On Energy-Efficient Trap Coverage in Wireless Sensor Networks

Transcription

1 On Energy-Efficient Trap Coverage in Wireless Sensor Networks JIMING CHEN, JUNKUN LI, and SHIBO HE, Zhejiang University TIAN HE, University of Minnesota YU GU, Singapore University of Technology and Design YOUXIAN SUN, Zhejiang University 2 In wireless sensor networks (WSNs), trap coverage has recently been proposed to trade off between the availability of sensor nodes and sensing performance. It offers an efficient framework to tackle the challenge of limited resources in large-scale sensor networks. Currently, existing works only studied the theoretical foundation of how to decide the deployment density of sensors to ensure the desired degree of trap coverage. However, practical issues, such as how to efficiently schedule sensor node to guarantee trap coverage under an arbitrary deployment, are still left untouched. In this article, we formally formulate the Minimum Weight Trap Cover Problem and prove it is an NP-hard problem. To solve the problem, we introduce a bounded approximation algorithm, called Trap Cover Optimization (TCO) to schedule the activation of sensors while satisfying specified trap coverage requirement. We design Localized Trap Coverage Protocol as the localized implementation of TCO. The performance of Minimum Weight Trap Coverage we find is proved to be at most O(ρ) times of the optimal solution, where ρ is the density of sensor nodes in the region. To evaluate our design, we perform extensive simulations to demonstrate the effectiveness of our proposed algorithm and show that our algorithm achieves at least 14% better energy efficiency than the state-of-the-art solution. Categories and Subject Descriptors: C.2.1 [Computer-Communication Networks]: Network Architecture and Design Wireless communication General Terms: Design, Algorithms, Performance Additional Key Words and Phrases: Wireless sensor networks, trap coverage, energy-efficient, scheduling ACM Reference Format: Chen, J., Li, J., He, S., He, T., Gu, Y., and Sun, Y On energy-efficient trap coverage in wireless sensor networks. ACM Trans. Sensor Netw. 10, 1, Article 2 (November 2013), 29 pages. DOI: 1. INTRODUCTION While recent advances in wireless communication and hardware device have posed a bright blueprint for Wireless Sensor Network (WSN) applications in a large range of fields, including military affairs, healthcare, and environment surveillance [Ko et al. A preliminary version of this article was presented in Proceedings of the IEEE Real-Time Systems Symposium (RTSS) [Li et al. 2011]. Research was supported in part by the NSFC under grants and , the SRFDP under grant , the 111 Program under grant B07031, the 863 High-Tech Project under grant 2011AA , SUTD-ZJU/RES/03/2011, Singapore-MIT International Design Center IDG , and SUTD SRG ISTD Authors addresses: J. Chen (corresponding author), J. Li, S. He, and Y. Sun, State Key Laboratory of Industrial Control Technology, Zhejiang University, Hangzhou , China; s: {jmchen, yxsun}@iipc.zju.edu.cn, {lijunkun, ferrer}@zju.edu.cn; T. He, Computer Science and Engineering, University of Minnesota; tianhe@cs.umn.edu; Y. Gu, Pillar of Information System Technology and Design, Singapore University of Technology and Design, Singapore; jasongu@sutd.edu.sg. Permission to make digital or hard copies of part or all of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies show this notice on the first page or initial screen of a display along with the full citation. Copyrights for components of this work owned by others than ACM must be honored. Abstracting with credit is permitted. To copy otherwise, to republish, to post on servers, to redistribute to lists, or to use any component of this work in other works requires prior specific permission and/or a fee. Permissions may be requested from Publications Dept., ACM, Inc., 2 Penn Plaza, Suite 701, New York, NY USA, fax +1 (212) , or permissions@acm.org. c 2013 ACM /2013/11-ART2 $15.00 DOI:

2 2:2 J. Chen et al. Fig. 1. An example of trap coverage; D is the largest diameter among all coverage holes. 2010; Cao et al. 2008; Zahedi et al. 2010], most practical implementations are restricted to small-scale experiments or applications with dozens or hundreds of sensors. One of the major reason for such relatively small-scale deployment is the prohibitively high cost of deploying thousands/millions of sensor nodes for large-scale applications. Consequently, designing a sensor network where the number of sensor nodes does not increase quickly (e.g., exponentially) with the deployment size while maintaining desired system performance is a fundamental challenge. Partial coverage is introduced in Liu and Liang [2005] to address the limited quantity of sensor nodes in large-scale applications, as it is prohibitively expensive to guarantee the full coverage of the Region of Interest (RoI) [Cardei et al. 2005; He et al. 2012a; Slijepcevic and Potkonjak 2001; Chen et al. 2010]. Partial coverage allows for coverage holes [Jeong et al. 2009], and its quality of coverage is mainly indicated by the ratio of uncovered area to the whole region [Liu and Liang 2005; Wang et al. 2004; Liu and Towsley 2004]. By adopting a partial coverage scheme, a significant number of sensor nodes could be saved and thus scale well with the network size. However, for the partial coverage, it is difficult to evaluate the actual network performance by the ratio of uncovered area, as the size of coverage holes could be extremely large or even unbounded. For applications such as intrusion detection, this implies that the moving target could travel an arbitrarily long distance and time in certain area of the RoI without being detected. Therefore, the simple partial coverage has a very narrow spectrum of practical applications that has been well recognized by many researchers. To trade off the scalability and network performance, Balister et al. have recently proposed a new kind of coverage, called trap coverage [Balister et al. 2009], based on the concept of partial coverage. In trap coverage, the size of each coverage hole is incorporated as the indicator of quality of sensing. For the coverage hole, its size is indicated by its diameter D, which is defined as the largest Euclidean distance between any two points in the coverage hole. A set of sensor nodes is said to provide D-Trap Coverage to the RoI A, if the diameter of every coverage hole in A is smaller than or equal to D. Although there may exist lots of coverage holes in the network according to this definition, the largest area of coverage holes is no more than π D 2. An example of trap coverage is illustrated in Figure 1. A target is trapped in a coverage hole, as it will be detected within a maximal moving distance of D. Compared with the partial coverage which takes coverage ratio as an indicator, trap coverage guarantees the quality of coverage in the worst case. By carefully controlling the parameter D, network performance such as connectivity or delay of detecting intrusions can be ensured in trap coverage. Balister et al. study for the first time the trap coverage of randomly deployed sensor networks. They consider the fundamental problem of how to design reliable and explicit

3 On Energy-Efficient Trap Coverage in WSNs 2:3 deployment density required to achieve D-trap coverage. Their work is concerned with conceptual network design, however, practical implementation scheme, such as how to simultaneously guarantee trap coverage and energy efficiency, is left uninvestigated. As sensor nodes could be deployed in an arbitrary manner, the required number of sensor nodes to ensure D-trap coverage is usually more than the optimal value. In this article, we fill in this gap by considering the energy-efficient scheduling of sensor nodes in the randomly deployed sensor networks to achieve D-trap coverage for the first time. In fact, this problem is extremely difficult, and we have proved that it is an NP-hard problem. To effectively solve this problem, we design an approximation algorithm called Trap Cover Optimization and implement the algorithm in a localized way. Specifically, the main intellectual contributions of this work are as follows. (1) To the best of our knowledge, we are the first to study how to schedule the activation of sensors to maximize network lifetime while guaranteeing D-trap coverage in the randomly deployed sensor networks. The problem of scheduling the activation of sensors while achieving D-trap coverage is first formulated in this article. Efficient algorithms have been designed to solve the problem in polynomial time. (2) We first introduce a centralized solution and theoretically prove that the performance of this centralized algorithm is not greater than O(ρ) times of the optimal solution, where ρ is the density of sensors scattered in the RoI, that is, ρ is defined as the ratio of the number of sensor nodes N to the size of RoI S. Our algorithm attains a provable guarantee in the worst case which is only related to ρ. The approximation ratio will be more desirable when our algorithm is applied to largescale wireless sensor networks. (3) To make our design practical for real sensor network systems, we introduce a localized trap coverage coverage protocol that can be implemented at individual sensor nodes. (4) Extensive simulations are performed to demonstrate that our proposed algorithm is effective and much more energy efficient than a naive approach to the optimal lifetime scheduling for trap coverage problem, as well as the state-of-the-art solutions. The rest of the article is organized as follows. We discuss related work in Section 2, followed by the formulation of Minimum Weight Trap Cover Problem in Section 3. Section 4 presents the details of our algorithm design. The approximation ratio of the algorithm is obtained theoretically in Section 5. We design a localized protocol in Section 6, perform extensive simulations to verify the effectiveness of algorithm in Section 7 and conclude in Section RELATED WORK Sensing coverage has been attracting considerable attention in WSNs [Berman et al. 2005; Cardei et al. 2005; Slijepcevic and Potkonjak 2001; He et al. 2012b; Meguerdichian et al. 2001]. Most of existing works concentrate on duty-cycling sensors to achieve full coverage as well as energy efficiency during the network operation [Berman et al. 2005; Slijepcevic and Potkonjak 2001]. Centralized and distributed algorithms are both proposed to dynamically activate a subset of sensors to ensure coverage, connectivity, and optimize energy consumption [Yang et al. 2005; Kasbekar et al. 2009; Dong et al. 2010]. Requiring every point in the RoI to be covered is infeasible in many real-life largescale deployments. Liu and Liang [2005] demonstrate that in some application scenarios, full coverage is either impossible or unnecessary. They indicate that a partial coverage with a certain degree of guaranteed coverage is acceptable and analyze its

4 2:4 J. Chen et al. corresponding properties for the first time. In most of the existing literatures, the percentage of uncovered area to the RoI mainly acts as the indicator of quality of partial coverage [Berman et al. 2005; Wang et al. 2004; Liu and Towsley 2004]. These solutions provide approaches to achieve desirable percentage of uncovered area with a guaranteed approximation ratio. Nevertheless, the fraction of uncovered region to the whole region only indicates quality of coverage at average. A target moving in a WSN can remain undetected for an arbitrarily long distance even though a large fraction of RoI is covered. This is because the area of the coverage hole in partial coverage can be arbitrarily large or even unbounded. On the other hand, for many applications, the quality of sensing coverage needs to be guaranteed in the worst, instead of average case. The quality of partial coverage in the worst case has been studied [Ren et al. 2007; Balasubramanian et al. 2006; Sankararaman et al. 2009; Gui and Mohapatra 2004; Cao et al. 2005]. In Ren et al. [2007], wave protocol is designed to improve the coverage performance in the worst case. For any specified continuous curve with two ends on opposite borders of the field, it always finds a subset of sensors whose sensing ranges completely cover this curve. The curve is allowed to move so that every geographical point on the field can be scanned at least once in one wave scanning period and the detection period in the worst case is guaranteed. In Gui and Mohapatra [2004], the quality of partial coverage is defined as the expected travel distance before a moving object is detected. In Balasubramanian et al. [2006], κ-weak coverage is first introduced to characterize the worst case in partial coverage. The problem of maximum network lifetime of κ-weak coverage is presented in Sankararaman et al. [2009], and an algorithm based on square and hexagonal tiling is proposed to solve the problem by dividing disjoint sensor sets as many as possible. The disadvantage of this approach is that it heavily depends on the uniformity of sensor deployment, which may not be applicable in a randomly deployed WSN of large scale. The framework of optimal sleep scheduling for rare event detection is described in Cao et al. [2005], and event detection delay is treated as an indicator of quality of partial coverage. Trap coverage is proposed in Balister et al. [2009], which defines the size of coverage hole as the indicator of quality of coverage. In Balister et al. [2009] the expected density of sensors to achieve the desired quality of coverage is studied. Balister and Kumar [2009] also study the effect of random failures of sensor nodes when sensor nodes are deployed randomly versus deterministically. Our work differentiates from the aforementioned works in that we focus on maximizing the network lifetime based on trap coverage. To our best knowledge, we are the first to address the problem by rotating active state of sensor nodes to guarantee trap coverage and prolong the network lifetime in large scale WSNs. An approximation algorithm called Trap Cover Optimization is proposed to solve the problem in polynomial time, based on which the network lifetime is maximized efficiently. We rotate sensor nodes at each time slot in a localized way and thus balance the energy consumption among sensors to extend the network lifetime. There also exist other kinds of sensing coverage in WSN besides full/blanket coverage. Barrier coverage was first proposed in Kumar et al. [2005]. The goal of barrier coverage is to detect crossing across a certain area with a barrier of sensors instead of providing full/blanket coverage to the area. In random sensor deployments, scheduling the activation of sensors alternatively to form barriers can help prolong the network lifetime. Several sensor scheduling algorithms for barrier coverage have been proposed in the literature [Kumar et al. 2005, 2010; Li et al. 2012]. Sweep coverage utilizes the mobile sensors to cover the area while guaranteeing the worst time delay. The nodes have locomotion capabilities and spread out to maximize the area covered by the network (e.g., [Cheng et al. 2008]).

5 On Energy-Efficient Trap Coverage in WSNs 2:5 3. PRELIMINARY AND PROBLEM FORMULATION 3.1. Network Model We consider a large-scale WSN in the RoI A. We assume A is a rectangle of size S = l 1 l 2 for simplicity. The WSN consists of N sensor nodes. Assume that the location of each sensor is known, either by GPS or other localization techniques [Mao et al. 2007; Patwari et al. 2005]. The location of individual sensor nodes is denoted by P i, where i = 1, 2,...,N. Each sensor can only detect targets in a certain range, which we refer to as sensing region, denoted by R. Each sensor is assigned with a unique ID number. We assume the sensing regions of sensor nodes are homogeneous, and all are unit open disc centered at the location of the sensor with radius of r, which does not include the boundary of sensing region. While we are aware that the actual sensing region is typically irregular dependent on the type of sensors and the environments [Hwang et al. 2007], we argue that the disk model can be regarded as the largest embedded circle of the actual sensing region [Yan et al. 2008]. By making such a simplified assumption, we can concentrate on our main problem and understand its intrinsical property, with a minor penalty of performance degradation in practical applications. The boundary of sensing region R i of sensor i is referred to as sensing border, which is essentially a circle of radius r centered at P i. For large-scale sensor network applications, controlled deployments of sensor nodes is normally infeasible, and therefore leads to the popular adoption of random deployment. For example, an airplane can be used to air drop sensor nodes in a forest. So for our network, we assume sensor nodes are randomly deployed with a density of ρ. Apparently, ρ can be approximated by N/S, where S = l 1 l 2 is the area of region A. While a classic Poisson point process is assumed in Balister et al. [2009], our solution can be applied to all potential deployment processes. We divide operation time of individual nodes into time slots. At each slot, a subset of sensors is activated to ensure trap coverage. We rotate active time of sensor nodes in different slots in order to extend network lifetime. Assume that each sensor has an initial energy of E units, E > 1, and consumes one unit per slot if it is active. For simplicity, if a sensor is put in sleep mode, we assume it consumes no energy. The sensor node with residual energy less than one unit can not be activated any more. In terms of communication, each sensor node can only communicate with other sensor nodes within a certain range, referred to as transmission range. As proved in Wang et al. [2003], if the transmission range of sensor node is at least twice of its sensing range, coverage implies connectivity of the network. This is to say if the sensing region of two sensors intersects with each other, they are connected. In trap coverage, the sensing regions of isolated sensors do not intersect with each other, meaning the isolated set of sensors must be trapped in a coverage hole with uncovered physical points around. If we neglect the detection measurements of isolated sensors, the main connected component of sensors can still provide required D-trap coverage (see the definition in the Section 3.2). We therefore assume that trap coverage also implies connectivity of the network Trap Coverage Model Trap coverage is a new coverage model allowing the existence of uncovered physical points in the RoI but restricts the size of coverage holes, as shown in Figure 1. In this section, we give a mathematical definition of trap coverage. Definition 1 (Coverage Hole). A set of uncovered points in RoI A forms a coverage hole H, if for any two points a and b in H, there always exists a curve ζ whose start

6 2:6 J. Chen et al. and end points are a and b, respectively, satisfying that ζ ( i C R i) =, where C is the set of active sensor nodes at that time. Obviously, H A. The diameter of coverage hole H is defined as the largest Euclidean distance between any two points in the coverage hole. Denote the diameter of coverage hole H by d(h) and dist(a, b) is the Euclidean distance between point a and b, then d(h) = max(dist(a, b)). (1) a,b H Definition 2 (D-Trap Coverage). Sensor set C provides D-trap coverage to RoI A, if the diameter of every coverage hole H in A is not greater than D, thatis, d(h) D, H A. (2) We call C a D-trap cover of RoI A. Obviously, if we set diameter threshold D to zero, D-trap coverage reverts back to full coverage Minimum Weight Trap Cover Problem To achieve energy efficiency, in this work we aim to design sensing scheduling algorithms that activate the minimal number of sensor nodes per time slot while guaranteeing D-trap coverage. To guarantee the balance of energy consumption among all sensors, we also require that the activated sensors should be those with more residual energy. Each sensor in the network is assigned with a weight based on its residual energy at the beginning of each time slot. We show an example of weight assigning in this article. Let E i denote the residual energy of sensor node i and γ i = 1 E i /E denote its energy consumption ratio, where, γ i is a variable between 0 and 1. The weight of sensor node i at time slot t, t = 1, 2,...,is assigned as an exponential function related to the residual energy, that is, w i (t) = θ γi(t) /E, 0 γ i < 1, (3) where θ is a constant value greater than one. Sensor i is specially marked by assigning weight θ N, which is greater than the sum of weights of other sensors with residual energy, if it has no residual energy, that is, γ i = 1. The weight θ N can also be viewed as an infinite high weight. After weight assigning, we formulate the Minimum Weight Trap Cover Problem in this section. Consider a sensor set C which provides D-trap coverage to RoI A, that is, each sensor i in C is associated with a weight w i. The weight of trap cover C is defined as the sum of weights of all sensors in C,thatis,w(C) = i C w i. Given the diameter threshold D, there exists a family of trap covers ℵ. Definition 3 (Minimum Weight Trap Cover). Given RoI A, a set of sensors {1, 2,...,N} with corresponding weights w 1,...,w N. A minimum weight trap cover C is a trap cover with minimum weight among all trap covers, that is, w(c ) = min w(c) = min w i. (4) C ℵ C ℵ Minimum Weight Trap Cover Problem. Given RoI A, a set of sensors {1, 2,...,N} with their corresponding weights w 1,w 2,...,w N, and sensing radius r. C is a subset of sensors. There are M coverage holes H 1,...,H M in RoI A if all sensors in C are activated while other sensors not in C are put into sleep. The minimum weight trap cover problem is to choose a minimum weight set C which can ensure that every i C

7 On Energy-Efficient Trap Coverage in WSNs 2:7 Fig. 2. A demonstration of diameter calculation. The diameter of coverage hole H equals the maximum Euclidean distance between the intersection points a and b in point set Hm,thatis,d(H) = max a,b Hm dist(a, b), hereby Hm ={P 1, P 2,...,P 8 } is the set of intersection points on the boundary hole H. coverage hole in A has a diameter no more than D, where D is a threshold set by applications. The problem can be formally formulated as follows. min C ℵ i C w i s.t. d(h m ) D, m = 1,...,M. 4. ALGORITHM DESIGN In this section, we show the concept of intersection point and then discuss how to calculate the diameters of coverage holes in the RoI. Based on the results, we further design our algorithm, Trap Cover Optimization, for minimum weight trap cover problem Finding the Diameter of A Coverage Hole We will show how to calculate the diameters of coverage holes in this section to prepare for the design of trap coverage optimization algorithm. An intersection point is one of the two points where two sensors sensing boundaries intersect with each other. We would like to introduce some basic knowledge on intersection points in previous literatures before presenting an efficient solution to the minimum weight trap cover problem. Let denote the set of intersection points of all sensors sensing boundaries in RoI. First, a sensor set covers every point in region A if and only if it covers all points in set. This theorem is first presented and proved in Kasbekar et al. [2009]. Considering the sensing region of a sensor is a disc, the problem of region coverage can be easily transformed into the problem of finding a vertex cover [Yang et al. 2005]. Note that the sensing region of a sensor is open which does not include the boundary of sensing region, the intersection points marked by red dot in Figure 2 are still not covered. Recall that the diameter of a coverage hole is defined as the largest Euclidean distance between any two points in the coverage hole. Let a denote the set of intersection points of all active sensors sensing boundaries. Without loss of generality, we also consider the situation on the edge of A by adding intersection points among all active sensors sensing boundaries and the boundary of A into set a. The set Hm denotes the intersection points on the border of H m. (5)

8 2:8 J. Chen et al. Second, the diameter of coverage hole H m equals to d( Hm ) if the sensing regions are convex, that is, d(h m ) = d( Hm ) = max dist(a, b), (6) a,b Hm where d( Hm ) denotes the largest Euclidean distance between any two points in set Hm. This theorem and detailed proof are first presented in Balister et al. [2009]. We can therefore calculate the diameter of a coverage hole in a convenient way. An example is demonstrated in Figure 2. In Figure 2, the boundary of coverage hole H is composed of sensors sensing boundaries, and Hm ={P 1,...,P 8 } are the intersection points of these sensing borders. The diameter of coverage hole H equals to the maximum Euclidean distance of the intersection points in Hm,thatis,d(H) = max a,b Hm dist(a, b) Algorithm Overview The threshold D in the Minimum Weight Trap Cover Problem, which indicates the quality of coverage, is determined by the applications, so we need to guarantee the threshold D in the algorithm design. First, we present the following theorem. THEOREM 1. Minimum Weight Trap Cover Problem is NP-hard. PROOF. Assume the minimum distance between any two intersection points in RoI is ɛ. Assume the parameter D in Minimum Weight Trap Cover Problem which satisfies ɛ>d > 0. If there exists a coverage hole with diameter M in the RoI, then M ɛ>d since M is the distance between two intersection points and it should be no less than ɛ. Thus, there exists no coverage hole in the RoI if D-trap coverage is guaranteed. We need to cover all intersection points with active sensors to provide D-trap coverage in this case. In this special case, that is, ɛ>d > 0, Minimum Weight Trap Cover Problem is reduced to a set covering problem which has already been proved to be NP-hard [Karp 2010]. As a conclusion, Minimum Weight Trap Cover Problem (for any D > 0), as a more general problem, should be no less hard than the set covering problem. So we can claim that Minimum Weight Trap Cover Problem is NP-hard. Since the problem is NP-hard and can not be precisely solved in polynomial time unless P = NP, we develop an efficient approximation algorithm Trap Cover Optimization (TCO) to solve minimum weight trap cover problem. Assume C is the minimum weight sensor cover [Berman et al. 2005], which is a set of sensors that provide full coverage to A, andc is the trap cover which provides D-trap coverage to A. We will derive C from set C in TCO, that is, C C. TCO is composed of two major steps. First, a minimum weight sensor cover C is selected to cover the whole RoI A, which is viewed as Minimum Weight Sensor Cover Problem [Berman et al. 2005]. Given RoI A, a set of sensors s 1...s n, and monitored subregion, and the weight w i for each sensor, the problem is to find a set of sensors with minimum total weight. Let denote the set of intersection points of all sensors sensing boundaries in RoI A. We regard all intersection points in as targets to be covered. Existing literatures have developed algorithms to efficiently solve this problem [Cardei et al. 2005; Yang et al. 2005; Cardei and Du 2005; Kasbekar et al. 2009]. We will introduce DSC proposed in Kasbekar et al. [2009] and use it to find a minimum weight sensor cover in Step 1 in Section 6.1. The output of TCO, C, isempty initially. We let represent the C C to simplify our description. Second, we remove each sensor in C successively and consider whether to put it into C. Consider a sensor i in C. Given the sensors in are active, sensor i is added into C when the maximum coverage hole diameter will exceed the threshold D if i is inactive.

9 On Energy-Efficient Trap Coverage in WSNs 2:9 Fig. 3. T (i) is an upper bound of increment of a coverage hole when i is removed. There exists a coverage hole with diameter d 1. The dots denote the intersection points. The sensor i denoted by the dashed circle is about to be removed. Assume the diameter of coverage hole after removing i is d 2 and d q is an auxiliary line. Finally, C is empty and C, the output of TCO, equals. TCO activates the sensors in C. As we can see in the procedure of TCO, given the sensors in are active, sensor i is removed from only when the maximum coverage hole diameter will not exceed the threshold D if i is set to be inactive. Our aim is to remove as much sensors with poor residual energy as possible in and activate only a few sensors which are rich in residual energy. The challenge is to design an optimal removal strategy and remove sensors in a proper order so that we can achieve a better performance Removal Strategy Design We will discuss how to design the removal strategy in this section. Again, We let represent the C C to simplify our description. To remove more sensors from, we consider the possible impact of removing a candidate sensor on the diameters of existing coverage holes. The diameters of coverage holes will increase or remain unchanged when we remove a sensor from. Intuitively, if we always remove sensors which will cause the diameters of existing coverage holes increase quickly, the largest diameter will soon approach the threshold D, thus only a few sensors can be removed. More sensors can be removed if we choose to remove sensors with less impact on the diameters of existing coverage holes. To quantify and bound the impact, we introduce T (i) and D (i) as the upper bounds of increase on the diameter of coverage hole when removing a sensor node i. At first, let (i) represent all intersection points which are covered by set but not covered by set ( i). Assume points in set (i) belong to boundary points of M i coverage holes. Accordingly, we divide (i)into 1 (i), 2 (i),..., Mi (i). Assume the diameters of these coverage holes are d 1 (i),...,d Mi (i), respectively. T (i) denotes the aggregate diameters of all coverage holes which are covered by sensor i but not covered by set ( i), that is, the sum of diameters of all newly emerging coverage hole when sensor i is removed. We set M i T (i) = d j (i). (7) j=1 Actually, T (i) is the largest possible increment of a coverage hole when i is removed from. We illustrate that T (i) is an upper bound of increase on the diameter of coverage hole in Figure 3. As shown in Figure 3, there exists a coverage hole with diameter d 1. The dots denote the intersection points. The sensor i denoted by the dashed circle is about to be removed and T (i) is marked in the figure. Assume the diameter of coverage hole after removing i is d 2. We introduce an auxiliary line d q which

10 2:10 J. Chen et al. Fig. 4. Amount of removed sensors by TCO and random approach vs. N, D = 25. connects two intersection points in the initial coverage hole in Figure 3. d 1 is greater than d q due to the definition of diameter of coverage hole. d 2 < d q + T (i) holds because of the triangle inequality. Thus, we have d 2 d 1 < T (i), which suggests that T (i) is an upper bound of increment of a coverage hole when i is removed. Note that T (i) should not be the maximum diameter among all newly emerging holes because these holes may merge into one large coverage hole when removing other sensors. Actually, the impacts should be bounded by summing up the diameters. We set D (i) asineq.(8). D (i) = min{t (i), 2r}. (8) As shown in Figure 3, T (i) is an upper bound of increment of a coverage hole. Meanwhile, the increment of a coverage hole when removing a sensor should not be greater than the diameter of sensing region. Thus, we have Eq. (8) as a more strict upper bound for the increase on diameter of coverage hole, which is important to determine how to remove sensors with less impact on the diameter of coverage holes. D (i) is an important metric because it denotes the possible impact of removing a candidate sensor on the diameters of existing coverage holes. Since D (i) represents the increment of diameters of coverage holes when sensor i is removed from C, we can preferably remove sensors with low D (i) whose effects on the existing coverage holes are bounded. Consequently, potentially more sensor nodes can be removed before the diameter of any coverage hole is beyond D. We adopt D (i) in TCO as an important factor. We also conduct simulation experiments to justify our design. Sensors with same weights are deployed randomly and a full cover set C is picked with aforementioned methods. We compare the amount of sensors removed from between TCO and a random approach which randomly selects a sensor to remove. The average results are plotted in Figure 4, which shows that TCO improves the amount of removed sensors significantly by considering D (i). Besides the amount of sensors in C, we also need to minimize the weights of sensors in C, so we consider normalizing the weights of sensors by D (i) to determine which sensor is to be removed. D (i) is a variable between 0 and 2r. To avoid zero in the denominator, we set the normalized factor as 1/(1 + αd (i)), where α = 1/(2r). Furthermore, the normalized weight G(i) ofsensori is defined, that is, G(i) = w i /(1 + αd (i)). (9) We remove the sensor with the greatest normalized weight G(i) each time. If there exist sensors with same G(i), we remove the sensor with the lowest ID number. In this way, sensors with less residual energy or less upper bound of increment of a coverage hole, that is, D (i), are supposed to be removed from with a higher priority. Every sensor i in set C is checked iteratively and added to C if the maximum coverage hole

11 On Energy-Efficient Trap Coverage in WSNs 2:11 Fig. 5. An illustration of Algorithm 1. diameter of ( i) exceeds the threshold D, while the uncovered intersection points and coverage holes are updated accordingly. TCO terminates when C is empty. The remaining set C is the output of TCO. Sensors with no residual energy are not involved in minimum weight sensor cover C since they have infinite weights, unless there are no set covers with residual energy to provide full coverage. Even if sensors with no residual energy are involved in C,they are removed first in TCO because they have infinite normalized weights. If the output of TCO, C contains sensors with no residual energy, it indicates that there are no trap covers with redundant energy to provide D-trap coverage any more, which means that the network reaches the end of its lifetime. Remark. Though we assume that the sensing radius of sensors are equal as a common scenario for simplicity, TCO can also be applied if the sensing ranges of sensors are unequal. Assume r max as the maximum sensing radius among all sensors in this case. We only need to replace the radius r by r max in TCO to guarantee the upper bound. That is to say, Eq. (8) should be D (i) = min{t (i), 2r max },andα in Eq. (9) should be 1/(2r max ). The rest of TCO does not need to be modified for an unequal sensing radius case Algorithm Illustration We illustrate TCO in Figure 5 to help understand the algorithm. The detailed algorithm is shown in Algorithm 1 and d( ) is used to represent the maximum diameter of coverage holes when only sensors in set are activated. ALGORITHM 1: Trap Cover Optimization. (1) Find a minimum weight cover set C which ensures the whole region A is covered. Let C =, α = 1/(2r). Let represent C C. (2) For every sensor i in set C,calculateG(i) = w i /(1 + αd (i)). If C =, return trap cover C. (3) Find the sensor i with maximum G(i) and remove i from C. (4) Update existing uncovered intersection points in A and the boundaries of coverage holes with respect to set. (5) Calculate d( ). If d( ) > D, thenletc = C {i}. (6) Back to step 2.

12 2:12 J. Chen et al. Fig. 6. An example of Algorithm 1. A simple example of TCO is presented in Figure 6. The four sensors in set C are deployed symmetrically which full cover the square region with side length a and set C is empty initially. The sensors are assumed to have the same weights. The threshold of the coverage hole is supposed to be less than a. We will show how TCO works then. At first, we assume TCO picks Sensor 1 to be removed from set C.Sinced( ) where = C C is not beyond the threshold, Sensor 1 will not be added into set C. Next, we find that Sensor 3 has the lowest D (i) amongc and only contains Sensor 2, 3, and 4 now. Considering the sensors have the same weights, we will remove Sensor 3 from C. After that, d( ) is still not beyond the threshold, so Sensor 3 will not be added into set C either. We then remove Sensor 2 from set C. The diameter of which only contains Sensor 4 can not provide required trap coverage any more, so we add Sensor 2 into set C. In the same way, we remove Sensor 4 from C and add it into set C. Finally, TCO terminates when C is empty and Sensors 2 and 4 in C are activated to provide required trap coverage. The time complexity of TCO is apparently polynomial, since we only traverse the elements in C once. Later on, we will prove the approximation ratio of TCO is only related to the sensor deployment density in Section 5. Simulations in Section 7 have confirmed that G(i)-based TCO always picks trap cover with higher average residual energy and lower energy consumption. 5. PERFORMANCE ANALYSIS 5.1. Theoretical Analysis We investigate the performance of our proposed algorithm TCO theoretically in this section. Before the derivation, we make assumptions as follows. Assumption 1. Given RoI A of size l 1 l 2, l 1 r + D and l 2 r + D, where r is the sensing range of each sensor and D is the diameter threshold of D-trap coverage. We will first prove the ratio bound of aggregate weight between the output C and the initial input C of TCO. Let N C denote the number of sensors in C. w C and w C denote the aggregate weight of sensors in C and C, respectively. LEMMA 1. w C 2N C 2N C +D/(2r) w C. PROOF. Let C = C C denote the set of sensors which are removed from set C by Algorithm 1. Here we use D(i) to represent D (i) for simplicity. Suppose at the (k+1)th iteration d(c k+1 ) exceeds threshold D for the first time. Let Q denote the set C C k,

13 On Energy-Efficient Trap Coverage in WSNs 2:13 where C k denotes C at the kth iteration. Thus, C C k. Obviously, Q C, which means w Q w C. Since TCO always selects to remove sensor i from C with maximum G(i) = w i /(1 + αd(i)), we get that { w Q i Q (1 + αd(i)) max j C w C w j 1 + αd( j) j C (1 + αd( j)). According to the definition of set Q and D(i), the upper bound of the incremental of maximum coverage hole diameter, we have D(i) D 2r. (11) i Q With Eqs. (10), (11), and α = 1/(2r), With w C = w C + w C,wehave w C w C w Q w C i Q (1 + αd(i)) j C (1 + αd( j)) αd. (1 + 2αr)N C } (10) (12) w C w C (1 + 2αr)N C (1 + 2αr)N C + αd (13) 2N C w C 2N C + D/(2r) w C, (14) which concludes the proof. We denote the optimum minimum weight trap cover which provide D-trap coverage as OPT.LetN 1 denote the number of sensors in OPT. LEMMA 2. The number of sensors providing D-trap coverage to RoI A of size l 1 l 2 2S must be greater than 3, where S = l 3(r+D) 2 1 l 2. PROOF. Suppose that there exists a point P in a coverage hole. The distance between P and the nearest boundary of detection area is less than D. Assume the boundary is belonged to sensor i. Then the distance between P and sensor i is less than (r + D) according to the triangle inequality and P will be covered if the sensing distance of i increases to (r + D). For all sensors, if the sensing radius r is increased to (r + D), the sensor set will provide full coverage to A. It is well known that it is optimal to deploy sensor nodes of disk sensing model on the vertices of equilateral triangles to cover a plane [Bai et al. 2006]. If l 1 r + D and l 2 r + D, according to the property of equilateral triangles, the minimum number of 2S sensors with sensing range r + D which provide full coverage to the RoI A is 3. 3(r+D) 2 We have 2S N 1 3 3(r + D). (15) 2 This concludes the proof.

14 2:14 J. Chen et al. Fig. 7. Optimal deployment on the vertices of equilateral triangles. Assume that w OPT denotes the aggregate weight of sensors in set OPT. According to Eq. (3), the weight of energy-redundant sensor i, w(i) satisfies that θ/e >w(i) 1/E. We have w OPT N 1 /E. (16) w C w OPT < Nθ/E N 1 /E θ N 3 3(r + D) 2. (17) 2S = ρθ 3 3(r + D) 2 2 We have the following main result for TCO, which theoretically guarantees the performance of TCO even in the worst case. Based on Lemma 1 and Eq. (17), we have the following theorem. 2N THEOREM 2. w C /w OPT < C 2N C +D/(2r) ρθ, where = 3 3(r+D) 2. 2 As θ, r and D are constants, the approximation ratio of TCO is only related to the density ρ. As the number of sensors in a full cover set N C increases, the approximation ratio approaches ρθ, which is treated as O(ρ). The bound guarantees the approximation ratio of TCO compared with optimal solution even in the worst case. A better ratio is possible for special cases, but in the worst case, our approximation ratio holds. If more and more sensor nodes are placed, the optimal solution improves quickly since more options are available. Compared with optimal solution, our algorithm relatively improves slower. Thus, the worst bound may deteriorate as ρ increases. But the performance of our algorithm is still desirable, since the density of nodes will not be extremely high for real deployment. We also plot a figure to show the magnitude of the ratio intuitively. If we set the average amount of sensors to cover the region N C is 20, D = 5, r = 15, θ = 1.01, we have the approximation ratio curve in Figure 8. For the sensing radius r which is 15, ρ above 0.01 is relatively high in reality, since there are almost seven sensors within the sensing range of a sensor whose area is πr 2. Remark. As we remark in Section. 4.3, TCO can be applied even if the sensing ranges of sensors are unequal. Assume that r max is the maximum sensing radius among the sensing ranges of all sensors. In this case, the approximation ratio of TCO is 2N C 2N C +D/(2r max ) ρθ, where = 3 3(r max +D) 2. The induction is very similar to the proof of 2 Theorem 2, so we do not repeat here.

15 On Energy-Efficient Trap Coverage in WSNs 2:15 Fig. 8. Approximation ratio vs. ρ, N C = 20, D = 5, r = 15, θ = Network Lifetime Analysis Given a minimum weight trap cover with an approximation ratio O(ρ) acquired by TCO, we can induct the approximation ratio bound of network lifetime if TCO is performed every time slot. We refer to the framework of proof of network lifetime approximation ratio in Kasbekar et al. [2009] to induct our approximation ratio of network lifetime below. Assume L is the network lifetime by TCO and L is the theoretical maximum lifetime by optimal algorithm. If TCO activates sensors with residual energy, the network is said to be alive; else if TCO attempts to activate sensors with no residual energy, the network is dead. We define ς ={1,...,L} as the set of time slots when the lifetime of network is alive. Since L is not greater than L, we define ς ={L + 1,...,L } as the set of slots when the network is dead under TCO algorithm but might still stay alive if running optimal algorithm. THEOREM 3. L is at most an O(ρ) factor greater than L. PROOF. Assume trap cover set C(t) is selected by TCO at each slot t and C (t) is the trap cover which is set to be active by optimal maximum lifetime algorithm. w C (t) is the sum of the weights of sensors in set C(t) atslott and wc (t) is the sum of the weights of sensors in set C (t) during the period of running TCO each time slot, w C (t) = i C(t) w i(t) andwc (t) = i C (t) w i(t). As proved in Theorem 2, we have w C (t) ρθ w COPT (t) ρθ w C (t). If the network is not alive, t ς,wehavew C (t) θ N. Thus, w C (t) N ρ t ς. Define the function X{i C (t)} as follows. X{i C (t)} = { 1, if i C (t); 0, otherwise.

16 2:16 J. Chen et al. We can derive that N ρ (L L) wc (t) t ς = w i (L + 1) t ς i C (t) = N w i (L + 1)X{i C (t)}. (18) t ς = i=1 N w i (L + 1) X{i C (t)} t ς i=1 E N w i (L + 1) i=1 Because the network is dead when t ς, any sensor i, i C (t) does not consume energy, which means w i (L + 1) = w i ( j), t ς. Inequality t ς X{i C (t)} E holds, because active sensor i under theoretical optimal lifetime algorithm must not cost more energy than E. At any slot t ς, sensori / C(t) will not consume any energy, thus w i (t) = w i (t + 1). The aggregate weight C(t) i C(t) θl i (t) Nθ, t ς. Hence we get E E N (w i (t + 1) w i (t)) = E (w i (t + 1) w i (t)) i=1 = i C(t) i C(t) log 2 θ ( ) θ l i(t) 2 log 2 θ E 1 i C(t) Nθ log 2 θ. θ l i(t) E We can know that N E w i (L + 1) i=1 L N = E (w i (t + 1) w i (t)). (19) t=1 i=1 N θ log 2 θ LN + E w i (1) = N(Lθ log 2 θ + 1) i=1

17 On Energy-Efficient Trap Coverage in WSNs 2:17 Fig. 9. The performance of TCO, D = 40. By combining Inequalities (18) and (19), we get L L(ρθ log 2 θ + 1) + ρ, (20) where and θ can be seen as constants. The proof completes Simulation Performance TCO removes as many sensors with high weight as possible. We conduct simulations to validate the performance of TCO, that is, how much aggregate weight can be removed. The ratio of the aggregate weight of removed sensors to the aggregate weight of initial full cover sensor set is viewed as the indicator of the performance of TCO. We present the boxplot in Figure 9(a), which shows the statistics of running for 300 times to test the performance of TCO in average. As we can see, TCO performs well both in the average case and in the worst case. The removed aggregate weight ratio is even above 0.45 in the worst case, which guarantees the effect of employing TCO. We also record the status of maximum hole diameter and aggregate weight of sensors in during a period of TCO running in Figure 9(b). The results illustrate the running status of TCO. The maximum hole diameter increases very slowly when it approaches D. That is because we remove sensors not just dependent on G(i). We enumerate each candidate sensor i successively according to the magnitude of G(i) when the maximum hole diameter is about to exceed D. Candidate sensor i is removed only if it will not cause violation against restraint of D. Hence, many sensors are removed with no significant effect on maximum hole diameter at the end. 6. LOCALIZED PROTOCOL 6.1. Protocol We have proposed and analyzed our algorithm TCO in Sections 4 and 5. It is significant to implement our algorithm in a real WSN. Due to the nature of WSN, we design a localized protocol, Localized Trap Coverage Protocol, to implement TCO in WSN, which avoids global communication and thus improves energy efficiency of network. Each sensor has two modes: sleep and active. At the beginning of each time slot, every sensor wakes up and performs distributed algorithm DSC [Kasbekar et al. 2009] to determine whether to be active initially. DSC is a distributed protocol that guarantees full coverage. Each sensor in DSC calculates the so-called activation preference ratio based on local information and determines whether to sleep according to the activation preference ratio. The upper bound of approximation ratio of the distributed algorithm is O(log n), the same as the well-known centralized algorithm on set cover problem [Chvatal 1979]. The active sensors then perform Localized Trap Coverage

18 2:18 J. Chen et al. Protocol to find a minimum D-trap cover. We focus on the design of Localized Trap Coverage Protocol next. Each sensor communicates with other sensors within a distance of (r + D) ina multihop way, where r is the sensing range and D is the diameter threshold. The sensors within a distance of (r + D) from sensor i is defined as neighbor sensors of i, denoted by set C i. Suppose that set V i contains active sensors in C i. Every active sensor broadcasts an initial-message, including location information at the beginning to recognize its neighbor sensors. Since the location of each sensor is known by itself, either by GPS or other localization techniques [Mao et al. 2007; Patwari et al. 2005], it is not difficult for sensor i to find its active neighbor sensors. In TCO, the sensor with the greatest G(i) in Eq. (9) has the highest priority to be removed, or to say, switch to sleep. If there exist sensors with same G(i), we remove the sensor with the lowest ID number. Since the protocol is designed as localized implementation of TCO, we also introduce priority to schedule the sensors. Define pr i ={G(i), ID i } as the priority of sensor i, where ID i is a unique number of sensor i and G(i) is calculated by sensor i as Eq. (9). pr i > pr j if (i) G(i) > G( j) or (ii) G(i) = G( j) and ID i < ID j.sensori packs the priority pr i into the initial message, and shares the information with its neighbor sensors. Thus, every sensor knows the initial priority of its active neighbor sensors after initialization. Sensor i will send mode-message to broadcast its decision after it has chosen to sleep or stay active. There exist two kinds of mode message: Mode-Message sleep and Mode- Message active.ifi receives Mode-Message sleep ( j) from sensor j in V i, it removes j from V i. Suppose set M i contains sensors in V i with higher priority than sensor i which have not decided to sleep or stay active. If i receives any kind of Mode-Message from sensor j in M i, it removes sensor j from M i. Sensor i starts to make decisions when M i is empty. If there does not exist a coverage hole whose diameter is greater than D in the area around sensor i within the range of (r + D) which is covered by set V i, it means the set V i can guarantee D-trap coverage within the area. Then sensor i chooses to sleep to save energy. Note that sensor i is not in set V i.ifv i is not sufficient to cover the area with no hole greater than the threshold, sensor i has to stay active since D-trap coverage can not be guaranteed without sensor i. Sensor i broadcasts the mode message and stays active or turns into sleep accordingly after decision making. However, for sensor i, G(i) maybe varies when a sensor whose sensing region overlaps with that of i chooses to sleep (see the definition of G(i) ineq.(9)).wheng(i) varies, sensor i needs to broadcast the updated priority to its neighbor sensors in an update message, which is used to convey updated priority of sensors. There is a useful property of the priority of sensors. Since D (i) will never decrease and w i remains unchanged during a time slot, G(i) can only decrease if it varies. Thus the priority of sensor i will not increase during a time slot. Based on the property of priority, sensor i only needs to consider update messages from sensors in M i since updated priority is always lower than original value. If sensor i receives an update message from sensor j in set M i, it compares the updated priority with pr i to decide whether to remove j from M i. The set M i may also be updated due to the change of pr i to guarantee that M i only contains sensors with higher priority than pr i. To help understand the procedure of the protocol, we illustrate the process of a normal sensor i in Figure 10 for Localized Trap Coverage Protocol. At the first state, sensor i receives initial messages from its neighbor sensors. Sensor i also broadcasts its own initial message and other received messages whose senders are within a certain range. After a certain period of time, referred to as t w,sensori turns into the second state, which suggests that the initialization process ends. At the second state, sensor i receives mode messages and update messages from other sensors, and broadcasts its