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Joint routing and compression in sensor networks: from theory to practice

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Joint routing and compression in sensor networks: from theory to practice

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Content JOINT ROUTING AND COMPRESSION IN SENSOR NETWORKS: FROM THEORY TO PRACTICE by Sundeep Pattem A Dissertation Presented to the FACULTY OF THE USC GRADUATE SCHOOL UNIVERSITY OF SOUTHERN CALIFORNIA In Partial Fulllment of the Requirements for the Degree DOCTOR OF PHILOSOPHY (ELECTRICAL ENGINEERING) August 2010 Copyright 2010 Sundeep Pattem Dedication To Sameera ii Acknowledgements My work at USC owes a great deal to collaborations with and help from several faculty and colleagues: Prof. Bhaskar Krishnamachari, Prof. Antonio Ortega, Prof. Ramesh Govindan, Prof. Gaurav Sukhatme, Prof. Kristina Lerman (USC/ISI), Sameera Poduri, Avinash Sridharan, Ying Chen, Alexandre Ciancio, Godwin Shen, Sungwon Lee, Matt Klimesh (JPL), Maheswaran Sathiamoorthy, Aaron Tu, Aram Galstyan (USC/ISI). It has been a privilege to be associated with Prof. Bhaskar Krishnamachari for all these years. It is no exaggeration to say that I would not be writing a thesis but for Bhaskar's help - the extra-ordinary patience and kindness, the ability to empathize, en- thuse and inspire, and the passion for helping students realize their potential, not just in research, but as well-rounded people. I hope my life will re ect what I imbibed from his emphasis on values and service. My roommates and buddies, Apoorva Jindal, Rahul Urgaonkar, Avinash Sridharan, Sonal Jain, Ankit Singhal, made time y. The wise old(er) folks, Narayanan Sadagopan, Venkata Pingali, Amit Ramesh, Srikanth Saripalli, Krishnakant Chintalapudi, Karthik Dantu, made life easy. ANRG members - Marco Zuniga, Ying Chen, Shyam Kapadia, Divya Devaguptapu, Joon Ahn, Hua Liu, Pai-Han Huang, Kiran Yedavalli, Jung-Hyun iii Jun, made working in the lab a pleasure. I will miss Shane Goodo and his edgy sense of humor. My mother, Sri Shobha Rani, is my rst teacher. I owe all my learning to her. My father, Sri Rajeswara Rao, toiled hard so we could dream. I hope he thinks his harvest has been good. My grandmother, Sri Kalavathi, introduced me to the power, and pleasures, of the imagination. My sister, Santoshi, is my biggest (and only) fan. We'll always have each other. My friend, Rakesh, helped open my eyes to the vast and the blissful. My wife, Sameera, has shared in all the delight and the despair. I hope she thinks it worthwhile. For me, she has been a blessing. iv Table of Contents Dedication ii Acknowledgements iii List Of Figures viii Abstract xi Chapter 1: Introduction 1 1.1 Data gathering sensor networks . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Thesis and Research Summary . . . . . . . . . . . . . . . . . . . . . . . . 3 1.2.1 Impact of spatial correlations on optimal routing . . . . . . . . . . 4 1.2.2 Algorithms for achieving distributed compression . . . . . . . . . . 5 1.2.3 Architecture and system implementation for distributed compression 5 Chapter 2: Background on in-network compression 7 2.1 Distributed aggregation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 2.2 Distributed compression . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 2.2.1 Distributed source coding . . . . . . . . . . . . . . . . . . . . . . . 9 2.2.2 Analysis of impact of correlations on routing . . . . . . . . . . . . 9 2.2.3 Spatial transforms . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 2.2.4 Compressed sensing . . . . . . . . . . . . . . . . . . . . . . . . . . 13 2.3 Distributed conguration . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 2.4 System implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 Chapter 3: Modeling of joint routing and compression 17 3.1 Assumptions and Methodology . . . . . . . . . . . . . . . . . . . . . . . . 19 3.1.1 Note on Heuristic Approximation . . . . . . . . . . . . . . . . . . . 23 3.2 Routing Schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 3.2.1 Comparison of the schemes . . . . . . . . . . . . . . . . . . . . . . 25 3.3 A Generalized Clustering Scheme . . . . . . . . . . . . . . . . . . . . . . . 28 3.3.1 Description of the scheme . . . . . . . . . . . . . . . . . . . . . . . 29 3.3.1.1 Metrics for evaluation of the scheme . . . . . . . . . . . . 29 3.3.2 1-D Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 3.3.2.1 Sequential compression along SPT to cluster head . . . . 31 v 3.3.2.2 Compression at cluster head only . . . . . . . . . . . . . 37 3.3.3 2-D analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 3.3.3.1 Opportunistic compression along SPT to cluster head . . 41 3.3.3.2 Compression at cluster head only . . . . . . . . . . . . . 43 3.4 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 3.4.1 Communication and Topology models . . . . . . . . . . . . . . . . 49 3.4.1.1 Random geometric graphs . . . . . . . . . . . . . . . . . 49 3.4.1.2 Realistic Wireless Communication model . . . . . . . . . 50 3.4.2 Joint entropy models . . . . . . . . . . . . . . . . . . . . . . . . . . 52 3.4.2.1 Linear and convex functions of distance . . . . . . . . . . 53 3.4.2.2 Continuous, Gaussian data model . . . . . . . . . . . . . 53 3.4.3 Summary of results . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 3.5 Summary and Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 Chapter 4: Practical schemes for distributed compression 58 4.1 Wavelet transform design for wireless broadcast advantage . . . . . . . . . 58 4.1.1 Wavelet basics: The 5/3 lifting transform . . . . . . . . . . . . . . 59 4.1.2 Wavelets for sensor networks . . . . . . . . . . . . . . . . . . . . . 60 4.1.2.1 Unidirectional 1D wavelet . . . . . . . . . . . . . . . . . . 60 4.1.2.2 2D wavelet for tree topologies . . . . . . . . . . . . . . . 60 4.1.3 2D wavelet for wireless broadcast scenario . . . . . . . . . . . . . . 62 4.1.3.1 Augmented neighborhoods . . . . . . . . . . . . . . . . . 62 4.1.3.2 New transform denition . . . . . . . . . . . . . . . . . . 63 4.1.3.3 Performance of new transform . . . . . . . . . . . . . . . 63 4.2 Compressed sensing for multi-hop network setting . . . . . . . . . . . . . 64 4.2.1 Combining routing with known results in compressed sensing . . . 65 Chapter 5: SenZip: Distributed compression as a service 68 5.1 SenZip architecture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 5.1.1 SenZip Specication . . . . . . . . . . . . . . . . . . . . . . . . . . 70 5.1.1.1 Compression Service . . . . . . . . . . . . . . . . . . . . . 70 5.1.1.2 Networking components . . . . . . . . . . . . . . . . . . . 72 5.1.2 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 5.2 Mapping algorithms to architecture . . . . . . . . . . . . . . . . . . . . . . 73 5.2.1 Algorithm details . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 5.2.1.1 DPCM . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 5.2.1.2 2D wavelet . . . . . . . . . . . . . . . . . . . . . . . . . . 75 5.2.2 Relating algorithms to SenZip . . . . . . . . . . . . . . . . . . . . . 76 5.2.2.1 Initialization . . . . . . . . . . . . . . . . . . . . . . . . . 76 5.2.2.2 Data forwarding and compression . . . . . . . . . . . . . 78 5.2.2.3 Reconguration . . . . . . . . . . . . . . . . . . . . . . . 78 5.3 System implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 5.3.1 TinyOS code . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 5.3.1.1 Interfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . 80 5.3.1.2 AggregationP component . . . . . . . . . . . . . . . . . . 80 vi 5.3.1.3 CompressionP component . . . . . . . . . . . . . . . . . . 81 5.3.1.4 Changes to CTP . . . . . . . . . . . . . . . . . . . . . . . 83 5.3.1.5 Application . . . . . . . . . . . . . . . . . . . . . . . . . . 83 5.3.2 Experimental Results . . . . . . . . . . . . . . . . . . . . . . . . . 84 5.3.2.1 Static topologies . . . . . . . . . . . . . . . . . . . . . . . 84 5.3.2.2 Dynamic topologies . . . . . . . . . . . . . . . . . . . . . 86 Chapter 6: Conclusion 88 6.1 Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88 6.2 Future work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 References 91 vii List Of Figures 1.1 (a) Illustration of a distributed phenomena and data gathering using sensor network. (b) Hardware - Telosb mote device. . . . . . . . . . . . . . . . . 2 1.2 (a) Software abstraction from application developer perspective. (b) Pos- sible t for compression. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.3 Software abstraction for compression as a service . . . . . . . . . . . . . . 3 3.1 Empirical data (from the rainfall data-set) and approximation for joint entropy of linearly placed sources separated by dierent distances . . . . . 21 3.2 Illustration of routing for the three schemes: DSC, CDR, and RDC. H i is the joint entropy of i sources. . . . . . . . . . . . . . . . . . . . . . . . . . 25 3.3 Comparison of energy expenditures for the RDC, CDR and DSC schemes with respect to the degree of correlation c. . . . . . . . . . . . . . . . . . . 27 3.4 Illustration of clustering for a two-dimensional eld of sensors . . . . . . . 30 3.5 Comparison of the performance of dierent cluster-sizes for linear array of sources(n =D = 105) with compression performed sequentially along the path to cluster heads. The optimal cluster size is a function of correlation parameter c. Also, cluster size s = 15 performs close to optimal over the range of c . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 3.6 Illustration of the existence of a static cluster for near-optimal performance across a range of correlations. The sources are in a linear array and data is sequentially compressed along the path to cluster heads. . . . . . . . . . 37 3.7 Performance with compression only at cluster head with nodes in a linear array(n = D = 105). Cluster sizes s = 5; 7 are close to optimal over the range of c . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 viii 3.8 Illustration of the near-optimal cluster size with compression only at cluster head with nodes in a linear array. The performance of cluster sizes near s = 7( q 105 2 ) is close to optimal over the range of c values . . . . . . . . 40 3.9 Routing in a 2-D grid arrangement. (a) Calculation of joint entropy. Using the iterative approximation joint entropy of k nodes forming a contiguous set is the same as the joint entropy of k sensors lying on a straight line. This is illustrated along the diagonal. This also illustrates opportunistic compression along SPT to cluster head. (b) Intra-cluster, shortest path from source to cluster head routing with compression only at cluster head. The routing from cluster heads to sink is similar. . . . . . . . . . . . . . . 41 3.10 Comparison of the performance of various cluster sizes for a network with 10 6 nodes on a 1000x1000 grid when compression is possible only at cluster heads. The performance for s = 5; 10 is observed to be close to optimal over the range of c values. . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 3.11 Illustration of the existence of a near-optimal cluster size. The network size is nn = 1000 1000 and compression is possible only at cluster heads. The performance of cluster side values near s = :6487n 1 3 is quite close to optimal for all values of c ranging from 0.0001 to 10000 . . . . . . 48 3.12 Random geometric graph topology. Performance of clustering with density = 1 node=m 2 , communication radius = 3m for network of size (a) 24x24 (b) 84x84 (c) 200x200. Near-optimal cluster sizes are (a) 3,4 (b) 4,7 (c) 8,10. 51 3.13 Realistic wireless communication topology. Performance of clustering in 48mx48m network with density = .25 nodes=m 2 for power level (a) -3dB (b) -7dB (c) -10dB. Cluster sizes 6, 8 are near-optimal. . . . . . . . . . . . 52 3.14 (a) Example forms of joint entropy functions for 2 sources. The entropy of each source is normalized to 1 unit. The convex and linear curves are clipped when the joint entropy equals the sum of individual entropies. The curves shown are for correlation parameter c = 2. Performance of clustering in 72m 72m network with density = .25 nodes=m 2 for (b) linear model (c) convex model of joint topology. Cluster size 6 is near- optimal. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 3.15 Performance of clustering in 48m48m network with density = .25nodes=m 2 with a continuous, jointly Gaussian data model and quatization step (a) = 1 (b) = 0.5 (c) = 0.05. Cluster size 6, 8 are near-optimal. . . . . . . 56 4.1 Example (a) signal and (b) 5/3 wavelet coecients . . . . . . . . . . . . . 59 ix 4.2 Illustration of odd (green) and even (blue) nodes in a subtree for 2D wavelet (a) with unicast and (b) exploiting broadcast nature of wireless communi- cations. The solid arrows are part of the tree routing paths. The dashed arrows are the wireless links not part of the tree. The arrows crossed o in red denote disallowed interactions for transform invertibility and unidi- rectionality. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 4.3 (a) Sample tree topology (b) With additional broadcast links in the aug- mented neighborhoods at each node (c) Performance gain in terms of SNR vs. cost for new transform compared to 2D wavelet for tree topologies . . 63 4.4 Compressed sensing performance in multi-hop setting. Plot of SNR vs cost for dierent schemes. The black and green curves are for Sparse Random Projections (SRP). The blue and red curves are for two variations of computing projections over shortest path routing. . . . . . . . . . . . . 67 5.1 The SenZip architecture. A completely distributed compression service is enabled by having the interacting components shown here at each network node. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 5.2 Aggregation table example. The recursive entry structure allows the same denition for dierent compression schemes. . . . . . . . . . . . . . . . . . 71 5.3 Partial computations for 2D wavelet. Gray (white) circles denote even (odd) nodes. Operations at each node are done in the order listed. . . . . . . . . . . . 77 5.4 Code structure of (a) CTP and (b) SenZip compression service over CTP 79 5.5 (a) Distributed compression and (b) Centralized reconstruction . . . . . . 81 5.6 Experiments on static trees with 2D wavelet transform and xed quanti- zation. (a) Two xed tree topologies, tree 1 and tree 2, for same set and locations of nodes. Raw measurement (dashed red) and reconstruction (solid blue) for node 7 with 2 bits allocated per sample for (b) tree 1 and (c) tree 2, for node 12 with 3 bits per sample for (d) tree 1 and (e) tree 2. Histogram of RMS error at all nodes with 3 bits per sample for (f) tree 1 and (g) tree 2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 5.7 (a) Average RMS error for tree 1 with increasing bit allocation per sample for DPCM and 2D wavelet. (b) Normalized cost wrt. to raw data gathering with CTP for increasing bit allocaton per sample. . . . . . . . . . . . . . . 86 x Abstract In-network compression is essential for extending the lifetime of data gathering sensor networks. To be really eective, in addition to the computations, the conguration re- quired for such compression must also be achieved in a distributed manner. The thesis of this dissertation is that it is possible to demonstrate completely distributed in-network compression in sensor networks. Establishing this thesis requires studying several aspects of joint routing and compression. First, our analysis of the impact of spatial correlations on optimal routing shows that there exist correlation-independent schemes for routing that are near-optimal. This im- plies that static routing schemes may perform as well as sophisticated ones based on learning correlations. Next, we develop novel and practical algorithms for distributed compression that take into account the routing structure over which data is transported. Finally, we argue that lack of work on (a) distributed conguration for compression oper- ations and (b) reusable software development, is the primary reason why compression has not been widely adopted in sensor network deployments. Our solution to address this gap is SenZip, an architectural view of compression as a service that interacts with standard networking services. A system implementation based on SenZip and results from experi- ments concretely demonstrate distributed and self-organizing in-network compression. xi Chapter 1 Introduction The sensor networks vision arose in the late 1990s, with the emergence of a new class of devices that allow ne-grained sensing of the physical world. Technological advances made it possible to integrate computation, communication, sensing, and even actuation, on the same platform, while keeping the form factor small and cost low. Potentially large numbers of these devices could be distributed in space to form a collaborating wireless network capable of achieving complex global tasks. It was apparent that such networks will enable several new applications of benet to society. The last ten years have seen a great amount of research activity in this area in both academia and industry. 1.1 Data gathering sensor networks Sensor networks are aiding the evolution of monitoring systems for earth and space science applications [GBR, TMEC + 10, WALJ + 06]. Frequently, these systems require continuous data gathering from a distributed eld to a central base station. A typical scenario is illustrated in Figure 1.1(a). The image shows the temperature of the ocean surface o the Los Angeles coast. A sensor network has been deployed to collect the temperature 1 (a) (b) Figure 1.1: (a) Illustration of a distributed phenomena and data gathering using sensor network. (b) Hardware - Telosb mote device. (a) (b) Figure 1.2: (a) Software abstraction from application developer perspective. (b) Possible t for compression. measurements and transport them to a base station on the coast. The hardware de- ployed could be, for instance, Telosb motes shown in Figure 1.1(b). In software, the data gathering application interfaces with sensors to receive sensor measurements and sends them to a networking \black box" that will perform operations necessary to transport the measurements to the sink. A crude abstraction for the software at each node from an application developer perspective is shown in Figure 1.2(a). The phenomena of interest being sensed evolves in space and time. For most naturally occurring phenomena, it can be expected that the signal will be correlated in both of these 2 Figure 1.3: Software abstraction for compression as a service dimensions. In-network compression or multi-node fusion is considered as a necessity due to the energy constraints of sensor nodes. Since the energy cost is directly related to the number of bits transmitted, it would be more ecient to exploit the correlations in the data to compress it inside the network. Where should this compression be performed? Perhaps as part of the application, as is the case in the Internet? The abstraction would then look something like Figure 1.2(b). However, in this situation, the 'spatial image' is not available at any single node. The compression needs to be performed as the data is routed to the sink. 1.2 Thesis and Research Summary From an application developer perspective, compression needs to be provided as a ser- vice. Given such a service, as shown in Figure 1.3, the application now sends the sensor measurements to a \compression plus networking black box". In addition to the regular networking functions, this \black box" will be capable of achieving both the computations 3 and conguration required for in-network compression in a distributed manner. Is it pos- sible to dene such a service? What is inside the box? Our thesis is that it is possible to practically demonstrate completely distributed in-network compression in sensor networks. Establishing this thesis to arrive at our goal of \compression as a service\ requires studying several aspects of joint routing and compression - What is the impact of spatial correlations on optimal routing? What algorithms can be used for distributed en-route compression? What issues need to be addressed in going from the theory to a widely adopted system design for distributed compression? 1.2.1 Impact of spatial correlations on optimal routing In considering the impact of spatial correlations on routing, since energy-eciency is the prime motivation for compression of correlated data, it makes sense to route along paths which allow for more compression. However, the increased routing costs for deviating too much from the original shortest paths might overwhelm the gains from compression. We build models and perform analysis to explore this tension. Clustering is a natural way of trading o progress towards the sink and opportunities for compression close to data sources. The optimal cluster size can be expected to depend on the degree or level of correlation in the data. Our analysis conrms this but also throws up two surprising results. First, when every node is capable of compression computations, the optimal cluster consists of the whole network i.e shortest path routing is optimal. Second, when compression operations are performed only at cluster heads, there exists a near- optimal cluster size that works well over a range of correlation levels. The implication 4 for correlated data gathering is that simple, non-adaptive routing schemes can perform as well as sophisticated, adaptive ones. 1.2.2 Algorithms for achieving distributed compression In the second part, we focus on the design of distributed compression algorithms. We consider two dierent views of structure in data: one based on wavelet transforms and the other on compressed sensing. Shen and Ortega [SO08a] have developed lifting based wavelet transforms that can operate over any 2-D tree routing topology. Their algorithms assume unicast communications between nodes in the network. We extend their work by designing a new transform to take advantage of the broadcast nature of wireless communication [SPO09]. This transform allows for better compression of data and hence energy eciency. The second approach is to extend the recent results in compressive sensing for the multi-hop routing scenario. Our work is the rst to consider this problem. 1.2.3 Architecture and system implementation for distributed compression In this part, we focus on software development and system implementation issues for dis- tributed compression. Earlier work has mostly dealt with specic schemes and optimiza- tions and has not led to reusable software development, which is the crucial step in wide adoption in deployments. Another important issue that earlier work has not addressed is that of distributed conguration and re-conguration. Along with the computations which have to be performed in a distributed fashion at the nodes, the conguration of compression operations i.e. which "roles" nodes play in the transform, which other nodes they receive data from and perform computations over, the topology-specic parameter 5 settings in the transform etc., (along with re-conguration in the face of network dynam- ics) also has to happen in a distributed manner. Finally, to avoid a re-design of the stack, the compression should be able to work over standard networking (esp. routing) compo- nents. Our solution incorporates these issues to propose SenZip, an architectural view of compression as a service that works over standard networking components. To establish that SenZip is a working architecture, we have implemented a nesC/TinyOS system that provides a compression service based on the SenZip architecture that works on top of the Collection Tree Protocol [tos]. The resulting system demonstrated distributed con- guration and computations and good reconstruction for compression with two dierent schemes, DPCM and 2D wavelet over both static and dynamic routing topologies. This system adapts to changes in the network topology using the tools provided by CTP. When the topology changes,the local aggregation tree is re-congured in a distributed manner and both compression and reconstruction continue smoothly. The software modules are available for download on tinyos-contribs. The rest of the dissertation is organized as follows. Chapter 2 provides background on in-network compression for sensor networks by discussing related literature. Chapter 3 presents the modeling and analysis of the impact of spatial correlations on routing. Chapter 4 presents new schemes for distributed compression. Chapter 5 presents the SenZip architecture and details of the system implementation based on it. Chapter 6 concludes the thesis with a summary of contributions and future work. 6 Chapter 2 Background on in-network compression In-network compression or multi-node fusion is essential for data gathering sensor net- works due to the energy constraints of the nodes. We discuss the several approaches that have been proposed to exploit the correlations for ecient and long-lived operation. There is some limited work on in-network compression in wired networks. A set of stan- dards has been developed for header compression inside the network [rfc90, rfc99, rfc01]. Obviously, in this case, the payload of the packets is not altered. Work on active networks looked at performing operations on data inside the network to tradeo computation and communication [BK01, TW96]. However, the vision of an ActiveNet that will succeed and replace the Internet did not materialize. The Internet is based on the end-to-end paradigm with only end hosts performing operations on the data. In this section, we begin by looking at schemes for distributed aggregation and com- pression for sensor networks. These works primarily focus on achieving the computations required for compression in a distributed manner. We then discuss the problem of dis- tributed conguration and reconguration required at the nodes to perform compression 7 operations. Finally, we describe work on system design and software development for energy-ecient data gathering. 2.1 Distributed aggregation Aggregation schemes aim to avoid redundancy at packet level. Some examples are dupli- cate suppression and nding statistics such as the minimum, maximum, average, count etc. for the measurements of distributed sensors. Krishnamachari et al. [KEW02] presented models and performance analysis for simple aggregation (duplicate suppression, min, max) and illustrated the gains when compared to end-to-end routing. They also studied the eects of network topology and the nature of optimal routing for such aggregation. Aggregation via a minimum Steiner tree is shown to be optimal and hence NP-hard, and some sub-optimal structures are then considered. Intanagonwiwat et al. [IEGH02] observed that greedy aggregation based on directed diusion [IGE + 03] can do better than opportunistic aggregation in high density scenarios. Madden et al. [MFHH] argued that aggregation should be provided as a core service for sensor network applications. They proposed the TAG (Tiny AGgregation) service for answering declarative queries over a routing tree. 2.2 Distributed compression In this section, we discuss literature on distributed compression. 8 2.2.1 Distributed source coding These works involve constructive approximations to distriuted Slepian-Wolf. Several works with little or no interaction between encoders. Typically, these approaches require knowledge of global correlation structure at the sink or at all nodes. Multi-hop routing is not considered. The techniques proposed by Pradhan et al. [PR99] suggest mechanisms to compress the content at the original sources in a distributed manner without explicit routing-based aggregation. The sink has complete knowledge of the correlation structure, which it uses to arrive at the optimal coding rates at each node and then disseminates the same to them. No inter-sensor communication is required for compression purposes. Gastpar et al. [GDV06] present the distributed K-L transform that has applications for distributed compression problems. The authors consider the optimal local operations at distributed agents, such as sensor nodes, to provide a locally compressed version of the data to a central base station which will then reconstruct the whole eld with minimum error. In general, the solution needs knowledge of global correlation structure and is shown to be globally convergent for the Gauss-Markov data case. In DSC techniques, the correlation between data captured by dierent nodes has to be known, which in practice will require data exchange between nodes. Practical application and deployment of such techniques for sensor network data gathering has not been attempted. 2.2.2 Analysis of impact of correlations on routing Work by Scaglione and Servetto [SS02] was the rst to explicitly consider the problem of joint routing and compression. By considering the joint entropy of sources as the data 9 metric and routing for compression within localized partitions (or clusters), it is shown that the network broadcast problem in multi-hop networks is feasible. Work by Enachescu et al. [EGGM04] presents a randomized algorithm which is a constant factor approximation (in expectation) to the optimum aggregation tree simulta- neously for all correlation parameters. A notion of correlation is introduced in which the information gathered by a sensor is proportional to the area it covers and the aggregate information generated by a set of sensors is the total area they cover. The performance of aggregation under an arbitrary, general model is considered by Goel and Estrin [GE03]. In this thesis, we analyze the relative performance of various routing and compression schemes based on using an empirically motivated model for the joint entropy as a func- tion of inter-source distances [PKG04, PKG08]. The optimal routing structure is then analyzed using this approximation. The analysis demonstrates that the optimal routing structure depends on where the actual data compression is performed; at each individual node or at \micro-servers" acting as intermediate data collection points. In both cases, we show that there exist ecient correlation independent routing schemes. The correlated data gathering problem and the need for jointly optimizing the coding rate at nodes and routing structure is also considered in [CBLV04]. The authors provide analysis of two strategies: the Slepian-Wolf or DSC model, for which the optimal coding is complex (needs global knowledge of correlations) and optimal routing is simple (always along a shortest path tree) and a joint entropy coding model with explicit communication for which coding is simple and optimizing routing structure is dicult. For the Slepian- Wolf model, a closed form solution is derived while for the explicit communication case 10 it is shown that the optimization problem is NP-complete and approximation algorithms are presented. In [vRW04], \self-coding" and \foreign-coding" are dierentiated. In self-coding, a node uses data from other nodes to compress its own data, while in foreign-coding a node can also compress data from other nodes. With foreign-coding, the authors show that energy-optimal data gathering involves building a directed minimum span- ning tree (DMST). For self-coding, it is shown in [CBLV04] that the optimal solution is NP-complete. Work by Enachescu et al. [EGGM04] presents a randomized algorithm which is a constant factor approximation (in expectation) to the optimum aggregation tree simulta- neously for all correlation parameters. A notion of correlation is introduced in which the information gathered by a sensor is proportional to the area it covers and the aggregate information generated by a set of sensors is the total area they cover. The performance of aggregation under an arbitrary, general model is considered by Goel and Estrin [GE03]. Zhu et. al [ZSS05] have shown that under many network scenarios, a shortest path tree has performance that is comparable to an optimal correlation aware routing struc- ture. While [GE03] takes a more general view of aggregation functions rather than as compression of spatially correlated sources and results in [ZSS05] are contingent on a lim- ited data compression model - compression gain independent of number of neighbors and distances between nodes, our nding that there exists a near-optimal clustering scheme that performs well for a wide range of correlations is in keeping with the results presented in these works. 11 2.2.3 Spatial transforms The design of spatial transforms involves separating the spatially distributed signal into low and high pass portions. One particular class of methods only send trend data or data models within a given cluster. In Ken [CDHH06] and PAQ [TM06] nodes are separated into clusters and assigned roles as cluster head or non-cluster head. Then, nodes forward data to cluster heads on some aggregation graph, model parameters for data in each cluster are estimated at cluster heads and only model parameters are forwarded to the sink along a routing tree. Note that an ordering of communications is implicit in this process. Another simple form of distributed data compression is dierential encoding. For example, in DOSA [ZCH07], nodes are assigned roles as either correlating (C) or non-correlating (NC) nodes, NC nodes forward data to C nodes and C nodes compute and forward dierentials of their NC neighbors. More sophisticated techniques have also been proposed [LTP05]. Distributed computation of dierentials must be done in a predened order on an aggregation graph. Dierentials are forwarded along a routing tree. Ciancio and Ortega [CO05] developed a distributed scheme for removing spatial cor- relations using wavelet transforms via lifting steps. This scheme was for 1D paths and was later extended by Ciancio et al. [CPOK06] to handle the merging of multiple paths. A further enhancement by Shen and Ortega [SO08a] designed a transform to work over any given tree. We describe a new transform [SPO09] that exploits the broadcast nature of wireless transmission to achieve better SNR vs. cost performance. 12 2.2.4 Compressed sensing Wavelet transform techniques are essentially critically sampled approaches, so that their cost of gathering scales up with the number of sensors, which could be undesirable when large deployments are considered. Compressed sensing (CS) has been considered as a potential alternative in this context, as the number of samples required (i.e., number of sensors that need to transmit data), depends on the characteristics (sparseness) of the signal [CRT06, Don06]. In addition CS is also potentially attractive for wireless sensor networks because most computations take place at the decoder (sink), rather than encoder (sensors), and thus sensors with minimal computational power can eciently encode data. CS theoretical developments have focused on minimizing the number of measurements (i.e., the number of samples captured), rather than on minimizing the cost of each mea- surement. In many CS applications (e.g., [DDT + 08, LDP07]), each measurement is a linear combination of many (or all) samples of the signal to be reconstructed. It is easy to see why this is not desirable in the context of a sensor network: the signal to be sampled is spatially distributed so that measuring a linear combination of all the samples would entail a signicant transport cost to generate each aggregate measure- ment. To address this problem, sparse measurement approaches (where each measure- ment requires information from a few sensors) have been proposed. Wang et al. [WGR07] look at such an approach in a single hop network. We consider multi-hop sensor net- works [LPS + 09, PLS + 09]. Compared with state of the art compressed sensing techniques 13 for sensor networks, our experimental results demonstrate signicant gains in reconstruc- tion accuracy and transmission cost. 2.3 Distributed conguration Only limited eorts have been devoted to understanding the problems associated with distributed node conguration for compression. For eciency and scalability, only a small amount of \local" communications should be needed to determine which nodes exactly perform which compression computations, over what data and how the data is then routed to them. Distributed conguration is also desirable as it can help reduce initialization and reconguration times since it is not necessary for a sink node to rst gather information about all nodes. From a purely architectural viewpoint, it is well understood that addressing the re- source constraints in sensor network operation requires cross-layer designs. However, this exibility has led to a proliferation of monolithic and vertically integrated systems. In the absence of an agreement on the decomposition of services provided by system components and their interactions, interfacing such designs for a practical deployment is more or less infeasible. Culler et al. [CDE + 05] advocate the need for an overall sensor network architecture. In a follow up paper by Tavakoli et al. [TDJ + 07], a set of design principles is proposed for the development of elements of the networking software architecture. In addition to the traditional goals of code reuse and interoperability, these include extensibility. This requirement arises in view of the relative immaturity of the eld, where a rigid 14 and complete modularization sti es innovation. They recommend a hybrid approach, with modularity for low level components (underlying infrastructure) and exibility and extensibility at the higher layer (programming paradigm). We address the problem of distributed conguration for compression by proposing the SenZip architecture. SenZip species a compression service that can encompass dier- ent compression schemes and its modular interactions with standard networking services such as routing. This architecture enables a distributed node conguration for compres- sion, just as existing systems make it possible for sensors to congure themselves for routing in a distributed manner. The architecture proposal is based on (a) lessons from overall architectural principles for sensor networks [TDJ + 07], (b) our own experience in implementing a practical wavelet-based distributed compression system, and (c) identify- ing common patterns in existing compression schemes. Work by Tarrio et al. [TVSO09] GSN '09 considers the design of simple wavelet-like techniques for distributed compression which are exlicitly designed to work over and take advantage of conguration mechanisms provided the Collection Tree Protocol [tos]. 2.4 System implementation Most earlier work has focused on theory and simulations to understand performance limits. These studies, and some limited system implementations (e.g., [ZCH07]), have therefore had limited impact on technology adoption and sensor network software devel- opment because they have not yielded modular and inter-operable software. 15 Previous eorts to implement simpler kinds of aggregation mechanisms in sensor net- works. These include the aggregation services in TAG/TinyDB [MFHH], application inde- pendent distributed aggregation (AIDA) [HBSA04], and the dierential encoding-based distributed compression scheme whose implementation is described in [ZCH07]. There has also been some prior work [GGP + 03] on implementing traditional non-distributed wavelets to compression for multi-resolution storage and querying in sensor networks. To demonstrate the utility and practicality of SenZip, we have implemented a system to achieve compression over the Collection Tree Protocol [tos]. The resulting system demonstrated distributed conguration and computations and good reconstruction for compression with two dierent schemes, DPCM and 2D wavelet over both static and dynamic routing topologies [PSC + 09]. The software modules were designed to be re- usable and extensible and are available on tinyos-contribs [sen]. 16 Chapter 3 Modeling of joint routing and compression In order to understand the space of interactions between routing and compression, we study simplied models of three qualitatively dierent schemes. In routing-driven com- pression data is routed through shortest paths to the sink, with compression taking place opportunistically wherever these routes happen to overlap [IEGH02] [KEW02]. In compression-driven routing the route is dictated in such a way as to compress the data from all nodes sequentially - not necessarily along a shortest path to the sink. Our analy- sis of these schemes shows that they each perform well when there is low and high spatial correlation respectively. As an ideal performance bound on joint routing-compression techniques, we consider distributed source coding in which perfect source compression is done a priori at the sources using complete knowledge of all correlations. In order to obtain an application-independent abstraction for compression, we use the joint entropy of sources as a measure of the uncorrelated data they generate. An empirical The work described in this section was published as follows: Sundeep Pattem, Bhaskar Krishnamachari and Ramesh Govindan, \The Impact of Spatial Correlation on Routing with Compression in Wireless Sensor Networks", Transactions on Sensor Networks (TOSN), Volume 4, Number 4, August 2008. Sundeep Pattem, Bhaskar Krishnamachari and Ramesh Govindan, \The Impact of Spatial Correlation on Routing with Compression in Wireless Sensor Networks," Third Symposium on Information Processing in Sensor Networks (IPSN), 2004 17 approximation for the joint entropy of sources as a function of the distance between them is developed. A bit-hop metric is used to quantify the total cost of joint routing with compression. Evaluation of the above schemes using these metrics leads naturally to a clustering approach for schemes that perform well over the range of correlations. We develop a simple scheme based on static, localized clustering that generalizes these techniques. Analysis shows that the nature of optimal routing will depend on the number of nodes, level of correlation and also on where the compression is eected; at the individ- ual nodes or at intermediate aggregation points (cluster heads). Our main contribution is a surprising result that there exists a near-optimal cluster size that performs well over a wide range of spatial correlations. A min-max optimization metric for the near-optimal performance is dened and a rigorous analysis of the solution is presented for both 1-D (line) and 2-D (grid) network topologies. We show further that this near-optimal size is in fact asymptotically optimal in the sense that, for any constant correlation level, the ratio of the energy costs associated with the near-optimal cluster size to those associated with the optimal clustering goes to one as the network size increases. Simulation experiments conrm that the results hold for more general topologies - 2-D random geometric graphs and realistic wireless communication topology with lossy links, and also for a continuous, Gaussian data model for the joint entropy with varying quantization. From a system-engineering perspective, this is a very desirable result because it elim- inates the need for highly sophisticated compression-aware routing algorithms that adapt to changing correlations in the environment (which may even incur additional overhead for adaptation), and therefore simplies the overall system design. 18 3.1 Assumptions and Methodology Our focus is on applications which involve continuous data gathering for large scale and distributed physical phenomena using a dense wireless sensor network where joint routing and compression techniques would be useful. An example of this is the collection of data from a eld of weather sensors. If the nodes are densely deployed, the readings from nearby nodes are likely to be highly correlated and hence contain redundancies, because of the inherent smoothness or continuity properties of the physical phenomenon. To compare and evaluate dierent routing with compression schemes, we will need a common metric. Our focus is on energy expenditure, and we have therefore chosen to use the bit-hop metric. This metric counts the total number of bit transmissions in the network for one round of gathering data from all sources. Formally, let T = (V;E; T ) represent the directed aggregation tree (a subgraph of the communication graph) corre- sponding to a particular routing scheme with compression, which connects all sources to the sink. Associated with each edge e = (u;v) is the expected number of bits b e to be transported over that edge in the tree (per cycle). For edges emanating from sources that are leaves on the tree, the bit count is the amount of data generated by a single source. For edges emanating from aggregation points, the outgoing edge may have a smaller bit count than the sum of bits on the incoming edges, due to aggregation. For nodes that are neither sources or aggregation points but act solely as routers, the outgoing edge will contain the same number of bits as the incoming edge. The bit-hop metric T is simply: T = X e2E b e : (3.1) 19 There are two possible criticisms of this metric that we should address directly. The rst is that the total transmissions may not capture the \hot-spot" energy usage of bottleneck nodes, typically near the sink. However, an alternative metric that better captures hot-spot behavior is not necessarily relevant if the a priori deployment and energy placement ensure that the bottlenecks are not near the sink or if the sink changes over time. The second possible criticism is that this does not incorporate reception costs explicitly. However, the use of bit-hop metric is justied because it does in-fact implicitly incorporate reception costs. If every bit transmission incurs the same corresponding reception cost in the network, the sum of these transmission and reception costs will be proportional to the total number of bit-hops. To quantify the bit-hop performance of a particular scheme, therefore, we need to quantify the amount of information generated by sources and by the aggregation points after compression. For this purpose we use the entropyH of a source, which is a measure of the amount of information it originates [CT91]. In this paper, we consider only lossless compression of data. In order to characterize correlation in an application-independent manner, we use the joint entropy of multiple sources to measure the total uncorrelated data they originate. Theoretically, using entropy-coding techniques this is the maximum possible lossless compression of the data from these sources. We now attempt to construct a parsimonious model to capture the essential nature of joint entropy of sources as a function of distance. The simplicity of this approximation model enables the analysis presented in Sections 3 and 4. In general, the extent of correlation in the data from dierent sources can be expected to be a function of the distance between them. We used an empirical data-set pertaining 20 0 50 100 150 200 250 300 350 400 450 Distance (km) Entropy (bits) actual data approximation H 2 H 3 H 4 H 1 2H 1 3H 1 4H 1 [c = 25, RMS error = .03] [c = 25, RMS error = .09] [c = 25, RMS error = .055] Figure 3.1: Empirical data (from the rainfall data-set) and approximation for joint en- tropy of linearly placed sources separated by dierent distances to rainfall 1 [WB99] to examine the amount of correlation in the readings of two sources placed at dierent distances from each other. Since rainfall measurements are a contin- uous valued random variable and hence would have innite entropy, we present results obtained from quantization. The range of values was normalized for a maximum value of 100 and all readings `binned' into intervals of size 10. Fig.3.1 is a plot of the average joint entropy of multiple sources as a function of inter-source distance. The gure shows a steeply rising convex curve that reaches saturation quickly. This is expected since the inter-source distance is large (in multiples of 50km). From the empirical curve, a suitable model for the average joint entropy of two sources (H 2 ) as a function of inter-source distance d is obtained as: H 2 (d) =H 1 + [1 1 ( d c + 1) ]H 1 : (3.2) 1 This data-set consists of the daily rainfall precipitation for the pacic northwest region over a period of 46 years. The nal measurement points in the data-set formed a regular grid of 50km x 50km regions over the entire region under study. Although this is considerably larger-scale than the sensor networks of interest to us, we believe the use of such \real" physical measurements to validate spatial correlation models is important. 21 Herec is a constant that characterizes the extent of spatial correlation in the data. It is chosen such that when d = c, H 2 = 3 2 H 1 . In other words, when inter-source distance d =c, the second source generates half the rst node's amount in terms of uncorrelated data. In Fig.3.1, a value of c = 25 matches the H 2 curve well. Finally, this leaves open the question of how to obtain a general expression for the joint entropy ofn sources at arbitrary locations. As we shall show later, this is needed in order to study the performance of various strategies for combined routing and compression. To this end, we now present a constructive technique to calculate approximately the total amount of uncorrelated data generated by a set of n nodes. From Eqn.3.2, it appears that on average, each new source contributes an amount of uncorrelated data equal to [1 1 ( d c +1) ]H 1 , where we take the d as the minimum distance to an existing set of sources. This suggests a constructive iterative technique to calculate approximately the total amount of uncorrelated data generated by a set of n nodes: 1. initialize a set S 1 =fv 1 g where v 1 is any node. We will denote by H(S i ) the joint entropy of nodes in set S i ; where H(S 1 ) =H 1 . Let V be the set of all nodes. 2. Iterate the following for i = 2 :n (a) Update the set by adding a nodev i wherev i 2VnS i1 is the closest (in terms of Euclidean distance) of the nodes not in S i1 to any node in S i1 , i.e. set S i =fS i1 ;v i g. (b) Let d i be the shortest distance between v i and the set of nodes in S i1 . Then calculate the joint entropy as H(S i ) =H(S i1 ) + [1 1 ( d i c +1) ]H 1 . 3. The nal iteration yields H(S n ) as an approximation of H n . 22 In the simple case when all nodes are located on a line equally spaced by a distance d, this procedure would yield the expression: H n (d) =H 1 + (n 1)[1 1 ( d c + 1) ]H 1 : (3.3) That this closed-form expression provides a good approximation for a linear scenario is validated by our measurements from the rainfall data set, as seen in Fig.3.1. The curve forH 3 was obtained by considering all sets of grid points (p1;p2;p3) such that they lie in a straight line with the distance between two adjacent points plotted on the x-axis. The curve for H 4 was similarly obtained using all sets of 4 points. 3.1.1 Note on Heuristic Approximation We note that the nal approximationH(S n ) is guaranteed to be greater than the true joint entropyH(v 1 ;v 2 ;::::;v n ). Thus it does represent a rate achievable by lossless compression. The approximation roughly corresponds to a rate allocation ofH(v i = v i ) at every nodev i , where v i is the nearest neighbor of v i . A more precise information-theoretic treatment in terms of the rate allocations at each node is possible, for instance, as in [CBLV04, CBLVW06]. We relinquish some rigor with the objective of gaining practical insight. This approach makes the problem more tractable and is the basis for analysis in subsequent sections. Another point of contention is the need for such a heuristic approach instead of using a continuous data model and using analytical expressions for the joint entropy for this model. In this regard, we note that (a) our model matches the standard jointly Gaussian 23 entropy model for low correlation [Appendix ??] and (b) since the standard expression is in covariance form, it cannot be used for high correlation values, necessitating a reasonable approximation. 3.2 Routing Schemes Given this framework, we can now evaluate the performance of dierent routing schemes across a range of spatial correlations. We choose three qualitatively dierent routing schemes; these schemes are simplied models of schemes that have been proposed in the literature. 1. Distributed Source Coding (DSC): If the sensor nodes have perfect knowledge about their correlations, they can encode/compress data so as to avoid transmitting re- dundant information. In this case, each source can send its data to the sink along the shortest path possible without the need for intermediate aggregation. Since we ignore the cost of obtaining this global knowledge, our model for DSC is very idealized and provides a baseline for evaluating the other schemes. 2. Routing Driven Compression (RDC): In this scheme, the sensor nodes do not have any knowledge about their correlations and send data along the shortest paths to the sink while allowing for opportunistic aggregation wherever the paths overlap. Such shortest path tree aggregation techniques are described, for example, in [IEGH02] and [KEW02]. 3. Compression Driven Routing (CDR): As in RDC, nodes have no knowledge of the correlations but the data is aggregated close to the sources and initially routed so 24 Routing and Aggregation in Distributed Source Coding sources sink routers 2n−1 n1 H 2n−1 Routing and Aggregation in Routing Driven Compression sources sink routers 2n−1 n1 H 1 H 1 H 1 H 1 H 1 H 1 H 1 H 1 H 1 H 1 H 1 H 1 H 1 H 1 H 1 H 1 H 1 H 1 H 1 H 1 H 1 H 3 H 5 H 2n−1 H 2n−1 H 2n−1 Routing and Aggregation in Compression Driven Routing sources sink routers 2n−1 n1 H 1 H 1 H 2n−1 H 2n−1 H 2n−1 H 2 H 3 H 2 H 3 H 2n−1 Figure 3.2: Illustration of routing for the three schemes: DSC, CDR, and RDC. H i is the joint entropy of i sources. as to allow for maximum possible aggregation at each hop. Eventually, this leads to the collection of data removed of all redundancy at a central source from where it is sent to the sink along the shortest possible path. This model is motivated by the scheme in [SS05]. 3.2.1 Comparison of the schemes Consider the arrangement of sensor nodes in a grid, where only the 2n 1 nodes in the rst column are sources. We assume that there aren 1 hops on the shortest path between the sources and the sink. For each of the three schemes, the paths taken by data and the intermediate aggregation are shown in Fig.3.2. 25 In our analysis, we ignore the costs associated for each compressing node to learn the relevant correlations. This cost is particularly high in DSC where each node must learn the correlations with all other source nodes. However the bit-hop cost still provides a useful metric for evaluating the performance of the various schemes and allows us to treat DSC as the optimal policy providing a lower-bound on the bit-hop metric. Using the approximation formulae for joint entropy and the bit-hop metric for energy, the expressions for the energy expenditure (E) for each scheme are as follows. For the idealized DSC scheme, each source is able to send exactly the right amount of uncorrelated data, and each source can send the data along the shortest path to the sink, so that: E DSC =n 1 H 2n1 : (3.4) Lemma 3.2.1. E DSC represents a lower bound on bit-hop costs for any possible routing scheme with lossless compression. Proof: The total joint information of all (2n 1) sources is H 2n1 . As discussed before, no lossless compression scheme can reduce the total information transmitted below this level. Each bit of this information must travel at least n 1 hops to get from any source to the sink. Thus n 1 H 2n1 , the cost of the idealized DSC scheme, represents a lower bound on all possible routing schemes with lossless compression. In the RDC scheme, the tree is as shown in Fig.3.2 (middle), with data being com- pressed along the spine in the middle. It is possible to derive an expression for this scenario: E RDC = (n 1 n)H 2n1 + 2H 1 n1 X i=1 (i) + n2 X j=0 H 2j+1 : (3.5) 26 10 −1 10 0 10 1 10 2 0 1000 2000 3000 4000 5000 6000 7000 8000 Performance with a convex function for joint entropy vs distance Correlation parameter in log scale log(c) Energy usage in bit−hops DSC RDC CDR Figure 3.3: Comparison of energy expenditures for the RDC, CDR and DSC schemes with respect to the degree of correlation c. For the CDR scheme, the data is compressed along the location of the sources, and then sent together along the middle, as shown in Fig. 3.2. It can be shown that for this scenario: E CDR =n 1 H 2n1 + 2 n1 X i=1 H i : (3.6) The above expressions, in conjunction with the expression for H n presented earlier, allow us to quantify the performance of each scheme. Fig.3.3 plots the energy expenditure for the DSC, RDC and CDR schemes as a function of the correlation constant c, for dierent forms of the correlation function. For these calculations, we assumed a grid with n 1 = n = 53 and 2n 1 = 105 sources. From this gure it is clear that CDR approaches DSC and outperforms RDC for higher values of c (high correlation) while RDC performance matches DSC and outperforms CDR for low c (no correlation). This can be intuitively explained by the tradeo between compressing close to the sources and transporting information toward the sink. CDR places a greater emphasis on maximizing the amount of compression close to the sources, at the expense of longer routes to the 27 sink, while RDC does the reverse. When there is no correlation in the data (small c), no compression is possible and hence it is RDC that minimizes the total bit-hop metric. When there is high correlation (large c), signicant energy gains can be realized by compressing as close to the sources as possible and hence CDR performs better under these conditions. Interestingly, it appears that neither RDC nor CDR perform well for intermediate correlation values. This suggests that in this range a hybrid scheme may provide energy- ecient performance closer to the DSC curve. CDR and RDC can be viewed as two extremes of a clustering scheme, with CDR having all data sources form a single aggre- gation cluster before sending data towards the sink while RDC has each source acting as a separate cluster in itself. A hybrid scheme would be one in which sources form small clusters and data is aggregated within them at a cluster head, which then sends data to the sink along a shortest path. This insight leads us to an examination of suitable clustering techniques. 3.3 A Generalized Clustering Scheme The idea behind using clustering for data routing is to achieve a tradeo between aggre- gating near the sources and making progress towards the sink. In addition to factors like number of nodes and position of sink, the optimal cluster size will also depend on the amount of correlation in the data originated by the sources (quantied by the value of c). Generally, the amount of correlation in the data is highest for sensor nodes located close to each other and can be expected to decrease as the separation between nodes 28 increases. Once an optimal clustering based on correlations is obtained, aggregation of data is required only for the sources within a cluster, after which data can be routed to the sink without the need for further aggregation. As a consequence, none of the scenarios considered henceforth will resemble RDC exactly. 3.3.1 Description of the scheme We now describe a simple, location-based clustering scheme. Given a sensor eld and a cluster size, nodes close to each other form clusters. The clusters so formed remain static for the lifetime of the network. Within each cluster, the data from each of the nodes is routed along a shortest path tree (SPT) to a cluster head node. This node then sends the aggregated data from its cluster to the sink along a multi-hop path with no intermediate aggregation. This is illustrated in Fig. 3.4. The intermediate nodes on the SPT may or may not perform aggregation. Data aggregation in the form of compression is computationally intensive. All nodes in a network might not be capable of performing compression, either because it is too expensive for them to do so or the delays involved are unacceptable. It is conceivable that there will be a few high power nodes or micro- servers [HCJB04] which will perform the compression. Nodes form clusters around these nodes and route data to them. In this case, data aggregation takes place only at the cluster head. 3.3.1.1 Metrics for evaluation of the scheme E s (c) is dened as the energy cost (in bit-hops) for correlation c and cluster size s. The optimal cluster size s opt (c) minimizes the cost for a given c. Let E (c) = E sopt (c) 29 0 1 2 3 4 5 6 0 1 2 3 4 5 6 Intra−cluster routing Extra−cluster routing of compressed data to sink Cluster−head Source Sink Figure 3.4: Illustration of clustering for a two-dimensional eld of sensors represent the optimal energy cost for a given correlationc. For simplifying system design, it is desirable to have a cluster size that performs close to the optimal over the range of c values. We quantify the notion of `being close to optimal' by dening a near-optimal cluster size s no as the value of s that minimizes the maximum dierence metric, i.e. s no = arg min s2[1;n] max c2[0;1) fE s (c)E (c)g: (3.7) In the following sections, we analyze the performance of the clustering scheme for both 1-D and 2-D networks when aggregation is performed at intermediate nodes on the SPT, and only at the cluster-heads. 3.3.2 1-D Analysis We begin with an analysis of the energy costs of clustering for a setup involving a linear array of sources to better understand the tradeos. Consider n source nodes linearly placed with unit spacing (i.e. d = 1) on one side of a 2-D grid of nodes, with the sink on 30 the other side, and assuming the correlation model, H n =H 1 (1 + (n1) 1+c ). We consider n s clusters each consisting ofs nodes. Since all sources have the same shortest hop distance to the sink, the position of the cluster head within a cluster has no eect on the results. Within each cluster, the data can either be compressed sequentially on the path to the cluster head or only when it reaches the cluster head. The cluster head then sends the compressed data along a shortest path involving D hops to the sink. The total bit-hop cost for such a routing scheme is therefore E s (c) = n s (E intra s;c +E extra s;c ); (3.8) whereE intra s;c andE extra s;c are the bit-hop cost within each cluster and the bit-hop cost for each cluster to send the aggregate information to the sink respectively. 3.3.2.1 Sequential compression along SPT to cluster head At each hop within the cluster, a node receives H i bits, aggregates them with its own data and transmitsH i+1 bits. This is done sequentially until the data reaches the cluster head. We have, E intra s;c = s1 X i=1 H i = s1 X i=1 1 + i 1 1 +c H 1 = s 1 + (s 2)(s 1) 2(1 +c) H 1 : 31 Since the cluster heads get aggregated data from s sources and send it to the sink using a shortest path of D hops, E extra s;c = H s D = 1 + s 1 1 +c H 1 D )E s (c) = nH 1 s 1 s + (s 2)(s 1) 2s(1 +c) + D s + (s 1)D s(1 +c) : (3.9) The optimum value of the cluster sizes opt can be determined by setting the derivative of the above expression equal to zero. It can be shown that s opt = 1; if c 1 2(D 1) = p 2c(D 1); if 1 2(D 1) <c< n 2 2(D 1) = n; if c n 2 2(D 1) : Note that s opt depends on the distance from the sources to the sink 2 and the degree of correlation c. Fig.3.5 shows (based on the analysis) how dierent cluster sizes perform across a range of correlation levels, based on the analysis presented above for a set of 105 linearly placed nodes. As expected the small cluster sizes and large cluster sizes perform well at low and high correlations respectively. However, it appears that an intermediate cluster size near 15 would perform well across the whole range of correlation values. The curve with s = 105 corresponds to CDR and the DSC curve is also plotted for reference. 2 It is, however, assumed that Dn, so there is an implicit dependence on n. 32 Theorem 3.3.1. For E s (c) given by Equation.3.9, the near-optimal cluster size s no de- ned by Equation.3.7 exists, and is given by s no = (min( p D;n)): The following lemma is required for proving the theorem. Lemma 3.3.2. To solve the optimization problem in Eqn.3.7 for E s (c) given by Eqn.3.9 it suces to nd s =s no such that E sno (0)E (0) =E sno (1)E (1): (3.10) Proof. We rst show that for any arbitrary s, this dierence is maximum at one of the two extremes (i.e. at c = 0 and c!1). Let E d s (c) = E s (c)E (c) =E s (c)E sopt (c) = nH 1 (ss opt ) ss opt 2c(D 1) 2ss opt (1 +c) @E d s (c) @c = nH 1 (s 1) s + 2(D 1) 2s(1 +c) 2 ; if c 1 2(D 1) = nH 1 s p 2c(D 1) s + q 2(D1) c 2s(1 +c) 2 ; if 1 2(D 1) <c< n 2 2(D 1) = nH 1 (sn) sn + 2(D 1) 2sn(1 +c) 2 ; if c n 2 2(D 1) : E d s (c) and its derivative vanish for the same values ofc and sinceE d s (c) is non-negative, the minimum is achieved at these values of c. 33 The derivative is continuous for all s2 [1;n], and for a particular value of s2 (1;n), it is zero only for one value of c. for s = 1, it is zero only for c2 [0; 1 2(D1) ]. for s =n, it is zero only for c2 [ n 2 2(D1) ;1). From the non-negativity of E d s (c) and the above properties of its derivative, we can conclude that: for s2 (1;n), E d s (c) is convex for s = 1, it is monotonously increasing for s =n, it is monotonously decreasing. This implies that E d s (c) is maximum either for c = 0 orc =1 and Eqn.(3.7) reduces to min s2[1;n] max(E s (0)E (0);E s (1)E (1)): (3.11) From Eqn. (3.9), we can derive the following expressions for energy costs of clustering schemes for the two extreme correlation values: E s (0) = nH 1 ( s 1 2 +D) E (0) = nH 1 D E s (1) = nH 1 (1 + D 1 s ) E (1) = nH 1 (1 + D 1 n ): (3.12) 34 Substituting Eqn. (3.12) in Eqn. (3.11) and disregarding common factors, we obtain: min s2[1;n] max( s 1 2 ; D 1 s D 1 n ): (3.13) Let f 1 (s) = s1 2 ;f 2 (s) = D1 s D1 n . We have max s=1 (f 1 ;f 2 ) = f 2 (1) max s=n (f 1 ;f 2 ) = f 1 (n): For s2 (1;n);f 1 ;f 2 are continuous, f 1 is increasing and f 2 is decreasing. Therefore, max(f 1 ;f 2 ) achieves its minimum for s =s no such that f 1 (s no ) = f 2 (s no ) i:e: E sno (0)E (0) = E sno (1)E (1): Proof of Theorem 3.3.1: Solving for f 1 (s no ) =f 2 (s no ), we get s no 1 2 = D 1 s no D 1 n ) s 2 no + ( 2(D 1) n 1)s no 2(D 1) = 0 ) s no = r 2(D 1) + ( D 1 n 1 2 ) 2 ( D 1 n 1 2 ) = (min( p D;n)): 35 10 −3 10 −2 10 −1 10 0 10 1 10 2 10 3 0 2000 4000 6000 8000 10000 12000 14000 16000 18000 correlation parameter in log scale log(c) Transimission cost E s (c) (bit−hops) s = 1 s = 3 s = 7 s = 15 s = 35 s = 105 (CDR) DSC Figure 3.5: Comparison of the performance of dierent cluster-sizes for linear array of sources(n =D = 105) with compression performed sequentially along the path to cluster heads. The optimal cluster size is a function of correlation parameterc. Also, cluster size s = 15 performs close to optimal over the range of c This is illustrated in Fig.3.6, in which the costs are plotted with respect to the cluster sizes for a few dierent values of the spatial correlation. The gure shows clearly that although the optimal cluster size does increase with correlation level, the near-optimal static cluster size performs very well across a range of correlation values. In this gure, D =n = 105 and the near-optimal cluster size obtained from Theorem.3.3.1, s no = 14 is indicated by the vertical line in the plot. Intersections of the dotted lines and the nearest c curve with this vertical line show the dierence in energy cost between the near-optimal and optimal solutions. 36 1 10 14 100 0 2000 4000 6000 8000 10000 12000 14000 16000 18000 cluster size in log scale log(s) Transmission cost E s (c) (bit−hops) s = s opt (c) s = s no c = .01 c = 1 c = 2 c = 5 c = 10 c = 100 Figure 3.6: Illustration of the existence of a static cluster for near-optimal performance across a range of correlations. The sources are in a linear array and data is sequentially compressed along the path to cluster heads. 3.3.2.2 Compression at cluster head only In this case, each source within a cluster sends data to the cluster head using a shortest path. There is no aggregation before reaching the cluster head. We have, E intra s;c = s1 X i=1 iH 1 = s(s 1) 2 H 1 E extra s;c = 1 + s 1 1 +c H 1 D )E s (c) = nH 1 s 1 2 + D s + (s 1)D (s)(1 +c) : (3.14) It can be shown that s opt = 1; if c 1 2D 1 = n; if c> n 2 2Dn 2 ; 2D>n 2 = r 2Dc c + 1 ; else : 37 Fig.3.7 shows that for a linear array of sources (with n =D = 105), the performance for cluster sizess = 5; 7 are close to optimal over the range ofc. The DSC curve is plotted for reference. Theorem 3.3.3. For E s (c) given by Equation.3.14, the near-optimal cluster size s no dened by Equation.3.7 exists, and is given by s no = (min( p D;n)) . The following lemma is required for proving the theorem. Lemma 3.3.4. The near-optimal cluster sizes =s no forE s (c) given by Eqn.3.14 satises the condition E sno (0)E (0) =E sno (1)E (1): Proof. The proof is similar to proof of Lemma 3.3.2 with f 1 (s) = E s (0)E (0) nH 1 = s 1 2 ; and f 2 (s) = E s (1)E (1) nH 1 = s 2 + D s p 2D if 2Dn 2 = sn 2 + D s D n else. 38 10 −3 10 −2 10 −1 10 0 10 1 10 2 10 3 0 2000 4000 6000 8000 10000 12000 14000 16000 18000 correlation parameter in log scale log(c) Transmission cost E s (c) (bit−hops) s = 1 s = 3 s = 5 s = 7 s = 15 s = 105 DSC Figure 3.7: Performance with compression only at cluster head with nodes in a linear array(n =D = 105). Cluster sizes s = 5; 7 are close to optimal over the range of c Proof. of Theorem 3.3.3: Using Lemma 3.3.4 and solving E sno (0)E (0) =E sno (1)E (1) for E s (c) given by Eqn.3.14, we get s no = 2D 2 p 2D 1 ( r D 2 ) if 2D<n 2 = 2Dn 2D +n(n 1) else. It can be veried that s no = ( p D) if D =o(n 2 ) = n if D = (n 2 ): 39 1 7 10 100 0 2000 4000 6000 8000 10000 12000 14000 16000 18000 cluster size in log scale log(s) Transmission cost E s (c) (bit−hops) c = 0.01 c = 0.5 c = 1.0 c = 2.0 c = 10 c = 100 c = 10000 s = (n/2) 1/2 s = s opt (c) Figure 3.8: Illustration of the near-optimal cluster size with compression only at cluster head with nodes in a linear array. The performance of cluster sizes near s = 7( q 105 2 ) is close to optimal over the range of c values The existence of a near-optimal cluster size is illustrated in Fig. 3.8. The performance of cluster sizes near s = 7 is close to optimal over the range of c values. 3.3.3 2-D analysis Consider a 2-D network in which N =n 2 nodes are placed on a nn unit grid and are divided into clusters of size ss. We assume that each node can communicate directly only with its 8 immediate neighbors. The routing pattern within a cluster and from the cluster-heads to the sink is similar and is illustrated in Fig.3.9. Note that using the iterative approximation described in Section 3.1, the joint entropy of k adjacent 3 nodes on a grid is the same as the joint entropy of k sensors lying on a straight line. Fig.3.9(a) illustrates this along the diagonal. The results for the linear array of sources do not extend directly to a two-dimensional arrangement where every node is both a source and a router. In the 1-D case, the optimal 3 nodes forming a contiguous set 40 H 1 H 2 H 1 H 2 H 1 H 1 H 1 H 4 H 9 H s 2 cluster head to sink H s 2 H 1 H 1 H 1 H 1 H 1 2H 1 2H 1 4H 1 9H 1 cluster head to sink (a) (b) Figure 3.9: Intra-cluster routing in a 2-D grid arrangement. (a) Opportunistic com- pression along shortest path to cluster head. For calculation of joint entropy, using the iterative approximation, joint entropy of k nodes forming a contiguous set is the same as the joint entropy of k sensors lying on a straight line. This is illustrated along the diagonal. (b) Compression only at cluster head. The routing from cluster heads to sink is similar to this case. aggregation tree is dierent from the shortest path tree (except for the case with zero correlation). This is because moving towards the sources allows greater compression than moving towards the sink. In the 2-D case however, there are opportunities for compres- sion in all directions. Hence, it is always possible to achieve compression while making progress towards the sink. 3.3.3.1 Opportunistic compression along SPT to cluster head According to the approximation we have been using for the joint entropy, the contribution of a nodev isH(v= v ), where v is the nearest neighbor ofv. If we assume thatH(v= v ) is the xed rate allocation for every node v, it follows 4 that a network-wide SPT is the 4 see [CBLV04] for a formal proof 41 optimal routing structure. In other words, the optimal cluster size s =n for all values of correlation parameter c. There is no incentive for data to deviate from a shortest path to the sink. The result is established more precisely in the following lemma. Lemma 3.3.5. For a 2-D grid with opportunistic compression along an SPT to cluster head, the optimal cluster size is s =n for any value of correlation parameter c2 [0;1]. Proof. Consider a cluster of size sxs. The routing within the cluster is as shown in Fig. 3.9a and routing from cluster head to sink is as shown in Fig. 3.9b. The routing costs are obtained as follows: E intra s;c = n s 2 s1 X i=1 2(si)H i +H i 2 = n s 2 s1 X i=1 (2(si) 1 + i 1 1 +c H 1 + 1 + i 2 1 1 +c H 1 = n s 2 (s 1) s + 1 + (s 2)(4s + 3) 6(1 +c) H 1 E extra s;c = n s 1 X i=0 n s 1 X j=0 maxfsi;sjgH s 2 = s( n s 1 X i=0 i X j=0 i + n s 1 X i=0 n s 1 X j=i+1 j) 1 + s 2 1 1 +c H 1 = n 6 ( n s 1) 4n s + 1 1 + s 2 1 1 +c H 1 : The total cost is E s (c) =E intra s;c +E extra s;c 42 The routing cost for a network-wide SPT i.e. with s =n is E n (c) = E intra n;c + 0 = (n 1) n + 1 + (n 2)(4n + 3) 6(1 +c) H 1 : now for any s<n and any value of c consider the dierence E s (c)E n (c) = n 6(1 +c) ns n s s 2 + 1 + c s 2 4n 2 3nss 2 6n + 6s 2 n : (3.15) It can be veried that the two terms ns n s s 2 + 1 and 4n 2 3nss 2 6n + 6s 2 n are positive for any value of s<n. Hence the dierence in Eqn. 3.15 is always positive. This implies that for all values of c2 [0;1], E s (c) is minimum for s =n. It should be noted that the optimality of a network-wide SPT obtained above is contingent on two of our assumptions: 1. a grid topology, and 2. routing within clusters is along an SPT. Cristecu et al [CBLV04] and Rickenbach et al [vRW04] show results for general graph topologies. 3.3.3.2 Compression at cluster head only When compression is possible only at cluster heads, there is a denite tradeo in progress towards the sink and compression at intermediate points. Since there is no compression before reaching and after leaving the cluster-heads, shortest-path routing is optimal within 43 clusters and from cluster-heads to sink (Fig.3.9(b)). Let E s (c) be the total cost for a network with cluster sizess and correlation parameterc. E intra s andE extra s are dened as the combined intra-cluster costs and the overall cost for routing from cluster heads to the sink respectively. From Fig.3.9, a node at (i;j) will take maxfi;jg hops to reach the cluster head at (0; 0). Since there are ( n s ) 2 clusters, we have E intra s;c = n s 2 s1 X i=0 s1 X j=0 maxfi;jgH 1 = n s 2 s1 X i=0 i X j=0 i + s1 X i=0 s1 X j=i+1 j H 1 = n s 2 s1 X i=0 i(i + 1) + s1 X i=0 (i + 1) + (i + 2) +::: + (s 1) H 1 = n s 2 s1 X i=0 i(i + 1) + s1 X i=0 (s 1)s 2 i(i + 1) 2 H 1 = n 2 6s (s 1)(4s + 1)H 1 : (3.16) Now, the shortest route between adjacent cluster-heads is s hops. Hence, E extra s;c = n s 1 X i=0 n s 1 X j=0 maxfsi;sjgH s 2 =s n s 1 X i=0 n s 1 X j=0 maxfi;jg 1 + s 2 1 1 +c H 1 = n 6 n s 1 4n s + 1 1 + s 2 1 1 +c H 1 : (3.17) [using the expression for PP maxfi;jg from Eqn.3.16] E s (c) = E intra s;c +E extra s;c = h n 2 6s (s 1)(4s + 1) + n 6 n s 1 4n s + 1 1 + s 2 1 1 +c i H 1 : (3.18) Fig.3.10 shows the performance of the scheme for various cluster sizes for a 10001000 network. While the optimal cluster size depends on the value of c, we again nd that 44 there are certain intermediate cluster sizes (s =5, 10, 25) that perform near optimally over a wide range of spatial correlations. It can be shown that s opt (c) = 8c 4c + 1 n 1 3 +o(n 1 3 ): Setting the partial derivative of E s (c) w.r.t s to zero, @E s (c) @s = n 6(c + 1) 2s + (4c + 1)n + (c 2) n s 2 8c n 2 s 3 H 1 = 0 )2s 3 +ns 2 +n = 0; if c = 0 )2s 4 + (4c + 1)ns 3 + (c 2)ns 8cn 2 = 0; if c6= 0: (3.19) Dierentiating again w.r.t s @E 2 s (c) @ 2 s = 2n s 2 + 2 H 1 ; if c = 0 (3.20) = n 3(c + 1)s 4 (12cn 2 s 4 (c 2)ns)H 1 ; if c6= 0: (3.21) Ifc = 0, the second derivative in Eqn.3.20 is always negative and hence the minimum is achieved at the two extremities s = 1 and s =n. Therefore, s opt (0) =f1;ng: (3.22) 45 If c > 0, for s = o(n 1 2 );s 4 = o(n 2 ) and (c 2)ns = o(n 2 ). Solving Eqn.3.19 with this constraint, (4c + 1)ns 3 8cn 2 +o(n 2 ) = 0 )s opt (c) = 8c 4c + 1 n 1 3 +o(n 1 3 ): (3.23) It can be veried that a minimum is achieved since the second derivative in Eqn.3.21 is positive for this value of s. If c> 0, for s = (n 1 2 ), it can be veried that Eqn.3.19 has no solution for sn. Lemma 3.3.6. The near-optimal cluster sizes =s no forE s (c) given by Eqn.3.18 satises the condition E sno (0)E (0) =E sno (1)E (1): The proof is similar to proof of Lemma 3.3.2 with f 1 (s) = E s (0)E (0) n 6 H 1 n s (s 1)(4s + 1) = s 2 3ns + 3n + 1;and f 2 (s) = E s (1)E (1) n 6 H 1 n s (s 1)(4s + 1) = 4n 2 s 2 3n s 6 2 1 3 n 4 3 + 3n + 2 2 2 3 n 2 3 : 46 Theorem 3.3.7. For E s (c) given by Equation.3.18, the near-optimal cluster size s no = (n 1 3 )( 0:6847n 1 3 ): Proof. From Eqns. 3.22 and 3.23, s opt (0) = 1;n and s opt (1)! (2n) 1 3 . Using Lemma 3.3.6, the near-optimal cluster size s =s no satises: E s (0)E (0) =E s (1)E (1) ) h n 2 6s (s 1)(4s + 1) + n 6 n s 1 4n s + 1 s 2 i h n 6 (n 1)(4n + 1) i = h n 2 6s (s 1)(4s + 1) + n 6 n s 1 4n s + 1 i h n 2 6(2n) 1 3 (2n) 1 3 1 4(2n) 1 3 + 1 + n 6 n (2n) 1 3 1 4n (2n) 1 3 + 1 i : (3.24) Rearranging Eqn.3.24 and factoring out n 6s 2 , we get the condition: s 4 + 3ns 3 (6 2 1 3 n 4 3 + 3n + 2)s 2 3ns + 4n 2 +o(n 2 ) = 0: (3.25) Since s 4 =o(ns 3 );ns =o(n 2 ), by factoring out n, Eqn.3.25 reduces to 3s 3 6 2 1 3 n 1 3 s 2 + 4n +o(s 3 ) +o(n) = 0: (3.26) It can be veried that Eqn.3.26 has only one non negative solution, s no = 0:6487n 1 3 +o(n 1 3 ). 47 10 −3 10 −2 10 −1 10 0 10 1 10 2 10 3 0 1 2 3 4 5 6 7 8 x 10 8 correlation parameter in log scale log(c) Transmission cost E s (c) (bit−hops) s = 1 s = 5 s = 10 s = 100 s = 200 s = 500 Figure 3.10: Comparison of the performance of various cluster sizes for a network with 10 6 nodes on a 1000x1000 grid when compression is possible only at cluster heads. The performance for s = 5; 10 is observed to be close to optimal over the range of c values. 1 10 13 100 1000 0 1 2 3 4 5 6 7 8 x 10 8 cluster side in log scale log(s) Transmission cost E s (c) (bit−hops) c = 0.0001 c = 10 c = 100 c = 10000 c = 1.0 c = 0.1 s = s opt (c) s = .6487N 1/3 s = (2N) 1/3 Figure 3.11: Illustration of the existence of a near-optimal cluster size. The network size isnn = 10001000 and compression is possible only at cluster heads. The performance of cluster side values nears =:6487n 1 3 is quite close to optimal for all values ofc ranging from 0.0001 to 10000 48 Fig.3.11 illustrates the existence of the near-optimal cluster size for a network of 10 6 nodes on a 1000 1000 grid. Clearly, the transmission cost with cluster side values near s = 7(=d:6487n 1 3 e) is quite close to the optimal for a large range of correlation coecient c values. 3.4 Simulation Results The analysis in Section 3.3 is based on simple and restricted communication, topology and joint entropy models. To verify the robustness of the conclusions from analysis, we present results from extensive simulation experiments with more general models. As before, the network is deployed in a NN area which is partitioned into grids of size ss, for s2 [1;N]. All nodes which are located within the same grid form a cluster. 3.4.1 Communication and Topology models We consider more general communication and topology models, while using the same entropy model as in the analysis. Nodes are deployed uniformly at random within the network area. Each node is assumed to transmit 1 bit of data. The joint entropy of nodes within the cluster are calculated using the iterative, approximation technique described in Section 3.1. 3.4.1.1 Random geometric graphs In this model, all nodes that are within the communication radius can communicate with each other over ideal, lossless links. Since each link has a unit cost, the routing cost is calculated as: 49 intra-cluster cost = P all nodes in cluster (node depth in cluster SPT) extra-cluster cost = P all clusters in network (cluster-head depth in network SPT) (cluster joint entropy) total cost = intra-cluster cost + extra-cluster cost. The simulation parameters are as follows: network sizes 24mx24m, 84mx84m, 240mx240m density of deployment = 1 node/m 2 communication radius = 3m Figures 12 (a), (b), (c) show performance of clustering for the network sizes considered. As predicted by the analysis, for a network of N nodes, N 1 3 is a good estimate of the near-optimal cluster size. 3.4.1.2 Realistic Wireless Communication model We consider the model for lossy, low power wireless links proposed in [ZK04a]. Link costs are the average number of transmissions required for a successful transfer and these are used as weights for obtaining the shortest-path tree. The routing cost is calculated as: intra-cluster cost = P all nodes in cluster (node cost in cluster SPT) extra-cluster = P all clusters in network (cluster head cost in network SPT) (cluster joint entropy) 50 10 −2 10 0 10 2 0 1000 2000 3000 4000 5000 6000 correlation parameter in log scale transmission cost s = 1 s = 2 s = 3 s = 4 s = 6 s = 8 s = 12 10 −2 10 0 10 2 0 0.5 1 1.5 2 2.5 x 10 5 correlation parameter in log scale transmission cost s = 1 s = 2 s = 4 s = 7 s = 12 s = 28 s = 42 10 −2 10 0 10 2 0 0.5 1 1.5 2 2.5 3 3.5 4 x 10 6 correlation parameter in log scale transmission cost s = 2 s = 4 s = 8 s = 10 s = 20 s = 40 (a) (b) (c) Figure 3.12: Random geometric graph topology. Performance of clustering with density = 1 node=m 2 , communication radius = 3m for network of size (a) 24x24 (b) 84x84 (c) 200x200. Near-optimal cluster sizes are (a) 3,4 (b) 4,7 (c) 8,10. The authors have made code available online for a topology generator based on the model [ZK04b]. The parameters used in the simulations are as follows: network size = 48mx48m , density of deployment = .25 nodes/m 2 random node placement NCSFK modulation, Manchester encoding PREAMBLE LENGTH = 2, FRAME LENGTH = 50, NOISE FLOOR = -105.0; Power levels: -3dB, -7dB and -10dB. Figures 3.13 (a), (b) (c) show performance of clustering for the dierent power val- ues. For lower power, there is an increase in the routing cost since links become more 51 10 −2 10 0 10 2 500 1000 1500 2000 2500 3000 3500 4000 4500 5000 5500 6000 correlation parameter in log scale transmission cost s = 2 s = 4 s = 6 s = 8 s = 12 s = 24 10 −2 10 0 10 2 500 1000 1500 2000 2500 3000 3500 4000 4500 5000 5500 6000 correlation parameter in log scale transmission cost s = 2 s = 4 s = 6 s = 8 s = 12 s = 24 10 −2 10 0 10 2 500 1000 1500 2000 2500 3000 3500 4000 4500 5000 5500 6000 correlation parameter in log scale transmission cost s = 2 s = 4 s = 6 s = 8 s = 12 s = 24 (a) (b) (c) Figure 3.13: Realistic wireless communication topology. Performance of clustering in 48mx48m network with density = .25 nodes=m 2 for power level (a) -3dB (b) -7dB (c) -10dB. Cluster sizes 6, 8 are near-optimal. lossy. However, since proximity relationships between nodes do not change drastically, the relative routing costs for dierent cluster sizes remain similar. 3.4.2 Joint entropy models We now consider more general models for the joint entropy of sources while using the realistic lossy link model from Section 5.1.2. The routing cost is calculated using the same equations and simulations are performed with power level of -3dB, all other parameters remaining the same. 52 3.4.2.1 Linear and convex functions of distance In the empirically obtained model, the joint entropy is a concave function of the distance between sources. We also look at a linear function, for which H 2 (d) =H 1 +min(1; d c )H 1 and a convex function, for which H 2 (d) =H 1 +min(1; d 2 c 2 )H 1 . Fig 3.14 (a) illustrates the three forms of joint entropy functions for 2 sources. The entropy of each source is normalized to 1 unit. The convex and linear curves are clipped when the joint entropy equals the sum of individual entropies. Figures 3.14 (b) and (c) show performance of clustering. 3.4.2.2 Continuous, Gaussian data model In order to verify that the results from analysis and all earlier simulations is not an artifact of the simple approximation models for joint entropy, we now consider a continuous, jointly Gaussian data model and use its entropy as the metric for uncorrelated data in 53 0 2 4 6 8 10 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2 Inter−node distance Joint entropy concave linear convex 10 −1 10 0 10 1 10 2 10 3 2000 4000 6000 8000 10000 12000 14000 correlation parameter in log scale transmission cost s = 2 s = 4 s = 6 s = 9 s = 18 s = 36 10 −1 10 0 10 1 10 2 2000 4000 6000 8000 10000 12000 14000 correlation parameter in log scale transmission cost s = 2 s = 4 s = 6 s = 9 s = 18 s = 36 (a) (b) (c) Figure 3.14: (a) Example forms of joint entropy functions for 2 sources. The entropy of each source is normalized to 1 unit. The convex and linear curves are clipped when the joint entropy equals the sum of individual entropies. The curves shown are for correlation parameter c = 2. Performance of clustering in 72m 72m network with density = .25 nodes=m 2 for (b) linear model (c) convex model of joint topology. Cluster size 6 is near-optimal. the routing cost calculations. The data is assumed to have a zero-mean jointly Gaussian distribution XN N (0;K), with unit variances ii = 1: f(X) = 1 p (2)jKj 1 2 e 1 2 (X) T K 1 (X) : , whereK is the covariance matrix ofX, with elements depending on the distance between the corresponding nodes and the degree of correlation, K ij = e d ij c , where d ij is the distance between nodes i and j and c is the correlation parameter. For this distribution and with quantization step size , entropy of a single source is [CT91]: H 1 = 1 2 log 2 (2e)log 2 () 54 and joint entropy of n sources is: H n = 1 2 log 2 ((2e) n jKj)nlog 2 (): SincejKj becomes singular for large c values, we clip H n by using max n 1 2 log 2 (2e); 1 2 log 2 ((2e) n jKj) o nlog 2 () . Figures 3.15 (a), (b) and (c) show performance of clustering for quantization step = 1, 0.5 and .05. The cluster sizess = 6; 8 are near-optimal. In Figures 3.15 (d), (e) and (f) , the same curves are presented but the transmission cost is normalized to make the highest value equal to 1. For lower values of , the quantization cost dominates and the gains from removing inter-source correlations in data are diminished. Accordingly, the relative gains from optimizing cluster size are also reduced. 3.4.3 Summary of results Overall, the results presented in this section show that the basic conclusions from the analysis hold even when the limiting assumptions of the analysis regarding node place- ment, communication link quality, exact form of the correlation model, quantization, are relaxed. In all cases, we observe the existence of small cluster-sizes that provide near-optimal performance over a wide range of correlation settings. 55 10 0 10 1 10 2 1000 2000 3000 4000 5000 6000 7000 8000 9000 correlation parameter in log scale transmisssion cost s = 2 s = 4 s = 6 s = 8 s = 12 s = 24 10 0 10 1 10 2 5000 6000 7000 8000 9000 10000 11000 12000 correlation parameter in log scale tranmission cost data1 data2 data3 data4 data5 data6 10 0 10 1 10 2 1.8 1.9 2 2.1 2.2 2.3 2.4 2.5 2.6 x 10 4 correlation parameter in log scale trnsmission cost data1 data2 data3 data4 data5 data6 (a) (b) (c) Figure 3.15: Performance of clustering in 48m48m network with density = .25nodes=m 2 with a continuous, jointly Gaussian data model and quatization step (a) = 1 (b) = 0.5 (c) = 0.05. Cluster size 6, 8 are near-optimal. 3.5 Summary and Conclusions We study the correlated data gathering problem in sensor networks using an empirically obtained approximation for the joint entropy of sources. We present analysis of the optimal routing structure under this approximation. This analysis leads naturally to a clustering approach for schemes that perform well(in terms of energy-eciency) over the range of correlations. The optimal clustering depends on the level of correlation and also on where the actual data compression is performed; at each individual node or at intermediate data collection points or cluster heads. Remarkably, however, there exists a static, near-optimal cluster size which performs well over the range of correlations. The notion of near-optimality is formulated as a min-max optimization problem and rigorous analysis of the solution is presented for both 1-D and 2-D network topologies. For a linear arrangement of N sources, the near-optimal cluster size is ( p D) irrespective of where 56 compression occurs, whereD(N;O(N 2 )) is the shortest hop distance of each source to the sink. For a 2-D grid deployment, with N sources and unit density, a network-wide shortest path tree is optimal if every node compresses its data using side information from its neighbors. If compression is possible only at cluster-heads, a (N 1 6 ) cluster size is shown to be near-optimal. The robustness of the conclusions from analysis is established using extensive simulations with more general communication and entropy models. The practical implication of these results for sensor network data gathering is that a simple, static cluster-based system design can perform as well as sophisticated adaptive schemes for joint routing and compression. 57 Chapter 4 Practical schemes for distributed compression The details of how exactly compression will be achieved were ignored in the earlier analysis for reasons of tractability. We now consider the design of practical schemes for achieving distributed compression based on two dierent views of structure in data. First, we build on work by Ciancio, Shen and Ortega [CO05, SO08a, SO08b] to obtain a transform to take advantage of the broadcast nature of wireless communications. Next, we extend the ideas of Candes et al., [CRT06], Donoho [Don06] and Wang et al. [WGR07] to the multi-hop routing scenario. 4.1 Wavelet transform design for wireless broadcast advantage Ciancio, Shen and Ortega [CO05, SO08a, SO08b] have developed lifting based wavelet transforms that can operate over tree routing topologies. Their algorithms assume unicast The work described in this section was published as follows: Godwin Shen, Sundeep Pattem, Antonio Ortega, \Energy-Ecient Graph-Based Wavelets for Distributed Coding in Wireless Sensor Networks", 34th International Conference on Acoustics, Speech, and Signal Processing (ICASSP), April 2009. Sungwon Lee, Sundeep Pattem, Maheswaran Sathiamoorthy, Bhaskar Krishnamachari, Antonio Ortega, \Spatially-Localized Compressed Sensing and Routing in Multi-Hop Sensor Networks", 3rd International Conference on Geosensor Networks, July 2009. 58 0 5 10 15 20 25 30 0 2 4 6 signal x(n) 5 10 15 20 25 0 2 4 6 5 10 15 20 25 30 0 2 4 6 smooth coefficients s(n) predict coefficients p(n) Figure 4.1: Example (a) signal and (b) 5/3 wavelet coecients communications between nodes in the network. In thsi section, we extend their work by designing a new transform to take advantage of the broadcast nature of wireless communication. This transform allows for better compression of data and hence energy eciency. 4.1.1 Wavelet basics: The 5/3 lifting transform We start by presenting an intutive explanation of de-correlation using lifting steps for the 5/3 wavelet transform. For a rigorous treatment of wavelets and liting, see Vetterli and Kovacevic [VK91] and Daubechies [DS98], respectively. Consider a discrete-time signalx(n). The basic idea is to separate the low pass and high pass components ofx(n). Each even-time sample x(2t) can be decomposed into an estimate, using the adjacent odd-time samplesx(2t 1) andx(2t + 1), plus a residual value. Given smoothness in the time-evolution of the signal, or temporal correlations, the residuals have a much smaller magnitude as compared to the original samples and will require signicantly less number of bits to represent. This is how compression can be achieved. The 5/3 lifting wavelet transform for signal x(n) is dened as follows: 59 even \predict\ coecients, d(2k) =x(2k) x(2k1)+x(2k+1) 2 odd "smooth\ coecients, s(2k + 1) =x(2k + 1) + d(2k)+d(2k+1) 4 An example signal and its coecients are illustrated in Figure. 4.1. 4.1.2 Wavelets for sensor networks We now discuss existing schemes for computing wavelet transforms in a distributed man- ner at sensor nodes. 4.1.2.1 Unidirectional 1D wavelet Ciancio and Ortega [CO05] proposed wavelet transforms for use in a sensor network scenario. For a linear array of sensor nodes transporting data hop by hop to a sink at one end, nodes at odd and even depth provide the odd and even samples for the spatial signal. The 5/3 wavelet computations are modied in a way that ensures that data always makes unidirectional progress i.e. towards the sink. This scheme was extended for tree topologies [CPOK06] by considering heuristic (and sub-optimal) ways of handling merging of 1D paths in the tree. 4.1.2.2 2D wavelet for tree topologies Shen and Ortega [SO08b, SO08a] proposed a lifting transform that works for any tree topology. As before, the sink is at the root of the tree and nodes at odd depth provide the "smoothing\ coecients and the nodes at even depth the "predict\ coecients. The dierence from the 1D transform is that at each node thare could be more than just two "adjacent" samples. This is illustrated in Figure. 4.2 (a). It was shown that the following 60 (a) (b) Figure 4.2: Illustration of odd (green) and even (blue) nodes in a subtree for 2D wavelet (a) with unicast and (b) exploiting broadcast nature of wireless communications. The solid arrows are part of the tree routing paths. The dashed arrows are the wireless links not part of the tree. The arrows crossed o in red denote disallowed interactions for transform invertibility and unidirectionality. computations over a tree topology T for the set of vertices V , result in an invertible transform: For i2V , let (i) be the parent in T,C i be the set of children in T For node m at even depth in T , "predict" coecient d m =x m 1 (jCmj+1) P k2Cm x k 1 (jCmj+1) x (m) For node n at odd depth in T , "smooth" coecient s n =x n + 1 2(jCnj+1) P k2Cn d k + 1 2(jCnj+1) d (n) Note that the above computations implicitly impose a schedule or ordering on the transmissions at nodes. Transmissions begin at the leaf nodes in the tree and every non- leaf node is constrained to hold its transmission until all nodes in the subtree rooted at itself have nished transmission. 61 4.1.3 2D wavelet for wireless broadcast scenario The 2D wavelet just described treats the transmissions along the routing tree as unicasts i.e. destined only for a particular node, in this case the parent in the tree. However, in the context of sensor networks, the wireless transmissions at each node can be potentially heard by many nodes in its neighborhood, based on the topology and the transmission power. In the earlier 2D wavelet, a node contributed to de-correlation operations only at its parent in the tree. Taking advantage of the broadcast nature of wireless transmissions, a single transmission at a node can be used for de-correlation operations potentially at all nodes that can receive it. We consider the design of a wavelet transform that exploits broadcast advantage. The routing tree is assumed to be known. The key issue is to decide which of all avail- able broadcast links and the data they provide can be incorporated into de-correlation operations while still ensuring a invertible and unidirectional transform. 4.1.3.1 Augmented neighborhoods Starting with a given tree topologyT over a set of verticesV , we consider an \augmented" neighborhood at each node. For node i2 V , dene the augmented neighborhoodN i according to the following constraints: avoid odd-odd and even-even pairs (for invertibility) send only to nodes with lower hop-count (for unidirectionality) compute only over data from earlier time-slots (for timely and correct computations) 62 0 100 200 300 400 500 600 0 100 200 300 400 500 600 0 100 200 300 400 500 600 0 100 200 300 400 500 600 0.06 0.08 0.1 0.12 0.14 0.16 0.18 5 10 15 20 25 30 35 40 45 total energy consumption (Joules) SNR (dB) tree transform graph transform (a) (b) (c) Figure 4.3: (a) Sample tree topology (b) With additional broadcast links in the augmented neighborhoods at each node (c) Performance gain in terms of SNR vs. cost for new transform compared to 2D wavelet for tree topologies 4.1.3.2 New transform denition Given (V;T;T AUG ), the new transform is dened as follows: For node m at even depth in T , "predict" coecient d m =x m + P k2Nm p m (k)x k For node n at odd depth in T , "smooth" coecient s n =x n + P k2Nn u n (k)d k It can be shown that the above conditions are provably necessary for invertibility and unidirectionality of the transform [SPO09]. 4.1.3.3 Performance of new transform A sample tree topology T and the augmented graph T AUG are shown in Figures. 4.3 (a) and (b) respectively. Figure. 4.3 (c) shows plots of SNR vs. cost for the new transform and the unicast based 2D wavelet. It can be seen that there is a signicant gain in 63 performance. A higher SNR is obtained for the same cost or the same SNR is obtained at a lower cost. 4.2 Compressed sensing for multi-hop network setting The results of our earlier studies are for traditional data compression and transport. Com- pressed sensing is a recent advance that allows a dierent solution for eld reconstruction. While the results from this area apply only for specic classes of signals, we investigate the implications for joint routing and compression in multi-hop sensor networks. Pioneering work by Candes, Romberg and Tao [CRT06] and Donoho [Don06] es- tablished that given an n-dimensional vector that is k-sparse in a certain basis, it can be reconstructed from O(klogn) random projections and showed that near-optimal re- construction can be obtained by solving a linear program. Tropp and Gilbert [TG07] subsequently showed that similar reconstruction can be achieved through a greedy algo- rithm, namely orthogonal matching pursuit (OMP). The number of projections required for reconstruction depends on incoherence between the sparsity inducing basis and the measurement matrix. The projection matrices used by Candes and Donoho are dense random 1 Bernoulli matrices or Gaussian matrices. Wang et.al. [WGR07] showed that the remarkable results of compressed sensing could also be obtained using sparse random projections. They showed that in a distributed network scenario, CS in its original for- mulation would require each node to transmit O(n) packets while using sparse random projections, similar results can be obtained with O(logn) packets per node. However, 64 this scheme is still very expensive in a multi-hop scenario. We present an extension to obtain SRPs in a distributed manner with shortest path routing. We use the following notation: : measurement matrix whose rows are projection vectors : sparsity inducing basis whose columns are the basis vectors H: the holographic basis H = 4.2.1 Combining routing with known results in compressed sensing Consider a network of n sensor nodes with diameter d hops. The average distance of nodes from the sink is also O d) hops. If every node sends its raw sensor measurement to the sink (independently) via the shortest path tree, then the average cost per reading for the network is Cost rawSPT =O(nd): Now consider compressed sensing and assume a spanning tree topology. Nodes route data to the sink along this tree. Each node adds its own reading multiplied by1 to the value received from all its children in the tree and sends this new value to its parent. The sink can add values received from each of its children to obtain one complete projection. Since each node in the tree transmits exactly once, the cost per projection isn. Assuming that the projection matrix is known to sink and nodes (each node only needs its column vector) in advance, the cost for obtaining O klogn projections is Cost CSDRP =O n:klogn =O knlogn : 65 The measurement matrix for sparse random projections is dened as [WGR07]: ij = 8 > > > > > > > > > < > > > > > > > > > : +1 if p = 1 2s 1 if p = 1 2s 0 otherwise For obtaining sparse random projections, each node decides to send with probability 1 s = logn n and the measurement is routed along the shortest path. The sink generates the row of the measurement matrix by placing 1 at positions for nodes from which data was received and 0 for all others. Since node choice is random, the average path length remains O d and the cost using O klogn SRPs is Cost CSSRP = O d:logn:klogn = O k:d:log 2 n : (4.1) This is a bound on the cost that any new scheme based on CS must better. We make the following propositions for using CS in the multi-hop scenario. Proposition 4.2.1. CS with time-domain sparsity is ineective in multi-hop scenario. Reasoning: At most k out of N sensors set o alarms when they sense a value greater than a threshold. In this case, the sparsity inducing basis =I, the identity matrix. If we use sparse random projections, Cost CSSRP = O kdlog 2 n . However, if only nodes that set o alarms route their measurements via shortest path to the sink, the cost is O kd . 66 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 0 10 20 30 40 50 60 70 80 Cost ratio to Raw data transmission SNR (dB) SPT 256 APR SRP 2 SRP 4 Figure 4.4: Compressed sensing performance in multi-hop setting. Plot of SNR vs cost for dierent schemes. The black and green curves are for Sparse Random Projections (SRP). The blue and red curves are for two variations of computing projections over shortest path routing. Proposition 4.2.2. In a multi-hop scenario, shortest path routing is optimal for com- pressed sensing via sparse random projections. Reasoning: Each node decides to send its measurement to the sink with probability 1 s = logn n . Since the distribution of theO logn (in expectation) nodes that choose to send measurements is random, there can be no coordination in the routing. When individual nodes route data independently, they have no incentive to move away from the shortest path. Figure. 4.4 shows the comparative performance of schemes computing projections along routing paths to sink and sparse random projections computed at the sink. Several routing schemes and their performance are considered by Lee et al. [LPS + 09]. 67 Chapter 5 SenZip: Distributed compression as a service Our work up to this point, and in general, work on correlated data gathering in sensor networks in the literature, has focused on theory and simulations to understand perfor- mance limits. These studies, and some limited system implementations (e.g., [ZCH07]), have therefore had limited impact on technology adoption and sensor network software development because they have not yielded modular and inter-operable software. We move towards addressing this problem by (i) proposing a novel architecture, SenZip, that ts into the overall networking software architecture for sensor networks and (ii) demon- strating that a practical design based on this architecture can be deployed on motes and can achieve distributed conguration and modularity. The SenZip architecture species a compression service that can encompass dierent compression schemes and its modular interactions with standard networking services such as routing. This architecture enables a distributed node conguration for compression, just as existing systems make it possible for sensors to congure themselves for routing The work described in this section was published as follows: Sundeep Pattem, Godwin Shen, Ying Chen, Bhaskar Krishnamachari, Antonio Ortega, \SenZip: An Architecture for Distributed En-Route Compression in Wireless Sensor Networks", Workshop on Earth and Space Science Applications (ESSA), April 2009. 68 Figure 5.1: The SenZip architecture. A completely distributed compression service is enabled by having the interacting components shown here at each network node. in a distributed manner. The architecture proposal is based on (a) lessons from overall architectural principles for sensor networks [TDJ + 07], (b) our own experience in imple- menting a practical wavelet-based distributed compression system, and (c) identifying common patterns in existing compression schemes. To concretely illustrate the utility of the architecture, we show how it can incorporate two dierent compression schemes, DPCM and 2D wavelets and present results from mote experiments for data gathering in which nodes can congure themselves for compression under dierent routing conditions. 5.1 SenZip architecture We propose and detail SenZip, an architecture for distributed en-route compression in sensor networks. The primary goals of SenZip are are exibility, modularity, and dis- tributed conguration and reconguration. In addition to the lessons from the principles 69 of an overall architecture for sensor networks and common abstraction identied for ex- isting compression schemes, our design of the SenZip architecture is based on a system implementation eort. 5.1.1 SenZip Specication The SenZip architecture species: 1. a compression service that can encompass dierent compression schemes and, 2. its interactions with standard routing and other networking services. Figure 5.1 is a block diagram representation of the SenZip architecture. It needs to be emphasized that a system based on SenZip would be completely distributed and com- ponents shown in Figure 5.1 would reside on each network node. Of course, compressed data from all nodes in the network nally reaches the base station where it is jointly reconstructed. We now describe the services, their responsibilities and interactions. 5.1.1.1 Compression Service The compression service consists of the aggregation module and the compression module. Aggregation module: The aggregation module disseminates and gathers information for maintaining the local aggregation tree by exchanging messages. This information is collated in an aggregation table. The aggregation graph abstraction allows the denition of a generic table that works for dierent compression schemes. Pseudo-code for such a table is shown in Figure 5.2. 70 struct attributesf int upstreamOnehopNeighborhoodSize; int downstreamOnehopNeighborhoodSize; . . . g weight attributes; struct entryf int node id; weight attributes weights; int further hops; tableEntry *neighborEntry[MAX NHOOD SIZE]; g tableEntry; AggregationTable tableEntry[MAX NHOOD SIZE]; Figure 5.2: Aggregation table example. The recursive entry structure allows the same denition for dierent compression schemes. Compression module: This module has the following functions: (a) From the aggre- gation tree structure provided by routing, this module obtains the role played by the node - which computations to perform and for which nodes, the parameters involved in computation and ordering information - the sequence in which nodes process and forward data. (b) It receives raw measurements from the application and packets with data that needs further processing from forwarding. (c) This module performs further processing over the partially processed data in storage and initiates processing for data of the node itself. The computations will be specic to the compression scheme and based on the role and parameter information. (d) Data that is still partially processed is packetized and sent to forwarding. For data that is fully processed, it checks if enough has been buered in storage to ll a packet. If yes, performs quantization and bit reduction operations, and sends the packet to forwarding. 71 5.1.1.2 Networking components SenZip introduces small changes to standard networking components as follows: Routing engine: In addition to the standard routing functionality, this component in SenZip has an extra interface to the compression service. It reports information of path routing that is relevant for the local aggregation, for example, the parent and hop count in a tree topology. Optionally decisions on changing parent can be coordinated with the compression service, which can also provide a specic metric for the routing cost. Forwarding engine: While partially processed data from nodes in the local aggrega- tion tree is allowed to be intercepted by the compression service, fully processed data is forwarded directly along the route to the sink. Optionally, it might apply dierent settings, such as power, number of retries etc., for the dierent types of packets. Link estimator: Ecient link estimation requires a limited choice of links to moni- tor [FGJL07]. To remove a link (or node in the neighbor table) that is part of the current aggregation tree, joint decision has to be made with the compression service to maintain consistency in the data processing. 5.1.2 Discussion We emphasize that the conguration of roles, parameters and ordering is to be achieved purely locally from the aggregation graph and based on the compression scheme. There is no centralized decision and dissemination. This is a design criteria for compression schemes that can t into the architecture. There is an overhead cost for the exchange of beacons to maintain the aggregation table. Whether the overhead is acceptable or not depends on the relative frequency of measurement versus the frequency of topology 72 changes. If the frequency of topology changes is very high, the potential gains from compression might be overwhelmed by the cost of packet exchanges to maintain the table. Which component is best suited for constructing and maintaining the local aggregation graph? One option is to give this additional responsibility to the routing engine, which already generates and receives messages to setup path routing. However, we believe it is much better for the compression service to handle the aggregation graph operations. This will aid code-reuse and exibility by restricting the changes to the routing engine to providing a single extra interface. To ensure exibility and extensibility, important goals for an overall sensor network ar- chitecture [CDE + 05, TDJ + 07], SenZip only details the interactions between compression and networking services and not the interfaces. The components within the compression service also follow the larger goal of \meaningful separation of concerns". The abstrac- tion helps avoid over-specication, by ensuring that the compression components are required by most existing schemes. Overall, the specication of SenZip has the features of a desirable programming paradigm described by Tavakoli et al. [TDJ + 07]. 5.2 Mapping algorithms to architecture We now discuss two compression schemes that work over tree routing topologies - a simple dierential encoding scheme, DPCM, and a more sophisticated 2D wavelet scheme developed by Shen and Ortega [SO08a]. We describe how these schemes t into the SenZip architecture. 73 table entry element DPCM 2D wavelet weight attributes not needed upstreamOnehopNeighborhoodSize number of children in tree downstreamOnehopNeighborhoodSize 1 (for parent in tree) further hops 1 (upstream only) 2 for upstream node, 1 for downstream neighborEntry[].further hops 0 1 upstream node, 0 for downstream Table 5.1: Aggregation table initialization 5.2.1 Algorithm details Assume a given graphG(V;E) with vertices dened by node locations and edges dened by communication links between nodes. Assume a tree graph T (V;R) (R E) rooted at a single sink node. Suppose every node is indexed by an integer n2V ,C n is the set of child indices of n, and (n) is the parent index of n in T . We say that that node n has depth k when it is k-hops from the sink. Also let x n denote the data measured at noden. For simplicity, we assume data is forwarded and compressed along the same tree T , i.e., the aggregation graph isT . In both schemes, we dene the following transmission schedule. Initially, nodes without any children (leaf nodes) forward raw data to their parents in T . Then, every node n waits until it receives data from all children m2C n before it transmits its own data. This induces an ordering of the communications which is necessary for nodes to compress data as it is forwarded to the sink. 5.2.1.1 DPCM Leaf nodes rst forward raw data to their parents. Each node n waits to receive raw measurements from all its children in T and then computes residual prediction errors as dierences from its own measurement as follows: 74 d m = x m x n 8m2C n s n = x n : (5.1) Node n then forwards the compressed prediction residuals of its children (and other descendants) and its own raw measurements to its parent (n). 5.2.1.2 2D wavelet This transform is constructed as follows for a single level of decomposition. First, vertices ofG are assigned roles by being split into disjoint sets of predicts (odd depth) and updates (even depth) based on depth in T . Next, a high-pass \detail" coecient d m for each predict node m is computed by subtracting from the data at node m, x m , a prediction that is based on information available at neighboring nodes (where neighbors are dened as nodes that are 1-hop away in the aggregation graph): d m =x m 1 (jC m j + 1) X k2Cm x k 1 (jC m j + 1) x (m) (5.2) Finally, a low-pass \smooth" coecient s n for each update node n is computed by adding to x n a correction term based on the detail coecients of neighboring nodes: s n =x n + 1 2(jC n j + 1) X k2Cn d k + 1 2(jC n j + 1) d (n) (5.3) 75 Under the given transmission schedule, each node only has access to data from its descendants and only forwards its own data and data from its descendants. Since each node n uses data from its parent, transform computations for n cannot be completed at n. However, note that terms corresponding to childrenC m and parent(m) are explicitly separated in the computations. This allows us to compute partial wavelet coecients and to update partial coecients as data ows towards the sink to make them full wavelet coecients as described in [CPOK06, SO08a]. This process is summarized as follows. Leaf nodes rst forward raw data. Each predict nodem waits to receive data from its children, then generates a partial coecient d p (m) using data from its children asd p (m) =x m 1 (jCmj+1) P k2Cm x k . Thenm forwards its partial d p (m) (and data from descendants) and (m) completes the computation as d(m) = d p (m) 1 (jCmj+1) x (m) . Each update node performs similar operations. This process is illustrated in Figure 5.3. Note that this induces an ordering of the computations. 5.2.2 Relating algorithms to SenZip We now describe the operation overview for SenZip based systems deploying the two algorithms. 5.2.2.1 Initialization The aggregation component congures aggregation table entries and initiates message exchanges (with its neighbors) in order to gather information needed to build the aggre- gation table. The specics of table entries for each scheme are shown in Table 5.1. This is shared with the compression component which can then identify their role, parent in the 76 1 3 2 4 5 6 Nodes 5 and 6 forward raw data x 5 and x 6 to node 4 Node 4: (a) Generate partials d p (4), s p (5) and s (b) Forward [d p (4) s p (5) s p (6)] to node 3 Node 2 forwards raw data x 2 to node 1 Node 3: (a) Complete partial 4 to get d(4) (b) Complete partials 5, 6 to get s(5), s(6) (c) Generate partial s p (3) (d) Forward [d(4) s(5) s(6) s p (3)] to node 1 Node 1: (a) Generate partials s p (2) and d p (1) (b) Forward [d p (1) s p (2) s p (3) d(4) s(5) s(6)] Figure 5.3: Partial computations for 2D wavelet. Gray (white) circles denote even (odd) nodes. Operations at each node are done in the order listed. tree and children in the tree, and ordering of computations, to congure each compression scheme as follows: DPCM: The roles are uniform i.e. all nodes have the same role. The ordering is that leaf nodes start forwarding and intermediate nodes wait for all one-hop upstream descendants (children) in aggregation tree. 2D wavelet: The roles are decided based on depth in tree from root, odd depth nodes are predicts nodes and even depth nodes are updates. The parameters in computation are equal to the weights, the number of one-hop (children) and two-hop (grandchildren) 77 upstream descendants. The ordering is that leaf nodes start forwarding and intermedi- ate nodes wait for partial coecients of one-hop (children) and two-hop (grandchildren) upstream descendants in the aggregation tree. 5.2.2.2 Data forwarding and compression DPCM: At each node n, the partially processed data to be received is raw data from children and that to be sent is raw data for node n and fully processed data of the children is the dierentials according to Equation 5.1. 2D wavelet: At each node, partially processed data is raw data from children and grandchildren. Sent partially processed data is raw data for node n and all children, and fully processed data is the coecients for all grandchildren according to Equations 5.2 and 5.3. 5.2.2.3 Reconguration The routing engine informs aggregation component of a change in parent (and hop count) in the tree. DPCM: When parent changes at node n, send an explicit parent change message to the old parent old (n) and initiate a message to the new parent. When a parent change message is received by old (n), remove child form table. The number of children is decremented, so waiting criteria in ordering changes. 2D wavelet: When parent of node n changes, send explicit delete message to ex- parent old (n) and add message to new parent. If the hop count changes parity from before, propagate the change to all upstream nodes (descendants in subtree). When a 78 (a) (b) Figure 5.4: Code structure of (a) CTP and (b) SenZip compression service over CTP parent change message is received by old (n), remove child from table. The number of children and grandchildren is decremented, so waiting criteria in ordering changes. old (n) sends a grandparent change message to ( old (n)) where changes in ordering are made. 5.3 System implementation We have implemented a SenZip compression service in nesC/TinyOS [Tin] to run over the Compression Tree Protocol [CTP, TinyOS Enhancement Proposal (TEP) 123] [tos]. This implementation eort has informed the design of the SenZip architecture and in turn, concretely demonstrates it in software. 5.3.1 TinyOS code The code structure of CTP and the SenZip extension are illustrated in Figure.5.4. We now present some details of the code for components, interfaces, changes to CTP and application. 79 5.3.1.1 Interfaces The following new interfaces have been dened for the interactions of the new components with other parts of the system. AggregationInformation: interactions between routing and aggregation component. AggregationTable: interactions between aggregation and compression component. StartGathering: interactions between application and compression component. 5.3.1.2 AggregationP component The aggregation component maintains the local aggregation tree. The routing component signals changes in parent in routing tree. At this point, the aggregation component sends an ADD beacon to the new parent and a DELETE beacon to the old parent. The old and new parents update their aggregation tables accordingly. Events: 1. Routing.parentChange: Signalled from the Routing engine to indicate a change in the parent in routing tree. (a) Send ADD beacon to new parent and DELETE message to the old parent. (b) Signal change to Compression component. 2. AggBeaconReceive.receive: (a) Update table for ADD/DELETE beacons from neighbors. Commands: 80 (a) (b) Figure 5.5: (a) Distributed compression and (b) Centralized reconstruction 1. Table.contactDescendant: Called by the Compression component to directly contact neighbors in the table from which expected packets have not been received. 5.3.1.3 CompressionP component When the aggregation component signals changes in the aggregation table, the compres- sion component allocates and de-allocates memory for storing the data of children in the aggregation tree. When the forwarding engine presents packets with data arriving from children, the data is stored, transformed, compressed and packetized to be handed back to the forwarding engine to transport it to the sink. Figure. 5.5 shows the sequence of operations for compression at each node. Currently the DPCM transform is applied and xed quantization encoding is used for compression. Given the overheads, per packet payload available for compressed data is 10 bytes or ve 16-bit measurements. Events: 81 1. Table.tablePointer: Signalled from the Aggregation component to provide a pointer to the table for the local aggregation neighborhood. 2. Table.change: Signalled from the Aggregation component to inform of changes in the local aggregation neighborhood. 3. Intercept.forward: Signalled from the forwarding engine to lter out packets meant for in-network processing i.e. compression. 4. AllRxTimer.red: Internal timer setup to check if all expected packets from the local aggregation neighborhood were received. If not, currently use the readings from the previous epoch. Commands: 1. StartGathering.isStarted: Called from application to check if compression has already started. 2. StartGathering.getStarted: Called from application to get compression started. 3. Measurements.set: Called from the application to transfer sensor measure- ments. Tasks: 1. changesTask: Posted to update internal table according to changes signalled by Aggregation component. 2. encodeCoecientsTask: To encode and compress the coecients generated by the transform. Currently using xed quantization encoding. Functions: 82 1. computeTransform: To apply the transform on data received from the local aggregation neighborhood. Currently DPCM or dierential computation. 5.3.1.4 Changes to CTP Some small changes are introduced in CTP components to account for and aid in-network compression. RoutingEngine: include AggregationInformation interface and inform Aggregation component of changes in parent. ForwardingEngine: obtain the next hop for forwarding packets from Compression component rather than Routing. 5.3.1.5 Application The current application is written for a TMote Sky mote with an on-board temperature sensor. Events: 1. StartGathering.startDone: Signal from Compression component to begin sen- sor measurements. 2. SubReceive.receive: At the sink node, the Compression component transfers all packets to application. They are then sent over the air to the base station attached to a pc/laptop. 83 5.3.2 Experimental Results An in-lab testbed with Tmote Sky motes [tmo] is used for the evaluation. Ambient tem- perature is the sensed phenomenon and we introduce temperature gradients by switching hot lamps on and o. 5.3.2.1 Static topologies We use xed topologies with 15 nodes for this set of experiments. The setting and two sample topologies are illustrated in Figure 5.6 (a). The spatial transforms used are DPCM and 2D wavelet and the bit reduction is via xed quantization. We assume a uniform bit allocation for all nodes. The same experiments (sequence of switches) are repeated for the two dierent tree topologies in Figure 5.6 (a) with dierent bit allocations per sample. On initialization, all nodes in the network self-congured the roles, parameters and ordering according to the topology. Figures 5.6 (b) and (c) show the reconstruction with 2 bit allocation at node 7 which has dierent depth and hence roles, in the two trees. Similarly, Figures 5.6 (d) and (e) show the reconstruction for node 12 with 3 bit allocation. Figures 5.6 (g) and (h) compare the reconstruction error at the nodes for each topology for 3 bit allocation. The RMS error ranges between .01 o C to .16 o C over the temperature range of 20 o C to 28 o C for 3 bit quantization of coecients for original sample of 16 bits. Since good and similar reconstruction is obtained, it is veried that the compression operations were correctly congured in a completely distributed manner. Figures 5.7 (a) shows the average RMS error for compression for tree 1 with varying bit allocation. As expected, better reconstruction is obtained for higher bit allocations. 84 (a) 0 50 100 150 200 21 22 23 24 25 26 27 sample number temperature (centigrade) original signal reconstruction 0 50 100 150 200 21 22 23 24 25 26 27 28 sample number temperature (centigrade) original signal reconstruction (b) (c) 0 20 40 60 80 100 120 140 160 180 200 21 21.5 22 22.5 23 23.5 24 24.5 25 25.5 26 0 20 40 60 80 100 120 140 160 180 200 21 21.5 22 22.5 23 23.5 24 24.5 25 25.5 (d) (e) 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 node id RMS error (centigrade) 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 node id RMS error (centigrade) (f) (g) Figure 5.6: Experiments on static trees with 2D wavelet transform and xed quantization. (a) Two xed tree topologies, tree 1 and tree 2, for same set and locations of nodes. Raw measurement (dashed red) and reconstruction (solid blue) for node 7 with 2 bits allocated per sample for (b) tree 1 and (c) tree 2, for node 12 with 3 bits per sample for (d) tree 1 and (e) tree 2. Histogram of RMS error at all nodes with 3 bits per sample for (f) tree 1 and (g) tree 2. 85 2 3 4 5 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 bit allocation per sample average RMS error 2D wavelet DPCM 2 3 4 5 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 allocation per sample (bits) cost (normalized wrt CTP) 2D wavelet, tree 1 DPCM, tree 1 2D wavelet, tree 2 DPCM, tree 2 asymptotic bound (a) (b) Figure 5.7: (a) Average RMS error for tree 1 with increasing bit allocation per sample for DPCM and 2D wavelet. (b) Normalized cost wrt. to raw data gathering with CTP for increasing bit allocaton per sample. Figures 5.7(b) shows the cost gain over raw data collection with CTP. In these experi- ments, DPCM has lower cost since the partially processed data only travels one hop while for 2D wavelet, this is 2 hops. The cost gains compared to raw data are relatively limited due to small network size, particularly small average depth which is 3.27 for tree 1 and 4.07 for tree 2. It can be shown in general that with increasing average depth, the cost for both schemes approaches the ratio of bit allocation to raw measurement size. For the same number of bits, the wavelet scheme has better reconstruction but as just discussed, a higher cost. 5.3.2.2 Dynamic topologies In these experiments, we send explicit messages to nodes to alter their parent in the routing tree while data gathering with compression is in progress. The compression settings are to use DPCM transform and Golumb-Rice encoding. These results verify (a) correct updating of aggregation table and conguration of storage, transform computations and packetization at node that adds a new child to its 86 aggregation table, (b) correct handling of coecients \pending" packetization at node that deletes a child from its aggregation table and (c) correct reconstruction of altered topology during reconstruction at base station. 87 Chapter 6 Conclusion This work studied several aspects of the joint routing and compression problem in sensor networks to arrive at a comprehensive solution. We have made signicant progress to- wards demonstrating completely distributed in-network compression in sensor networks. We conclude with a discussion of the contributions and future work. 6.1 Contributions The main contributions of this thesis are as follows: Theoretical understanding of interplay between routing and in-network compression: Two dierent scenarios, homogeneous and heterogeneous, are shown to have dierent near-optimal routing. Subsequently, this problem was addressed by other researchers, primarily for the homogeneous case and while they use dierent models, their results agree with our basic conclusions. When the spatial correlation is uniform, for the homogeneous case where every node is capable of compression computations, shortest-path routing is order-optimal. 88 Design of algorithms for spatial compression: First, a wavelet compression algorithm to take advantage of broadcast nature of wireless communications. This algorithm works for any type of data over any connected 2D topology. The second is a compressed sensing based scheme that extends the classical framework to the multi-hop scenario. This scheme works when data is known to be sparse in some known spatial basis. SenZip, architectural view of distributed compression as a service: A new \compres- sion layer\ is dened to interact with standard networking components to achieve the conguration (and dynamic reconguration) and computations required for compression in a completely distributed fashion. System design and software development: Software modules for a SenZip compression service that works on top of the Collection Tree Protocol (which provides the networking components). This concretely demonstrates that SenZip is a working architecture. The code has been released to tinyos-contribs. 6.2 Future work Some directions for future work on the analysis, algorithm design and system development for distributed compression follow. There is very little work on analyzing the case when regions that have correlated data are not geographically proximate [DBF07]. The analysis presented in this thesis is limited to the case of uniform spatial correlations. It will interesting to extend it for the scenarios with non-uniform spatial correlations. 89 In the design of wavelet based algorithms, we have assumed that the optimal routing is known and that the compression operations are congured for the chosen routes. The design of algorithms that jointly optimize routing and compression needs more attention. Our analysis and algorithms are focused primarily on ways to exploit spatial correlations. The implicit understanding is that temporal correlations can be handled at each node in- dividually. However, algorithms that account for temporal correlations across nodes, and for the general space of spatio-temporal correlations need further research. Further work is needed on studying distributed compression algorithms to understand how they might t into the SenZip architecture. Simplications and modications to these algorithms might be needed for them to correspond to the abstraction used for SenZip design. With some extensions, the SenZip based system can allow for distributed compression to be widely adopted in data gathering sensor networks. We are working on a TinyOS Enhancement Proposal (TEP) for standardization of SenZip. The system currently pro- vides a few options in terms of spatio-temporal transforms and encoding schemes. It will be useful to develop and provide a suite of compression schemes in TinyOS. Further, there is a need for a manual for helping users make the correct choice of which scheme to use based on domain and application specic knowledge. Improvements need to be made for ensuring robustness. The current distributed initialization is based on a simple broadcast ooding scheme. 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Abstract (if available)

Abstract In-network compression is essential for extending the lifetime of data gathering sensor networks. To be really effective, in addition to the computations, the configuration required for such compression must also be achieved in a distributed manner. The thesis of this dissertation is that it is possible to demonstrate completely distributed in-network compression in sensor networks. Establishing this thesis requires studying several aspects of joint routing and compression.

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Creator Pattem, Sundeep (author)

Core Title Joint routing and compression in sensor networks: from theory to practice

School Viterbi School of Engineering

Degree Doctor of Philosophy

Degree Program Electrical Engineering

Publication Date 08/10/2010

Defense Date 05/10/2010

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

Tag distributed compression,distributed transforms,OAI-PMH Harvest,optimal routing,sensor networks,SenZip architecture

Language English

Contributor Electronically uploaded by the author (provenance)

Advisor Krishnamachari, Bhaskar (committee chair), Ortega, Antonio (committee member), Sukhatme, Gaurav S. (committee member)

Creator Email pattem@gmail.com,pattem@usc.edu

Permanent Link (DOI) https://doi.org/10.25549/usctheses-m3381

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Document Type Dissertation

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