for 14. t =t+1; 15. until All edges inE k whose receivers lie in cells of colorj are scheduled. 16. end for 17. end for Theorem 8. Given a routing treeT on an arbitrarily deployed network in 2-D, andK orthogonal frequencies, Algorithm 4 gives a O(g(E T )) approximation on the optimal schedule length. Proof. Let the schedule length of an optimal algorithm beOPT and that produced by Algorithm 4 beΓ. The proof depends on the choice of a critical grid cell ˆ c for which the number of edges on the same frequency, as assigned by an optimal load-balanced fre- quency assignment strategym ∗ , is maximum across all length classes. This corresponds to the maximum load on any frequency over all length classes, i.e., L m ∗ ˆ c = max k,c L m ∗ c∈C k (4.12) 114 We first prove that an optimal algorithm can schedule at mostμ edges simultaneously that are on the same frequency in any cell of any given length class, whereμ is given by μ = 2( √ 2·δ +1) α β . (4.13) and δ as defined in (4.5). This will imply OPT will take at least L m ∗ ˆ c /μ time slots to schedule the whole network. We prove this by contradiction. Consider an edgee i = (s i ,r i )∈E k whose receiver lies in cellc∈C k , and supposeμ more edges (numbered1,2,...,μ) of the same length class whose receivers also lie in cellc are scheduled simultaneously; thus, there areμ+1 edges scheduled simultaneously inc. Ignoring noise, theSINR at receiverr i is SINR r i = P/d(s i ,r i ) α P μ j=1 P/d(s j ,r i ) α (4.14) The length of each edge inE k is at least2 k , i.e.,2 k ≤d(s i ,r i )< 2 k+1 , and the length of each grid cell inC k isδ·2 k . Therefore, the maximum distance between two receivers r i andr j that lie in cellc is at most2 √ 2·δ·2 k , and sod(s j ,r i )< 2 √ 2·δ·2 k +2 k+1 . Substituting these inequalities andμ from (4.13) in (4.14), we get SINR r i < P/2 kα μP/ 2 √ 2·δ·2 k +2 k+1 α = 2( √ 2·δ +1) α μ = β (4.15) 115 Thus, OPT≥L m ∗ ˆ c /μ (4.16) Now, since Lemma 3 from Chapter 3 still holds, and each length class is processed separately in Algorithm 4, the total number of time slots required is Γ ≤ 4·max k,c {|γ c∈C k |}·g(E T ) ≤ 8·max k,c n L φ c∈C k o ·g(E T ) ≤ 8·max k,c 4 3 − 1 3K ·L m ∗ c∈C k ·g(E T ) ≤ 8μ· 4 3 − 1 3K ·g(E T )·OPT = O(g(E T )), (4.17) where we have used similar notations from Chapter 3, thatγ c∈C k denotes the set of time slots required to schedule all the edges from cellc in length classE k , andφ denotes a fre- quency assignment strategy according to FREQUENCY-GREEDY Therefore, the theorem follows. 4.4 Summary In this chapter, we have considered a realistic interference model, the so called SINR model, and extended the multi-channel scheduling algorithms presented in Chapter 3. By using the notion of link diversity, which measures the number of non-empty link 116 classes in the network, and by appropriately choosing grid sizes, which is a function of the path-loss exponentα and the hardware-dependent SINR thresholdβ, we have shown that slight modification to the algorithms presented earlier retains a good performance ratio even under SINR. In particular, the extended algorithms scale in proportional to O(g(E T )), whereg(E T ) is defined as the number of non-empty length classes, which is typically a small constant. 117 Chapter 5 Optimal Spanning Trees for Maximizing Throughput and Minimizing Packet Delays 5.1 Motivation In the previous chapter, we showed that multiple frequencies can reduce the number of interfering links and enable more concurrent transmissions, thereby enhancing the data collection date for aggregated convergecast in tree-based sensor networks . Evaluation results of our proposed multi-channel scheduling algorithms indicate that increasing the number of frequencies beyond a certain point has diminishing returns. For instance, on shortest path trees (SPT), the schedule lengths remain almost similar when utilizing 3 or 5 frequencies. The reason behind this trend is that, on shortest path trees, nodes choose their parents based on minimum hop counts to the sink, and so in a densely populated network the number of children for each node is enormous, resulting in high Parts of this chapter are based on [52] and [51]. 118 0 50 100 150 200 0 20 40 60 80 100 120 140 160 180 200 Figure 5.1: Shortest path tree (SPT): High node degrees and minimum hop distances to the sink. Dark lines represent tree edges and dotted lines represent interfering links. node degrees on the routing topology. This leads to a large number of primary conflicts, because those children need to be scheduled at different time slots due to the single half-duplex transceiver on the nodes. Recall that multiple frequencies cannot eliminate primary conflicts. Figure 5.1 shows a sample routing topology based on shortest path trees on a network of 800 nodes randomly deployed in a region of size 200×200. The sink is located at the center, and a link between any two nodes exists if they are within distance 25 of each other. We observe that the tree has high node degrees but minimum hop distances to the sink. In order to further verify that high node degrees indeed create bottlenecks and yield diminishing returns with increasing number of frequencies, and that routing topologies 119 0 50 100 150 200 0 20 40 60 80 100 120 140 160 180 200 (a) u v (b) Figure 5.2: (a) Minimum interference tree (MIT): Low node degrees but large number of hops to the sink. Dark lines represent tree edges, dotted lines represent interfering links. (b) Cost of an edge (u,v) is equal to the number of nodes lying within the union of the two disks centered at nodesu andv, each of radiusd(u,v); here cost is11. with low node degrees might further enhance the data collection rate, we consider a par- ticular type of spanning tree, known as the minimum interference tree (MIT) [17], which has very low node degrees and perhaps could achieve a better scheduling performance with multiple frequencies. An MIT is basically a minimum spanning tree (MST) with a particularly defined edge cost. It is constructed as follows: Given a graphG = (V,E), define the cost of an edge e = (u,v)∈ E as the number of nodes covered by the union of the two disks centered at nodesu andv, each of radius equal to their Euclidean distanced(u,v). This particular cost function (cf. Figure 5.2(b)), therefore, gives a measure of interference by counting the number of nodes affected byu andv communicating with just enough transmission 120 100 200 300 400 500 600 700 800 5 10 15 20 25 30 35 40 45 50 Number of Nodes Schedule Length MIT, K=1 MIT, K=3 MIT, K=5 Figure 5.3: Schedule length on minimum interference trees with different network sizes forK = 1,3, and5 frequencies. power to exactly reach each other. Figure 5.2(a) shows an MIT on the same deployment of nodes as in Figure 5.1. Clearly, it has low node degrees but large number of hops to the sink. In Figure 5.3, we plot the schedule length with increasing network sizes for 1, 3, and 5 frequencies on minimum interference trees by running the same multi-channel scheduling algorithm proposed in the previous chapter. To minimize statistical variation, we construct MITs on the same network topologies that were also used to construct SPTs (cf. Figure 3.9). We observe a significant reduction in the schedule length as compared that on shortest path trees. This is because an MIT has small node degrees and large number of hops to the sink, which gives rise to a lot of secondary conflicts but only a very few primary conflicts. A typical path on an MIT from any node to the sink looks almost 121 like a linear network, where every non-adjacent edge can be scheduled simultaneously with only two frequencies, and this is the best one could achieve on linear networks. Note that, with one frequency, only distance-2 edges, i.e., edges which are neither adjacent nor whose end points are incident on a common edge, can be scheduled simultaneously. The discussion so far points out to the fact that routing topologies with low node degrees are perhaps best suitable for maximizing the network throughput in aggregated convergecast. However, many large-scale sensor networks, particularly those intended for mission critical applications, such as disaster early warning systems, or tracking and surveillance systems, also require timeliness of data collection in addition to maximizing the throughput. In other words, such applications are not delay-tolerant and need close to real-time data delivery, failure of which might lead to catastrophes. One of the network properties that contributes to packet delays is the hop count on the routing topology from any node to the sink. The further away a nodes is from the sink, the longer it takes for its packets to reach the sink. Thus, a minimum interference tree, which is usually characterized by large number of hops, will incur unacceptable delays for such mission-critical applications. On the other hand, shortest path trees, although have minimum hop counts to the sink, are characterized by high node degrees that tend to lower the data collection rate. Therefore, if shortest path trees are more suitable for minimizing delays, minimum interference trees are better fit for maximizing through- put, but neither of them can yield the best throughput-delay guarantee, for instance, in terms of minimizing the maximum delay while guaranteeing a given lower bound on the 122 throughput. This calls for the need of constructing special type of routing topologies that have low node degrees as well as small hop counts to the sink. In addition to minimizing delay, lower hop counts are also useful to achieve better network reliability [84]. For example, consider a network where each edgee fails with probabilityp e , and that all failures occur independently. Then the probability that a path e 1 →e 2 →...→e k is operational is(1−p e 1 )×(1−p e 2 )×...×(1−p e k ). Thus, given a certain threshold value for the desired reliability, there is a corresponding parameter γ such that the radius of the network defined as the maximum of the number of edges on any path to the sink is required to be at most γ. Therefore, reliability constraint is transformed into a radius constraint. In a similar fashion, besides enhancing convergecast throughput, node degree con- straints also appear naturally in many graph-theoretic abstractions of communication network design problems [164]. As an example, consider the IP multicast problem where we would want to disseminate centrally stored information from a server node to a set of client hosts. The standard solution is to construct a tree rooted at the server node and spanning all client nodes. Data packets are then sent from the root along each of its incident edges in the tree. An internal node forwards incoming packets to its chil- dren in the tree. The number of children of a node has is therefore proportional to the amount of work done by the node, and hence it is natural to compute spanning trees of low maximum node degree. 123 In this chapter, we design an algorithm to construct such routing topologies, known as the Bounded-Degree Minimum-Radius Spanning Trees (BDMRST) [51], and show that multi-channel scheduling on these trees can help in balancing mutually conflicting performance objectives, in particular, an optimal throughput-delay trade-off. The rest of the chapter is organized as follows. In Section 5.2, we describe our model and problem formulation. In Section 5.3, we present an approximation algorithm and its analysis to construct a BDMRST. Section 5.4, we evaluate the multi-channel scheduling algorithm on different types of spanning trees. 5.2 Bicriteria Problem Formulation Given a network modeled as a graph G = (V,E), where V is the set of nodes and E is the set of communication links, we define the radiusR(T) of a spanning treeT ofG rooted at a distinguished vertexs∈ V as the maximum number of hops from any node tos. An edgee = (u,v)∈ E exists between any two nodesu andv if their Euclidean distanced(u,v) is no more than the transmission rangeR. We formulate the problem of constructing routing topologies as a bicriteria optimization problem [125] where, given an upper bound on the maximum node degree, our goal is to minimize the tree radius. We call such a tree a Bounded-Degree Minimum-Radius Spanning Tree (BDMRST), and formally define the routing tree construction problem as follows. Routing Tree Construction Problem: Given a graph G and a constant parameter Δ ∗ ≥ 2, we want to construct a Bounded-Degree Minimum-Radius Spanning TreeT on 124 G rooted at sinks such that the radius ofT is minimized, subject to the constraint that the degree of any node inT is at mostΔ ∗ . We note that such bicriteria optimization formulations [103] are quite generic and robust, as the quality of approximation is independent of which of the two criteria the budget is imposed on, and it subsumes the case where one wishes to optimize a func- tional combination of the two objectives, such as, maximizing the sum or product of the maximum node degree and the tree radius. Bicriteria optimization problems on spanning trees are often NP-hard on general graphs, and sometimes even on geometric graphs. Extending the notion of performance guarantee of approximation algorithms for unicri- terion optimization problems, here we want to design an(α,β) bicriteria approximation algorithm to the routing tree construction problem, such that the maximum node degree is at mostΔ ∗ +α, and the radius is at mostβ times the minimum possible radius subject to the degree constraint, whereα andβ are positive constants. 5.3 Routing Tree Construction In this section we present an approximation algorithm for constructing a BDMRST, and show that it gives a constant factor approximation on both maximum node degree and tree radius. 125 5.3.1 An Approximation Algorithm We tessellate the 2-D deployment region into a set of hexagonal grid cells{c j }, each of side lengthR/2, as shown in Figure 5.4. We associate each node to a unique cell whose center is closest to the node, breaking ties arbitrarily. We define a cell to be non-empty if it has at least one node, and two cells to be neighbors of each other if they share at least one common side. We also define a cell to be adjacent to a node u if a circle of radiusR centered atu intersects the cell. The basic idea of the spanning tree construction algorithm is as follows. The algorithm runs in two phases. • Phase 1: In Phase 1, we construct a backbone tree T B = (V B ⊆ V,E B ⊆ E) from the original graph G by choosing one representative node arbitrarily from each non-empty cell and connecting them in a Breadth-First-Search (BFS) order starting from the sink, while ensuring that the hop distances alongT B is not to long compared to a shortest path tree inG. This backbone tree determines the global structure of our solution. We call the representative nodes the local roots. • Phase 2: In Phase 2, we construct a local spanning tree of minimum radius within each cell from the remaining nodes inV\V B lying in that cell while respecting the degree constraint Δ ∗ . This is always possible because the nodes within each cell form a complete graph, as the diameter of the circumcircle for each hexagonal cell isR. Finally, we construct the overall spanning treeT by taking the union of the backbone tree and all the local spanning trees. 126 During the execution of the algorithm, we mark a cell if its local root has been included inT B ; otherwise the cell is unmarked. We now describe the two phases in detail below. Phase 1 - Backbone Tree Construction: • In the beginning, all the cells are unmarked. We initialize the backbone tree with sinks, and mark the cell to whichs belongs. Choose one local root arbitrarily from each non-empty cell; let this set of nodes beR ={r 1 ,...,r n }. • Consider all the unmarked adjacent cells ofs in some arbitrary order. In Figure 5.4, these are the shaded cells. • Let r j be the local root in the unmarked adjacent cell c j of s. One of the three possibilities could occur: – ifr j lies within the transmission range ofs, connect it directly tos and mark c j . Update the backbone tree by addingr j toV B , and the edge (r j ,s) toE B . Nodesr 1 ,r 2 , andr 6 in Figure 5.4 are such nodes. – ifr j lies outside the transmission range ofs, but there is some other nodew k lying within one of the adjacent cells ofs, and is within the range of bothr j ands, connectr j tos viaw k . We call such a nodew k a helper node. Mark c j and update the backbone tree by adding bothr j andw k toV B , and the two edges(r j ,w k ) and(w k ,s) toE B . In Figure 5.4, nodesr 3 andr 5 are connected tos viaw 1 , and noder 4 viaw 2 . 127 r 1 r 3 w 1 r 4 r 2 s w 2 r 5 r 6 R R/2 r 7 R R r 8 r 9 r 10 Figure 5.4: Backbone tree construction: Filled black circles represent local roots (chosen arbitrarily from each non-empty cell), and shaded cells are non-empty adjacent cells of s. Iteration 1: Local rootsr 1 ,r 2 ,r 3 ,r 4 ,r 5 , andr 6 from non-empty adjacent cells ofs are connected. Nodesr 3 andr 5 are connected to sinks via helper nodew 1 , and noder 4 via helper nodew 2 . Noder 7 is out of range of any helper node, and is not connected in the first iteration. 128 w k r j v 1 R/2 Figure 5.5: Local tree construction on an induced complete graph within each cell for maximum node degreeΔ ∗ = 4; filled black circle represents the local root. – ifr j lies outside the range ofs, and there is no helper node, leaver j as it is. Noder 7 in Figure 5.4 is such a node. • Consider the local roots whose cells have been marked in a BFS order, and re- peat the second and third steps. Continue until all the local roots inR have been connected. We implement the BFS processing of the local roots in a queue data structure. Phase 2 - Local Spanning Tree Construction: Since the transmission range of every node isR, nodes lying within each cell induce a complete graph, i.e., every node has an edge to every other node. Consider the local root r j in cell c j . Let the set of nodes in c j that are not yet connected to the tree be V j ={v 1 ,...,v n j }⊂ V\V B . Connectv 1 tor j treatingr j as its parent. Then, connect at most Δ ∗ − 1 nodes (if those many exist) from V j to v 1 ; these constitute the direct neighbors ofv 1 . Next, treating these direct neighbors as parents, connect at mostΔ ∗ −1 129 nodes to each one of them, if those many exist. Figure 5.5 shows an illustration of this phase. Continue this until there is no isolated node left inV j , and repeat the procedure for each of the non-empty cells. At the end of this phase, eachc j contains a local spanning treeT j rooted atr j , with each node (except the leaves and the last parent) having degree Δ ∗ . The overall spanning treeT is the union of the backbone treeT B and all the local spanning treesT j . A formal description of the algorithm is given in Algorithm 5. 5.3.2 Algorithm Analysis Theorem 9. Algorithm 5 gives a constant factor (α,β) bicriteria approximation to the Bounded-Degree-Minimum-Radius Spanning Tree, whereα = 10 andβ = 5. The proof unfolds in the following lemmas. Lemma 8. Letr i andr j be any two local roots on the backbone treeT B . LetP G (r i ,r j ) be the shortest path on the original graph G between r i and r j consisting of m hops. Then, the length of the unique simple pathP T B (r i ,r j ) betweenr i andr j onT B is at most 4m. Proof. LetP G (r i ,r j ) ={r i =u 0 ,u 1 ,...,u m−1 ,u m =r j } be the shortest path on graph G, andP T B (r i ,r j ) ={r i = v 0 ,v 1 ,...,v h−1 ,v h = r j } be the unique simple path on the backbone treeT B . Note that eachv k is either a local root or a helper node. We will show that for every edge(u k ,u k+1 ) inP G we add at most a constant number of(v k ,v k+1 ) edges inP T B . 130 r k r k+1 u k u k+1 c k c k+1 R P G Figure 5.6: Traversing the edge (u k ,u k+1 ) along the shortest pathP G in graphG. Local rootsr k andr k+1 are at most distance3R away from each other. 131 Consider the nodesu 1 ,...,u m−1 , and traverse them in the order as they appear inP G . At the same time, traverse the path inP T B by tracking the progress inP G , i.e., for every edge traversed inP G , we traverse a certain number of edges inP T B . Both traversals start from the same cell (and the same noder i ). Suppose at any give point, we are about to traverse the edge(u k ,u k+1 ). One of the two possibilities could occur: (i)u k andu k+1 lie in the same cell, or (ii)u k andu k+1 lie in different cells, sayc k andc k+1 , respectively, as shown in Figure 5.6. In the first case, we do not traverse any edge inP T B . In the second case, we traverse along the local roots and helper nodes inP T B , such that the end point of the last edge traversed in P T B is a local root r k+1 that lies in the same cell as u k+1 . This is always possible because each non-empty cell contains a local root. Since the Euclidean length of the edge (u k ,u k+1 ) is at mostR,u k+1 must lie in one of the adjacent cells ofu k . Also, since the side length of each cell isR/2, the distance between the local roots r k and r k+1 lying in cells c k and c k+1 , respectively, is at most 3R. Now, during the backbone tree construction phase in Algorithm 5, we first connect the local roots from all the non-empty adjacent cells of an already connected local root before exploring other cells. Since connecting a local root to the backbone tree takes at most 2 edges (when helper nodes are needed), the number of edges traversed onP T B for case (ii) is at most 4. Therefore, the length of the pathP T B is at most 4 times the length of the pathP G . Lemma 9. The degree of any local root or helper node in the backbone tree is at most a constant 12 in the worst case. 132 Proof. Recall that during the backbone tree construction, a new local root from each non-empty cell adjacent to an already chosen local rootr j is connected either directly or via a helper node tor j . This increases the degree ofr j by at most one. Since the side length of each cell isR/2, the maximum number of cells that are adjacent tor j is at most 11; this will happen when r j lies near one of the corners within its cell c j . There will be 6 cells that are neighbors toc j (i.e., share exactly one side withc j ) and 5 other cells that are not neighbors toc j . Also, recall that during constructing the local spanning trees within each cell, at most one node is connected to the local root of that cell. Therefore, the degree of any local root in the backbone tree is at most 12 in the worst case. A similar argument also holds for the degree of any helper node. Proof. (Theorem 9) (Bound on the Radius): Let the radius of an optimal spanning tree whose maximum node degree is Δ ∗ on graph G be OPT , and let that produced by Algorithm 5 beR(T). Supposev ∗ be the node farthest from sinks, andm be the hop distance on a shortest path tree (no degree bound) from s to v ∗ on graph G. Then, OPT≥m, because a degree constraint can only increase the radius of a tree. Now the node v ∗ can either be a local root, or a helper node, or a node in a local spanning tree. • ifv ∗ is a local root, then by Lemma 8,R(T)≤ 4m≤ 4·OPT . • ifv ∗ is a helper node, then the shortest path on the tree comprises a path froms to the local root of the cell containingv ∗ , plus an additional hop from the local root tov ∗ . Thus,R(T)≤ 4m+1≤ 4·OPT +1. 133 • ifv ∗ is a node in a local spanning treeT j , then the shortest path on the tree com- prises a path froms to the local root of the cell containingv ∗ , plus at mostR(T j ) hops from the local root tov ∗ . Thus, R(T) ≤ 4m+R(T j ) ≤ 4·OPT +OPT = 5·OPT The second inequality follows because the radius of each local spanning tree T j constructed on the complete graph within each cell is minimum, respecting degree constraintΔ ∗ , and soOPT≥R(T j ). (Bound on the Degree): From Lemma 9, the maximum node degree of any node in the backbone tree is at most 12. Also, the degree of any node is a local spanning tree is at most Δ ∗ . Thus, the degree of any node in the overall spanning tree is bounded by max(Δ ∗ ,12)≤ Δ ∗ +10, for anyΔ ∗ ≥ 2. Therefore, the theorem follows withα = 10 andβ = 5. 5.4 Evaluation In this section, we evaluate the performance of the multi-channel scheduling algorithm presented in the previous chapter, and the routing tree construction algorithm presented in this chapter using MATLAB simulations on networks modeled as random geometric 134 graphs. We generate connected networks by uniformly and randomly placing nodes in a square region of size200×200, and connecting any two nodes that are at most distance R = 25 apart. 5.4.1 Schedule Length and Maximum Delay We run the multi-channel scheduling algorithm on three different kinds of spanning trees – BDMRST, SPT, and MIT. We first present the results for a single frequency, and then discuss the case with multiple frequencies. Figure 5.7(a) and 5.7(b) show the schedule length and the maximum packet delay, respectively, with increasing network size on three different types of trees for single fre- quency scheduling. Each point in the plot is averaged over20 iterations. In each iteration, we deploy the nodes uniformly and randomly, construct the spanning trees, and then run the scheduling algorithm. In other words, we keep the node deployment fixed in a given iteration for all the three types of trees in order to preserve the underlying communica- tion graph and minimize any statistical variation. The maximum degree bound on the BDMRST is taken asΔ ∗ = 4. We observe that the schedule lengths on a BDMRST and MIT are very close to each other, and are much lower than that on an SPT. This differ- ence becomes more predominant with increasing network size, because the maximum node degrees on an SPT go up rapidly, as shown in Figure 5.8. On the other hand, the maximum packet delays on an SPT are minimum, and those on a BDMRST are very close. However, the delays on an MIT are much higher due to its very small and almost 135 100 200 300 400 500 600 700 800 5 10 15 20 25 30 35 40 45 50 Number of Nodes Schedule Length BDMRST SPT MIT (a) 100 200 300 400 500 600 700 800 0 20 40 60 80 100 120 Number of Nodes Maximum Delay (Tree Radius) BDMRST SPT MIT (b) Figure 5.7: (a) Schedule Length, and (b) Maximum Delay (tree radius) with increas- ing network size on three different types of trees (BDMRST, SPT, and MIT) for single frequency scheduling. 136 100 200 300 400 500 600 700 800 0 5 10 15 20 25 30 35 40 Number of Nodes Maximum Node Degree BDMRST SPT MIT Figure 5.8: Maximum Node Degree with increasing network size on three different types of trees (BDMRST, SPT, and MIT) for single frequency scheduling. constant node degrees throughout, as shown in Figure 5.8, which give rise to more num- ber of hops. Thus, scheduling on a BDMRST achieves the best of both worlds in terms of having a small schedule length as well as very close to smallest possible maximum delay. 5.4.2 Multiple Frequencies on Schedule Length Since multiple frequencies can eliminate interfering links and reduce the schedule length, we now evaluate their effects on BDMRST for our proposed multi-channel scheduling algorithm. Figure 5.9 show the schedule lengths with increasing network size for 1, 3, and5 frequencies on a BDMRST. We observe that the schedule length does not improve with multiple frequencies. This is due to almost constant maximum hop distances to 137 100 200 300 400 500 600 700 800 5 10 15 20 25 30 35 40 45 50 Number of Nodes Schedule Length BDMRST, K=1 BDMRST, K=3 BDMRST, K=5 Figure 5.9: Effect of multiple frequencies on schedule lengths for BDMRST with differ- ent network sizes forK = 1,3, and5 frequencies. the sink and nearly constant maximum node degrees, as shown in Figure 5.7(b) and 5.8. In order to gain insights on the effects of node degrees on the schedule length, we plot the degree distribution of BDMRST, SPT, and MIT for two different network sizes with N = 150 and800 nodes in Figure 5.10(a) and 5.10(b), respectively. The bar graphs show the number of nodes that have degrees of particular values averaged over 20 iterations for each tree type. For all network sizes, we observe that most of the nodes on an SPT have degree one, whereas few have degrees very high. This is because large groups of degree one nodes (leaf nodes) are connected to common parents, giving rise to a lot of primary conflicts, and thereby being more resistant to improving the schedule length with multiple frequen- cies. In MIT, we see that most of the nodes have degree two, and no node has degree 138 1 2 3 4 5 6 7 8 9 10 11 12 0 10 20 30 40 50 60 70 80 90 Node Degree Number of Nodes BDMRST SPT MIT (a) 0 5 10 15 20 25 30 0 100 200 300 400 500 600 700 Node Degree Number of Nodes BDMRST SPT MIT (b) Figure 5.10: Node Degree Distribution of BDMRST, SPT, and MIT for two different network sizes with (a)N = 150, and (b)N = 800 nodes. more than five. This is because most of the paths from any node to the sink on an MIT 139 0 50 100 150 200 0 20 40 60 80 100 120 140 160 180 200 Figure 5.11: A BDMRST constructed on the same deployment of800 nodes of Figure 5.1 and 5.2(a). The node degrees are more uniform compared to those on an SPT and MIT. look like a linear topology. We also observe that an MIT has a lot of parent nodes com- pared to an SPT, thus further explaining the reason for much more improvement in the schedule length with multiple frequencies. Lastly, the degrees on a BDMRST are more evenly distributed for all network sizes, as illustrated in Figure 5.11 by a sample tree constructed on the same deployment of800 nodes of Figure 5.1. 5.5 Summary In this chapter, we discussed the trade-off between aggregated convergecast throughput and packet delays for fast and timely data collection in sensor networks. We showed that multi-channel scheduling combined with routing over bounded-degree minimum-radius spanning trees can help in achieving the best of both worlds in terms of maximizing 140 the throughput and minimizing the maximum packet delay. To this end, we designed a spanning tree construction algorithm that gives a constant factor bicriteria approximation guarantee on minimizing the tree radius under a given maximum node degree constraint. 141 Algorithm 5 Approximation algorithm for Bounded-Degree Minimum-Radius Spanning Tree 1. Input:G = (V,E); sinks; degree boundΔ ∗ ≥ 2 2. Output: BDMRSTT ofG 3. Tessellate the 2-D region into hexagonal grid cells, each of side lengthR/2. 4. Associate each node to a unique cell whose center is closest to the node. 5. Phase 1: Backbone Tree 6. All cells are unmarked. 7. InitializeT B :V B ←{s},E B ←φ, mark cell ofs. 8. Choose one local root arbitrarily from each non-empty cell; letR ={r 1 ,...,r n } be the set of local roots. 9. Q←φ; 10. ENQUEUE(Q,s); 11. whileQ6=φ do 12. u← DEQUEUE(Q); 13. for all unmarked cellsc j adjacent tou do 14. r j ← local root inc j ; 15. ifd(u,r j )≤R then 16. V B ←V B ∪{r j }; 17. E B ←E B ∪{(u,r j )}; 18. Markc j ; 19. ENQUEUE(Q,r j ); 20. else ifd(u,r j )>R and∃ helper nodew k then 21. V B ←V B ∪{r j ,w k }; 22. E B ←E B ∪{(u,w k ),(w k ,r j )}; 23. Markc j ; 24. ENQUEUE(Q,r j ); 25. end if 26. end for 27. end while 28. Phase 2: Local Spanning Tree 29. for all non-empty cellsc j do 30. r j ← local root inc j ; 31. Let V j ={v 1 ...v n j } be the set of not yet connected nodes in c j (V j induces a complete graph). 32. Construct local spanning treeT j of minimum radius with nodes inV j such that no node exceeds degreeΔ ∗ . 33. end for 34. return T =T B ∪{T j }. 142 Chapter 6 Efficient Distributed Topology Control in 3-Dimensional Wireless Networks 6.1 Motivation With growing interest in diverse applications of sensor networks, such as structural health monitoring [156], underwater marine life and coral reef monitoring [2], smart homes, and industrial automation, a large number of these networks embedded in the physical world will be three dimensional . Current literature, however, is heavily focused on two- dimensional networks, and there is a tendency to believe that the results from 2-D will directly extend to 3-D. Unfortunately, this is not always true and gives rise to several challenges in terms of computational complexity of the protocols when applied in a 3-D setting [119]. For instance, consider the problem of deploying and configuring a network in order to guarantee complete coverage and connectivity of a region in 3-D space. In Parts of this chapter are based on [53]. 143 2-D, our assumption about uniform random deployment of nodes with high density is usually well accepted to achieve these goals. However, such high density deployment might not be practical or even possible due to geometric constraints in 3-D. Forn nodes deployed randomly in a unit cube, [0,1] d , ind dimensions, it is known that the critical transmission radius [86] for connectivity is O logn n 1/d . For n = 1000, the critical radius in 2-D is therefore 0.07, while in 3-D it is 0.2, resulting in a critical average node degree of15(≈π·0.07 2 ·1000) in 2-D and34 ≈ 4π 3 ·0.2 3 ·1000 in 3-D. This implies that with 1000 nodes, a uniform random deployment in 3-D that is almost surely connected will have more than double the number of communication neighbors compared to its 2-D counterpart. Similarly, in terms of coverage, if the sensing region covered by a node is a ball of radius R s around it, then under uniform random deployment, the ratio of the region covered to the total region is 1−e λVs , where λ is the density of deployment andV s is the volume of the ball of radiusR s . For the region to be covered with high probability, say more than 0.99, we must have 1−e λVs ≥ 0.99 which, after simplification, gives λV s ≥ 4.6. Since the sensing region of a node will intersect with those of all nodes that are within distance 2R s away, the number of such nodes, on an average, will be 37 ≈λ· 4π 3 ·(2R s ) 3 =λV s ·2 3 ≥ 4.6·2 3 in 3-D, while the corresponding number in 2-D is only18. In many distributed protocols, the amount of computation on a node depends on the number of neighbors that are within its communication range and/or sensing range, and both these numbers are very high in 3-D as compared to 2-D. This signifies that simple 144 extensions of 2-D protocols might not be practical in 3-D due to very high computational requirements, and need to be designed explicitly. In addition, such high node degrees in 3-D could result in excessive interference, thus preventing nodes from transmitting con- currently and reducing network throughput. In such cases, controlling the transmission powers of nodes to form sparser yet connected topologies could substantially alleviate the effects of interference. Moreover, transmitting at lower power levels in a multi-hop network results in more relaying through intermediate nodes, which potentially could save energy consumption. Such power control schemes, which are commonly known as topology control, therefore play an important role in optimizing network throughput and prolonging lifetime. In this chapter, we address the problem of efficient distributed topology control in 3-D wireless sensor networks. We note that, this problem falls under the broader class of problems that guarantee a global network property by satisfying cer- tain local constraints. In this case, this global property is network connectivity, and the local constraints, as we will see, are the existence of certain number of neighbors within each node’s communication range. Although the problem of efficient topology control has been well researched in 2-D, its extension to 3-D brings in several challenges that have not been adequately addressed. For instance, there is a natural ordering of nodes in 2-D in terms of angular directions, based on which many topology control algorithms, such as Cone-Based Topology Con- trol (CBTC) [150], [93], are designed. However, no such ordering of angles is possible in 3-D; we only have the notion of solid angles that does not lend itself to any particular 145 ordering of nodes. There have been extensions of CBTC to 3-D networks, such as the one proposed by Bahramgiri et al. [12] using the notion of 3-D cones instead of angles, but their algorithm requires very high computational overhead –O(d 3 logd) compared to O(dlogd) for 2-D CBTC, where d is the average number of communication neigh- bors of a node. Given that uniform random deployment of nodes in 3-D requires very high average node degrees for a network to be almost surely connected, an increase by a factor of evend 2 leads to very high computational overhead. This underlines the need for explicitly designing topology control algorithms for 3-D networks. In this chapter, we focus on one particular aspect of 3-D networks – high node de- grees – and employ transmission power control to construct sparse topologies and reduce interference [53]. This brings in additional leverage in optimizing the throughput-delay trade-off. Contrast this with the scenario when there is no flexibility in controlling the transmission power levels, and nodes always transmit at their maximum power. Under this setting, even if we apply our earlier spanning tree construction algorithm that guar- antees constant factor approximations on the node degree as well as on the tree radius, the amount of interference caused because of high power levels is far more than in the case with power control. Keeping that mind, our goal here is to design a distributed algo- rithm, especially catered to 3-D networks with lower computational overhead, in order to minimize the transmission power levels while keeping the network connected, and then 146 create a spanning tree on the resulting network. In essence, we make use of power con- trol techniques, in addition to routing tree construction and multi-channel scheduling, to optimize the throughput-delay performance in 3-D networks. Our first approach is based on orthographic projections in 2-D that is simple to im- plement and runs inO(dlogd) time, whered represents the average node degree. This approach borrows concepts from the 2-D CBTC technique and performs very well in practice, although it does not guarantee a connected network in theory. Our second ap- proach is based on the computational geometry construct, called Spherical Delaunay Triangulation [119], [126] which also runs inO(dlogd) time but is always guaranteed to produce a connected network. The rest of the chapter is organized as follows. In Section 6.2, we describe preliminaries and our solution approach. Section 6.3 discusses the multi-dimensional scaling (MDS) technique in 3-D that we use as a primitive. We present our heuristic based on 2-D orthographic projections in Section 6.4, and the more rigorous SDT based algorithm in Section 6.5. Section 6.6 presents detailed simulation results, and we summarize the chapter in Section 6.7. 6.2 Preliminaries and Approach We first introduce some notations for the ease of exposition. Given a network modeled as a graphG = (V,E), we associate a three tuple (x i ,y i ,z i )∈ℜ 3 denoting the location coordinates for each nodeu i ∈ V . We denote the spherical ball of radiusR and center atu i asB(u i ,R). Given any three non-collinear pointsp i ,p j , andp k on the surface of 147 p i p j p k r h n u i R B (u i , R) Cap (p i , p j , p k ) q 2 Figure 6.1: Three non collinear pointsp i ,p j , andp k on the surface of a sphere uniquely determine a spherical cap. The radius of the base of the cap isr, andh is its height. ~ n denotes normal to the cap. a sphere, we define a spherical cap as the smaller (in terms of volume) portion of the sphere that is cut-off by a plane passing through these three points. Note that, a plane can be uniquely determined by a set of three non-collinear points in 3-D. We denote such a spherical cap byCap(p i ,p j ,p k ). The height of the cap is denoted byh and the radius of its base is denoted byr, as illustrated in Figure 6.1). In general, the transmission range of a node is a monotonic function of its power level. We denote byR(P) the transmission range when the power level isP . For a givenP , we denote the set of neighbors of nodeu i byN i (P). Since a node can change its neighbor set depending on its transmission power, thus causing changes in the network topology, we assume that the maximum power graph formed when all the nodes transmit at their maximum power P max is connected. Our goal is to construct a connected spanning 148 subgraph of the maximum power graph using only local geometric information, such that it is locally optimal in terms of power levels, i.e., even if a single node transmits at a power level lower than that dictated by the subgraph, then the network will become disconnected. Our approach in developing a topology control algorithm that works in 3-D with sub- stantially lower computational complexity consists of two phases. In the first phase, we use MDS to find the relative locations of all the neighbors of each node when the trans- mission power isP max . In the second phase, we propose two different strategies. The first one is heuristic-based, and uses the well known CBTC algorithm. The heuristic per- forms extremely well in practice, although theoretically it does not guarantee a connected network. The second strategy, which is more rigorous and always guarantees a connected network, uses the properties of spherical Delaunay triangulation. In the following, we describe these two approaches. 6.3 Phase 1: Multi-Dimensional Scaling in 3-D Multi-dimensional scaling [1] is a statistical method that has been widely used to discover spatial structures and relationships among sets of objects from their observed similarity or dissimilarity data sets. The technique basically transforms a pairwise distance matrix among a set of objects into a set of coordinates, such that the pairwise Euclidean distances derived from these coordinates approximate the original distances as closely as possible. The distance matrix, however, cannot be analyzed directly using eigen-decomposition, 149 because distance matrices are not positive semi-definite. But if it can be converted into an equivalent cross-product matrix then eigen-decomposition is possible, which gives a principal component analysis (PCA). MDS precisely does that. Each object is repre- sented as a point in a multi-dimensional space, and the points are so arranged that their pairwise distances have the strongest possible relation to the similarities among the pairs of objects. That is, two similar objects are represented by two points that are closer to each other, and two dissimilar objects are represented by two points that are further apart. Finding out the appropriate dimension is also part of the problem in MDS. However, in our case, since we know that the space is 3-D, we can get much better approximations of the relative location maps. The general MDS technique works in any dimensions and even in non-Euclidean space. MDS has been applied for localization in 2-D [75] to calculate relative sensor loca- tions based on their pairwise Received Signal Strength Intensity (RSSI) values. In our work, based on link quality indicator (LQI), we use MDS in 3-D as a primitive for finding relative location maps of all possible neighbors,N i (P max ) for each nodeu i . Note that, this is slightly different from the earlier approaches as a localization technique. Here, we are interested only in relative locations of each node’s neighbors. 6.4 Phase 2: Orthographic Projections The basic idea here is to simplify the original problem from 3-D by reducing it to mul- tiple similar problems in 2-D using orthographic projections, and then solving the 2-D 150 y z x u i u i ’ L 1 L 2 L 3 p q p’ q ’ q B(u i , R(P)) q Figure 6.2: An empty sector of angleθ aroundu i ’s projected location on thexy plane. problems using CBTC. The 2-D CBTC results says that, if every node adjusts its trans- mission power level such that there exists at last one neighbor in every sector of angle θ = 2π/3 around it, then network connectivity can be guaranteed so long as the MPG is connected. This is called theθ constraint. We begin with the following lemma. Lemma 10. Consider the projections of the locations of nodeu i and its neighbors for some power levelP (P≤P max ) on each of the three orthogonal planesxy,yz, andzx, as illustrated in Figure 6.2. If there is an empty sector of angleθ aroundu i ’s projection 151 on any of the planes, then there exist an infinite number of empty 3-D cones of angleθ aroundu i ’s location in the 3-D. Proof. The proof is by construction. In Figure 6.2, there exists an empty sector of angle θ around the projected location u ′ i of node u i on the xy plane for transmission range R(P). Consider the two intersection pointsp ′ andq ′ of the circle and the sector. If we raise the plane of the triangle△ u ′ i p ′ q ′ vertically upwards and parallel to thexy plane, it sweeps a triangular shaped volume bounded by three vertical planes, which are uniquely determined by three pairs of lines L 1 , L 2 ; L 1 , L 3 ; and L 2 , L 3 . By construction, it is clear that all the 3-D cones of apex angleθ contained within the region formed by the intersection of this triangular shaped volume and the spherical ballB(u i ,R(P)) are all empty. The lemma implies that if there is an empty sector of angle θ = 2π/3 on any of the three orthogonal projection planes, then there exist an infinite number of empty 3-D cones with apex angle2π/3 around nodeu i , which in turn implies by the results of [150] that the network will be disconnected ifu i chooses to transmit at the corresponding power level. We describe the algorithm in Figure 6. Each node u i starts off by transmitting a “Hello” message at its minimum transmission power level. Neighboring nodes upon hearing the “Hello” message acknowledge back with a “Reply” message. Nodeu i then projects the locations of those neighbors from which it heard the “Reply” message on to thexy,yz, andzx planes, as illustrated in Figure 6.3. Then, for each of the three planes, 152 y z x u Figure 6.3: Projected locations on xy and zx planes of node u i and its neighbors N i (P max ) whenu i transmits at maximum power. 153 Algorithm 6 ORTHOGRAPHIC PROJECTION-BASED ALGORITHM 1. N i (P)←φ 2. P←P min 3. D i ←φ: directions of projected neighbors 4. while(P≤P max andgap θ (D i )) do 5. Broadcast “Hello” message at powerP and gather “Reply” messages from neigh- bors 6. N i (P)←N i (P)∪{v | v replied} 7. Project locations ofu i andN i (P) onxy,yz, andzx planes 8. for (each plane) do 9. D i ←D i ∪{dir i (v)} 10. if(gap θ (D i )) then 11. P←increment(P); 12. break 13. end if 14. end for 15. end while node u i checks whether there is any empty sector of angle 2π/3 around it using the CBTC technique. If all the three planes satisfy theθ constraint, it stops and chooses the current power level. Otherwise, it increments its power to the next level, sends another “Hello” message and repeats the above process until there is no empty sector of angle 2π/3 around it on all the three projection planes, or until the maximum power level is reached. The minimum power that is required to guarantee theθ = 2π/3 constraint on all the three planes is chosen as the transmission power for that node. As in CBTC, we assume the existence of the following functions: (i)increment(P) that takes the current power level and increases it to the next level, (ii)dir i (v) that takes the projected locations on a plane ofu i ’s neighbors and sorts them with respect to some reference direction, and (iii)gap θ (D i ) that takes input as a set of directions and checks if there is an empty sector of angleθ around the projected location ofu i . 154 The novelty of the heuristic described above is that the algorithm runs inO(dlogd) time and does not assume directional information. However, it should be noted that the network topologies thus generated with transmission power levels as dictated by the algorithm are not always guaranteed to be connected. This can be intuitively seen by the following argument. Consider a particular nodeu i located at the origin and its set of neighbors that lies above thexy plane for a given power level. Project those neighbors on the three ortho- graphic planes. Next, consider a particular 3-D cone of angleθ < π/2 around nodeu i contained within the first quadrant (i.e., positivex,y,z) and project the cone on the three planes as well. This will form three sectors of angleθ aroundu i s projected locations on the three planes. Now, let there be a particular neighbor that lies just outside and above the surface of the 3-D cone at such a position, which when projected on the three planes, falls within the respective sectors formed by the cones projection on two of the planes (say, xy and xz). Note that, unless a neighbor lies inside the cone, its projection will not fall inside all the three projected sectors. Now with little thought we can convince ourselves that there could be another node(s) that does not lie within the cone but falls within the projected cones sector on the third plane (yz). Therefore, we observe that even though the projected sectors are not empty, that is, they satisfy theθ constraint on all the three planes, the 3-D cone can be empty. This implies that if we base our conclusion of network connectivity by satisfying the θ constraint on the three planes, it might be 155 incorrect at times for such degenerate cases. Here we restrainedθ<π/2 to illustrate one particular instance; however, the argument holds true forθ = 2π/3 as well. 6.5 Phase 2: Spherical Delaunay Triangulation The second approach in Phase 2 of our algorithm is based on the properties of Spherical Delaunay Triangulation. In computational geometry, Delaunay triangulation is the dual of V oronoi diagrams [8] which, for a set of points, tessellate the 2-D (3-D) region into a set of convex polygons (polyhedra), such that any point lying within a polygon (polyhe- dron) is closest to the point that is inside the polygon (polyhedron). This is known as the nearest neighborhood property of V oronoi diagrams. Likewise, Delaunay triangulation follows the dual of the nearest neighborhood property, called the empty circle property, as defined below. Definition 4. DELAUNAY EMPTY CIRCLE PROPERTY: Given a set of points lying on a plane such that no four points are co-circular (i.e., affinely independent), the circumcircle around each of the Delaunay triangles is empty, i.e., it does not contain any of the points in its interior. The empty circle property generalizes in 3-D in the form of empty spheres for Delaunay tetrahedrization. When Delaunay triangulation is carried out on points that lie on the surface of a sphere, it produces spherical triangles, and the empty circle property still holds. That is, for any three pointsa,b, andc that form the vertices of a spherical triangle, the spherical 156 a b c Figure 6.4: Spherical Delaunay Triangulation illustrating empty circle property: Spheri- cal capCap(a,b,c) is empty in its interior. capCap(a,b,c) is empty. This is illustrated in Figure 6.4. We use this empty spherical cap property as a primitive in our algorithm. Consider node u i and its set of neighbors N i (P) for a given power level P . We project the locations of these neighbors on the surface of a spherical ball centered atu i and of radiusR(P). This construction basically means drawing radial lines connecting u i and each of the neighbors until they intersect with the spherical surface. Then, we construct a spherical Delaunay triangulation with the projected points. This construction leads to the following lemma. Lemma 11. The 3-D cones that are formed withu i as the apex, and the spherical caps generated from Delaunay triangulation as their bases are empty. 157 p q r u i Figure 6.5: The 3-D cone withCap(p,q,r) as its base andu i as its apex is empty. Black dots show the actual locations of the neighbors, blue dots show their projected locations on the surface of the spherical ball.△ pqr is a spherical Delaunay triangle. Proof. The proof is by construction, as illustrated in Figure 6.5. Consider a particular 3D cone that has its base as the spherical cap formed by the verticesp,q, andr of a spherical triangle. Assume that the cone is not empty. This implies that there is some point which lies inside the cone, whose projection on the spherical surface by construction will lie in the interior ofCap(p,q,r). But this is a contradiction, because according to the Delaunay empty spherical cap property this cap is empty. Hence, the cone is empty. Theorem 10. Let each nodeu i construct a spherical Delaunay triangulation of its pro- jected neighbor locations on the spherical surface for a given power level P . If the largest surface area Ω max i of the spherical cap for nodeu i (except for boundary nodes) is no more than 2.72·R(P) 2 , then the network topology formed by the nodes with the corresponding transmission power levels is guaranteed to be at least one-connected. 158 Proof. Let the 3-D cone formed with apex at nodeu i , and base as the spherical cap with the largest surface areaΩ max i has an apex angleθ. From simple trigonometry, the surface area of the spherical cap is 2πrh, where r = R(P)· sin(θ/2) and h = R(P)· (1− cos(θ/2)). Therefore, Ω max i = 2πrh = 2πR(P) 2 ·sin(θ/2)(1−cos(θ/2)) (6.1) Now it is easy to verify that if Ω max i ≤ 2.72·R(P) 2 , thenθ≤ 2π/3. This implies that if the surface area of the largest cap is no more than 2.72·R(P) 2 , then there is no empty 3-D cone of apex angle greater than 2π/3 aroundu i . Since this is true for every node (except the boundary nodes), the network is at least one connected from the result of [12]. The implication of the above theorem is that if every node adjusts its power level to have enough neighbors, such that, none of the caps of the spherical Delaunay triangula- tion has a surface area greater than the threshold mentioned above, then the network will be at least one connected so long as the maximum power graph is connected. There is a subtlety with boundary nodes while checking for the largest spherical cap. We define a boundary node as one that lies outside the 3-D convex hull formed by all its neighbors when transmitting at maximum power. A node can identify itself as a boundary node in Phase 1 of the algorithm after running MDS and by constructing a 3-D 159 Figure 6.6: Spherical Delaunay triangulation using the Quickhull algorithm of a set of 100 points randomly distributed on the surface of a sphere of radius50. 160 Algorithm 7 SPHERICAL DELAUNAY TRIANGULATION-BASED ALGORITHM 1. N i (P)←φ 2. P←P min 3. while(P≤P max ) do 4. Broadcast “Hello” message at powerP and gather “Reply” messages from neigh- bors; 5. N i (P)←N i (P)∪{v | nodev replied}; 6. Project the locations ofN i (P) on the surface of the spherical ballB(u i ,R); 7. Construct SDT with the projected points; 8. Find the largest spherical cap surface area,Ω max i ; 9. ifΩ max i > 2.72·R 2 then 10. P←increment(P); 11. else 12. break; 13. end if 14. end while convex hull with all the neighbors that lie within its communication range. Each node can perform this in O(dlogd) time using the Quickhull algorithm [14], where d is its degree. Since a boundary node does not have neighbors in all directions around itself, its spherical Delaunay triangulation might form caps that are smaller than the threshold area but still have empty 3-D cones pointing outwards. In such cases, a boundary node calculates the difference in surface area of the spherical ball and the sum of the spherical triangles. If this difference is greater than the threshold then it further increases its power level. One advantage of this SDT based approach compared to the CBTC technique is that the boundary nodes do not end up with maximum power levels. Formally, the following steps shown in Algorithm 7 will be executed on each internal nodeu i . Constructing a spherical Delaunay triangulation in Step 7 of the algorithm is equiv- alent of finding a 3-D convex hull for the set of projected points on the sphere. This 161 can be done in O(dlogd) time using the Quickhull algorithm [14], where d is num- ber of neighbors (see Figure 6.6 for an illustration). Note that, the number of spherical caps thus generated is of O(d). Hence, the time complexity of the above algorithm is O(dlogd). This is a substantial improvement over the existing algorithms that run in O(d 3 logd) time. Since the average node degree in 3-D is very high compared to that in 2-D for a network to be connected with high probability under random deployment, an improvement ofd 2 implies a much faster algorithm. As we mentioned earlier, the orthographic projection based approach, although works very well in practice, does not theoretically guarantee a connected 3-D network, even when the θ constraint is satisfied on all the three planes. The degenerate cases can be identified by combining the two approaches in the following way. Before projecting the neighbors on the spherical surface in Step6 of Algorithm 7, each node first projects them on thexy,yz, andzx planes. Then it increments its power level until all the three planes satisfyθ = 2π/3 constraint, i.e., until the functiongap θ (D i ) returns false for all the three planes. Let this power level beP proj . Only after this, the neighbors are projected on the surface of a sphere of radius R = f(P proj ) and Steps 7, 8, 9 and so on are followed. If the algorithm terminates with the node choosing a power level that is smaller than its maximum power level, then this is a degenerate case. This is because had the node settled down with its maximum power levelP max , it would mean that either the largest spherical cap area is no less than the threshold value of Ω max = 2.72·R 2 . In the first case, the 3-D network will not be connected, whereas in the second case the network 162 finally formed is, in fact, the original maximum power graph, which was assumed to be connected. 6.6 Evaluation In order to understand the efficiency of our algorithms, we conducted simulations by gen- erating random networks in 3-D of different sizes inside a cuboid of size100×100×100. Before running the projection based or the spherical Delaunay triangulation based algo- rithms, we run MDS to get relative neighbor locations for each node at their maximum transmission power level. For the SDT based algorithm, we generated random network topologies for n = 200 nodes under different maximum power levels. The performance of our algorithm is measured in terms of the average node degree and average transmission power. As an example, we show one specific instance of the initial network topology (the connected maximum power graph) whenP max = 40 in Figure 6.7(a). In Figure 6.7(b), we show the same network after all the nodes have settled down with minimal transmission power according to the SDT algorithm. For the same instance, we plotted the initial and final node degrees in Figure 6.8(a). Finally, in Figure 6.8(b), we show the assigned minimal power levels for all the nodes. Note that, node degrees have drastically reduced. We also observe that only 7.5% of the nodes transmit at their maximum power level, and more than 25% of the nodes transmit at less than half the maximum power level (i.e., below 20). Next, we describe the general trends. 163 (a) Maximum power graph. (b) Final connected network topology based on SDT. Figure 6.7: (a) Network topology of the original maximum power graph. (b) Network topology after running the SDT-based algorithm. Heren = 200 andP max = 40. Node degrees have drastically reduced. 164 3 − 8 9 −14 15−20 21−26 27−32 33−38 39−44 45−50 51−56 57−62 0 10 20 30 40 50 60 Node degree Number of nodes Final topology Maximum power graph (a) 10 15 20 25 30 35 40 0 2 4 6 8 10 12 14 16 Power level Number of nodes (b) Figure 6.8: (a) Node degrees of the maximum power graph and that of the final topology forn = 200,P max = 40. (b) Final assigned minimal transmission power levels of nodes, forn = 200,P max = 40. 165 50 100 150 200 250 300 350 400 0 10 20 30 40 50 60 70 80 90 Number of nodes Average node degree Final topology Maximum power graph (a) 30 35 40 45 50 55 60 65 70 0 20 40 60 80 100 120 Maximum power Average node degree Final topology Maximum power graph (b) Figure 6.9: (a) Dependency of average node degree with network size. (b) Dependency of the average node degree with maximum power, for n = 200; node degree remains almost flat in the final topology based on the SDT algorithm. 166 We measured the dependencies between (i) average node degrees, and (ii) average transmission power, with different network sizes. In Figure 6.9(a), which shows the plot of average node degree with network size for P max = 40, we observe that as the network size (density) increases, the average node degree increases sharply for maximum power graphs, whereas increases very slowly for network topologies formed based on the SDT algorithm. Figure 6.10 shows the plot of average transmission power levels with increasing network density for topologies generated based on SDT. We observe that as the network gets denser, the average transmission power decreases. This is expected because as the density increases, nodes can use lower power levels to form more number of multihop paths to guarantee network connectivity. Note that, in Figure 6.9(a) when the number of nodes increases to 400, the average node degree reaches≈ 15, which is in accordance with the percolation theory of critical average node degree for network connectivity ( 4π 3 ·(0.2) 3 ·(400)≈ 13.4). In Figure 6.9(b), we plot average node degrees for random topologies of 200 nodes with increasing maximum power levels. We observe that average node degrees almost remain flat in the final topology generated based on the SDT algorithm as compared to that of the maximum power graph. This is because of the fact that as the maximum power varies, only the very small number of boundary nodes get affected, because they are more likely to transmit at or close to the maximum power compared to the internal nodes in order to guarantee a minimal number of neighbors. The average transmission power level chosen by the nodes in the final topology is observed to vary from 26 to 21 167 50 100 150 200 250 300 350 400 20 22 24 26 28 30 32 34 36 38 Number of nodes Average transmission power Final topology Figure 6.10: Dependency of average transmission power with network size. as the maximum power increased from 30 to 70 (not plotted). Thus, the effectiveness of the algorithm is predominant at higher maximum power levels. To compare the SDT based algorithm with the one described in [12] based on the proceduregap−3D α (), both of which check for empty 3-D cones, we simulated the al- gorithms and measured the CPU running times using the MATLAB profile tool. Our measurements show that the respective proceduressdtcheck() (part of our implementa- tion) andgap−3D α () take most percentages (80%−90%) of the total execution times. In Figure 6.11, we plot of CPU execution times for these two functions on random network topologies of different sizes with maximum power set to40. We observed that SDT per- forms much faster than the other one, and with increasing network size (or equivalently, 168 50 100 150 200 250 300 350 400 0 50 100 150 200 250 Number of nodes CPU time (seconds) SDT gap−3D α Figure 6.11: CPU execution time of the SDT based algorithm and the one [12] based on the proceduregap−3D α () for different random topologies withP max = 40. with increasing average node degree), the difference between execution times becomes more predominant. Finally, to measure the practicality of the orthographic projection based algorithm, we generated two random networks with n = 50 and 100 nodes, and P max = 100. Then we ran the projection based algorithm and generated different network topologies where the θ constraint is satisfied on one, two, or all three planes. For each of these cases, we checked the connectivity of the original 3-D graph. In Figure 6.12, we plot the probability of connectivity withθ. Each point in the plot is averaged over 200 runs. We observe that when theθ constraint is satisfied on only one plane, the original 3-D graph is actually never connected. However, when each node guaranteesθ = 2π/3 on all three planes, the 3-D topology generated with that power level is found to be connected at all 169 2 2.5 3 3.5 4 4.5 5 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 θ in radians Probability of 1−connectivity 2π/3 5π/6 π 7π/6 4π/3 3π/2 3 planes 2 planes 1 plane (a) Forn=50 nodes 2 2.5 3 3.5 4 4.5 5 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 θ in radians Probability of 1−connectivity 2π/3 5π/6 π 7π/6 4π/3 3π/2 3 planes 2 planes 1 plane (b) Forn=100 nodes Figure 6.12: Probability of network connectivity as theθ constraint is satisfied on 1, 2, or all3 orthogonal planes. 170 times. Moreover, even when the constraint is satisfied on only two planes, we find that the 3-D graph is disconnected only a very small number (less than 1%) of times. This shows that the heuristic based approach works very well in practice. 6.7 Summary In this chapter, we have designed efficient power control schemes for sensor networks deployed in 3-dimensions where very high density of nodes causes high interference which, in turn, lowers network throughput. We presented two efficient distributed topol- ogy control algorithms for 3-D networks. Our first approach is based on the idea of 2-D orthographic projections, by which we reduced and simplified the original 3-D problem into a set of three similar problems on 2-D, and used existing techniques from CBTC to show that network connectivity can be guaranteed almost at all times. In the second approach we used the properties of spherical Delaunay triangulation on the surface of a sphere to determine the existence of empty 3-D cones in an efficient way, and showed that the network topology generated based on this is always connected. Both the algo- rithms are computationally very efficient and scale asO(dlogd) in time, whered is the average node degree. Thed 2 improvement in running time implies a much faster algo- rithm especially in 3-D, asd is typically very high in 3-D compared to that in 2-D, and this is verified by measuring CPU execution times on topologies with increasing node degree. To substantiate our claims we conducted simulations on network topologies in 171 3-D for both the SDT based algorithm and the heuristic based approach. Doing a proba- bilistic analysis of network connectivity for the orthographic projection based approach and implementation of the SDT algorithm on real motes are part of our future work. 172 Chapter 7 Conclusions and Future Work Data collection is a fundamental operation in wireless sensor networks where sensor nodes measure attributes about a phenomenon of interest and transmit their readings to a common base station. In applications such as structural health monitoring, security surveillance, health care, environmental monitoring, etc., it is often important to de- liver data in a timely fashion. In this thesis, we focused on the algorithmic aspects of throughput-delay performance for fast data collection in tree-based sensor networks. We addressed the problem of jointly optimizing these two mutually conflicting performance objectives in the context of aggregated convergecast. Our approach comprised three tech- niques - (i) multi-channel TDMA scheduling, (ii) routing over optimal topologies, and (iii) transmission power control. Using a combination of these three mechanisms, we showed that it is possible to design efficient algorithms that have provably good worst- case performance bounds. In the following we briefly describe our contributions and discuss some future research directions and open problems. 173 7.1 Multi-Channel TDMA Scheduling In Chapter 3, we exploited the benefits of multiple frequency channels and designed ap- proximation algorithms for maximizing the aggregated convergecast throughput under a graph-based interference model. We considered two cases with respect to the knowledge of the routing topology. First, when the routing topology is known a priori, and second when the routing topology is unknown and can be arbitrary. We decoupled the joint fre- quency and time slot assignment problem into two separate subproblems of frequency assignment and time slot assignment, and showed that this decoupling still leads to good approximation ratios. For the case of known routing topology, we considered two random geometric graph models : (i) unit disk graphs, where nodes have uniform transmission range, and (ii) general disk graphs, where nodes have different transmission ranges. The first one corre- sponds to a network where all the nodes transmit at a uniform power level, whereas the second one might correspond to a network where nodes can control their transmission power. Benefits of controlling transmission have been described in detail earlier in Chap- ter 2. To assign frequencies in the UDG model, we proposed a receiver-based greedy frequency assignment strategy, where the receivers of the given routing tree are assigned frequencies in such a way that the maximum number of edges transmitting on the same frequency is minimized. This corresponds to a suboptimal load-balanced frequency as- signment which gives a constant factor approximation with respect to max-min load on any frequency. Our time slot assignment strategy is also greedy, where we showed that 174 the schedule length, which is an indicator of the aggregated convergecast throughput, is within a constant factor of the optimal for unit disk graph models. For the general disk graph model, we formulated the two subproblems as integer linear programs, and by us- ing a randomized rounding strategy we showed that the overall approximation ratio of the schedule length is logarithmic with respect to the number of nodes in the network. Lastly, for the case of unknown routing topology, we proved that the approximation ra- tios still hold so long as the maximum node degree of any node in the tree is bounded by a constant. Simulation results show significant improvement when using multiple frequencies and underscores their benefits in scheduling. These results are perhaps the first ones in their category of designing provably good approximation algorithms with multiple frequencies for arbitrarily deployed networks where fast data collection is the primary motivation. The algorithms proposed in Chapter 3, however, are limited to only graph-based in- terference models, which are somewhat idealistic in nature and does not capture the cu- mulative interference from many simultaneous transmitters that are located far away. To address this issue, in Chapter 4, these multi-channel scheduling algorithms are extended for the more realistic SINR-based interference model. By using the notion of link diver- sity, and by suitably reusing time slots, we showed that the modified algorithms still hold a good performance ratio that scales with the number of non-empty length classes. Al- though theoretically this number can be as large as the number of nodes in the network, in practice, it is a small constant. 175 7.2 Optimal Routing Topologies In Chapter 5, we considered delay as another performance objective, and addressed the joint problem of optimizing both throughput and delay in the context of aggregated con- vergecast. We defined the maximum delay in terms of the largest hop distance from any node to the sink, called the radius of the tree, and constructed routing topologies that minimizes the radius under a given degree constraint on the nodes. We noted that mini- mizing the radius and degree of a spanning tree at the same time are mutually conflicting objectives, and discovered that a high node degree creates a bottleneck in the network that limits the aggregated throughput. Accordingly, we formulated the joint optimization problem of both degree and radius as a bicriteria optimization problem with the goal to minimize the maximum hop distance in the tree such that the maximum node degree remains within a certain threshold. To this end, we designed an approximation algorithm that constructs a bounded-degree-minimum-degree spanning tree with constant factor bi- criteria approximation ratios on both the objectives. Although the notion of bicriteria formulation for network design problems is not new, we are the first ones to consider throughput and delay under the same bicriteria optimization framework and designed algorithms with provably good performance bounds. Once the routing topology is constructed, we evaluated the multi-channel schedul- ing algorithms proposed in Chapter 3 on such topologies. Our simulation results show 176 significant improvements on the delay as well as on aggregated throughput on bounded- degree-minimum-radius spanning trees; it gives the best of both worlds in terms of max- imizing throughput and minimizing delay as compared to some of the other more com- monly used routing topologies, such as minimum spanning trees where the radius is low but the average degree of a node is very high, and minimum interference trees where the radius is large but the average node degree is low. 7.3 Transmission Power Control In Chapter 6, we considered sensor networks deployed 3-dimensions and designed ef- ficient power control schemes from local geometric information in order to construct sparse topologies while maintaining network connectivity. Although the amount of liter- ature in topology control in 2-D is enormous, there exist few works that address under a 3-D setting. Moreover, the existing solutions require directional information and incur a lot of computational overhead. As the density of nodes required to maintain a connected network in 3-D is very high, it negatively impacts the achievable throughput due to in- creased interference, and so with a sparser topology is likely help. Our contribution in this respect is two fold. We extended some of the existing works on topology control for 2-D to be applicable in a 3-D setting. In particular, we used orthographic projections to extend the well known cone-based topology control algorithm, and showed that al- though network connectivity cannot be theoretically guaranteed, it performs very well in practice. In addition, we proposed a robust new technique based on spherical Delaunay 177 triangulation that guarantees network connectivity and is superior to existing techniques in terms of overhead and computational requirements. 7.4 Future Work 7.4.1 Real Testbed Implementation Although we evaluated our proposed algorithms using simulations, real implementation on sensor nodes where schedules are computed locally and are adaptive to network dy- namics are necessary to enhance the operation of sensor networks and to meet application requirements. For instance, we observe a trend in using WSNs to support more complex operations ranging from industrial control to health care, which require complex oper- ations like detection of events in real-time, or responsive querying of the network by collecting streams of data in a timely manner. Thus, supporting QoS metrics such as delay and reliability become more important. Therefore, distributed implementation and performance testing of the proposed algorithms on testbed or real deployments becomes essential. Additionally, real implementation and deployment will help in addressing the problems of intermittent connectivity and channel errors with unreliable links and han- dling asymmetric links. 178 7.4.2 Other Joint Objectives Although there exist some works that address multiple joint objectives, more detailed investigations to address the trade-offs between conflicting objectives will be beneficial. Most of the studies consider the trade-offs between energy efficiency and latency objec- tives. Different trade-offs can be identified between other objectives, such as minimiz- ing latency and maximizing reliability, or maximizing capacity and minimizing energy consumption. For instance, with the extension of WSNs in the visual domain where embedded cameras act as sensors, criteria such as reliability, QoS, and timeliness of the streamed data are becoming important. Solutions to address different objectives and trade-offs, for instance, consumed energy versus reconstructed image quality, should be explored within the perspective of data collection in WSNs. In addition, our proposed tree construction is applicable only when nodes transmit at a uniform power level. Ex- tension of these results when nodes transmit at different power can be useful. 7.4.3 Traffic Patterns We have considered fixed traffic patterns where every node generates a fixed number of packets (one in our case) in each data collection cycle. In real scenarios, some nodes may have a lot of packets that require more than one time slot per frame, while some others may not have any data to send in a time slot, thus wasting bandwidth. It will be interesting to explore the performance in such scenarios with random packet arrivals 179 and combining the solutions of TDMA scheduling with rate allocation algorithms, espe- cially in applications where high data rates are necessary. In the proposed algorithms, we have considered data collection under perfect aggregation where a node can aggre- gate all the packets coming from its children as well as its own into a single packet before transmitting to its parent. Another possibility is to investigate different levels of aggre- gation, i.e., how much of the data received from the children is forwarded to the parent node. Investigating different levels of aggregation was proposed in the literature where the efficiency of different tree construction mechanisms were analyzed in terms of la- tency and energy metrics. This study can be extended for TDMA-based data collection algorithms in WSNs. Lastly, we have considered tree-based sensor networks where the routing tree is fixed for every packet traversal to the sink. It will be interesting to evalu- ate the throughput-delay performance where packets from a given source nodes can take different paths depending on network conditions. 7.4.4 Cross Layer Solutions Our proposed algorithms provide cross layer solutions in some sense, where the sched- ules are computed together with transmission power control, optimal routing topolo- gies, and multi-frequency scheduling. It is indeed essential to address the problems from a cross layer perspective to achieve the target functions and offer better perfor- mance. Along this line of research, Chafekar et al. in [20] extended the work by 180 Moscibroda [112] in designing cross-layer protocols using the SINR model and pro- posed polynomial time algorithms with provable worst-case performance guarantee for the latency minimization problem. Their cross-layer approach chooses power level for all transceivers, routes for all connections, and constructs an end-to-end schedule such that SINR constraints are satisfied. A prominent research direction is to consider such cross-layer approaches from a theoretical point of view. More research can be done in this direction to combine the existing work with the solutions at different layers. In most studies, static topologies are assumed. Problems related to dynamic topologies, such as topological changes and addition of new nodes are open. In addition, the time complex- ity of data gathering under various network conditions, such as when some nodes have no packet to transmit, or when no buffering is allowed remain open. 7.4.5 Realistic Interference Models As was pointed by Moscibroda in [111], the type of interference model may heavily im- pact the achievable results. Use of realistic models for communication and interference is another direction that can be further investigated. Along this direction, Goussevskaia et al. [56] present the first NP-completeness proofs (by reducing from the Partition prob- lem) on two scheduling problems using the SINR interference model. The first problem consists in finding a minimum-length schedule for a given set of links. The second prob- lem receives a weighted set of links as input and consists in finding a maximum-weight subset of links to be scheduled simultaneously in one shot. In [111, 112], Moscibroda 181 et al. study a generalized version of the SINR interference model and obtain theoretical upper bounds on the scheduling complexity of arbitrary topologies. They prove that if signals are transmitted with correctly assigned transmission power levels, the number of time slots required to successfully schedule all links in an arbitrary topology is propor- tional to the squared logarithm of the number of nodes times a previously defined static interference measure. 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Abstract (if available)

Abstract Convergecast, namely the many-to-one flow of data from a set of sources to a common sink over a tree-based routing topology, is a fundamental communication primitive in wireless sensor networks. For real-time, mission-critical, and high data-rate applications, it is often critical to maximize the aggregated data collection rate (throughput) at the sink node, as well as minimize the time (delay) required for packets to get there. In this thesis, we look into the algorithmic aspects of jointly optimizing both throughput and delay for aggregated data collection in sensor networks. Our contributions are in designing efficient algorithms with provably good, worst-case performance bounds for arbitrarily deployed networks. To the best of our knowledge, we are the first ones to address these two mutually conflicting performance objectives -- throughput and delay -- under the same optimization framework and develop techniques to meet the stringent requirements for fast data collection.

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Asset Metadata

Creator Ghosh, Amitabha (author)

Core Title Algorithmic aspects of throughput-delay performance for fast data collection in wireless sensor networks

School Viterbi School of Engineering

Degree Doctor of Philosophy

Degree Program Electrical Engineering

Publication Date 08/26/2010

Defense Date 07/27/2010

Publisher University of Southern California (original), University of Southern California. Libraries (digital)

Tag algorithms,data collection,OAI-PMH Harvest,scheduling,sensor networks,throughput-delay

Language English

Contributor Electronically uploaded by the author (provenance)

Advisor Krishnamachari, Bhaskar (committee chair), Raghavendra, Cauligi S. (committee member), Sukhatme, Gaurav S. (committee member)

Creator Email amitabha.ghosh@gmail.com,amitabhg@usc.edu

Permanent Link (DOI) https://doi.org/10.25549/usctheses-m3413

Unique identifier UC1144831

Identifier etd-Ghosh-4054 (filename),usctheses-m40 (legacy collection record id),usctheses-c127-392230 (legacy record id),usctheses-m3413 (legacy record id)

Legacy Identifier etd-Ghosh-4054.pdf

Dmrecord 392230

Document Type Dissertation

Rights Ghosh, Amitabha

Type texts

Source University of Southern California (contributing entity), University of Southern California Dissertations and Theses (collection)

Repository Name Libraries, University of Southern California

Repository Location Los Angeles, California

Repository Email cisadmin@lib.usc.edu