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Exploring Redundancy in Sensor Deployment to

Maximize Network Lifetime and Coverage

Wei ShenFaculty of Informatics & Electronics

Zhejiang Sci-Tech University

Hangzhou, 310018, China

Email: [email protected]

Qishi WuDepartment of Computer Science

University of Memphis

Memphis, TN 38152, USA

Email: [email protected]

Abstract—Energy efficiency and fault tolerance are two impor-tant features required for sustained and reliable operations of wireless sensor networks deployed in unstructured environments.This paper investigates an approach to prolonging network life-time and ensuring sensing reliability by organizing the sensorsinto several disjoint subsets, each of which takes shift to cover theentire region. This strategy is made possible by the enormous re-dundancy in large-scale sensor network applications where manysmall and inexpensive sensors are deployed to achieve qualitythrough quantity. However, such energy savings through shifttaking in time and fault tolerance via redundant coverage requirean appropriate network partition in space: each on-duty subsetmust (i) cover the entire region, (ii) maintain its own connectivity,and (iii) cover every point with multiple sensors. Based on ageneral sensor network model, we formulate this problem asan NP-complete Connected M -SET k-Coverage problem. Werigorously derive a necessary and sufficient condition for checkingthe sensor coverage of a continuous two-dimensional space basedon geometric reasoning, and analytically derive the upper boundson both M and k for any given sensor network. We furtherpropose a heuristic approach to this problem and evaluate its

performance through extensive simulations. Index Terms—Sensor deployment; energy efficiency; fault tol-

erance

I. INTRODUCTION

Multiple sensor systems have been the target of active

research since the early 90s due to their widespread use in

many agricultural, civil, and military applications that involve

environmental monitoring and situational assessment. An im-

portant subclass of sensor network applications require the de-

ployment of sensors in vast unstructured geographical areas for

remote operations, where wireless networks are often the only

means of communication among the sensors that are typicallypowered by irreplaceable batteries of limited energy supply.

Minimizing network-wide energy consumption and providing

effective coverage within the region of interest are critical to

ensuring sustained and reliable operations of Wireless Sensor

Networks (WSNs) in these applications.

Recent developments in Micro-Electro-Mechanical Systems

(MEMS) make it now possible to deploy a large number

of inexpensive and small sensors to achieve quality through

quantity. Such redundancy in sensor deployment enables a

The network is divided into 3 subsets of sensors

represented by , , , respectively.

Fig. 1. Network partition in a sleep-wake scheduling strategy.

sleep-wake scheduling strategy that organizes the sensors into

a number of subsets, each of which takes turn to monitor the

entire region while the rest are put in a sleeping mode for

energy saving to prolong the network lifetime [1]–[3]. Ideally,

this network partitioning and shift taking strategy could extend

the network lifetime as many times as the number of subsets.

Determining network partitions to implement the sleep-wake

scheduling policy was first defined as the NP-complete SET K -Cover problem [4]. As illustrated in Fig. 1, the sensor network

is partitioned into three subsets of sensors, denoted by triangles,

squares, and circles, respectively, and one subset of sensors

are currently active covering the entire rectangular region of

interest. Note that this partitioning approach is fundamentally

different from the clustering process widely adopted in WSNs.

Fault tolerance is another important performance requirement

in WSNs, and is often associated with a k-Coverage goal where

every point in the region needs to be covered by at least ksensors. Finding a minimum subset that reaches k-Coverage

was defined as the k-Coverage problem, which is also NP-

complete [5], [6].

Each of the aforementioned problems has been well studied

in a separated context in the literature [7], [8]. In this paper,

we investigate a hybrid approach to achieving both energy

2011 8th Annual IEEE Communications Society Conference on Sensor, Mesh and Ad Hoc Communications and Networks

978-1-4577-0093-4/11/$26.00 ©2011 IEEE 557



savings through shift taking in time and fault tolerance via

multiple sensor coverage by exploring the redundancy in sensor

deployment. We formulate this problem as an NP-complete

Connected M -Set k-Coverage (CMSKC) problem, whose goal

is to partition the network appropriately into M subsets in

space such that each subset (i) covers the entire region, (ii)

maintains its own connectivity, and (iii) covers every point with

at least k sensors in the subset. Based on a general sensor

network model, we rigorously derive a necessary and sufficient

condition for checking the sensor coverage of a continuous two-

dimensional (2D) region based on geometric reasoning, and

analytically derive the upper bounds on both M and k for any

given sensor network. We further propose a heuristic approach

to this problem and evaluate its performance through extensive

simulation-based experiments.

The rest of the paper is organized as follows. In Section II,

we conduct a broad survey on related work. In Section III,

we present the network model and formulate the problem.

We derive the region coverage check condition and design the

partition algorithms in Section IV. We provide simulation-based

performance evaluation in Section V. We conclude our work and discuss future research in Section VI.

I I . RELATED WORK

There have been a significant amount of research efforts

devoted to maximizing the lifetime of WSNs by organizing

the network in different structures. Clustering is one of the

most commonly employed hierarchical organization strategies

to reduce energy consumption in large-scale WSNs. Note that a

cluster is a partial organization that monitors only a subregion

and all (or most) of the clusters must work simultaneously to

achieve a full coverage of the entire region.

In a sleep-wake scheduling approach, the sensors are also

organized into a number of subsets, but each of them takes turnto cover the entire region. Providing a full coverage of the entire

region using each subset of sensors in a round-robin manner

is feasible when there exits a superfluous redundancy in sensor

deployment [9], [10], and it has the potential to increase the net-

work lifetime by multiple folds. In [4], Slijepcevic et al. studied

the SET K -Cover problem, which is proved to be NP-complete

by reducing from the Minimum Cover problem. The SET K -Cover problem in [4] attempts to maximize the value of K such that each subset covers the entire region. We would like

to point out that the coverage check algorithm proposed in [4]

examines discrete points in the region under a finite resolution,

therefore the accuracy and complexity of the algorithm not

only depend on the number of sensor nodes, but also theresolution and area of the region. Gallais et al. [11] presented a

decentralized algorithm to solve the k-area-coverage problem,

which is similar to the CONNECTED SET K -Cover problem.

Their algorithm checks the grid point coverage to obtain the

value of K , and attempts to organize the sensors into K subsets,

each of which can cover the entire region. A variant of this

problem, where the requirement that each subset covers the

entire region is relaxed, was tackled using one randomized and

two deterministic greedy approximation algorithms [12].

The k-Coverage problem in WSNs has also been extensively

studied in the literature and most efforts along this line consider

a discrete target model. Gupta et al. proposed the minimal

connected cover set (MCCS) problem [13] whose goal is to

find a minimal subset of nodes that are connected and cover the

entire region. In their solution, the coverage area of a sensor is

approximated by a set of square units, resulting in a resolution-

dependent complexity. The MCCS problem has been shown

to be NP-complete [13], [14], and is a special case of the

Connected M -Set k-Coverage problem with M = 1, k = 1,

and discrete targets. Closely related to our problem is the one

investigated by Zhou et al. in [15] where they proposed an

approximation algorithm and derived the upper bound of the

size of the resultant subset. However, their discrete treatment

on the target or query region remains the same as [13], and the

computational complexity is not given.

The k-Coverage problem was further considered by Hefeeda

in [6] and Yang et al. in [5] where they attempted to find a

minimal subset of sensors to cover all the target objects that

are made of discrete points. The computational complexity

of the algorithm in [5] is O(n3

), where n is the number of sensors. Kumar et al. used a virtual grid to represent the target

region such that the target region was covered if all grid points

were covered [16], and they also derived the conditions for

1-coverage and k-coverage in three different types of sensor

deployment.

Funke et al. developed a solution to the Connected 1-

Coverage problem using grid placement algorithms [17], where

a simple grid-based approach was first employed in place of

the greedy algorithm to improve the performance without the

guarantee for full coverage. To overcome this limitation, they

also proposed a fine grid algorithm that divides coarse grids

into many small grids based on the intersection of sensing disks

until each small grid is covered by some sensors. Obviously,

the number of small grids is the key to the performance.

Simon et al. [18] designed centralized and distributed algo-

rithms to determine the appropriate number of sensors that

are sufficient to reach k-coverage in the region where sensors

can sleep during most of their lifetime. Zhang et al. [19]

presented an algorithm, referred to as Optimal Geographical

Density Control (OGDC), to maintain coverage as well as

connectivity using a minimum number of sensor nodes. The

problem in [19] is similar to the k -Coverage problem, and the

OGDC algorithm can guarantee almost full coverage, which,

however, is still based on grid points, leading to resolution-

dependent performance.So et al. [20] presented several theorems and algorithms

for a similar problem based on the Voronoi diagram. They

investigated two problems: one is to check if every point in

a target region is k-covered by all sensors and the other is to

determine the largest k such that every point in the target region

is k-covered by all sensors, which they referred to as Max-k-

Coverage problem. They proved that the first problem can be

solved in O(n log n + nk2), where n is the number of sensors.

Furthermore, they proved that Max-k-Coverage problem can be

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Fig. 2. A counterexample that shows SET K -Cover = k -Coverage.

solved in O(n3). Huang et al. designed a novel algorithm to

determine whether a WSN is k-covered based on the perimeter

coverage of the sensor’s sensing range [21] with complexity

of O(nd log d), where n is the number of sensors and d is

the maximum number of sensors whose sensing ranges may

intersect a sensor’s sensing range. Note that the above work on

Max-k-Coverage and k-Coverage determination considers the

entire set of sensors, not a subset.

If a WSN is SET K -Covered, it satisfies the requirement

of k

-coverage, but the reverse is not necessarily true. This can

be illustrated by a counterexample shown in Fig. 2, where the

rectangular target region located at the center is 2-covered, but

these three circles are not able to form two subsets, each of

which covers the target region. Obviously, the SET K -Cover

and k -Coverage problems are equivalent when K = k = 1.

Our work on CMSKC differs from the aforementioned

efforts [4]–[6], [13], [15]–[17] in the following aspects: (i)

we investigate the Connected M -SET k-Coverage problem

in a continuous two-dimensional space; (ii) our solution is

completely independent of the grid resolution and region size;

(iii) the SET K Cover and k-Coverage problems are special

cases of our problem where k = 1 and M = 1, respectively.

CMSKC is a generalization of the individual work on energyefficiency and fault tolerance in [7], [8]. Our work is also

beyond the Max-k-Coverage problem [20] and the k-Coverage

determination problem [21] in that we wish to find M minimal

subsets of sensors, each of which is connected and achieves

k-coverage.

III. NETWORK M ODEL A ND P ROBLEM F ORMULATION

We consider an ad-hoc homogeneous WSN deployed in a

two-dimensional continuous region A with width W and height

H . Each sensor v is equipped with an isotropic sensing device

that covers a disk-shaped sensing area of radius r, an isotropic

wireless transceiver that covers a disk-shaped communication

area of radius R, a processor of low computing capacity, and abattery with limited energy supply. We assume that the width

and height of the region monitored by the sensor network

are much larger than the sensing radius r, i.e. H ≫ r, and

W ≫ r. Since sensors are densely deployed in the region

with high redundancy to cover the entire region and typically

R > r, we assume that the initial network is connected. We

also assume that the sensor locations are distinct and can

be acquired through some range-based or range-free sensor

location discovery approaches [22]. We further assume that

most of the energy on a sensor node is spent on sensing or

communication, which is independent of the sensing or radio

distance, and the energy consumption of a sensor in idle or

sleeping mode is considered negligible.

Based on this general WSN model, we formulate the Con-

nected M -SET k-Coverage (CMSKC) problem as follows:

Given an ad-hoc connected WSN with an arbitrary topology

represented as graph G(V, E ) deployed in a two-dimensional

continuous rectangular region A, where V denotes the set of

N sensor nodes and E denotes the set of wireless links, the

deployment coordinates (x, y) and a disk-shaped sensing area

S (v) of sensor v (v ∈ V ), and two positive integers M and

k, does there exist M disjoint subsets V 1, V 2, . . . , V M ⊂ V ,such that for each point p ∈ A, there exists at least ksensor nodes vi1, vi2, . . . , vik ∈ V i, i = 1, 2, . . . , M , such that

p ∈k

j=1

S (vij)? We assume that the disk-shaped sensing area

S (v) of sensor v ∈ V includes the circular rim (circle) of the

disk, which is denoted as S (v). Similarly, we denote the border

of region A as A. We also define the distance of a simple path

in the network as the number of sensor nodes on the path.The SET K Cover problem has been proved to be NP-

complete by reducing from the Minimum Cover problem [4],

[23] and the k-Coverage problem that selects a minimum

number of sensors to achieve k-coverage has also been proved

to be NP-hard by a reduction from the minimum dominating

set problem in [5]. These two problems are special cases of

CMSKC that restricts problem instances to those where (i)

M = 1 and k = 1, (ii) the target points are limited to the

sensor locations, and (iii) the network is fully connected (any

subnetwork of a fully connected network is automatically con-

nected). The validity of NP-completeness proof by restriction

is established in [23], where “restriction” constrains the given

(i.e. target points and network topology), not the question of aproblem. Since the SET K Cover problem and the k-Coverage

problem are NP-complete, so is the CMSKC problem in a

continuous 2D region, whose solution is obviously verifiable

in polynomial time.

IV. TECHNICAL A PPROACHES

A. Definitions and Theorems

When multiple sensors are deployed, their sensing disks

intersect and divide the region into a certain number of parts,

which we refer to as divisions, as defined below:

Definition 1: A division is a minimum enclosed area within

which all points are covered by the same subset of V . I f aboundary of a division is an arc, i.e. a part of sensing rim, the

boundary is called a basic arc. If the chord of a basic arc of

a division is inside the division, the arc is a convex arc of the

division; otherwise, the arc is a concave arc of the division.

Obviously, how to evaluate the coverage of each division is

the key to the CMSKC problem. A general approach is to find

some points that can represent the divisions in region A. To

facilitate the explanation of our approach, we define below two

key terms, intersection point and neighborhood .

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Definition 2: An intersection point p is either an internal

point (inside region A) intersected by two or more circles, i.e.

circular rims of disk-shaped sensing areas, or a border point

intersected by the border of region A and one or more circles.

The set of internal intersection points is defined as:

Type 1 :N

i,j=1i=j S (vi) ∩ S (vj) ∩ A

, vi ∈ V, vj ∈ V,

and the set of border intersection points is defined as:

Type 2 :

N i=1

S (vi) ∩ A

, vi ∈ V.

We denote the set of all intersection points as P . If point

p ∈ P ∩ S (vi), we say that the circle (i.e. circular rim) S (vi)of sensor vi marginally covers point p.

Definition 3: A neighborhood of point p, denoted as δ ( p),

is a disk-shaped area (including the rim) centered at p with an

arbitrarily small radius ε > 0.

Based on the above definitions, we derive the necessary and

sufficient conditions to check whether or not region A canbe fully covered by a given WSN, which is essentially the

1-Coverage problem.

Theorem 1: Region A can be fully covered by a given

WSN, iff the following two conditions are satisfied:

1) P = ∅;

2) for ∀ p ∈ P , there exists a neighborhood δ ( p) with radius

ε > 0, and for an arbitrary point a ∈ δ ( p)∩ A, there exists

a sensor v ∈ V such that a ∈ S (v).

Proof. In Theorem 1, Condition 1) requires that there exists at

least one intersection point and Condition 2) requires that the

neighborhood of any intersection point is covered.

We first prove the necessity of these conditions in this

theorem. Suppose that region A with height H ≫ r and width

W ≫ r is fully covered by the WSN. Since one single sensor

is not able to cover the entire region, there must exist multiple

sensors with intersection points. For each intersection point p,

we define an arbitrary neighborhood δ ( p). Obviously, any point

a ∈ δ ( p) ∩ A must be covered by some S (v), v ∈ V , if the

region is fully covered by the WSN.

We use reduction to absurdity to prove the sufficiency of the

conditions in Theorem 1. Suppose that those two conditions are

satisfied but region A is not fully covered by the WSN. There

must exist a circle, denoted as S (v∗), v∗ ∈ V , which has at least

one intersection point and splits the region into some covered

and uncovered zones; otherwise, we can conclude that everycircle is isolated from each other and does not intersect with

the border A, which conflicts with Condition 1) in Theorem 1.

Let p be an intersection point on S (v∗) that falls on a basic

arc splitting between a covered zone and an uncovered zone.

Obviously, there must exist a neighborhood δ ( p) of point p that

contains some points in the uncovered zone, which conflicts

with Condition 2) in Theorem 1. This concludes the proof.

We define a few more terms below that will be used in the

rest of the derivation and analysis.

1

2

Fig. 3. Illustration of the rim intersection point in Definition 6.

Definition 4: An interior intersection point p of sensor v,

v ∈ V , is an intersection point that is located inside the

coverage area S (v) of v , i.e. there exists a neighborhood δ ( p)with radius ε > 0 such that δ ( p) ∩ A ⊂ S (v).

Definition 5: If p is an intersection point and falls within

the sensing range of each sensor v ∈ V ′, V ′ ⊂ V , i.e. p is

an interior intersection point of each sensor v ∈ V ′, then the

interior degree of p, denoted by I ( p), is defined as the number

|V ′| of sensors in V ′, where V ′ is the maximum subset of V .

Definition 6: Let p be an intersection point. If there exist a

subset of sensors V ′ ⊆ V, |V ′| ≥ 3, such that { p} = v∈V ′

S (v),

then p is a rim intersection point of V ′.

An example of the concept of rim intersection point in

Definition 6 is illustrated in Fig. 3, where there are two rim

intersection points p1 and p2. However, these two intersection

points differ in the following aspect: (i) for p1, there exists a

neighborhood of p1 marked as a dotted circle, which is fully

covered by those 3 circles intersecting at p1, while (ii) for p2,

there does not exist any neighborhood of p2, which is fully

covered by those 3 circles intersecting at p2. We propose the

concept of rim degree to describe the difference between these

two types of rim intersection points.

Definition 7: Suppose that p is a rim intersection point of

V ′ ⊂ V , d1, d2, . . . , dn are the divisions intersected by S (v),

v ∈ V ′ with common border point p, and ci is the number of

sensors in V ′ that cover division di, i = 1, 2, . . . , n. The rim

degree of intersection point p , denoted by R( p), is defined as

mini=1,2,...,n

(ci). If intersection point p is not a rim intersection

point, then R( p) = 0.

An example of the concept of rim degree in Definition 7 is

illustrated in Fig. 4, where there are 11 divisions d1, d2, . . . , d11created by 6 sensors v1, v2, . . . , v6 surrounding the common

border point p. Since divisions d1, d5 and d9 are each covered

by a minimum set of 2 sensors, R( p) is 2. Definition 8: The degree, denoted by D( p), of an intersec-

tion point p is defined as the sum of the interior degree and the

rim degree of p, i.e. D( p) = I ( p) + R( p).

We have the following lemma to determine if two non-

intersection points are located in the same division.

Lemma 1: For ∀a, b ∈ A, a,b /∈ v∈V

S (v), both a and b are

located in the same division iff a ∈ S (v) ⇐⇒ b ∈ S (v), for

∀v ∈ V .

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1

2 3

4

56

1

2

3

4

5

6

7 8

910

11

12

Fig. 4. Illustration of Definition 7 and Theorem 2.

The correctness of Lemma 1 is straightforward. It simply

means that if two non-intersection points a and b are in the

same division, they must be the interior points of the same set

of sensor coverage disks, and vice versa.

Lemma 2: If p is an interior intersection point of sensor v,

then any division with border point p is covered by v .Proof. Because p is an interior intersection point of v, there

must exist a neighborhood δ ( p) with certain radius ε > 0, such

that δ ( p) ∩ A ⊂ S (v), and δ ( p) intersects with each division

with border point p. It follows that, in δ ( p), there exist some

interior points of each division with border point p. Based on

the Lemma 1, those divisions must be covered by v .

We provide Theorem 2 that describes the relation between

the degree of intersection points and divisions.

Theorem 2: Suppose that an intersection point p is the

common border point of divisions d1, d2, . . . , dn. Let ci be the

number of sensors in V that cover division di, i = 1, 2, . . . , n.

We obtain the degree of p

as D( p) = mini=1,2,...,n(ci)

.

Proof. Based on the definition of degree: D( p) = I ( p)+R( p),

we know that there exist I ( p) sensors, denoted as set V 0, each

of which covers p as an interior point, and there also exist

some sensors, denoted as set V 1, such that { p} =

v∈V 1

S (v),

and V 0∩V 1 = ∅. Based on Lemma 2, we know that each v ∈ V 0covers divisions d1, d2, . . . , dn. Suppose that mi is the number

of sensors in V 1 covering division di, i = 1, 2, . . . , n. It follows

that ci = I ( p) + mi, i = 1, 2, . . . , n. Based on Definition 7, we

have D( p) = I ( p) + mini=1,2,...,n

(mi) = mini=1,2,...,n

(ci).

Calculating I ( p) is trivial. We design Algorithm 1 to calcu-

late R( p). Suppose that p1, p2, . . . , pk are intersection points

created by n sensors v1, v2, . . . , vn, which intersect at a com-

mon rim intersection point p, i.e { p} =ni=1

S (vi).

In Algorithm 1, lines 2 and 3 create a small circle that

intersects with all the divisions having p as a common border

point. Line 5 calculates the radial tangents that intersect with

those divisions with concave arc(s). Obviously, the number of

sensors covering a division without a concave arc is not the

minimum one. Since p is intersected by the sensing rims of nsensors, there exist 2n basic arcs with end point p. For each

Algorithm 1 Calculate R( p), Input: v1, v2, . . . , vn, p,

p1, p2, . . . , pk, Output: R( p)

1: Sort v1, v2, . . . , vn in anti-clock order;2: Calculate r ′ = 1

2 mini=1,2,...,k

(| pi p|);

3: Create circle X centered at p with radius r′;4: Create circle Y centered at p through v1, v2, . . . , vn;5: Calculate the radial tangent of each basic arc with end point p;6: for all radial tangent ti, i = 1, 2, . . . , 2n do

7: Calculate the intersection point a between ti and X ;8: Calculate two points a1, a2 on Y , such that |a1a| = |a2a| =

r;9: Calculate the number ni of sensors on the minor arc a1a2;

10: R( p) = mini=1,2,...,2n

(ni).

tangent, line 7 determines an interior point a inside a division

with concave arc(s). Lines 8 and 9 obtain a set of sensors that

cover a. Based on Lemma 1, these sensors cover the division

with interior point a. After checking all tangents from point p,

Algorithm 1 obtains R( p).

An example of Algorithm 1 is illustrated in Fig. 5. All

sensors marked as small gray squares having p as a commonrim intersection point are located on the rim of circle Y centered at p with radius r , which is marked as a dotted circle

going through all sensors. There exist some other intersection

points p1, p2, . . ., among which p2 is the closest intersection

point to p. Circle X denotes a small dotted circle centered at

p (created by lines 2 and 3). Obviously, circle X with a radius

smaller than | pp2| intersects with all the divisions with common

border point p. Here, for simplicity of calculation, we set the

radius of circle X to be 1

2| pp2|. We draw a radial tangent pa

from p that intersects circle X at point a. Two points a1 and

a2 on circle Y are then determined such that the length of line

segments a1a and a2a is the sensing radius r .

Sorting all sensors takes time of O(n log n), and determiningeach ni requires finding certain v in sorted v1, v2, . . . , vn twice,

each of which is of O(log n) complexity. Therefore, the total

complexity of Algorithm 1 is of O(n log n + 2n log n), or

O(n log n).

1

2

1 2

34

5

6

7 8

910

Fig. 5. Illustration of Algorithm 1 for calculating R( p).

Theorem 2 depicts the relation between the degree of in-

tersection points and the coverage of the region because the

intersection points can be used to represent the entire set

561



of points in the region for checking region coverage. Based

on Definition 8 and Theorem 2, we obtain Theorem 3 and

Corollaries 1 and 2 as follows.

Theorem 3: Region A is fully k-covered by a WSN iff

D( p) ≥ k, for all intersection points p ∈ P .Corollary 1: The maximum value of k in the k-Coverage

problem is upper-bounded by min p∈P

(D( p)).

Corollary 2: The maximum value of K in the SET K Coverproblem is upper-bounded by min

p∈P (D( p)).

Theorem 3 provides an efficient way to find a subset of

sensors that k-cover region A. Corollary 1 provides the exact

maximum value of k and solves the Max-k-Coverage problem

in [20]. Corollary 2 defines the upper bound of K in the

SET K Cover problem, which provides a comparison base for

evaluating the performance of our network partition heuristics.

B. Algorithm Design

We first consider a special case where R ≥ 2r and then

extend our solution to a general case without this constraint.

1) When R ≥ 2r: We first design an algorithm for the CM-SKC problem with constraint R ≥ 2r as shown in Algorithm 2.

When R ≥ 2r, a subset of sensors that fully cover the region

are automatically guaranteed to be connected.

Algorithm 2 Connected M -SET k-Coverage where R ≥ 2r,

Input: V , k , Output: V i, M

1: Calculate the intersection points on S (vi), i = 1, 2, . . . , N , andgenerate P ;

2: for all p ∈ P do3: Calculate D( p);4: if D( p) < k then5: return ∅;6: i = 1;7: while (∀ p ∈ P , D ( p) > 0 and P = ∅) do8: V i = V ; V = ∅; IP = ∅;9: while (T RUE ) do

10: if there exists v ∈ V i, where D( p) > k for ∀ p ∈ P inS (v) then

11: for all p ∈ P created by v intersecting with othersensors in V do

12: P = P − { p};13: for all p ∈ P in S (v) or on S (v) do14: Update D( p);15: V i = V i − {v}; V = V + {v};16: IP = I P + { p| p is created by v intersecting with

other sensors in V };17: for all p ∈ I P in S (v) or on S (v) do18: Update D( p);

19: else20: break.21: i = i + 1; P = I P ;22: M = i.

We create a new region A′ with the same size as the given

region A and take the following procedure to find a subset with

k-coverage:

1) move a sensor in A covering an intersection point with

degree larger than k into the new region A′;

(a) (b)

(c) (d)

Fig. 6. Procedure of Algorithm 3. (a): original V ′i

, (b): some nodes are addedinto V ′

i , (c): end of the while loop, (d): delete the redundant nodes.

2) update the intersection points and their degrees in both

regions.

When Step 1 finishes, the remaining sensors form a subset with

k-coverage. These steps are repeated until all subsets are found.

In Algorithm 2, the first for loop (starting from line 2)calculates D( p) for all p ∈ P and calculates the value of k.

Each iteration of the outer while loop (starting from line 7 to

line 21) calculates a subset with k -coverage and each iteration

of the inner while loop (starting from line 9 to line 20) moves

a sensor to the new region and updates the intersection points

and their degrees in both regions.

Suppose that the maximum number of overlapped circles is

d. We consider the worst case of Algorithm 2 to compute

the upper bound of its time complexity. In the worst case,

the complexity of generating P (line 1) is no more than

O(2N d) ∼ O(N d). Since calculating each D( p) has complex-

ity no more than O(d log d), in the worst case, calculating all

D( p) has complexity of O(2N d2

log d) ∼ O(N d2

log d). In theworst case, every sensor is traversed and all intersection points

in its sensing area (including those on the sensing circular

rim) are also traversed. Hence, the sensing area of a sensor

covers no more than d2 intersection points and creates no

more than 2d intersection points with other d − 1 sensors.

The complexity of the outer while loop is no more than

O(N · d2 + N · 2d) ∼ O(N d2). Therefore, in the worst case,

the complexity of Algorithm 2 is no more than O(N d2 log d).

2) A general case: Now we develop a solution to the

CMSKC problem without the constraint R ≥ 2r. Suppose

that V R1 = {V 1, V 2, . . . , V M } is the result produced by

Algorithm 2, and the set of remaining sensors L = V −M

i=1

V i.

We define a function F (V i, L) as follows:

F (V i, L) =

C, C ⊆ L, V i + C is connected, C is minimal,

∅, C does not exists.

Here, we consider the minimal C , which means that none

of the subsets of C is connected after combining with V i. We

design an algorithm to calculate F (V i, L) in Algorithm 3.

The procedure of Algorithm 3 is illustrated in Fig. 6. Each

connected component in V i is shrunk to a virtual black node,

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(a)

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K p e r c e n t a g e

(c)

Fig. 7. Simulation-based performance comparison between our algorithm marked by asterisks and the algorithm in [4] marked by circles: (a) time costcomparison, (b) K percentage comparison, and (c) connected K percentage.

Algorithm 3 Calculate F (V i, L), Input:V i, L, Output:C

1: Generate V ′i , each v ∈ V ′i corresponds to a connected componentin V i;

2: Generate G′

i(V ′i , E ′i), where E ′i = ∅;

3: C = ∅, V S = V i, LS = L, Found = false;4: Mark all nodes in V ′i , V S as black;5: while L = ∅ do6: Choose an arbitrary c ∈ L, and mark c as white;7: V ′i = V ′i + {c}, V S = V S + {c}, L = L − {c};8: if {(c, v)|(c, v) ∈ E , v ∈ V S } = ∅ then9: E ′i = E ′i + {(c, v′)}, where v ′ ∈ V ′i is the virtual node

corresponding to v ∈ V S ;10: while there exists a path between two black nodes in V ′i

do11: Mark white c ∈ V ′i on the path as black;12: Mark those c ∈ V S as black;13: Update G′

i(V ′i , E ′i), shrink all black nodes in aconnected component to a virtual node;

14: if |{v|v ∈ V ′i , color of v is black }| = 1 then15: Found = true;16: Break;17: if Found then18: L = L + {v|v ∈ V S , color of v is white};19: C = LS − L;20: else21: C = ∅;

as shown in Fig. 6(a). The nodes in L are added into V ′i one by

one, represented by white squares (lines 6 and 7), as shown in

Fig. 6(b). If an added node is connected to an original node, an

edge is added to E ′i. If there exists a path between two virtual

black nodes, the white squares on the path are also marked

black (lines 11 and 12), and then all black nodes are shrunk toa virtual node. When there is only one virtual black node in

V ′i including all added or original black nodes, we find all the

sensors in C . Note that each added black node is essential to

the connectivity of V ′i for a minimal C .Suppose that |V i| = n, |E i| = m, |L| = l, where n > m, V i

may not be connected, and E i is the set of edges. Generating

all connected components is of O(n). In the worst case, each

node in L has some edges with nodes in V i. Adding each node

is of O(n). Therefore, calculating F (V i, L) is of O(nl).

The algorithm for the CMSKC problem without constraint

R ≥ 2r is shown in Algorithm 4. Suppose that the result

of Algorithm 2 contains M subsets. Since in the worst case,

calculating F (V i, L) is of O(N 2), and the second for loop runs

no more than M times, the complexity of Algorithm 4 is of O(N 2M ).

Algorithm 4 Extended algorithm without constraint R ≥ 2r,

Input:V R1, L, Output:V R2

1: V R2 = ∅;2: for all V i ∈ V R1 do3: Calculate all connected components in V i;4: while V R1 = ∅ do5: for all V i ∈ V R1 do6: Calculate F (V i, L);7: if F (V i, L) = ∅ then8: V i = V i + F (V i, L);9: L = L − F (V i, L);

10: V R1 = V R1 − {V i};11: V R2 = V R2 + {V i};12: Choose V i ∈ V R1 having maximal number of nodes;13: V R1 = V R1 − {V i};14: L = L + V i;

V. PERFORMANCE EVALUATION

We evaluate the performance of the proposed algorithms

in solving the Connected M -Set k-Coverage problem using a

large set of simulated sensor networks. The simulations were

conducted on an Intel T5500 laptop running Windows XP

Professional with 3 GBytes of RAM. We randomly place N

sensor nodes in a 500×500 region. In each test case, a differentrandom seed is used. The number N of nodes varies from

100 to 500. We use a pair of (R = 200, r = 100) and

another pair of (R = 200, r = 200) to evaluate Algorithms 2

and 4, respectively. Every measurement point plotted in the

performance figures is the average of 100 test cases.

We compare the performance of our algorithm with that of

the algorithm in [4]. Fig 7(a) shows the comparison of the time

cost of these two algorithms running on the same computer to

find K -COVER subsets in the designated square region, i.e.

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i n t h e s u b s e t

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300 sensors

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05

1015 0

5

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100

150

200

k MinDegree

N u m b e r o f s e n s o r s

i n t h e s u b s e t

100 sensors

200 sensors

300 sensors

400 sensors

500 sensors

(b)

Fig. 8. The results with M = 1 and different k values in sensor networks of different sizes and topologies: (a) Algorithm 2; and (b) Algorithm 4.

k = 1 in our problem. We observed that our algorithm (the

curve marked by asterisks) uses significantly less time than

the algorithm in [4] (the curve marked by circles) because the

complexity of our algorithm is completely independent of theresolution of the region. A resolution-dependent algorithm does

not scale well as the resolution increases. Fig. 7(b) shows the

comparison of the K percentage, which is defined as the ratio

of the average K , i.e. the average M when k = 1 to the

average analytical upper bound of K . We observed that our

algorithm consistently finds more subsets than the algorithm

in [4] and approaches the theoretical upper bound. We would

also like to point out that our algorithm satisfies the connectivity

requirement of each subset while the algorithm in [4] does not.

Fig. 7(c) plots the ratio of the connected M value to the average

analytical upper bound of M when k = 1, which shows that

our algorithm is able to find most of the connected subsets.

Figs. 8(a) and 8(b) show the sizes of the resultant subsetscalculated by Algorithms 2 and 4, respectively, with M = 1 and

different k values in networks of different sizes and topologies.

We define MinDegree as the value of min p∈P

(D( p)). We observed

that, for a given k value and M = 1, if k < min p∈IP

(D( p)), our

algorithms obtain a similar subset size, indicating the robustness

of our algorithms in different k-Coverage problem instances. As

shown in Fig. 8, our algorithms exhibit very stable optimization

performance in all the cases we studied.

1 2 3 4 5 6 7 8 9 101

2

3

4

5

6

7

8

9

10

Mindegree

k * A v g ( M )

k*Avg(M)

Fig. 11. The changing trend of Avg(M ) · k as MinDegree increases.

Figs. 9 and 10 show the comparison of the average M value

and the average MinDegree with r = 100 and r = 200,

respectively. Fig. 9(a) shows that when k = 1, the average M value obtained by our algorithms is very close to the average

MinDegree. Fig. 9(b) shows that when k = 2, the average M approaches half of the average MinDegree, and Fig. 9(c) shows

about one third when k = 3. We observed qualitatively verysimilar results in Fig. 10 with r = 200.

We plot in Fig. 11 the changing trend of Avg(M ) · k as

MinDegree increases, which represents the property of a sensor

network that can be partitioned into subsets with k-coverage.

We hoped that Avg(M ) · k could be consistently close to

MinDegree. However, the curves show that Avg(M ) · k departs

from MinDegree as MinDegree increases. This phenomena is

significant when MinDegree is a prime number. We believe that

this is a reasonable observation because the value of k changes

during the calculation of M , and if k is not a factor of Min-

Degree, the difference between M and MinDegree is relatively

large, which is even more significant when MinDegree is a

prime number. Fig. 11 illustrates the stability of our algorithms

from a different perspective.

VI . CONCLUSION

This paper generalized and investigated the Connected M -Set k-Coverage (CMSKC) problem. We derived the necessary

and sufficient condition for checking the coverage of a 2D

continuous region, based on which, we derived the upper

bounds of both M and k. The time complexity of the proposed

solution to network partitioning is completely independent of

the resolution of the region. The simulation results show that

our approach consistently outperforms other methods in search

of the best values of M and k .The trade-off between M and k is essentially the trade-

off between energy efficiency and fault tolerance, which is an

important performance consideration in many sensor network

applications. We will explore this trade-off and extend the

CMSKC problem to 3D space in our future work.

ACKNOWLEDGMENTS

This research is sponsored by National Science Foundation

under Grant No. CNS-0721980 with University of Memphis.

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(c)

Fig. 9. Comparison of M (curves marked by circles) and MinDegree (curves marked by asterisks): (a) k = 1 and r = 100, (b) k = 2 and r = 100, and (c)k = 3 and r = 100.

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Fig. 10. Comparison of M (curves marked by circles) and MinDegree (curves marked by asterisks): (a) k = 4 and r = 200, (b) k = 5 and r = 200, (c)k = 6 and r = 200.

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