Strong Minimum Energy Topology
in Wireless Sensor Networks:
NP-Completeness and Heuristics
Xiuzhen Cheng, Member, IEEE, Bhagirath Narahari, Rahul Simha, Member, IEEE,
Maggie Xiaoyan Cheng, Member, IEEE,andDanLiu
Abstract—Wireless sensor networks have recently attracted lots of research effort due to its wide range of applications. These
networks must operate for months or years. However, the sensors are powered by battery, which may not be possible to be recharged
after they are deployed. Thus, energy-aware network management is extremely important. In this paper, we study the following
problem: Given a set of sensors in the plane, assign transmit power to each sensor such that the induced topology containing only
bidirectional links is strongly connected. This problem is significant in both theory and application. We prove its NP-Completeness and
propose two heuristics: power assignment based on minimum spanning tree (denoted by MST) and incremental power. We also show
that MST heuristic has a performance ratio of 2. Simulation study indicates that the performance of these two heuristics does not differ
very much, but, in average, the incremental power heuristic is always better than MST.
Index Terms—Minimum energy topology, power control, wireless sensor networks, NP-Completeness, incremental power heuristic.
æ
1INTRODUCTION
T
HE research in wireless sensor networks has rapidly grown
in recent years, especially when the technologies of
sensor, actuator, and radio become more and more mature
[3], [11], [21]. Sensor network has many applications,
ranging from civil, such as smart classroom, medical
monitoring, and monitoring of natural habitats and eco-
systems, to military, such as security and surveillance, etc.
One of the most important features of wireless sensor
networks is the extreme energy constraint [19], [21]. In this
paper, we are going to consider the following problem:
What is the lower bound of the total energy consumed by all
sensors when they form a networked collaboration system? This
problem is significant in both theory and application.
A sensor network provides a global view of the
monitored object area based on many local observations.
The local information must be disseminated for further
processing to complete the job. Thus, global connectivity
should be maintained for proper data propagation. In a
sensor network or any other all-wireless network, the
connection between any two devices is controlled by their
transmit powers. If sensor A can correctly decode the signal
from sensor B, then there exists a unidirectional link from B
to A; otherwise, A is unreachable from B. If unidirectional
links from both A to B and B to A exist, then the link
between A and B is bidirectional. Usually, a sensor network
contains both unidirectional and bidirectional links. And,
these links must f orm a globall y connected network.
Minimizing the total transmit power to maintain global
connectivity is a network optimization problem that attracts a
lot of attention [6], [9], [17].
There exist several variations of the problem formula-
tion, based on whether unidirectional links are allowed or
not, and how strong we want the connectivity to be. For
example, we may require that all bidirectional links form a
strongly connected graph (a graph with at least one path
from any node to any other node) or we relax this
requirement to allow unidirectional links. We may want
to compute a k-connected graph (a graph with at least
k disjoint paths from any node to any other node), where
k>1 for network survivability; or, we simply ask for k ¼ 1.
We may even upper bound the degree of each node (the
number of one-hop neighbors) to balance the network load.
The pr oblem we study in this pap er considers only
bidirectional links and the resultant topology is strongly
connected. The formal definition of the problem is stated
below:
Definition 1 (Strong Minimum Energy Topology (SMET)
Problem). Given a set of sensors in the plane, compute the
transmit power of each sensor, such that there exists at least
one bidirectional path (containing only bidirectional links)
between any pair of sensors and the sum of all the transmit
powers is minimized.
Remarks.
1. Note that, in this paper, “sensor” and “node” are
exchangeable. By convention, in the context of
sensor networks, we use “sensor,” while in the
context of graph topology, we use “node.”
IEEE TRANSACTIONS ON MOBILE COMPUTING, VOL. 2, NO. 3, JULY-SEPTEMBER 2003 1
. X. Cheng, B. Narahari, and R. Simha are with the Department of
Computer Science, The George Washington University, Washington, DC
20052. E-mail: {cheng, narahari, simha}@gwu.edu}.
. M. Xiaoyan Cheng and D. Liu are with the Department of Computer
Science and Engineering, University of Minnesota, Minneapolis, MN
55455. E-mail: {xiaoyan, danliu}@cs.umn.edu}.
Manuscript received 23 Nov. 2002; revised 16 Apr. 2003; accepted 2 June
2003.
For information on obtaining reprints of this article, please send e-mail to:
tmc@computer.org, and reference IEEECS Log Number 11-112002.
1536-1233/03/$17.00 ß 2003 IEEE Published by the IEEE CS, CASS, ComSoc, IES, & SPS
2. We assume all sensors are located in the two-
dimensional plane.
3. The power function or the transmit power for
sensor u to reach sensor v is denoted by t d
,
where d is the distance between u and v, t is the
threshold which is a function of the signal-to-
noise ratio at v, and is a constant that is related
to path loss. In a typical application environment,
is from two to four [26].
4. For better exposition, we use fðdÞ to refer to the
power function assigned to a node with transmis-
sion distance d.
Note that our problem formulation is different than those
in [6], [9], [17], which focus on strong connectivity with
unidirectional links. We denote their problem by gSMET.In
other words, gSMET seeks a power assignment for an ad
hoc network such that the total power is minimized and the
topology is strongly connected. The resultant topology in
gSME T may contain unidirectional links. The optimal
solution of SMET is no less than that of gSMET, for the
same given network. This is because only bidirectional link
will be considered in SMET . Chen and Huang [6] and
Clementi et al. [9] have proven that gSMET is NP-Complete.
Both references show that the minimum cost spanning tree
(with edge length as the cost) has a performance ratio of 2.
While gSMET and SMET are two variations of the general
minimum energy topology problem mentioned above, their
NP-Completeness proofs are totally different. The reduction
schemes used in [6] and [9] are not applicable to SMET. We
are going to use the “unidirectional graph model” [17] in
our NP-Completeness proof as we only consider bidirec-
tional links, while [6] and [9] use the “directed graph
model” [17], as they consider both unidirectional and
directional links.
Our problem formulation is also based on the following
observations:
1. Even though unidirectional links exist in sensor
networks, current MAC layer protocols such as
IEEE 802.11 [15] and S-MAC [31] only take bidirec-
tional links into consideration. And, it seems that a
MAC layer protocol, which efficiently supports
asymmetric links, may not exist in the near future
[22]. The promising network routing protocols for
multihop system such as DSR [14] and AODV [20]
assume 802.11 as their MAC layer protocol. AODV
itself even only works with bidirectional links.
2. A higher transmit power incurs higher interference
to the surrounding devices, thus decreasing network
throughput [26]. A transmit power strong enough
for the receiver to correctly decode data signal is
ideal in multihop wireless networks.
3. The solution to our problem and other variations of
the problem can provide reference to sensor network
deployment. For example, the heuristics proposed in
this work can serve as a theoretical guide for
determining strongly related key parameters such
as sensor density and transmit power [28].
4. Our work also provides reference to scientists who
depend on network simulators for their research. For
example, with either GloMoSim [2] or NS2 [1], the
two most popular wireless network simulators,
users need to determine the coverage area for each
network node if uniform transmission range is not
suitable [25]. Our heuristic can provide the lower
bound for the transmission range to maintain global
connectivity.
5. The minimum energy topology actually forms an
energy-efficient backbone. This backbone can be
used to facilitate control message dissemination in
an efficient way [7].
6. Recently, minimum energy broadcast/multicast is a
hot research topic [5], [29]. In these research works,
multicasting is realized through pruning a broadcast
tree, while a minimum energy broadcast tree is a
spanning tree constructed based on the following
assumption: Each inner (nonleaf) node transmits
with the minimum power to reach all of its children
and each leaf node does not transmit and, thus, does
not consume energy. According to [23], a sensor
consumes almost the same amount of energy when
transmitting, idle, and receiving. Thus, actually our
strong minimum energy topology is a better model
for energy-aware broadcasting and multicasting.
This paper is organized as follows: In Section 2, we first
prove that our SMET problem is NP-Complete. Then, we give
a very simple incremental power heuristic in Section 3 and
compare it with a minimum spanning tree based algorithm
by simulation. Finally, in Section 4, we briefly summarize
some related work and conclude our paper in Section 5.
2 NP-COMPLETENESS IN COMPUTING A SMET
A graph is planar if it can be drawn in a plane with no edge
crossing. Given an undirected graph G ¼ðV;EÞ,avertex
cover C is a subset of V such that each edge in E is incident
to at least one vertex in C.Avertex cover problem asks for a
vertex cover with minimum cardinality in G. Vertex cover,
in general graphs is NP-Complete [10]. It is also NP-
Complete in planar graphs with no vertex degree exceeding
four [12]. For convenience, we denote this problem by
4PlanarVC. We are going to show that a vertex cover in planar
graphs with degree at most 3, denoted by 3PlanarVC, is also
NP-complete.
Lemma 2.1. 3PlanarVC is NP-complete.
Proof. 3PlanarVC is in NP since its superclass, VC in
general graphs, is in NP. Now, we show the Turing
reduction from 4PlanarVC to 3PlanarVC.
For a given instance of 4PlanarVC G ¼ðV;EÞ,we
construct an instance of 3PlanarVC G
0
¼ðV
0
;E
0
Þ in the
following way: Replace each degree 4 vertex v 2 G with a
widget shown in Fig. 1a. The resultant graph is G
0
, with a
2 IEEE TRANSACTIONS ON MOBILE COMPUTING, VOL. 2, NO. 3, JULY-SEPTEMBER 2003
Fig. 1. Each degree-4 vertex (vertex v in (a)) is replaced with a widget (b).
degree at most 3. This construction takes polynomial
time.
Note that, for each widget, the minimum number of
vertices to cover the six edges formed by v
1
;v
2
; ;v
7
,is
either 3 (fv
2
;v
4
;v
6
g), if none of fv
1
;v
3
;v
5
;v
7
g is included; or
4, if at least one of fv
1
;v
3
;v
5
;v
7
g is included. No vertices
other than v in G has influence on the covering of these six
edges. In other words, at least four vertices are needed to
cover any of the edges in fðb; v
1
Þ; ða; v
3
Þ; ðd; v
5
Þ; ðc; v
7
Þg and
all of the edges in fðv
1
;v
2
Þ; ðv
2
;v
3
Þ; ; ðv
6
;v
7
Þg, and at
least three vertices are needed to cover all of the edges in
fðv
1
;v
2
Þ; ðv
2
;v
3
Þ; ; ðv
6
;v
7
Þg. Thus, we can compute a
vertex cover C
0
for G
0
from a vertex cover C of G in the
following way: If v 2 C,putfv
1
;v
3
;v
5
;v
7
g into C
0
,
otherwise, put fv
2
;v
4
;v
6
g into C
0
. Put all vertices in C
with degree < 4 into C
0
. From the aboveanalysis, C
0
is a VC
of G
0
if C is a VC of G. The size of C
0
is
k k
1
þ 4k
1
þ 3ðm k
1
Þ¼k þ 3m;
where k is the size of C, m is the number of degree-4
vertices in G, and k
1
is the number of degree-4 vertices
in C.
Conversely, assume we are given a VC C
0
of G
0
with size k þ 3m, where m is the number of widget (as
shown in Fig. 1) in G
0
. Let C ¼ . For any widget with
vertices fv
1
;v
2
; ;v
7
g in G
0
, find the corresponding
vertex v in G. Note that as elaborated in the previous
paragraph, if any of fv
1
;v
3
;v
5
;v
7
g is in C
0
, then four of
the seven vertices fv
1
;v
2
; ;v
7
g must be in C
0
. Let k
1
be the number of widgets with this feature. If none of
fv
1
;v
3
;v
5
;v
7
g is in C
0
, fv
2
;v
4
;v
6
g must be in C
0
. Thus,
we can compute C in the following way: For any
widget in G
0
with corresponding vertex v in G,ifat
least one of fv
1
;v
3
;v
5
;v
7
g is in C
0
, put v into C. Put all
vertices in C
0
not belonging to the seven vertices of
any widget into C. It is clear that C is a VC of G with
size k þ 3m 3ðm k
1
Þ4k
1
þ k
1
¼ k. tu
Now, we consider a restricted version of the 3Pla-
narVC problem: Given a planar graph G ¼ðV;EÞ satisfying
the following two constraints: 1) each node has degree at most
3 and 2) the shortest path (a path with minimum number of
edges) between any two degree-3 vertices is at least 3. Compute
a vertex cover with minimum size. We denote this problem
by 3rPlanarV C. Note that, in an instance of 3rPlanarVC,
the shortest path between two degree-3 vertices contains
at least two degree-2 vertices.
Lemma 2.2. 3rPlanarVC is NP-Complete.
Proof. 3rP lanarV C 2 NP is trivial. Now, we show that
3PlanarVC is polynomial time Turing reducible to
3rPlanarVC.
Consider any instance of 3PlanarVC, denoted by
G ¼ðV;EÞ. For each edge e 2 E, place two Steiner points
which divide e into three equal-sized edges, as shown in
Fig. 2. With this modification, we get G
0
¼ðV
0
;E
0
Þ,an
instance of 3rPlanarVC. This construction takes polyno-
mial time. Now, we show that G has a VC of size k iff G
0
has a VC of size k þjEj.
Let C be a VC of G with size k. Let C
0
¼ . For each
edge ðu; vÞ2G, we have three edges ðu; aÞ; ða; bÞ; ðb; vÞ in
G
0
as shown in Fig. 2. For each edge ðu; vÞ2G,ifu 2 C,
put b to C
0
, otherwise, put a to C
0
. (Note that we only
consider one node u in edge ðu; vÞ.) After that, put all
vertices in C into C
0
. It is clear that C
0
is a VC of G
0
with
size jC
0
j¼k þjEj.
Conversely, suppose we are given a VC C
0
of G
0
with
size k þjEj.LetC ¼ . For every thr ee edges like
ðu; aÞ; ða; bÞ; ðb; vÞ in Fig. 2, C
0
must contain either a or b
or both. If a 2 C
0
and b=2 C
0
, then v 2 C
0
to cover edge
ðb; vÞ.Ifb 2 C
0
and a=2 C
0
, then u 2 C
0
to cover edge
ðu; aÞ. If both a and b are 2 C
0
, then neither u nor v will be
in C
0
since C
0
is minimal. Based on this analysis, we
compute C in the following way: let C ¼ C
0
\ V . For
every three edges like ðu; aÞ; ða; bÞ; ðb; vÞ in Fig. 2, if both a
and b are included in C
0
, then set C ¼ C [fug. It is clear
that C is a VC of G with size k . tu
2.1 SMET is NP-Hard
Now, we are going to show that 3rPlanarVC is polynomial
time Turing reducible to our SMET problem. The decision
version of 3rPlanarVC is denoted by P
1
and is stated below:
P
1
. Given a planar graph G ¼ðV;EÞ and a constant k,
with G satisfying the following two constraints: 1) G has a
degree at most 3 and 2) the shortest path between any two
degree-3 vertices is at least 3. Is there a vertex cover of G
with a size at most k?
We use P
2
to denote the decision version of our SMET
problem:
P
2
. Given a set of vertices V
0
and a real number P , is there
a power assignment such that the total energy is at most P?
We begin with a brief review of several terms in graph
theory. Each closed area in a planar graph is called a face.
According to Leonhard Euler formula, f ¼jEjjV jþ1,
where f is the number of faces in a planar graph. Each face
F is associated with F
x
edges and F
x
vertices, which form
the boundary of the face. We denote the vertex set of F by
fv
F
1
;v
F
2
; ;v
F
x
g. In an instance of P
1
, if a face has y
neighboring faces, then there are at least 2y degree-2
vertices on its boundary. Two neighboring faces share at
least two degree-2 vertices.
Now, we first construct an instance of P
2
(denoted by V
00
)
from any instance of P
1
(denoted by graph G ¼ðV;EÞ). This
construction procedure is relatively complicated. First, we
compute a Steiner tree to connect all vertices in G. The
superposition of and G is a planar graph G
0
. Then, we use
slightly different strategies to Steinerize edges in and
edges in G. After this Steinerization, we get an instance of P
2
which contains all Steiner points and original vertices in G.
2.1.1 Constructing Steiner Tree
This procedure contains two steps, as shown below:
1. Initally, F¼. Start from any face F. Add a vertex g in
the center of F. Draw edges ðg; v
F
1
Þ; ðg; v
F
2
Þ; ; ðg; v
F
x
Þ
CHENG ET AL.: STRONG MINIM UM ENERGY TOPOLOGY IN WIRELESS SENSOR NETWORKS: NP-COMPLETENESS AND HEURISTICS 3
Fig. 2. Each edge e ¼ðu; vÞ in G ¼ðV;EÞ is replaced by three edges
ðu; aÞ; ða; bÞ; ðb; vÞ in G
0
¼ðV
0
;E
0
Þ.
to connect g and all vertices associated with F . Put F
into F .
2. Choose a face F
0
=2F satisfying the following
constraints: a) F
0
shares a common edge with some
face F
00
in F and b) F
0
has maximum number ( 1)
of new vertices not on the boundary of any face in F .
Add a vertex g
0
inside F
0
. Connect g
0
with each of the
new vertices on the boundary of F
0
. Choose a
degree-2 vertex q which is located on the boundary
of F
0
and F
00
. Note that we can always find this
vertex q, since F
0
and F
00
share at least three edges as
their common boundary. Connect q and g
0
. Repeat
this step until no F
0
exists.
The Steiner tree connects all vertices in G. No edge in
overlaps with any edge in G. The superposition of and G
forms planar graph G
0
. The degree of non-Steiner vertices
(vertices in V )inG
0
is at most 4. Each vertex v 2 V is
incident to exactly one edge in ,ifv has a degree of 3 in G;
or one or two edges in ,ifv has a degree of 2 in G. The
example in Fig. 3 demonstrates this construction procedure.
2.1.2 Steinerizing the Planar Graph G
0
Before the Steinerization, we need to make some modifica-
tion to G
0
. Note that, with this procedure, bent or curved
edges may be introduced. But, the graph remains planar.
This modification is stated below:
1. If v 2 G
0
has a degree of at most 4, as shown in Fig. 4,
we adjust the edges such that angles formed by any
two neighboring edges are the same.
2. Other vertices in G
0
must belong to and each has a
degree of at least 5. Replace each of these vertices by
a widget as shown in Fig. 5. After this modification,
each new vertex has a degree of at most 4. We now
adjust edges such that angles formed by any two
neighboring edges are the same, as shown in Fig. 4.
Let L
1
0
be the smallest edge length in the modified G
0
. Let
L
2
0
be the shortest Euclidean distance between two edges.
Set L
0
¼ minfL
1
0
;L
2
0
g. Select two numbers L
1
and L
2
such
that L
1
<
1
3
L
0
, L
2
<L
1
. We choose L
1
¼
1
4
L
0
. Choose L
2
such that it is the largest common divisor, which is <L
1
,of
all jðu; vÞj 2 L
1
, where ðu; vÞ2E, and all jðu; vÞj, where
ðu; vÞ =2 E. Thus, both jðu; vÞj and jðu; vÞj 2 L
1
are multi-
ples of L
2
. (Remember that E is the edge set of the original
graph G.)
. For each e ¼ðu; vÞ2E, place equally spaced Steiner
points in the middle part of e , as shown in Fig. 6a.
4 IEEE TRANSACTIONS ON MOBILE COMPUTING, VOL. 2, NO. 3, JULY-SEPTEMBER 2003
Fig. 3. The step by step demonstration for constructing a Steiner tree to connect all vertices in G. In this figure, solid lines represent edges in G, while
dashed lines represent edges in .
Fig. 4. Modify nodes with a degree less than 5 in G
0
. (a) v 2 G
0
has a degree of two, ¼ 180 degrees, (b) v 2 G
0
has a degree of three, ¼ 120
degrees, and (c) v 2 G
0
has a degree of four, ¼ 90 degrees.
Fig. 5. Widget for transforming a node in with a degree of 5. (a) Vertex v has a degree of exactly 5. (b) Widget when v has a degree of x 6.
Here, t ¼d
x6
2
eþ2.
Fig. 6. (a) e ¼ðu; vÞ is an edge in the original graph G. (b) e ¼ðu; vÞ is an edge in .
. For each e ¼ðu; vÞ =2 E, place equally spaced Steiner
points on e, as shown in Fig. 6b.
We denote this modified and Steinerized graph by G
00
.
All the vertices in G
00
form an instance of P
2
, our SMET
problem. We denote the vertex set in G
00
by V
00
. This
construction takes polynomial time.
2.1.3 Transmit Powe r Assignm ent
Let C be any VC of G with size k. For each edge
ðu; vÞ2E, choose u if u 2 C,orv if v 2 C, but not both, as
its cover. In other words, each edge in G has exactly one
cover in C. Let p
v
be the power assigned to v. Note that,
any edge ðu; vÞ2E in G is divided into many short edges
in G
00
: ðu; u
1
Þ; ðu
1
;u
2
Þ; ; ðu
x
;vÞ, where u
x
¼
ðjðu;vÞj2L
1
Þ
L
2
þ 1
is the number of Steiner points on ðu; vÞ.Ifu is a cover of
ðu; vÞ in G, we also say u is a cover of edge ðu; u
1
Þ in G
00
.
Now, we assign energy to each vertex v
00
2 V
00
according
to the following rule:
p
v
00
¼
fðL
1
Þðv
00
2 V ^ v
00
2 CÞ_
ðv
00
2V
00
V;u2V ^u is a cover of edge ðu;v
00
Þ in G
00
Þ
fðL
2
Þ o:w:
8
>
<
>
:
ð1Þ
It is clear that this assignment generates a strong
topology with a total power of
P ¼ðk þjEjÞ fðL
1
ÞþðjV
00
jk jEjÞ fðL
2
Þ:
Conversely, if 8v
00
2 V
00
is assigned a power based on (1),
the set C ¼fv
00
j v
00
2 V \ V
00
^ p
v
00
¼ fðL
1
Þg is a VC of G
with size k.
Theorem 2.3. SMET is NP-Complete.
Proof. It is obvious that SMET is in NP. From the above
analysis, we know that SMET is NP-hard. tu
3TWO HEURISTICS TO APPROXIMATE SMET
SMET is NP-Complete, thus, no efficient exact algorithm
exists. The best we can do is to seek fast approximation
heuristic with good performance ratio. In this section, we
are going to analyze the performance of two heuristics. The
first one is based on minimum spanning tree (MST). We
assign power to each sensor such that it can reach the
farthest neighbor in the tree. The topology generated by this
power assignment (only bidirectional links are considered)
is globally connected. The second one is an incremental
power heuristic. We build a globally connected topology
from one node, selecting a node with minimum incremental
power to add to the topology at each step. At any time in
this heuristic, all bidirectional links in the partial topology
form a connected graph. These two heuristics are approx-
imation algorithms to SMET. We will compare them
through simulation.
3.1 Minimum Spanning Tree-Based Heuristic
A naive heuristic is to compute a minimum (edge-cost)
spanning tree (MST) and let each sensor transmit at a power
that can reach its farthest neighbor in the tree. We call this
power assignment based on MST and call this heuristic MST.
Lemma 3.1. Power assignment based on MST has performance
ratio 2.
Proof. Let P
MST
, P
SMET
, and P
gSMET
denote the total
transmit power assign ed based on any minimum
spanning tree of the problem instance, in the optimal
solution of SMET, and in the optimal solution of gSMET,
respectively. Since P
MST
2 P
gSMET
(as proven by [6],
[9]) and P
gSMET
P
SMET
, we see P
MST
2 P
gSMET
. tu
We claim that this performance analysis is tight. An
example with a tight bound of 2 is shown in Fig. 7. In this
example, is a sufficiently small positive real number.
There are 4n þ 2 nodes in the plane. They form n -shaped
structures connected by edges with a length of 1. The total
power assigned to each node based on the minimum
spanning tree (represented by solid and dashed edges) is
4n fð1Þþ2 fð1Þ, while the optimal topology generated by
optimal power assignment (represented by dotted and
dashed edges, and solid edges with length ) has a power of
2n fð1 þ Þþ2 fð1Þþ2n fðÞ.Whenn !1,
P
MST
P
OP T
¼ 2.
This indicates that power assignment based on MST has a
performance ratio of exactly 2.
Lemma 3.2. The MST heuristic takes time Oðn
2
log nÞ, where n
is the number of nodes in the network.
Proof. We need time OðnÞ to assign transmit power (there
are n vertices and n 1 edges in the MST, and we need
to examine each edge in the tree twice) after the MST is
constructed. The famous Kruskal’s algorithm and Prim’s
algorithm both take time Oðn
2
log nÞ to construct a
minimum spanning tree. Thus, MST heuristic takes time
Oðn
2
log nÞþOðnÞ¼Oðn
2
log nÞ. tu
3.2 An Incremental Power Greedy Heuristic
We propose the following Incremental Power heuristic to
approximate the SMET problem. Interestingly, this heuristic
gives optimal solution to the example in Fig. 7.
Input: A set V of n sensors in the plane.
Output: A power assignment to each sensor such that the
resultant topology is c onnected and there is at le ast one
bidirectional path between any pair of sensors.
Incremental Power Heuristic:
CHENG ET AL.: STRONG MINIM UM ENERGY TOPOLOGY IN WIRELESS SENSOR NETWORKS: NP-COMPLETENESS AND HEURISTICS 5
Fig. 7. An example to demonstrate that power assignment based on MST has a performance ratio of exactly 2.
