Computer Science Technical Report Archive

USC Computer Science Technical Reports, no. 814 (2004)

(USC DC Other)

USC Computer Science Technical Reports, no. 814 (2004)

PDF

Download a page range

Download transcript

Copy asset link

Request this asset

Transcript (if available)

Content Analysis of Wired Short Cuts in Wireless Sensor Networks
Rohan Chitradurga Ahmed Helmy
Department of Electrical Engineering, University of Southern California
email: {chitradu, helmy}@usc.edu

Abstract: In this paper we investigate the use of wired
short cuts in large-scale location aware sensor networks.
This new paradigm augments a sensor network with a very
limited wired infrastructure to improve its overall energy-
efficiency. A wire can be thought of as a short cut between
two nodes in the sensor networks. Energy-efficiency is
obtained mainly by reducing the average path length by
the introduction of the short cuts. This borrows from the
concept of small worlds in which adding a few random
short cuts to highly clustered networks decreases the
degrees of separation (i.e., the average path length)
drastically, resulting in a degree of separation similar to
that of random graphs. In our work we show how to
systematically add these short cuts in order to construct a
small world. The aim of the study is to find the limits on
the gain that can be achieved by using these short cuts in
sensor networks with a single sink. This would also give us
an understanding about how these short cuts can be
placed, and how many are required. We have developed an
analytical model to analyze the gain in path length
reduction by using short cuts, for arbitrary positions of the
sink. We also conducted extensive simulations to validate
our analysis. Our results show that there is an optimal
wire length for which the path length reduction is at its
maximum, beyond which it decreases. The optimal length
is only a small fraction (37.8-50%) of the network
diameter. In a network with 1000 nodes uniformly
distributed on a disk the path length reduction saturates at
60-70% with 5-24 wires, depending on the location of the
sink. Also, we find that restricting the knowledge about the
wires to 2 hops does not degrade the performance from the
case when we have global knowledge of all wires. These
results show promise of the new paradigm.

Keywords: Sensor networks, hybrid networks, small
worlds

I. INTRODUCTION

In this paper, we propose and study a new paradigm for
sensor networks in which a partial wired infrastructure
would augment the wireless sensor network, and in which
wires may act as short cuts to create a small world [1]. The
initial work on small worlds pointed out that for regular
relative graphs (that are highly clustered) adding a few
random short cuts decreases the degrees of separation (i.e.,
the average path length) drastically, resulting in a degree
of separation similar to that of random graphs [2][3]. Our
earlier work on small worlds in wireless networks showed
that a similar relationship exists between spatial graphs
(including wireless networks) and small worlds [4]. The
short cuts need not be random but can be limited to only a
fraction of the network diameter. In our previous work we
treated the short cuts as logical contacts to which the path
may traverse several hops [5] [6].
This work investigates the use of wires as physical short
cuts to reduce the average hop count of the network. This
in turn can increase the energy-efficiency of the sensor
network as it reduces the number of transmissions in the
network. In many applications like remote surveillance, it
may not be possible to augment the network with wires.
Also, for some networks, the duration of deployment
makes the use of wires infeasible due to the cost. But for
some applications like ecological monitoring in which the
sensors monitor the environment for long durations, it can
be economically feasible to do so. Such a network is being
developed and deployed as a part of a project started by
researchers at UCLA and partner universities. The
Network Infomatical Systems (NIMS) [7] infrastructure
consists of a collection of steel cables, each attached to any
two points - buildings, trees, or other natural structure -
that serve as suspension points. Nodes suspended on the
cable collect data about the environment through a range
of sensors which can be lowered or elevated, and also
move, activate, and recover fixed nodes set along the cable
pathway. They also have the ability to dock when
necessary to recharge their energy source, removing
energy constraints that have hobbled other sensor networks
in the past. The cables can also be used for communication
purposes.
In our network model, we focus on the class of sensor
networks in which the data is routed towards a single sink.
Traffic from the sensors is routed to the sink using greedy
geographic routing. We assume that the nodes at the ends
of the wires are simply more powerful sensors with much
larger battery power, or have a mechanism by which they
can replenish their power like in the case of the NIMS
infrastructure. Also, we concentrate on networks in which
the data generated by the nodes is low rate, which implies
that we can safely assume that the wires are not bandwidth
limited for the traffic they carry. Thus in our model, we
have not considered the cost of the wired transmission.
Using analytical modeling and simulations, we have
focused on the following questions:
• How many wires are required to achieve
maximum reduction in the average path length?
• What is the maximum attainable reduction?
• What is the minimum investment that leads to
significant reduction in path length?
• How should the wires be placed?
• What happens if we restrict the information to
some hops from the wire (instead of global
knowledge of all wires)?

We have developed analytical model to obtain theoretical
values for the average path length reduction. The analytical
expression for the path length provides the reduction that
can be obtained for any position of the sink. The
expression shows that for our model, there is an optimal
length of the wires for which the average path length is at
its minimum. We further validate our analysis using
extensive simulations. We compare the average path
length obtained by using wires with the case having no
wires. Our results show that the maximum reduction in
average path length that can be obtained is 70% for the
sink placed at the center and is over 60% when the sink
placed at the edge of the network. The length of wire
needed for the same reduction is higher when the sink
placed at the edge. The maximum reduction is achieved
with 24 wires when the sink is placed at the center. For a
similar experiment, 5 wires give near saturation results for
the sink placed at the edge. Thus results show that we need
more number of wires when the sink is at the center than
when at the edge, but the length of the wire required for is
much larger for the sink positioned at the edge. Restricting
the knowledge of the wire does not result in deterioration
of performance for small wire lengths.
Overall, our results show the promise of this new paradigm
of augmenting a sensor network with wires, in achieving a
drastic reduction in path length (and hence significant
improvement in energy-efficiency). A relatively small
number of wires are needed, and the knowledge about
locations of the wires need only be propagated a couple of
hops away from the ends of the wires. To our knowledge
no such work has been done studying the effect of adding
wires to wireless sensor networks.
The rest of the paper is organized as follows. In section II,
we discuss related work to place our contributions in
context. Section III discusses the problem formulation,
evaluation metrics considered and the analytical model
developed. In Section IV we present our experimental
setup, simulation results and its analysis. Section V
discusses possible future extensions to the work presented
in this paper. Concluding comments are presented in
Section VI.

II. RELATED WORK

Previous work has shown that wireless networks are
spatial graphs that tend to be much more clustered than
random networks and have much higher path length
characteristics. It has been observed that by adding only
few random links, path length of wireless networks can be
reduced drastically [2], [3]. Our earlier work on small
worlds in wireless networks showed that a similar
relationship exists between spatial graphs (including

The new paradigm: Sensor network augmented by a few wires or cables
to be used in communications as short cuts

wireless networks) and small worlds [4]. In our previous
work, we had used this knowledge to use short cuts as
logical contacts to develop efficient resource discovery
techniques for large scale wireless networks [5], [6]. The
work showed promising results for the use of short cuts as
logical contacts. It further motivated research on the use of
short cuts as physical contacts (either using wires or large
range wireless). In this paper we focus on the use of short
cuts as physical contacts for a subset of sensor networks
which have a single sink.
Recently researchers at UCLA and partner universities
have developed a new class of aerial, suspended robotic
sensors able to monitor their own performance as they
move themselves along a network of cables. The
technology, known as Networked Infomechanical Systems
(NIMS) [7] can be used to monitor a mountain stream
ecosystem from the ground to the treetops for global
change indicators, or observe coastal wetlands and urban
rivers for biological pathogens. The NIMS infrastructure
consists of a collection of steel cables. Each cable is
attached to any two points – building, trees etc that can
serve as suspension points. The cables can also be used to
move robots that can monitor these sensors, and replenish
resources like power. Such infrastructure support removes
previous limitations of on site sensor networks, as nodes
can be replaced, relocated and replenished. Our work can
complement the efforts of the NIMS project as it gives us
an insight into the use of wires for carrying data. By
understanding the minimum investment required and the
fundamental limitations we can gain insight into how the
wires could be used as short cuts in sensor networks.
Various energy-aware routing protocols for sensor
networks have been developed. Some of the protocols use
data centric routing in which the nodes first exchange
some metadata information before the actual data
transmission [8, 9]. In [10], the authors propose LEACH,
in which the nodes are organized into clusters and the
lifetime of the network is increased by randomly choosing
the cluster heads. These routing protocols explore one of
the dimensions of increasing the energy efficiency of
sensor networks. We have explored an orthogonal
dimension in which energy efficiency is obtained by the
use of wires. Our work is complementary where adding
short cuts may work in conjunction with those schemes to
even further improve the energy efficiency drastically.
Base station repositioning has been suggested in [11] for
increasing the network lifetime. Our work can be extended
to develop schemes for wire placement when the sinks or
the sensor nodes are mobile, which can complement the
work in [11]. However, using mobility for energy
efficiency may not work in some scenarios (like rugged
terrains) where it may not be feasible (from a robotics
perspective) to control the mobility of the base station and
overcome natural obstacles. In such cases, it may be more
feasible to install wires.

III. PROBLEM FORMULATION AND MODELS

Network Model

We consider a disk-shaped network topology. Sensors are
distributed on the disk in a uniform random manner. Every
node knows about its neighbors. The network is location-
aware; that is, sensor nodes have some mechanism by
which they can find out their location and their neighbors’
locations (e.g., Hello messages). The network consists of
only one sink. So the traffic from all the nodes in the
network is routed towards the sink. The location of the
sink can be anywhere in the network. The sensors and the
sink are not assumed to be mobile.

Wire Model

The two ends of a wire are equipped with transceivers that
enable only the ends of a wire to communicate with the
wireless sensor network and with the other end of that
wire.

Fig 1.Placement of 4 wires in the topology. The wires end up one hop
from the Sink, and are equally spaced.

O
S
R

L
A B

Fig 2.Figure showing how the wires are placed for a sink (S) placed at (0,
-L).The wires will lead from the sink to nodes on the arc A-B. In the
figure, k is the wire length and R is the radius of the topology.
The ends of the wires have information about their
locations and they share that information with nodes in the
wireless sensor network up to h hops away. More will be
said about this in the routing model.
The wires are placed with one end away one hop from the
sink. The other end of the wires lie on a circle which has
its center at the sink and the radius of the circle is the
length of the wire. The nodes on the circle are equidistant
from each other. Figure 1 shows the placement of 4 such
wires. The 4 wires are equally spaced (inter-wire angle of
90
0
). One end of each wire is one hop from the sink, and
the other end lies on a circle of radius k, where k is the
length of the wire. When the sink is placed at an arbitrary
position, the circle with radius k may not lie completely
inside the topology. In that case, the wired nodes are
placed equally spaced out on the arc of radius k lying
within the topology. Figure 2 illustrates the placement of
wires for an arbitrary position of the sink. If we consider
wires of length k, then a circle of radius k centered at the
sink intersects the topology at points A and B. One end of
each wire is still one hop from the sink while the other end
of the wires is placed equidistant on the arc A-B.
We assume that the bandwidth of the wires is large enough
so that there are no bandwidth constraints on the wire.
Also, we ignore MAC issues as well as the cost of sending
packets on the wire. Another assumption we have made in
our simulations and analytical models is that the
communication between any two nodes A, B is symmetric,
i.e. if A is in radio range of B, then B is in radio range of A.

Routing Model

The routing policy used is greedy geographic routing [12].
Thus when a node A has packets for a node B, it is
assumed that it has the location of node B. Node A then
transmits the packet to their neighbor who is
geographically closest to B. Every node does the same
until the packet reaches node B. If a node X receives a
packet for node B, and it finds that none of its neighbors
are closer to the destination than itself, then the packet is
dropped. Modified geographic routing like the one
discussed in [13] deal with such cases, but we have not
used such techniques for simplicity.
ℓ
When there are no wires in the network, then the packets
are routed to the sink using greedy geographic routing.
When wires are introduced in the network, the packets
may either be routed to the wire, or directly to the sink
depending on the wire knowledge model. For our study we
consider two cases about the knowledge of the wires
locations. In one, we assume that all nodes in the network
have full knowledge of all the wires. The second assumes
that knowledge about wire locations is propagated only h
hops away from the wire end. So only nodes h hops from
the end points of each wire know about the wire.
Depending on the value of h, some nodes may have the
location of more than one wire, whereas some nodes may
not have the location of any wire. Also note that in our
model, we assume that the sinks (S) must be either the

S
ℓ
ℓ
ℓ
r
R
w
ie1
A
B
w
ie2

Fig 3.Routing policy when all nodes have knowledge of all the wires. The
decision is shown for an arbitrary node A, where w
ie1
– w
ie2
is the wire
and B is the sink. A computes d(A,w
ie1
) + d(B,w
ie2
) and d(A,B). If the
former is smaller, then the packet is forwarded towards A else towards B.
This decision is taken for all wires in the topology, and the packet is sent
to the appropriate destination.

source or the destination for all communications (i.e., the
traffic flows from the sink to the sensors (in terms of
queries for example) or from the sensors to the sink (in
terms of reports)).
Let us first consider the routing decision for the case when
all nodes in the network know about all the wires. Let the
source and destination be A and B, respectively. Let there
be n wires, denoted by w
i
, 1 ≤i ≤n. Let the two ends of the
wire be denoted by w
ie1
and w
ie2
, with w
ie2
representing the
end closer to the destination. Let d(x,y) represent the
distance between any two points x and y. For each packet,
the source computes:
1. d
wi
= d(A,w
ie1
) + d(B,w
ie2
) for each wire w
i
, i =
1 to n.
2. d(A,B)
The source then computes the minimum of the above n + 1
distance. If the minimum is d(A,B), then the packet is sent
towards the destination directly over the wireless links
without using the wired short cut. If d
wi
is minimum for
some i, then the packet is sent from A towards w
ie1.
It then
travels over the wired short cut to w
ie2
and then finally
reaches B from w
ie2
over one hop wireless. This routing
policy is illustrated in Fig 3.
When we have partial information about a wire, we
assume that the knowledge about the locations of the two
edges of a wire is propagated to only h hops from the wire.
In that case, nodes take the routing decision in the
following manner:
1. The source computes:
1. d
wi
= d(A,w
ie1
) + d(B,w
ie2
) for each wire w
i
, i =
1 to k.
2. d(A,B)
It may be possible that k is zero for some nodes. In that
case, the nodes send the packet towards the sink (B) by
sending it to the neighbor geographically closest to B. If k
is greater than 0, then the nodes compute (d
wi
)
minimum
and
compare it to d(A,B). If (d
wi
)
minimum
< d(A,B) then the
packet is sent towards w
ie1
(to use the wire) else it is sent
toward B.
2. An intermediate node checks to see if the destination of
the packet is the sink. This is possible as we assume every

l/2
l
Y
B
A
S
X
Z

Fig 4.Routing policy when wire information is restricted to k hops from
the wire. In the above figure, S is the sink; A-B is the wire of length ℓ. We
have two nodes, X and Y wanting to send packets to S, both of which are
outside the coverage of the wire A-B. Y starts by sending the packet in the
direction of S. When the packet reaches Z, which knows about the wire A-
B, Z computes d(Z,B) + d(A,S) and d(Z,S). If using the wire is shorter, it
redirects the packet towards B. The other node X is outside the coverage
of A-B. If there was complete knowledge of the wire, X would have sent
its packet to B (as it is closer). X now sends its packets to S without using
the wire.

node knows the location of the sink, and that the network
has a single sink. If it is, then the node does the same
computation as the sink and may re-direct the packet to go
through a wired short cut if the policy favors that route.
This is illustrated in Fig 4.

The Energy-efficiency Metric

We define the energy-efficiency metric in terms of the
average energy consumed in the network to get 1 bit
consumed by the network to be the sum of radio energy
spent in transmitting and forwarding the packets, then the
energy will be directly proportional to the number of hops
traversed by the traffic. Alternatively this is represented in
our work by the average path length (or degrees of
separation in the small world terminology). Hence
reducing the average path length, reduces the energy
consumed by the network, and subsequently increases the
energy-efficiency of the overall sensor network. Let us call
average path length to go to/from the sink to any other
node in the network in the pure wireless case as ‘l(0)’.
Also, let us call the average path length in case of using
wires of length i meters as ‘l(i)’. We define the path length
ratio PLR(i)=l(i)/l(0). Hence, the reduction in path length
due to wires of length i is 1-PLR(i).

Another aspect of energy efficiency is load balancing. In
pure wireless sensor networks, nodes near the sink are
used more often than other nodes and are susceptible to
early energy depletion, sometimes disconnecting the
network. By using distributed wires in the sensor network
the traffic concentration on the nodes near the sink is
alleviated. Hence we expect the lifetime (or useful time) of
the network to be also extended in our paradigm. We shall
study this aspect in our future work.

Analytical Model

In this section, we develop the analytical model for the
average path length by adding short cuts. Through this
model we shall reason about the maximum attainable gain
(as reduction in the average path length) and the optimal
wire length for the short cuts. Let the center of the
topology be the point (0, 0). Let the sink be placed at
location (0,-L). Let the radius of the topology be R and the
range of the nodes be r.

C1 (0,0)
C2 (0,-L)
R
r
i
r
A B

Fig 5.Figure showing the intersection area of the topology with a disk
with centre at the sink and radius r
i.
The intersection points are A and B.

We can draw concentric rings in the topology of varying
radii with center at the sink. Let the radii of these
concentric rings be r
i
. Let us choose r
i
as follows:
r
i
= i.r , i = 1 to
r
L R +
(1)
Let Area
i
be the area of the circle of radius r
i
that lies
inside the topology (Fig 5). As the rings are multiples of
the range of the nodes, the regions between these rings
represent the area in which all nodes have the same hop
count to the sink using purely wireless communication. Let
us denote such regions by RCH (Region of Constant Hop
Count). Further, nodes in RCH
i
are i hops from the sink.
The expression for the area of RCH
i
(ARCH
i
) is given by
ARCH
i
= Area
i
– Area
(i-1)
, i = 2 to
r
L R +
(2)
ARCH
1
= Area
1
(3)
As nodes are distributed uniformly in the network, the
number of nodes in a RCH is proportional to the area of
the RCH.

The average hop count for the pure wireless case is given
by:
2
1
.
R
RCH i
AHC
r
L R
i
i
i
π
∑
+
=
=
=
(4)

Now, let us introduce wires in the topology. Let us assume
that the wires are of length k, and we have infinite number
of wires connecting the sink to nodes k meters away. This
means that there are infinite nodes on the other end of the
wires. All these nodes are placed equidistant from one
another on a ring of radius k.
Let us divide the topology into 3 regions centered at the
sink– R1, RII and RIII. R1 represents the region of the
topology within radius k/2 from the sink. RII represents the
region of the topology lying within arcs of radius k/2 and
k. RIII represents the region of the topology outside the arc
of radius k. Now, since we are using greedy geographic
routing, the nodes in region RI will not use the wire as the
sink is closer to the nodes than the other end of the wire.
Thus packets from all nodes in this region will be routed to
the sink using wireless transmissions. The nodes in region
RII and RIII will use the wire to reach the sink. Thus these
nodes will send their packets using the wireless medium to
the end of the wire away from the sink. The packets will
then travel over the wire and be transmitted to the sink
from the other end. In the presence of wires, we no longer
have the fact that nodes in RCH
i
are i hops from the sink.
This is because now some nodes send the packets to one
end of the wire rather that the sink. The expressions for the
hop count of nodes in regions RI, RII and RIII are given
by:
RI:
2
2
1
1
R
iARCH
H
r
k
i
i
i
π
∑
=
=
= (5)
RII:
2
1
2
). 1 (
2
R
ARCH i
r
k
H
r
k
r
k
i
i
π
∑
+ =
− +
=
(6)
RIII:
2
1
). (
3
R
ARCH
r
k
i
H
r
L R
r
k
i
i
π
∑
+
+ =
−
=
(7)

Note that the expression for H1 is similar to equation (4).
This is because in region RI nodes do not use the wire. The
expression for RII and RIII are different as the average hop
count is now the average hops taken by packets from
nodes in RII and RIII to the wire and not the sink. The
expression for the average hop count in the presence of
wires is simply the sum of equations (5), (6) and (7):
2
1
2
1
2
1
). ( ). 1 ( .
R
ARCH
r
k
i ARCH i
r
k
ARCH i
AHC
r
k
r
k
i
r
L R
i
r
k
i
i i
r
k
i
i
i
π
∑∑ ∑
+ =
+
=
+ =
=
=
− + − + +
=
(8)
Note that the expression for the AHC depends purely on
ARCH
i
. That in turn, depends on computing the values of
Area
i
. For an arbitrary position of the sink, Area
i
does not
yield a closed form solution. This is because the arcs
centered at the sink and having a radius r
i
intersect the
topology for some values of r
i
, while may lie completely
inside the topology for other values of r
i
. The detailed
derivation of Area
i
for an arbitrary position of a sink is
provided in Appendix I. Once we have that, we can
compute the average hop count with and without wires for
different lengths of the wire.

Fig 6.Path length ratio obtained for the reduction with the analytical
model

We have plotted the path length ratio for varying wire
length and three different sink locations in Figure 6. In the
next section, we will compare the results obtained through
simulation with the values obtained by using this model.
The graphs show that the path length ratio decreases
rapidly with increase in the wire length, after which the
path length increases. We see this happening for each of
the three positions of the sink. The rapid decrease is shown
by the fact that path length ratio reaches 0.5 for wire length
of 400m. For the sink placed at the edge, the path length
ratio is at its minimum for wire length equal to 0.5D, while
it reaches the minimum at 0.375D for the sink placed at the
center, D being the diameter of the topology. This result is
quite significant, as it puts an upper limit on the
recommended length of the wire to be placed in the
network.

There is one case which yields an analytical expression for
the length of the wire required for minimum average path
length. This is the case when the sink is placed at the
center. The minimum path length for this case is found at
the wire length of R
R
756 . 0
7
2
= , where R is the radius
of the topology. This means that we get maximum
reduction at wire length of 37.8 % of the diameter. In the
following discussion, we provide the derivation of the
above result.

For the sink placed at the centre, the disk centered at the
sink having radius r
i
and the topology do not intersect as
they have a common center. Thus, the value of Area
i
is
simply
, i = 1 to
2
i i
r Area π =
r
R
(9)
Thus the value of the area of RCH
i
can be written as:
) 1 2 (
2
− = i r ARCH
i
π , i=1 to
r
R
(10)
Substituting the value of ARCH
i
in equations (5), (6) and
(7) and simplifying we obtain the values of H1, H2 and H3
as:
()
2
2
12 8 12
2 3
1
R
r
H
x x x
− + = (11)
( )
2
2
6
) 1 4 )( 1 (
2 2
2
R
r
H
x
x x
+ +
= (12)
2
2
3
) 5 . 0 2 )( 1 )( (
3
R
r z x x z x z
H ⎟
⎠
⎞
⎜
⎝
⎛ − + + − −
=
(13)

Where
r
R
z = and
r
k
x = . By summing up (11), (12), (13)
and obtaining the first derivative with respect to x and
equating it to 0, we get )
6
1
(
7
2
2
− = z x . For most
practical values of z, we can approximate
7
2z
x ≈
without any significant error. In terms of length, this is
equivalent to saying
7
2R
k ≈ .
Next, we discuss our simulation environment and our
simulation results, and compare the simulation results with
the theoretical values generated by the model in this
section.

IV. SIMULATION AND ANALYSIS

A. Simulation Setup

We use a disk topology similar to the analytical model. To
provide uniform random distribution of nodes, the disk is
divided into cells, with the density in each cell being
constant. Within the cell, the nodes are placed randomly.
The cells are used only for placing nodes in the network
such that the nodes cover the entire network. We have
made no assumption of the nodes knowing other nodes in
their cell. While computing the average path length in the
presence of wires, we assume that the cost of sending and
receiving packets on the wire is negligible.
The simulation was written in C. Since we measure the
path length reduction, we do not consider MAC layer
issues. The parameters of the simulation were as follows.
The range of all nodes was assumed to be the same, and
was taken as r=55m. The radius of the topology was
R=1000m. The number of nodes is 1000 nodes uniformly
randomly distributed as described above. The network is
divided into 100m x 100m cells, with the node density in
each cell taken as 10. The traffic was generated randomly
by the sensors and sent to the sink. Each experiment
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 200 400 600 800 1000 1200 1400 1600
Wire length (i) [meters]
Path length ratio
center
edge
midway
generated 200000 data packets. Geographic routing is used
as described in Section III
1
.

B. Experimental description

We have conducted various experiments to investigate the
different dimensions of the problem. Simulations have
been conducted for the following cases: (1) varying the
number of short cuts by varying the inter-wire angle, (2)
varying the length of the wires, (3) varying the position of
the sink, (4) limiting the information about wires locations
to nodes k hops from the wire. In this case, a node
forwards a packet towards the sink if it does not have any
wire information. Once a packet reaches an intermediate
node having the location of the wire, it makes a local
decision if the packet needs to be forwarded to the sink, or
to the short cut.

Fig 7.Path length ratio for sink placed at center. Graphs have been plotted
for number of wires from 2-48.

C. Analysis and Results

Sink placed at center

Figure 7 shows the average path length ratio for the sink
placed at the center. The nodes in the topology are
assumed to have full knowledge of all the wires. The curve
we get from the analytical model has also been plotted. As
we can see from the curves, the curve for the number of
wires as 24 is very close to the theoretical result. This
means that going beyond 24 radial wires (or inter-wire
angle lesser than 15
o
) does not help in reducing path length
further. Thus we can say that the number of wires saturates
at 24. A mathematical analysis explaining why saturation
occurs with 24 wires is given in appendix II. The
difference between the theoretical curve and the actual

1
In greedy geographic routing a node forwards the packet to the neighbor
closest to the destination. This technique may fail with local maxima
[12][13] in which the forwarder itself is the closest node to the
destination. Also, network partitions may be created due to random node
placement. We reduce the effect of both of the above problems by using
the uniform node placement technique. In our simulations, greedy routing
failed to deliver less than 5% of the traffic for which we accounted by
slightly increasing the traffic sent (from 200k packets to 210k packets).
Thus, we ensure that our simulation results are not affected by local
maxima or partition problems.
results can be attributed to the node placement. We have
placed nodes in cells in a grid like manner, while the
theoretical curve assumes uniform density across
concentric rings.
The reduction with increasing length shown in Figure 7
indicates that the reduction we get increases rapidly with
increasing length of the wire, but starts to saturate after
some point. The maximum reduction is found at 75 % of
the radius if the topology, close to the value obtained using
the analytical model. The maximum path length reduction
we can get is around 68 % (path length ratio of 32 %). The
curve obtained mathematically shows the maximum
reduction that can be achieved at 70 %. Thus the
simulation results obtained are very close to the analytical
result. However, as the reduction improves rapidly with
increase in wire length, we do not need to add a large
length of wire to get significant reduction. For example, if
we have 3 wires, we can get reduction of 25 % by having
the lengths of the three wires as 300m or 30 % of the
Radius. Thus given a restriction on the total length of wire,
we can use these curves in Figure 7 and 8 to determine if
we need short cuts of smaller lengths, but more of them in
number, or do we need smaller number of short cuts, but
each of those shortcuts will be of large lengths. Obviously
this decision is also affected by other factors like the
capacity of the wires, traffic concentration around the
wires in addition to applicability of wire installation
according to the terrain of the sensor network. We do not
claim that this decision can be made solely on these
curves, just that they can be one of the factors in making
the decision. We plan to investigate these other factors in
the future.

Fig 8.Path length ratio obtained by limiting the wire information to k
hops. K is varied from 1 to 6.

Figure 8 shows the effect of limiting the wire information
to k hops from the wires. The value of k is varied from 1 –
8. For the experiment, the number of wires was taken as
24, the saturation value obtained from the previous
experiment. For this case, we see that performance is the
same for small lengths of the wire. Beyond 300 meters,
restricting the knowledge for 1 hop degrades the
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
Path length ratio
0.9
1
0 200 400 600 800 1000
Wire length (i) [meters]
1
2
3
4
6
Complete
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 100 200 300 400 500 600 700 800 900 1000
Path length ratio
2
3
4
5
6
8
12
24
48
Theoretical
Wire length (i) [meters]
performance. The curve for 2-hop knowledge starts
deviating from the curve for complete knowledge after
about 600 m. For all lengths, 3 hops give the same
performance as the complete knowledge case. Note that
this is a function of the number of wires we use, because
the smaller the number of wires, the more will be the
degradation by restricting the information of the wires.
There are two reasons for the gain to degrade beyond a
certain length when we restrict the knowledge of wires:
1. For small lengths, the coverage of the wired nodes in the
azimuthal angle overlaps. As the length increases, the
coverage no longer overlaps and there are regions

Fig 9.The sink is placed at the center. We have shown 1 wire of length ℓ
in the topology (S-C). When they have complete knowledge of the wires,
we have regions R1, RII and RIII. RII reduces to RII’ when knowledge of
the wires is restricted to 1 hop. In the figure, SC = ℓ, SA = ℓ/2, SB = ℓ -
r, with r being the radio range of the nodes. If we have infinite number of
such wires, then the grey region represents nodes that have information of
the wire.

perpendicular to the radial direction in which nodes do not
have information of any wired nodes.
2. In the radial direction, we can divide the topology into
three regions – R1, RII and RIII. When we had complete
knowledge, nodes in regions II and III used the wire. Now
by restricting the knowledge, region II, has reduced from
RII to RII
’
(Figure 9). For nodes in region III, the
direction of the sink is same as the direction of the wired
nodes. So the packets are forwarded hop by hop towards
the sink, until it reaches a node which has the information
about the wired nodes, and the packet is then redirected
towards the appropriate wired node. If we have enough
wires for azimuthal coverage, this should happen for
almost every packet originating in region III. Thus the
overhead of restricting the knowledge in this case is the
extra distance the packet travels due to the redirection, and
this overhead is small when the number of wires is large.
The real overhead comes because of region II transforming
to region II
’
. Let’s take the case when we restrict to 1 hop.
Packets 2 hops from the wired nodes and x hops from the
sink (x > 2) which earlier used to take 2 hops, now take x
hops. This overhead becomes significant for larger lengths,
as the difference between RII and RII
’
becomes significant.
Even with the degradation for larger lengths, for large
number of wires and lengths up to 30 % of the radius of
the topology, restricting the knowledge to k hop does not
have any effect on performance. This is very significant as
now we have a case for keeping a large number of wires,
with each wires being of small length. This way we
conserve energy by not propagating the knowledge of the
wire more than k hop from the wire.

Sink placed at edge

Figure 10 shows the average path length reduction for
varying number of wires when the destination is placed at
the edge. The average path length reduction saturates at
around 60 % (path length ratio of 40 %). This saturation
occurs at a large value of wire length – around 50 – 60 %
of the diameter of the topology. The saturation value is
about 10% higher than the case when the sink is placed at
the center. We see that the curves for the average path

0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 200 400 600 800 1000 1200 1400 1600
Wire Length (i) [meters]
Path Length Ratio
1
3
5
7
11
Th

Fig 10.Path length ratio obtained by for the sink placed at the edge. The
number of wires is varied from 1 – 11. The curve labeled “Th” is the
curve obtained using the analytical model

length ratio are almost identical for number of wires 5or
greater. This value is much lesser than that for the sink
placed at the center (~ 24). This can be attributed to the
fact that the wires in this case need not cover the entire 360
degrees, but only the arc of radius ℓ that lies within the
topology. This can be achieved with fewer wires. Even for
this scenario, the gain we achieve increases rapidly as the
length increases initially, after which we get diminishing
returns. Another point of observation is the difference
between the simulation results and the analytical result.
This is because we place nodes in the network in a grid
like manner. The analytical model though, assumes that
the nodes within the concentric rings are proportional to
the area of the region, which is not exact.
Figure 11 shows the effect of restricting the information of
each wire to k hops from the wire. The value of k is varied
from 1-6. The graph shows that for small lengths
restricting the wire information does not affect the gain we
can achieve. As the length increases, the different curves
start to branch out. This can again be understood by
dividing the entire topology into regions RI, RII and RIII
(Figure 9), centered at the sink. Restricting the knowledge
to k hops means that the RII changes to RII
’
. This shifts the
division between RI and RII from being at ℓ/2 from the
sink to ℓ – k from the sink, ℓ being the length of the wire.

Fig 11.Effect of restricting wire information to k hops for sink placed at
edge. K is varied from 1-6.

For large length, the difference between RII and RII
’
is
much larger than that for the sink placed at the center.
Consequently, we do not see the curves for k=6 and
k=complete knowledge match for very large lengths ( ℓ >
R). To illustrate this point, for a length of 500m, restricting
to 2 hops is as good as complete knowledge. On the other
hand, for a length of 1200, even 6 hops are not sufficient to
give the same path length reduction as the complete
knowledge case. As networks will typically have wire
lengths to be a small percentage of the diameter of the
network, we can restrict knowledge of wires to a minimum
of 2 hops without performance degradation.

V. FUTURE WORK

In this paper we have investigated some of the issues
related to the use of wired short cuts in sensor networks.
We have also tried to model the effect of adding wires in
sensor networks mathematically. To our knowledge, this is
the first that investigates the use of wires in sensor
networks. The focus of this paper has been on path length
reduction. A lot of work needs to be done to understand
the application of short cuts to solving other problems.
Possible extensions to this work include:
• Multiple Sinks: This work assumes that there is one sink
and that it is stationary. The wires end up one hop from the
sink. This cannot be the case if we have multiple sinks. In
those networks, the question of where to place the wires
becomes important. This is one of the extensions to this
work we envision.
• Energy balancing and data extraction: In sensor
networks, the data gathered is transmitted via wireless
transmission towards the sink. This means that the nodes
close to the sink are heavily involved in packet forwarding
and thus they can lose energy very rapidly. This can finally
lead to the sink being partitioned from the rest of the
network when all the nodes close to the sink have died,
thus rendering the network useless. One solution of the
problem has been suggested in [11]. A variant of the above
problem is the data extraction problem [14]. In this, the
sensors do not transmit their data continuously, but gather
all their data and send it in one short ‘en masse’. In such a
case, the problem becomes of maximizing the data
extracted from the network. For both of the above cases,
short cuts can be used to distribute the forwarding done by
the wireless nodes between nodes far away and close to the
sink. This can lead to energy balancing between nodes
close to the sink and far way from it. We plan to study the
use of short cuts for such applications in the future.
• Mobile sinks and sensors on wire: Mobility has not been
considered in this work yet. This is because most sensor
networks, once deployed, are non-mobile. The NIMS
project has introduced the concept of limited mobility in
sensor networks by having mobile robots on cables. We
plan to extend our work to address mobility in the future.
We plan to consider both scenarios – limited mobility of
the sensors and mobile sinks.

VI. CONCLUSIONS

In this paper we have investigated a new paradigm for
sensor networks. The paradigm uses wires in location
aware sensor networks with a sink for achieving path
length reduction. We have developed analytical models to
analyze the average path length reduction for arbitrary
position of the sink for differing lengths of the wire. The
analytical results show that for any position of the sink,
there is an optimal wire length for which the path length
ratio is at its minimum, beyond which it increases. Further,
for the sink placed at the center, this occurs for a wire
length which is a small fraction of the network diameter
(37.8 %). Simulations results show that the maximum
reduction in average path length that can be obtained is 70
% for the sink placed at the center and is around 60 % for
the sink placed at the edge of the network. The length of
wire for significant reduction is higher when the sink
placed at the edge. The maximum reduction is achieved
with 24 wires when the sink is placed at the center. For the
same experiment, 5 wires give near saturation results for
the sink placed at the edge. Thus results show that we need
more number of wires for the center than the edge, but the
length of the wire required for significant reduction is
much larger for the sink positioned at the edge. We have
also investigated the effect of restricting the knowledge of
the wires. Restricting the knowledge of the wire to 2 hops
does not result in deterioration of performance for small
wire lengths (less than 40 % of the radius). At wire lengths
close to the radius of the topology, restricting the
knowledge does show a decrease in performance. The
model developed in this paper can be applied to
investigating other issues in sensor networks like the data
extraction problem. To our knowledge no previous work
has addressed these issues of using wires in sensor
networks.

REFERENCES

[1] S. Milgram, “The small world problem”, Psychology Today 1,61,
1967.
[2] D. J. Watts, “Small Worlds, The dynamics of networks between
order and randomness”, Princeton University Press, 1999.
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 200 400 600 800 1000 1200 1400 1600
Wire Length (i) [meters]
Path length ratio
1
2
3
4
6
complete
[3] D. Watts, S. Strogatz, “Collective dynamics of ‘small-world’
networks”, Nature 393, 440 (1998).
[4] Ahmed Helmy, “Small Worlds in Wireless Networks”, IEEE
Communication Letters, pp. 490-492, Vol. 7, No. 10, October 2003.
[5] A. Helmy, "Mobility-Assisted Resolution of Queries in Large-Scale
Mobile Sensor Networks (MARQ)", Computer Networks Journal –
Elsevier Science - Special Issue on Wireless Sensor Networks, Vol.
43, Issue 4, pp. 437-458, November 2003.
[6] A. Helmy et. al, "Contact Based Architecture for Resource
Discovery (CARD) in Large Scale MANets", IEEE/ACM IPDPS
WMAN, pp. 219-227, April 2003.
[7] W. Kaiser et al., Network Infomechanical Systems (NIMS), demo
ACM SenSys, UCLA, 2000.
www.engineer.ucla.edu/stories/2003/nims.htm.
[8] W. Heinzelman et al., “Adaptive protocols for information
dissemination in wireless sensor networks,” in the Proceedings of
ACM MobiCom’99, Seattle, WA, August 1999.
[9] C. Intanagonwiwat et al., “Direct Diffusion: A scalable and robust
communication paradigm for sensor networks”, in the Proceedings
of ACM MobiCom’00, Boston MA, August 2000.
[10] W. Heizelman, A Chandrakasan and H Balakrishnan, “Energy
Efficient Communication Protocol for Wireless Microsensor
Networks”, in Proceedings of the Hawaii Conference on System
Sciences, Jan 2000.
[11] M. Younis, M. Bangad and K. Akkaya, “Base-Station Repositioning
For Optimized Performance of Sensor Networks,” in the
Proceedings of the IEEE VTC 2003 - Wireless Ad hoc, Sensor, and
Wearable Networks, Orlando, Florida, October 2003.
[12] Rahul Jain, Anuj Puri, and Raja Sengupta., “Geographical routing
using partial information for wireless ad hoc networks”, Technical
Report M99/69, University of California, Berkeley, 1999.
[13] Brad Karp and H. T. Kung, “GPSR: Greedy perimeter stateless
routing for wireless networks”, In Proc. ACM MobiCom, August
2000.
[14] N. Sadagopan, B. Krishnamachari, “Maximizing Data Extraction in
Energy-Limited Sensor Networks”, to appear in INFOCOM 2004.

Appendix I Derivation of intersection area

In the Analytical Model section, we discussed how we can
obtain the analytical expression for the average path length
reduction for a sink placed at an arbitrary location in a
circular topology. We saw that the expression for the
average path length depends on the area of the circle of
radius r
i
centered at the sink that lies inside the topology.
Here we derive the expression for the intersection area.
Once we have the expression for the intersection areas for
different values of r
i
, we can obtain the value of RCH
i
and
compute the average hop count of the network in the
presence and absence of wires using equations (4) and (8).
The symbols used in the following discussion are the same
that are used in the Analytical Model section.

Let the sink be placed at (0,-L). The range of the nodes is
r. The radius of the topology is given as R. We need to
calculate the intersection area of the circle centered at (0,
0) having radius R and circle centered at (0,-L) with radius
r
i
(Figure 5). We shall refer to the two circles using the
labels for their centre. So the two circles are called C1 and
C2. In order to get the area common between the 2 disks
we need to compute the point(s) of intersection (if any)
between the two circles originating from C1 and C2.

k
C1
C2
R
ri
y
i
x
i

β
α
B
A

Fig 12.Figure showing how we can calculate the intersection area
between the two discs.

Case I: ri < L

If ri < L then there is no intersection and the common area
is the area of the smaller circle.
Area = πr
i
2
, r
i
< L (14)

Case II: ri > L

We first get the intersection points (A and B) for the two
circles as follows.

(i) For the circle with center C1 the equation for that circle
becomes:
x
2
+ y
2
= R
2
(15)

(ii) For the circle with center C2 (0,-L) we have:

x
2
+ (y + L)
2
= r
i
2
(16)

(iii) Points on the intersection must satisfy both equations
(15) and (16) simultaneously. We get (16) – (15) and we
get:
2L y
i
+ L
2
= ri
2
– R
2

or
y
i
= (ri
2
– R
2
– L
2
) / 2L (17)

Also from (16) we get:
x
i
= √ (R
2
– y
i
2
) (18)

Thus the point A is (-x
i
,y
i
) and B is (x
i
,y
i
). This way we can
get the (x,y) coordinates of the intersecting points and start
calculating the common area.
From Figure (12), the common are between the two circles
is given by:
i i i
y x r R − + β α
2 2
(19)
Where α and β are given as follows:
) ( sin
1
R
x
i −
= α , ) ( sin
1
i
i
r
x
−
− = π β , y
i
< -L (20)
) ( sin
1
R
x
i −
= α , ) ( sin
1
i
i
r
x
−
= β -L < y
i
< 0 (21)
) ( sin
1
R
x
i −
− = π α , ) ( sin
1
i
i
r
x
−
= β 0 < y
i
< R (22)
Using the value of y
i
from (17) we transform the limits on
y
i
to limits on r
i
.
) ( ) (
2 2
L R r L R L y
i i
− < < − ⇒ − < (23)
) ( ) ( 0
2 2 2 2
L R r L R y L
i i
+ < < − ⇒ < < − (24)
) ( ) ( 0
2 2
L R r L R R y
i i
+ < < + ⇒ < < (25)

The condition r
i
> (R-L) is obtained by noting that the
intersection points A and B do not exist for r
i
< (R – L).
From equation (1) we know that r
i
=i.r. Thus, to
summarize, the intersection area as a function of i is given
as:
2
i
r π
,
r
L R
i
) ( −
<
(26)
i i
i
i
i
i
y x
r
x
r
R
x
R − − +
− −
)) ( sin ( ) ( sin
1 2 1 2
π
,
) (
) (
2
2 2
r
L R
i
r
L R −
< <
−
(27)
i i
i
i
i
i
y x
r
x
r
R
x
R − +
− −
) ( sin ) ( sin
1 2 1 2 ,
) ( ) (
2
2 2
2
2
2
r
L R
i
r
L R +
< <
−

(28)
i i
i
i
i
i
y x
r
x
r
R
x
R − + −
− −
) ( sin )) ( sin (
1 2 1 2
π
,
) ( ) (
2
2 2
r
L R
i
r
L R +
< <
+

(29)

x
i
and y
i
are given by equations (18) and (17) respectively.

Appendix II Improvement Saturation Analysis

We noticed that when the sink is placed at the center, the
improvement obtained by our simulations saturates with 24
wires (15
o
inter-wire angle). In order to understand why
15
o
inter-wire angle leads to saturation of improvement we
provide the following mathematical analysis. Let the inter-
wire angle be θ for wires of length ℓ. The question we
would like to ask is the following: What maximum value
of θ, will give us result closest to the case if we have
infinite radial wires of length ℓ. Refer to Figure 14 for the
following discussion. Let us denote the scenario that has
infinite radial wires as the ideal scenario. Consider any
node A that is positioned to use the wire to reach the sink.
In the ideal case, the node would have forwarded the
packet to B, the endpoint of the wire closest to it, or in the
radial direction. Instead, it now needs to forward the
packet towards one of the wires that has its end point at C
or D. We can make the following three observations:

1. The nodes placed on the line dividing the points C and
D will have greatest difference between d(A,B) and d(A,C).
2. Amongst the nodes on that line, the node at location B
will have the greatest difference in distance: d(B,C) –
d(B,B) = d(B,C). As the distance of a node from B
increases, the difference between d(A,B) and d(A,C)
diminishes.

C
Fig 13.Figure to show the difference in the distance traveled from the
ideal case by a packet from A, if there are wires placed θ degrees apart. A
lies on the line segment bisecting angle θ. The length of the wire is ℓ and
the radio range of the nodes is r. Here OC and OD are 2 such wires placed
θ apart. In the ideal case, there would have been a node at B, and A
would have sent its packets to B. Now packets from A travel to C. For A
placed 1 hop from B, and 2 hops from C, the packets from A experience
an increase in path length of 1 hop. For all nodes between A and B, it will
be greater than 1 hop. Since such nodes will a fraction of the total nodes,
increase in their path length will not affect the total by much. For any
point A more than 1 hop from B, the difference in path length for nodes
between A and B increases beyond 2, and the number of such nodes also
increases. Thus, the value of θ for which AB = r and AC = 2r is the
closest to the ideal case

Thus if the difference is of 1 hop when A and B are co-
located, it will definitely be less than 1 hop for nodes
farther away.
3. The second point needs to hold for the largest length of
the wire in the network in order to calculate the value of
θ
max
. If that is the case, then it will definitely hold for
smaller lengths.

The difference in distances will show up in a difference in
the average path length for difference in distances greater
than the range of the nodes. Thus one value of θ
max
can be
computed by considering the largest length of the wire that
can be placed in the network, and computing θ such that
the distance between B and C is less than the range (less
than one hop count) Mathematically the condition for θ
max
would be
r C B d < ) , ( (30)
The value of d(B, C) for ℓ (length of the wire) equal to R
is:
⎟
⎠
⎞
⎜
⎝
⎛
=
4
sin 2 ) , (
max
θ
R C B d (31)
Substituting (31) in (30) we get:
⎟
⎠
⎞
⎜
⎝
⎛
=
−
R
r
2
sin 4
1
max
θ (32)

For r = 55 and R = 1000, θ
max
can obtained as 6.3
o
. We
argue that this is a more conservative value of θ
max
and
performance close to the ideal case can be obtained at a
higher value of θ
max
due to two reasons:

R
A
B
ℓ
θ
D
r
θ/2
O
1. The region midway between the end points of two wires
will have very few nodes. Thus, for the above case, the
path length of packets originating from only a few nodes
will increase by one. This has almost no effect on the
overall average path length. In fact, this would be the case
till the point when A is placed 1 hop from the point B, but
is 2 hops from C. In this case, all nodes lying around the
line joining A and B will experience an increase in path
length of 1 – 2 hops. Although we have not proved this
mathematically, we can say that an increase of 1 -2 hops
can be tolerated for some small number of nodes without
affecting the overall average path length considerably. Any
inter-wire angle for which the difference between the hops
taken for packets to reach C and B increases beyond 2 will
start affecting the overall average path length.
2. The discussion in the analytical section proved that the
path length reduction has its minima for wire length at 75
% of the Radius. This means that the average path length is
at its minimum for wire of length 0.75R. Thus we would
not be deviating from the ideal case very much if we
optimize the inter wire angle for a maximum length of
wire of 0.75R instead of R.

We can thus compute θ
max
in the following manner. We
consider the length of the wire as 0.75R. Further, θ is such
that for a node 1 hop from B, it takes 2 hops to C. That is,
for a node A that lies on the line bisecting , d(A,B)
is r and d(A,C) is 2r.
COD ∠
Using ∆ OCB,
2
θ
= ∠COB (33)
4
sin 2
θ
l BC = (34)
Where ℓ = length of wire

In ∆ ABC,

CBA ∠ = 90 – θ/4 (35)
AC
2
= BC
2
+ BA
2
– 2 * BC * AB * cos (90 - θ/4) (36)

Plugging in AB=r, CA=2r and BC from (34) we get two
values of θ

⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
−
=
−
)
) (
3
(
2
sin 4
1
1
l r l
r
θ (37)
Similarly, for a node E one hop away from C, but away
from the center,

⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
+
=
−
)
) (
3
(
2
sin 4
1
2
l r l
r
θ (38)

For a maximum value of ℓ = 0.75 R, and values of R =
1000 and r = 55, we get θ
1
= 15.13
o
and θ
2
= 14.05
o
. Thus,
we see that the minimum angle that gives the same results
as the ideal case is ≈ 15
o
or 24 wires (360/15). Note that
this value will vary for different values of r and R.

Linked assets

Computer Science Technical Report Archive

Conceptually similar

PDF

USC Computer Science Technical Reports, no. 837 (2004)

PDF

USC Computer Science Technical Reports, no. 804 (2003)

PDF

USC Computer Science Technical Reports, no. 859 (2005)

PDF

USC Computer Science Technical Reports, no. 770 (2002)

PDF

USC Computer Science Technical Reports, no. 812 (2003)

PDF

USC Computer Science Technical Reports, no. 781 (2002)

PDF

USC Computer Science Technical Reports, no. 806 (2003)

PDF

USC Computer Science Technical Reports, no. 788 (2003)

PDF

USC Computer Science Technical Reports, no. 797 (2003)

PDF

USC Computer Science Technical Reports, no. 887 (2007)

PDF

USC Computer Science Technical Reports, no. 877 (2006)

PDF

USC Computer Science Technical Reports, no. 749 (2001)

PDF

USC Computer Science Technical Reports, no. 752 (2002)

PDF

USC Computer Science Technical Reports, no. 830 (2004)

PDF

USC Computer Science Technical Reports, no. 774 (2002)

PDF

USC Computer Science Technical Reports, no. 743 (2001)

PDF

USC Computer Science Technical Reports, no. 753 (2002)

PDF

USC Computer Science Technical Reports, no. 860 (2005)

PDF

USC Computer Science Technical Reports, no. 884 (2006)

PDF

USC Computer Science Technical Reports, no. 809 (2003)

Description

Rohan Chitradurga, Ahmed Helmy. "Analysis of wired short cuts in wireless sensor networks." Computer Science Technical Reports (Los Angeles, California, USA: University of Southern California. Department of Computer Science) no. 814 (2004).

Asset Metadata

Creator Chitradurga, Rohan (author), Helmy, Ahmed (author)

Core Title USC Computer Science Technical Reports, no. 814 (2004)

Alternative Title Analysis of wired short cuts in wireless sensor networks (title)

Publisher Department of Computer Science,USC Viterbi School of Engineering, University of Southern California, 3650 McClintock Avenue, Los Angeles, California, 90089, USA (publisher)

Tag OAI-PMH Harvest

Format 12 pages (extent), technical reports (aat)

Language English

Unique identifier UC16270252

Identifier 04-814 Analysis of Wired Short Cuts in Wireless Sensor Networks (filename)

Legacy Identifier usc-cstr-04-814

Format 12 pages (extent),technical reports (aat)

Rights Department of Computer Science (University of Southern California) and the author(s).

Internet Media Type application/pdf

Source 20180426-rozan-cstechreports-shoaf (batch), Computer Science Technical Report Archive (collection), University of Southern California. Department of Computer Science. Technical Reports (series)

Access Conditions The author(s) retain rights to their work according to U.S. copyright law. Electronic access is being provided by the USC Libraries, but does not grant the reader permission to use the work if the desired use is covered by copyright. It is the author, as rights holder, who must provide use permission if such use is covered by copyright.

Repository Name USC Viterbi School of Engineering Department of Computer Science

Repository Location Department of Computer Science. USC Viterbi School of Engineering. Los Angeles\, CA\, 90089

Repository Email csdept@usc.edu

Inherited Values

Title Computer Science Technical Report Archive

Description Archive of computer science technical reports published by the USC Department of Computer Science from 1991 - 2017.

Coverage Temporal 1991/2017

Repository Email csdept@usc.edu

Repository Name USC Viterbi School of Engineering Department of Computer Science

Repository Location Department of Computer Science. USC Viterbi School of Engineering. Los Angeles\, CA\, 90089

Publisher Department of Computer Science,USC Viterbi School of Engineering, University of Southern California, 3650 McClintock Avenue, Los Angeles, California, 90089, USA (publisher)