Target Coverage Management in WSN Babacar DIOP, Dame DIONGUE, Ousmane THIARE Department of Computer Science Faculty of Applied Sciences and Technology University Gaston Berger, BP. 234 Saint-Louis, Senegal Email:
[email protected], {ddiongue.ep2112812, ousmane.thiare}@ugb.edu.sn
Abstract—As an important issue reflecting the QoS of the sensing task, coverage problem impacts widely on the performance of wireless sensor networks. The target coverage lifetime maximization problem is yet a challenging problem, which tries to settle a compromise between managing the coverage of a set of targets and maximizing the lifetime of the network. This problem becomes more accurate when targets detection is distance dependent. In this paper, we address the target coverage lifetime maximization problem by considering a probabilistic coverage model, which takes into account the distance parameter. We propose an algorithm based on a modified version of the classical well-known weighed set cover which organizes sensors in disjoint and non-disjoint set covers. Performance evaluation of our solution indicated good performance in managing coverage of targets while extending the network lifetime.
Keywords: Wireless sensor networks, Target coverage, Greedy set cover, Probabilistic coverage. I. I NTRODUCTION The development of micro-electromechanical low power sensors have enable wireless sensor networks (WSN) to be a powerful instrument for physical phenomena monitoring. This facilitates the emergency of a wide range of potential applications in the area of WSN, ranging from target surveillance in friendly environment to disaster prevention in hostile and inaccessible areas. Wireless sensor nodes are usually randomly deployed in an AoI1 , to sense and process physical measures from the observed area, and forward data to a local processing center (BS2 ) which is responsible for relaying information to remote users via Internet or satellite network. Coverage is one of the major issues in WSN since it reflects the QoS of the sensing task. It refers to how well the AoI is monitored by the deployed sensors. Roughly speaking, we can distinguish three types of coverage namely, area coverage, target coverage and barrier coverage. In target coverage problem, the main challenge can be how to efficiently monitor a finite set of targets points in an AoI, while maximizing the lifetime of the network. A well-known mechanism to optimize energy consumption is to schedule sensors activities, which enables them to switch between active and sleep mode. This helps extending network lifetime due to the activation of a subset of sensors, that fulfill the coverage requirements. Several solutions in target coverage problem are based on the assumptions of perfect coverage between sensors and targets, and fail to clarify how well each target point is 1 Area 2 Base
of Interest. station.
detected by neighboring sensors. Another weak point is the assumption that target detection model is based on boolean coverage model, i.e a target is well covered by any sensor if its location is within the sensing range of the sensor, no matter how long is the distance between this sensor and this target. Our contribution in this paper is described as follow : i) We tackle the target coverage problem by using set cover approach, and we then propose a target coverage scheduling scheme which attempt to maximize disjoints and non-disjoints set covers such that each set cover can monitor all the targets. The proposed algorithm is a refinement of our previous works in [1], where produced sets were disjoints. ii) We aim to bring practical considerations in target coverage associated with lifetime maximization problem when organizing sensors in set covers. The detection of each target follows a probabilistic model, and its measure is a real positive value ranging from 0 to 1. A probabilistic profit function is discussed, that helps selecting sensors according to their cost effectiveness. The rest of the paper is organized as follow. In section II, we make a concise literature review by relating some interesting solutions and their approach. In section III, we present the probabilistic coverage model and the maximum set covers problem. Then we relate our solution in section IV and describe the pseudo-code of the proposed algorithm. Performance evaluation with simulation results of the proposed algorithm are presented in section VI. Finally section VII concludes the paper and gives some limitations and perspectives of this work. II. L ITERATURE R EVIEW Referring to the literature, the target coverage scheduling process in random deployment can be performed by organizing sensors into set covers that can be either disjoints or nondisjoints, and activate them one after another. In the first case, each sensor can not be part of more than one set cover, and the set covers’ number theoretical bound is fixed to the minimum number of sensors that cover each target after the deployment [2]–[4]. In non-disjoint set covers, sensors can be included in more than one set cover as long as its remaining battery lifetime can afford to it. In [5], the authors formulate the target coverage problem as the maximum set covers (MSC) problem such that each set cover can monitor all the targets. The decision version of the MSC problem is proved NPcomplete and the authors proposed an LP-based solution for
optimal set covers and a greedy heuristic as an approximation solution. Some other works focus on managing target coverage with connectivity constraint. Several solutions addressing this problem are based on graph algorithms approach and attempt to maximize connected set covers number [6]–[8]. In [9], the authors address the target coverage problem by introducing the concept of coverage pattern associated with a column generation based approach. A coverage pattern is a subset of sensors that can cover all the targets. The concept of coverage pattern helps to formally define the problem and the column generation based approach is to efficiently solve this formulation. They further introduce in [10] the idea of a time-dependent formulation for the problem of maximizing target coverage lifetime, and attempt to justify the method of continuous time discretization by proving that the timedependent formulation can be converted into the pattern-based formulation. III. P ROBLEM S TATEMENT A. System Model Let us assume that n sensors are randomly and redundantly deployed in an AoI to monitor m targets which locations in the sensing field are known before. Network parameters can be defined by the following variables : • • • • • • •
S0 = {sk , 1 ≤ k ≤ n}: the initial set of deployed sensors. Z0 = {zj , 1 ≤ j ≤ m}:the set of targets to be monitored. E k : the initial battery lifetime of one sensor sk . Rk : the maximum sensing range of one sensor sk . dk→j : distance between sk and zj . S j : subset of sensors that can monitor zj . Z k : subset of targets that can be monitored by sk .
We assume that the signal propagation from a target to a sensor follows a probabilistic model. By considering this assumption we take into account certain kind of sensors where the signal strength decays with the distance from the source. This work is based on the path loss log normal shadowing model [11] and is extendable to incorporate different signal decay models. Formally, the detection value is modeled as a detection probability namely pkj between sk and zj , with (zj ∈ Z k ), that decays exponentially with dk→j [12]. This model is defined in equation 1. { pkj =
exp−αdk→j 0
if dk→j ≤ Rk Otherwise
(1)
Where α is a parameter representing the physical characteristics of the sensor. Despite the existence of a maximum sensing range, pkj is negligible when dk→j is greater than a certain threshold, which value is tunable according to the application requirements. Thus pkj becomes a real positive value that range from 0 to 1. Furthermore each target in the AoI can be covered by more than a single sensor. If one target is detected by k sensors, (k ≥ 1), with different detection probabilities, the cumulative detection probability at this target point, namely Pj can be defined as the product of individual
detection probabilities of all its neighboring sensors [12]. Pj is defined by equation 2. ∏ ( ) Pj = 1 − 1 − pkj (2) sk ∈S j
The boolean coverage model considers that pkj equals either to 0 or 1. In this case, if dk→i ≤ Rk , then pkj = 1, no matter how long is the distance between zj and sk . Otherwise if dk→j > Rk , then pkj = 0. B. Maximum Set Covers Problem Given the system model above, the objective of the MSC Problem is to partition S0 into a maximum number of set covers C1 , ..., Cp , each set cover with time weight t1 , ..., tp such that: ∪ 1) sk ∈Ci Z k = Z0 , ∀Ci , 1 ≤ i ≤ p. (Coverage ) 2) t∑ 1 + ... + tp is maximized. (Lifetime) k k 3) Ci ∋sk ei ≤ E , 1 ≤ i ≤ p. (Non-disjointness) k Where ei represents the energy expense of sk in the set cover Ci . The decision version of the MSC Problem is NP-complete [5]. The third constraint states that a sensor can not be active for more than its battery lifetime. The lifetime constraint depends on the number of sensors that each set cover contains. The minimum the number of sensors contained in one set cover, the greater the probability to extend the lifetime of the network. IV. P ROPOSED A LGORITHM A. Used Approach The algorithm that we propose is based on a modified version of the classical weighed greedy set cover. When building set cover, the classical greedy set cover selects iteratively at each stage the sensor covering the largest number of uncovered targets, remove these targets from the list of uncovered targets, and repeat the same process until the list of uncovered targets becomes empty. The minimum subset of sensors that can cover a target places an upper bound on the number of computable set covers for an optimal solution. This bound is called the theoretical maximum. In this case, this target represents the most sparsely covered target often called the critical target. Formally if zt is the critical target, then |S t | represents the theoretical maximum in the case of disjoint set covers. This is due to the fact that each target has to be covered by at least one sensor in a set cover, and a sensor can be part of at most one set cover. When produced sets are non-disjoints, a sensor can be used at most d times to be part of d set covers, the theoretical maximum equals to d|S t |. To compute the closest number of set covers to the theoretical maximum, an efficient sensor selection strategy must be adopted, which avoids to include more than one sensor covering the critical target in one set cover. This can be achieved by designing a profit function which represents the contribution of each sensor. This function is applied in each stage of sensor selection, and the sensor having the greatest contribution is selected. We model this contribution by a profit function that evaluates the cost
effectiveness of selecting one sensor from those available at the expense of another.
removed from the set Z0 , the process is iterated until Z0 becomes empty.
B. Cost Effectiveness of Sensors
C. Algorithm Description
Since the deployment is carried out randomly, all targets are not monitored by the same proportion of sensors. Each sensor has a contribution or weight of selection that depends on how many targets it can monitor and how high is the detection probability associated to the coverage relationship between that sensor and its neighboring3 targets. The cost effectiveness takes into account the remaining battery lifetime of each sensor, which represents the weight associated to it. We define for each sensor sk and target zj the coverage relationship function (CRF) represented by wjk in equation 3.
1) Setup: Once the deployment is done, the sensors send to the base station useful information (sensors IDs, coordinates) for processing input data required for the execution of the algorithm. We remark that the calculation of the CRF and the detection probabilities between sensors and targets must be performed at each stage during the execution of the algorithm. To reduce the frequency of these{ calculation, we define a new } variable denoted by DSZ00 = wjk , ∀sk ∈ S0 and zj ∈ Z0 , which can be computed once during the setup phase of the algorithm. Each time a new set cover is built, the CRF of all the sensors that belong to this set are { updated. In the same way, the } detection probabilities PZS00 = pkj , ∀sk ∈ S0 and zj ∈ Z0 between sensors and targets are calculated once during the setup phase and do not need to be recomputed during the execution of the algorithm. The parameter α representing the physical characteristics of the sensors is fixed by performing a manipulation of other parameters such as sensors sensing range, distance and the maximum-minimum probability of detection between sensors and targets. 2) Pseudo-code Description: The pseudo-code of the algorithm is shown in algorithm 1. Since the goal is to compute a maximum number of sensors set covers such that each one can monitor all the targets, the algorithm starts by checking if all targets can be covered by the set of deployed sensors (line 4). If this is the case, a new set cover is initialized and the algorithm passes the control to the second loop (line 8) which is responsible for constructing the set cover. Between line 11 and 21 sensors cost effectiveness are evaluated iteratively and the one with the greatest cost is selected. The selected sensor is added to the set cover in construction and all targets covered by this sensor are removed from the set of uncovered targets. When Zcur is empty, Ccur becomes a complete set cover, then the lifetimes and CRF of all the sensors that have been selected to form Ccur are updated (line 27-34). If a sensor exhausts its energy, it becomes a dead sensor and then it is removed from the set of available sensors Scur . When all targets can not be covered by available sensors then the algorithm returns the collection of computed set covers.
pkj
wjk = ∑
sk′ ∈Z0
pkj
′
(3)
Ek E k′
The CRF between sk and zj equals to the ratio between the detection probability of sk and zj and the sum of detection probabilities of all sensors that can monitor zj . If these latter can monitor zj with a great coverage value, this implies zj is densely covered, resulting thus in a lower CRF for sk compared to the case where zj is sparsely covered. Furthermore wjk value equals to 0 if zj can not be monitored by any sensor. The largest value corresponds to the case where zj can be monitored only by a single sensor. Based on equation 3, we define in equation 4 the overall contribution of sk considering Z0 which represents its cost k effectiveness denoted as wZ . 0 k wZ = 0
∑ zj ∈Z0
mE
pkj ∑ k
′ pk j k′
sk′ ∈S0 E
=
1 ∑ k wj m
(4)
zj ∈Z0
The cost effectiveness of sk with respect to Z0 equals to the normalized sum of CRF between sk and all targets that belong to Z0 . Since targets detection follows a probabilistic model, sensors selection during set covers computation must be based first on the amount of critical targets they can monitor, secondly on the proportion of uncovered targets they can monitor associated with the remaining battery lifetime of the sensor and third how well these targets are covered. The profit function described in equation 4 is to favor sensors that cover the largest number of uncovered targets with high detection probability, without loosing the requirement of managing critical targets. When a boolean model is considered, many sensors can have the same cost effectiveness, while they do not cover their neighboring targets in the same way. Probabilistic cost effectiveness can easily break ties by distinguishing the one with the highest detection probability. Equation 4 is used to define a new algorithm for sensors selection. When constructing one set cover, the sensor with a largest cost effectiveness with respect to Z0 , denoted by sselected , is selected, and targets covered by sselected are 3 Targets
that are spread within the sensing range of that sensor.
D. Complexity Analysis Let us consider that we have n sensors randomly deployed in an AoI to monitor m targets. When constructing one set cover, sensors are iteratively selected until all targets are covered. When the set cover is completed, all selected sensors battery lifetimes are updated. If the remaining battery lifetime of one sensor allows it to be part of another set cover, this sensor can be selected further. Knowing that every sensor spends uniformly r energy unit when its set cover is activated, a sensor can be selected at most d times by the algorithm, with d = E k /r, ∀sk ∈ S0 , where E k represents the initial battery lifetime of sk . The worst case complexity would have included the complete sensors’ set of length n, each of them being
V. P ERFORMANCE E VALUATION
Algorithm 1: Weight-based Maximum Set Covers input : S0 , Z0 , ES0 , rt, α, max output: Ctotal 1 2 3 4 5 6 7 8 9 10 11 12 13 14
15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37
theo, DSZ00 , PZS00
In this section, we will evaluate the performance of our algorithm in terms of target detection and number of covers. We simulate both disjoints and non-disjoints version of the weight-based greedy algorithm, and we then compare our solution with some other works in the literature. Since none of these works considers a probabilistic coverage model, it is more relevant to compare our solution with each of these works in term of computed set covers.
Scur = S0 ; Zcur = ϕ; c = 0; while all targets can be covered by Scur do Ccur = ϕ; Zcur = Z0 ; c = c + 1; while Zcur ̸= ϕ do max = 0; sselected = none; foreach sk ∈ Scur do k wZ = 0; cur N EIGH =∑ zk ∩ Zcur ; sumDP = zj ∈N EIGH wjk ; k = sumDP wZ ; m cur k if wZcur ≥ max then k ; max = wZ cur sselected = max; end end end if sselected ̸= none then Zcvd = Z selected ; Zcur = Zcur − Zcvd ; Ccur = Ccur ∪ sselected ; end Cc = Ccur ; foreach sk ∈ Ccur ∩ Scur do E k = E k − rt; if E k = 0 then Scur = Scur − {sselected }; else recompute wjk , ∀zj ∈ Z k ; end end Ctotal = Ctotal ∪ Cc ; end Ctotal = C1 , ..., Cp ;
A. Simulation Environment The simulations that we present in this section have been performed using the Matlab4 environment. We consider a 100 × 100 quire meters planar space representing the sensing field where targets are deployed first, followed by a random deployment of sensors that redundantly cover all the targets. B. Simulation Results
used d times with m targets. The runtime of the algorithm is equivalent to the following summation : T (n) = dnm + d(n − 1)(m − 1 mod m) + . . . +d(n(n − 1))(m − (n − 1) mod m)
T (n) =
n−1 ∑
(n − k)(m − k mod m)
(5)
k=0
We can deduce that the worst case complexity of the weightbased greedy algorithm is O(dn2 m).
1) Scenario 1: The numerical measures presented in Table 1 describe the simulation results when we vary the number of sensors from 30 to 100, and fixing the number of targets to 10 and the sensing range to 30m. We drive the simulation by executing 100 times the weight-based greedy algorithm considering different random deployment for every sensors’ number value, and we estimate the minimum and maximum detection values by taking the arithmetic mean. Since the distribution of the sensors in the area of interest follows a uniform distribution and that we essentially deal with the target coverage management problem, the randomness effect on the detection value of set covers is neglected in the interpretation of our simulation results. Roughly speaking, we can remark a little degradation of the minimum detection values when the number of targets varies from 30 to 100 and in parallel a little improvement of the maximum detection values, for both disjoints and non-disjoints versions of the weight-based greedy algorithm. This can be explained by the hypothesis that when the number of sensors becomes larger, redundancy grows reflecting in more components in the sensing field. This implies a greater likelihood for each target to be redundantly covered, then to be highly detected. By considering the AVG of detection values, we notice that there is no major change as we can remark it with the standard deviation. 2) Scenario 2: We compare the disjoint version of our algorithm with B{GoP} and the MCMC-H introduced respectively in [2], [4] in term of computed set covers and we display the results in Table 2. The same parameters in scenario 1 are considered and the simulation is driven 10 times, then the average value of computed set covers is calculated for every sensors number value. Considering the disjoints set covers, we can remark that for every instance of sensors number value, the disjoint MPTC shows better performance in term of computed set covers compared to other algorithms, and acceptable performance in case of non-disjoint set covers. The profit function is more accurate when produced set covers are 4 www.mathworks.fr/products/matlab/
TABLE I AVERAGE OF THE MINIMUM ,
MAXIMUM AND MEAN TARGETS DETECTION VALUES FOR NON - DISJOINT WMSC.
Nb.Sensors 30 40 50 60 70 80 90 100 Standard Deviation (σ)
Disjoint Set Covers Min. Max. AVG 0,055 0,241 0,128 0,046 0,282 0,133 0,036 0,266 0,121 0,039 0,247 0,116 0,036 0,260 0,120 0,036 0,302 0,125 0,031 0,289 0,119 0,037 0,314 0,125 7,7E-03 2,6E-02 5,5E-03
100
TIMES INSTANCES EXECUTION BY DISJOINT AND
Non-disjoint Set Covers Min. Max. AVG 0,083 0,279 0,151 0,073 0,290 0,151 0,074 0,305 0,161 0,070 0,312 0,153 0,068 0,332 0,158 0,060 0,327 0,151 0,060 0,355 0,159 0,058 0,382 0,155 8,7E-03 3,4E-02 3,8E-03
TABLE II AVERAGE OF THE NUMBER OF SET COVERS COMPUTED BY THE DISJOINT AND NON - DISJOINT WMSC COMPARED WITH S LIJEPCEVIC [2] , Z ORBAS ( DISJOINT AND NON - DISJOINT B{G O P} [4], [13]), C ARDEI [5] AND WGA [1].
Nb.Sensors 30 40 50 60 70 80 90 100
Slijepcevic 6,6 6,9 8,5 10,5 12,5 16,2 17,8 18
Disjoint Set Covers B{GoP} WGA Disjoint MPTC 8,8 8,6 9,2 9,2 9,3 11,8 11,4 12,3 14,6 12,9 13 13,6 17,1 18 21,8 23,4 24,4 25,2 26,8 28,4 28,1 27,6 28,2 29,4
disjoints. Furthermore, the mean of non-disjoints set covers number is always larger than the mean of disjoints set covers number. This implies that the first one is susceptible to extend more the lifetime of the network. VI. C ONCLUSION This paper is devoted to the problem of managing target coverage in wireless sensor networks. When sensors are randomly and redundantly deployed in an AoI for monitoring a set of targets with known locations, and that the target detection is distance-dependent, how can we partition the set of available sensors into a maximum number of group of sensors called set covers, such that each set cover completely monitors all the targets. We achieve this by proposing an algorithm based on a modified version of the classical weighted greedy set cover, which attempts to organize sensors into a maximum number of sets. These sets will be activated successively, in a round robin fashion, until all sensors depleted their energy. Our proposed algorithm provides sub-optimal solution, with lacking of an approximation ratio; the connectivity issue has not been addressed in this paper, therefore sensors are supposed to be directly connected to the base station so that they can report information. Our future work will include the matter of managing both coverage and connectivity in the same formulation. R EFERENCES [1] Babacar Diop, Dame Diongue, and Ousmane Thiare. A weightbased greedy algorithm for target coverage problem in wireless sensor networks. In Proceedings of the IEEE International Conference on Computer, Communication and Control Technology Informatics. IEEE, 2014.
Non-disjoint Set Covers Non-disjoint B{GoP} Cardei Non-disjoint MPTC 16,4 16,4 12,8 20,2 19,2 19,2 24,3 23 18,4 28,3 28 26,8 33,7 32,8 32,8 41,9 41,6 37,8 48,5 48,2 43,6 58,4 58,2 47
[2] Sasha Slijepcevic and Miodrag Potkonjak. Power efficient organization of wireless sensor networks. In Communications, 2001. ICC 2001. IEEE International Conference on, volume 2, pages 472–476. IEEE, 2001. [3] Mihaela Cardei and Ding-Zhu Du. Improving wireless sensor network lifetime through power aware organization. Wireless Networks, 11(3):333–340, 2005. [4] Dimitrios Zorbas, Dimitris Glynos, Panayiotis Kotzanikolaou, and Christos Douligeris. B {GOP}: An adaptive algorithm for coverage problems in wireless sensor networks. In 13th European Wireless Conference, EW, 2007. [5] Mihaela Cardei, My T Thai, Yingshu Li, and Weili Wu. Energy-efficient target coverage in wireless sensor networks. In INFOCOM 2005. 24th Annual Joint Conference of the IEEE Computer and Communications Societies. Proceedings IEEE, volume 3, pages 1976–1984. IEEE, 2005. [6] Qun Zhao and Mohan Gurusamy. Lifetime maximization for connected target coverage in wireless sensor networks. IEEE/ACM Transactions on Networking (TON), 16(6):1378–1391, 2008. [7] M. Cardei, I. Cardei. Energy efficient connected coverage in wireless sensor networks. Int. J. Sen. Netw., 3(3):201210, 2008. [8] Dimitrios Zorbas and Christos Douligeris. Satisfying coverage and connectivity in bandwidth constrained sensor networks. In Communications and Information Technology, 2009. ISCIT 2009. 9th International Symposium on, pages 390–395. IEEE, 2009. [9] Yu Gu, Yusheng Ji, Jie Li, and Baohua Zhao. Qos-aware target coverage in wireless sensor networks. Wireless Communications and Mobile Computing, 9(12):1645–1659, 2009. [10] Yu Gu, Bao-Hua Zhao, Yu-Sheng Ji, and Jie Li. Theoretical treatment of target coverage in wireless sensor networks. Journal of Computer Science and Technology, 26(1):117–129, 2011. [11] T.S. Rappaport. Wireless communications : Principles and practice. In New Jersey: Prentice Hall, pages 1584–1592. IEEE, 1996. [12] Nadeem Ahmed, Salil S Kanhere, and Sanjay Jha. Probabilistic coverage in wireless sensor networks. In Local Computer Networks, 2005. 30th Anniversary. The IEEE Conference on, pages 8–pp. IEEE, 2005. [13] Dimitrios Zorbas, Dimitris Glynos, Panayiotis Kotzanikolaou, and Christos Douligeris. Solving coverage problems in wireless sensor networks using cover sets. Ad Hoc Networks, 8(4):400–415, 2010.
