Designing k-coverage schedules in wireless sensor networks

Transcription

1 J Comb Optim (2008) 15: DOI /s Designing k-coverage schedules in wireless sensor networks Yingshu Li Shan Gao Published online: 21 March 2007 Springer Science+Business Media, LLC 2007 Abstract Some sensor network applications require k-coverage to ensure the quality of surveillance. Meanwhile, energy is another primary concern for sensor networks. In this paper, we investigate the Sensor Scheduling for k-coverage (SSC) problem which requires to efficiently schedule the sensors, such that the monitored area can be k-covered throughout the whole network lifetime with the purpose of maximizing network lifetime. The SSC problem is NP-hard and we propose two heuristic algorithms under different scenarios. In addition, we develop a guideline for users to better design a sensor deployment plan to save energy by employing a density control scheme. Simulation results are presented to evaluate our proposed algorithms. Keywords Wireless sensor networks k-coverage Sensing range Network management 1 Introduction As communication technology, embedded computing technology and sensing technology are becoming more and more mature, sensors that have the capabilities of sensing, computation and communication emerge all over the world. Sensor networks which usually consist of a large number of sensors are attracting people s attentions. Sensor networks integrate sensing technology, embedded computing technology and distributed information processing technology. They can sense and collect information from all kinds of objects in the monitored area in a cooperated and timely manner. Furthermore, they can process the gathered information and send it Y. Li ( ) G. Gao Department of Computer Science, Georgia State University, 34 Peachtree Street, Atlanta, GA 30303, USA yli@cs.gsu.edu G. Gao sgao@cs.gsu.edu

2 128 J Comb Optim (2008) 15: back to users. A large number of sensors deployed densely in the monitored area construct a wireless sensor network. The deployment of sensors does not need to be engineered or pre-determined. Sensors could be deployed randomly by many types of carriers in inaccessible areas. Because of these features, sensor networks can help people gather a large amount of reliable information at anytime, any place and under any conditions. Therefore, they are being widely employed for military fields, national security, environmental monitoring, traffic control, health, industry, disaster prevention and recovery (Akyildiz et al. 2002). Some applications have strict coverage level requirements where the coverage level indicates how well the area is being monitored, i.e., what is the percentage of the area that is being monitored and how many sensors are monitoring a region. The coverage requirement is mainly for the purpose of reliability and fault tolerance in case of node failures. The coverage levels for different applications may vary. For example, for the application of home security, where the environment is friendly, the coverage level can be set to a low value. On the other hand, having a high coverage level is always a major concern if sensors work in a hostile environment such as battlefields or chemically polluted areas. Even for the same application, the coverage level requirements may vary. For example, for forest fire detections, the coverage level may be low in rainy seasons, but high in dry seasons. Therefore, having a user-defined parameter k to represent the coverage level is sometimes a mandatory requirement in designing a network. In this paper, we investigate the k-coverage problem for sensor networks. Energy is a restricted resource for sensors, which determines how long and how well sensor networks can work. Most energy-efficient approaches use the same way to save energy, reducing the number of sensors working simultaneously. By scheduling redundant sensors to go to sleep while the essential coverage has been satisfied, the network lifetime can be significantly prolonged. In this paper, we propose the PCL- Greedy-Selection (GS) algorithm which groups sensors into disjoint subsets (one sensor can only present in one subset). In each subset, there is no redundant sensor and every subset can k-cover the whole area. The subsets are scheduled to work successively. Therefore, the network lifetime is prolonged. Furthermore, we notice that it is not necessary to keep each sensor s sensing range the same. Therefore, reducing a sensor s sensing range without affecting the coverage level is another way to conserve energy. Thus, we propose another algorithm, PCL-Greedy-Selection-Adjustable (GSA). This algorithm constructs some non-disjoint subsets (one sensor may present in multiple subsets; meanwhile the total consumed energy equals its initial energy). And in each active subset, each sensor only works with its smallest possible sensing range. Our algorithms demonstrate good performances in simulations. From the simulation results, we provide users with a guideline about which algorithm is better for different applications. For the situations where the long-term running of sensor networks is not necessary, the sensing range of each sensor is fixed, or the quality and the stability of surveillance are the most considered factors, the GS algorithm is a good choice and the network lifetime can be 90% of the ideal network lifetime. For long-term running tasks where the sensing range of each sensor is adjustable and the network lifetime should be prolonged as much as possible, the GSA algorithm is a good choice and it has a 16.63% performance improvement on average over the GS algorithm.

3 J Comb Optim (2008) 15: In this paper, we also consider the boundary effect and develop a density control scheme for sensor deployment. We consider the Euclidean metric instead of the toroidal metric. The assumption of the toroidal metric technically eliminates the boundary effect under the Euclidean metric. Intuitively, there are fewer sensors close to the boundary than at the central region of the monitored area. When deploying sensors, users can apply the density control scheme to deploy sensors in a more uniform manner so that each region or target can be covered by almost the same number of sensors. This approach reduces the number of sensors which will never be used because there is no more living sensor working with them to provide k-coverage. The simulation results show that this density control scheme improves the performances of the GS and GSA algorithms. Our contributions are as following: (i) we define the problem of Sensor Scheduling for k-coverage (SSC) problem which is NP-hard; (ii) we design two heuristic algorithms for the SSC problem which divide the sensors into disjoint/non-disjoint subsets, such that a schedule can be worked out by activating these subsets successively to extend network lifetime, also the fixed/adjustable sensing range issue is considered; (iii) the energy of each sensor is considered by our algorithms so that the energy consumptions are efficient and balanced; and (iv) we propose a density control scheme for sensor deployment to reduce the number of unallocated sensors such that the network efficiency is improved. The rest of this paper is organized as follows. Section 2 surveys the related work. In Sect. 3, we define the problem of Sensor Scheduling for k-coverage (SSC). Section 4 presents a disjoint solution (GS) which divides sensors with fixed sensing ranges into disjoint subsets to the SSC problem. In Sect. 5, we present a non-disjoint solution (GSA) which divides sensors with adjustable sensing ranges into non-disjoint subsets. Section 6 proposes an approach to deploy sensors more rationally. The simulation results are illustrated in Sect. 7 and Sect. 8 concludes our paper. 2 Related work The coverage problem, a fundamental problem in sensor networks about how well an area is monitored by sensors, has attracted a lot of attention. Basically, there are three types of coverage problems (Cardei and Wu 2004) which are the target coverage problem, the area coverage problem and the breach coverage problem. The works in (Cardei et al. 2005a, 2005b; Cardei and Du 2005) addressed the target coverage problem where the purpose is to cover all the targets. In (Cardei et al. 2005a), Cardei et al. proposed a Linear-Programming-based algorithm which extends the network lifetime by organizing sensors into a maximal number of set covers. The sets are activated successively and only the active sensors are responsible for monitoring all the targets. The works in (Huand and Tseng 2003; Zhou et al. 2004; Wang et al. 2003; Guptaetal.2003; Jiand and Dou 2004; Abrams et al. 2004) addressed the area coverage problem. As the most studied problem, its main objective is to cover an area. Approaches on this problem are to divide sensors into disjoint or non-disjoint sets, such that each set can monitor every point in the surveillance area and all sets are activated successively. The breach coverage problem addressed in (Cheng et al. 2005) is about minimizing the number of

4 130 J Comb Optim (2008) 15: uncovered targets. The works in (Huang 2003; Lietal.2003; Mehtaetal.2003; Meguerdichian et al. 2001) tried to determine a maximal breach path (MBP) and the maximal support path (MSP). For the MBP (MSP) problem, the distance from each point on the path to the closest sensor is maximized (minimized). The authors in (Meguerdichian et al. 2001) proposed a centralized solution, based on the observation that MBP lies on the Voronoi diagram lines and MSP lies on Delaunay triangulation lines. None of the above works considers the k-coverage requirement for the purpose of quality of surveillance. Wang et al. (2003) first studied this problem. The coverage levels of all the intersection points are determined through verifying the coverage degrees of the area. They proposed a localized heuristic for constructing a cover set that can provide k-coverage. However, the size of the obtained subset cannot be guaranteed to be as small as possible. In (Zhou et al. 2004), the authors designed a greedy heuristic for the k-coverage problem and the size of their constructed cover set is claimed to be within O(log n) factor of the optimal. The main idea is to select a candidate path which has the maximum k-benefit value. Both of these two works only consider constructing one cover set instead of dividing sensors into subsets such that each of them can provide k-coverage. In (Kumar et al. 2004), the authors studied the sensor deployment problem so that k-coverage could be guaranteed. However, it only concerns the k-coverage, and the energy conservation was not considered. In (Yan et al. 2003), Yan et al. proposed an adaptable energy-efficient coverage protocol which can provide a differentiated surveillance service. Nodes are designed to decide their own working schedule dynamically to provide k-coverage. This protocol is energy efficient but it cannot guarantee k-coverage when k>2. In (Jie and Yang 2004), three energy-efficient models are proposed. The overall energy consumption is reduced by minimizing the overlapped sensing area of sensors, and at the same time the high ratio of coverage is reached. However, there are still some limitations. This algorithm cannot guarantee specific coverage level. Meanwhile, the sensors deployed in the area should be deployed densely enough to meet the basic requirements of the three energy-efficient models. Cardei et al. (2005b) introduced an energy-efficient approach which employs adjustable sensing range at each sensor. Both centralized and distributed heuristics are proposed to solve the Adjustable Range Set Covers (AR-SC) problem, using linear programming and greedy techniques. In (Cardei et al. 2005a), Cardei et al. have shown that non-disjoint sensor sets can provide longer network lifetime comparing with disjoint sensor sets. Our GSA algorithm validates this feature. In (Huand and Tseng 2003), the coverage problem is formulated as a decision problem, whose goal is to determine whether each point in the monitored region of a sensor network is covered by at least k sensors. The main idea is to check the Perimeter Coverage Level (PCL) of each sensor. They prove that the whole monitored region is k-covered if and only if each sensor in the monitored region is k-perimetercovered. Based on this work, we design two heuristic algorithms in this paper to divide sensors into subsets and each of these subsets can provide k-coverage, such that the network lifetime can be prolonged. The differences between the above algorithms and ours are: (1) our algorithms provide solutions to k-cover the surveillance area; (2) in our algorithms, k-coverage is 100% guaranteed; (3) there is no limitation on sensor s sensing range which could

5 J Comb Optim (2008) 15: vary in a range instead of several fixed values; (4) our algorithms have no limitation on the number of sensors and the sensor positions. 3 Sensor scheduling for k-coverage We consider a sensor network which monitors a two dimensional area and no two sensors are located at the same location. Every point in the area needs to be continuously covered (monitored) by at least k sensors. The network lifetime is defined as the total duration during which the whole area is k-covered. We assume the number of the deployed sensors is more than the required number of sensors that can provide k-coverage for the monitored area. To extend the network lifetime, instead of making all the sensors to be active throughout the whole network lifetime, a subset of the sensors can be turned on to provide k-coverage at any time, while the rest sensors are set to sleep mode. We also assume the transmission range of a sensor is at least twice the sensing range of a sensor so that connectivity is also guaranteed within each subset (Wang et al. 2003). Then the problem of sensor scheduling for k-coverage can be defined as following. Definition 1 Sensor scheduling for k-coverage (SSC): Given a sensor network with n sensors that can provide k-coverage for the monitored area, schedule the activities of the sensors such that at any time the whole area can be k-covered and the network lifetime is maximized. We believe the SSC problem is NP-hard since the MSC problem (Cardei et al. 2005a), which is NP-complete, can be reduced to SSC problem in polynomial time. Given n sensors, s 1,s 2,...,s n, denote Set i (i n) as the set of subarea targets (Note that area targets can be reduced to point targets.) that sensor s i can cover. Associated with each Set i is a weight T i that is the activate time for s i. The SSC problem is to seek a set of set covers where each of them can cover every subarea target at least k times and the total lifetime of these covers is maximized. Note that the total activated time of each sensor is no greater than its lifetime. For MSC problem, C is {Set 1, Set 2,...,Set n }, and R is {s 1,s 2,...,s n }. Denote S j (1 j m) as a set cover such that s 1,s 2,...,s n appear at least once in each S j. S j (1 j m) corresponds to S 1,...,S p in MSC. There is a weight t j associated with each S j and we want to maximize 1 j m t j. It can be seen that the MSC problem is polynomial time reducible to a sensor scheduling for 1-coverage problem which is a special case of SSC (k = 1). The scheduling decisions can be made at the Base Station (BS). The BS broadcasts the schedule to all the sensors so that each sensor can know when it should be active to monitor the area. To solve the SSC problem, we can divide the sensors into disjoint or non-disjoint subsets and each subset k-covers the whole area, where k-cover indicates for every point in the area, it is covered (monitored) by at least k sensors. These subsets can be scheduled to be active successively. For each subset, its lifetime is determined by the sensor which has the least power. The following notations are used to formulate the SSC problem and to describe our algorithm.

6 132 J Comb Optim (2008) 15: K: If all the sensors are active, all the points in the monitored area can be covered by at least K sensors. k: k (k K) is a user-specified parameter which specifies the required coverage level the sensor network must provide at any time. It is a measurement of the quality of surveillance. S: The set of all the sensors. n: The number of sensors. m: All the sensors can be divided into at most m subsets and each subset can k- cover the whole area. C i :Theith subset, 1 i m. R s : The maximum sensing range of a sensor. cov i : The coverage level of the C i, which means any point in the monitored area is covered by at least cov i sensors belonging to C i. E: The initial energy of a sensor. e j,i : The energy of sensor j used in C i. r j,i : The sensing range of sensor j in C i. If sensor j is not present in C i, r j,i is 0. l i : The lifetime of C i. l j,i : The remaining lifetime of sensor j whose sensing range is r j,i. Considering the case where each sensor has a fixed sensing range and all the sensors are divided into disjoint cover sets, our goal is to construct as many subsets as possible such that (i) each subset can k-cover the whole monitored area; (ii) the network lifetime is maximized. Then the SSC problem is formulated as Objective: Max m. Subject to: C i S, 1 i m C i C j =, 1 i, j m, i j, cov i k, 1 i m. Considering the case where each sensor can adjust its sensing range and all the sensors are divided into non-disjoint cover sets, our goal is to use the energy of each sensor as completely as possible such that (i) each subset can k-cover the whole monitored area; (ii) the network lifetime is maximized. Then the SSC problem is formulated as Objective: Subject to: Max 1 i m m l i. i=1 C i S, cov i k, 1 i m, e j,i [0,E], 1 j n, 1 i m, r j,i [0,R s ], 1 j n, 1 i m, m e j,i E, 1 j n, i=1 l i = min l j,i, 1 j n, 1 i m, r j,i > 0.

7 J Comb Optim (2008) 15: Disjoint cover sets with fixed sensing range In this section, we present a greedy heuristic which generates disjoint cover sets for the SSC problem. We assume each sensor has a fixed sensing range. One sensor can only participate in at most one set and all sensors sensing ranges are the same. In (Huand and Tseng 2003), the authors proposed and proved the following Theorem 1. Theorem 1 Suppose that no two sensors are located in the same location. The whole network area is k-covered if and only if each sensor in the network is k-perimetercovered. We say for any sensor s i it is k-perimeter-covered if all points on the perimeter of s i are covered by at least k sensors, which are in the same set with s i, other than s i itself. Similarly, a segment of s i s perimeter is k-perimeter-covered if all the points on the segment are covered by at least k sensors, which are in the same set with s i, other than s i itself. We define the Perimeter Coverage Level (PCL) of a sensor s i as the number of the senors in the same set with s i that cover any point on s i s perimeter of the sensing area. The lower the PCL, the smaller the node density (the number of nodes per unit area). According to Theorem 1, if all the sensors PCL values are greater than k, the area is k-covered. We design the PCL-Greedy-Selection algorithm as following. The main idea is to iteratively construct subsets C i by choosing sensors from the region with the lowest sensor density. When constructing an individual C i, at each step, the sensor with the smallest PCL value is added to C i. In this way, we can include as few sensors as possible in C i and these sensors are distributed in the area as widely as possible because they are from the regions with the lowest sensor density, such that more sensors can be left to join other subsequent subsets and the overlapped sensing regions in each subset are reduced as much as possible. This also indicates that when constructing a subset C i, the region with smaller node density is taken care of with higher priority. The PCL-Greedy-Selection algorithm is shown in Algorithm 1. The input of this algorithm includes k, the user-specified coverage level, and S, theset of all the sensors. The output is C, a set of subsets, and each subset can k-cover the whole area. To justify whether a subset C i can k-cover the entire surveillance area, we can use the method proposed in (Huand and Tseng 2003) and we call it getcoveragelevel(c i ). First, all the sensors in S are sorted in non-decreasing order based on their PCL values. Then sensors are added into a subset in a greedy manner. If at some iteration, the current subset C i can provide k-coverage, a new subset C i+1 will begin to be constructed in the same manner. PCL-Greedy-Selection stops when we can no longer construct a subset which can k-cover the whole surveillance area. Since each subset is constructed in a greedy manner, it is possible that there exist some redundant sensors in a subset. Therefore after finishing constructing a subset, we need to remove those redundant sensors and add them back to S so that they are still available to be added to the subsequent subsets. The algorithm for this operation is PruneGreedySelection which is described in Algorithm 2. In this algorithm, given a subset C i, we check for each sensor in C i to see whether removing it will make cov i smaller than the user-specified k. If a sensor is redundant (after removing this sensor from C i, cov i is still not samller than k), it will be added back to S. Based on Theorem 1, the correctness of our algorithm is guaranteed.

8 134 J Comb Optim (2008) 15: Algorithm 1 PCL-Greedy-Selection(k, S) 1: Sort S in non-decreasing order based on their PCL values 2: i 0 3: while S is not empty do 4: cov i getcoveragelevel(c i ) 5: if cov i <kthen 6: node the first sensor in S 7: Add node to C i 8: Remove node from S 9: else 10: PruneGreedySelection(k,S,C i ) 11: Add C i to C 12: i++ 13: end if 14: end while 15: Output C Algorithm 2 PruneGreedySelection(k,S,C i ) 1: for j 1 to C i do 2: s j the jth sensor in C i 3: Remove s j from C i 4: cov i getcoveragelevel(c i ) 5: if cov i k then 6: Add s j to S 7: else 8: Add s j back to C i 9: end if 10: end for Theorem 2 The time complexity of the algorithm PCL-Greedy-Selection is O(n 2 d log(d)), where n is the number of the sensors and d is the maximum node degree. Proof The time for sorting S is O(nlog n). There are n iterations in the while loop. At each iteration, the main part that dominates the time complexity is getareacoveragelevel or PruneGreedySelection. The function, getareacoveragelevel, isproposed in (Huand and Tseng 2003). Its time complexity is O( C i d log(d)), where C i is the size of a subset C i. The time complexity for PruneGreedySelection is O( C i 2 d log(d)). Therefore, the time complexity of the algorithm PCL-Greedy- Selection is O(n 2 d log(d)). The number of the subsets constructed by PCL-Greedy-Selection decides the network lifetime. The following theorem gives the bound for the number of the subsets in the ideal circumstance.

9 J Comb Optim (2008) 15: Theorem 3 Given some sensors k-covering an area, if each sensor s sensing range is fixed and the sensors are divided into disjoint subsets, the maximum number of subsets m is K k, where K (K k) is the minimum coverage level that the sensor network can provide if all the sensors are activated. Proof If the minimum coverage level provided by a sensor network is K, there exists some point a in the monitored region such that there are K sensors that can cover a. After the first subset is constructed, there are K k candidate sensors that can cover a. By repeatedly constructing subsets, ideally at most K k subsets can be constructed so that each of them can k cover a, to guarantee k-coverage for the whole monitored region. Thus, K k is the upper bound of m. The lower bound of m is K k as well. To prove this, without loss of generality, we assume m = K k α, where α is an integer and greater than 0. Then, after allocating sensors into K k α subsets, the remaining sensors should be able to (K K k k + αk)-cover the monitored area in ideal circumstances. Because (K K k k) 0, the remaining sensors could construct α more subset(s). This leads to a contradiction. Thus, the lower bound of m is K k. Based on the upper bound and the lower bound of m, we conclude that m = K k. 5 Non-disjoint cover sets with adjustable sensing range The algorithm PCL-Greedy-Selection deals with the case where sensors cannot adjust their sensing ranges. This may lead to unnecessary energy consumption in the circumstance where sensors have the ability to adjust their sensing ranges. As shown in Fig. 1, the dashed lines are the original sensing ranges of n2 and n3 and they can be reduced such that n2 and n3 can save more energy while still provide the required coverage to the monitored area. In this section, we propose the algorithm PCL-Greedy-Selection-Adjustable (GSA) to divide all the sensors into non-disjoint cover sets. Meanwhile, each sensor uses the least possible sensing range to provide coverage. The GSA algorithm is shown in Algorithm 3. First, we try to construct a subset to k-cover the region. At line 15, PruneGreedySelection removes all unnecessary nodes from the current constructed subset. Then, shrinkranges shrinks each node s sensing range as much as possible while considering the k-coverage requirement. The lifetime of a subset depends on the shortest lifetime of the node in this Fig. 1 Nodes with adjustable sensing range

10 136 J Comb Optim (2008) 15: Algorithm 3 PCL-Greedy-Selection-Adjustable(k, S) 1: if S is empty then 2: return null 3: end if 4: Set the sensing ranges of all the nodes in S to MaxSensingRange 5: i 0 6: while S is not empty do 7: updatepcl(s) 8: Sort nodes in S in non-decreasing order based on their PCL values 9: cov i getcoveragelevel(c i ) 10: if cov i <kthen 11: node the first sensor in S 12: Add node to C i 13: Remove node from S 14: else 15: PruneGreedySelection(k,S,C i ) 16: shrinkranges(k,c i ) 17: lifetime min(lifetimes of each sensor in C i ) 18: for each node in C i do 19: node.reducelifetime(lifetime) 20: if node still has energy then 21: node.sensingrange MaxSensingRange; 22: S.add(node) 23: end if 24: end for 25: TotalLifetime +=lifetime 26: Add C i to C 27: i++ 28: end if 29: end while 30: unallocatednodes.addall(c i ); 31: return C subset. The energy consumption of a sensor s i per unit time is defined as e i = μr α i (Cardei et al. 2005b), where μ is an application-related power consumption parameter and r i is the actual sensing range of s i. We adopt the quadratic energy consumption model where α = 2. After working for lifetime time, the remaining energy of a node is node.energy = node.energy e i lifetime. (1) The sensors still having energy are added back into S. At last, if the left sensors working with the maximum sensing range cannot provide k-coverage, they are added to the unallocated group. Algorithm 4 is used to shrink sensors sensing ranges such that the overlapped regions of the sensors sensing area are reduced as much as possible and the sen-

11 J Comb Optim (2008) 15: Algorithm 4 shrinkranges(k,c) 1: Sort nodes in C in non-decreasing order based on their PCL values 2: for i 0toC.size() do 3: node C.get(i) 4: while getareacoveragelevel(c) k do 5: node.senserange =RangeInterval; 6: end while 7: node.senserange +=RangeInterval; 8: i ++ 9: end for sors energy can be used the most efficiently. In this algorithm, RangeInterval is an application-dependant value which determines the accuracy of the sensing range adjustment. A larger RangeInterval value results in shorter execution time. A smaller RangeInterval value results in longer execution time, but the solution is better because the larger RangeInterval value results in bigger overlapped sensing regions. The complexity of shrinkranges is O( C i 2 d log(d)) determined by getarea- CoverageLevel whose complexity is O( C i d log(d)). The size of C i is n in the worst case. Therefore, the complexity of PCL-Greedy-Selection-Adjustable is O(n 3 d log(d)) determined by shrinkranges. 6 Density control of the sensor deployment From Theorem 3, we can see there is a linear relationship between K and the number of the constructed subsets. This is also validated through simulations in Sect. 7. As the network lifetime is decided by the number of the constructed subsets for disjoint cover sets and by how well sensors are used efficiently for non-disjoint cover sets, to have a longer network lifetime, K should be larger which indicates the total number of the sensors should be larger. Another factor that may affect the network lifetime is the sensor density which is defined as the number of sensors in each unit area. Different regions in a monitored area have different sensor densities. From the simulation results, we found that there always exist some sensors that cannot be allocated to any subset. This is due to the difference between the sensor density of the regions near the border of the monitored area and the sensor density of the regions at the center of the monitored area. The unallocated sensors are usually the ones close to the center of the monitored area. They cannot find partners to monitor the whole area cooperatively. The sensors near the borders have smaller PCL values and the sensors at the center have larger PCL values. Both of our algorithms choose sensors beginning from the sensors with smaller PCL values to add into a subset. Thus, it is possible (actually always) after all the sensors near the border have been allocated, there still exist some sensors close to the center region and no more subsets can be constructed to provide k-coverage for the whole monitored area. Therefore, to extend the network lifetime, the PCL values of the sensors need to be balanced, so that the closer to the border a region is, the more sensors this region should have. To guarantee balancing the PCL values, the number of the neighbors of the sensors close to the borders should be close

12 138 J Comb Optim (2008) 15: Fig. 2 Density computation to the number of the neighbors of the sensors close to the center region. We derive a relationship between the sensor density of the regions near the border and the sensor density of the central region in Theorem 4. We define a disk centered at c as D c.the sensor density of D c is denoted as ρ c, and ρ c = the number of sensors ind c D c, where D c is the area of D c. Theorem 4 Assume the sensor density at the center of the monitored area A is ρ c. To guarantee the number of the neighbors of the sensors close to the borders is equal to the number of the neighbors of the sensors close to the center, for a point p whose distance to the border of A is r, the sensor density at p should be ρ p = where R s is the sensing range of a sensor. 4πR 2 s 4(π arccos r 2R s )R 2 s + r 4R 2 s r2 ρ c Proof Assume the sensor density in a disk is uniform. As shown in Fig. 2, D c is the disk centered at c (center of the monitored region) with radius of 2R s and D p

13 J Comb Optim (2008) 15: is the disk centered at p with radius of 2R s minus area A. Since we assume the transmission range of a sensor is at least twice of the sensing range of a sensor to guarantee connectivity, the neighbors of the sensor located at c must be within D c. We desire that the number of the neighbors of the sensor located at c is the same as that of the sensor located at p. This indicates that the sensor density in D p and the sensor density in D c satisfy the following condition: ρ p = D c D p ρ c. We know D c =π(2r s ) 2 and D p =A 1 + 2A 2 ( ) 2 2π 2α 1 = π(2r s ) + 2 2π 2 r (2R s ) 2 r 2 = 4(π α)r 2 s + r 4R 2 s r2 where A 1 is the area filled with dashed lines, A 2 is the area filled with dotted lines and α = arccos 2R r s. Hence, 4πRs 2 ρ p = 4(π arccos 2R r s )Rs 2 + r 4Rs 2 ρ c. r2 Based on Theorem 4, users can develop a plan for deploying sensors such that any point in the monitored area could be covered by almost the same number of sensors. This scheme can reduce the number of the unallocated (or insufficiently used) sensors. In other words, the amount of wasted recourse can be minimized and the network lifetime can be extended. 7 Simulation results In this section, we evaluate our algorithms performances through conducting simulations measuring the network lifetime, the remaining energy and the number of sensors in each subset. The effect of density control is shown in Sect Networks are randomly generated in a fixed area of We assume the sensing area of a sensor is circular. Each set of experiments are conducted for k = 1, 2 and 4. All data are averages from 50 times experiments. The energy model used in our simulations is the quadratic model e = μ rs 2.Wesetμ to 0.01 and each sensor s initial energy E to The simulations were carried out on our own simulation environment created by Java. All algorithms were compared under the same conditions. 7.1 Simulations without density control We first evaluate the performances of GS and GSA without employing the density control scheme. These two algorithms are compared with the ideal case discussed in Theorem 3 (Note that this ideal case is for disjoint sets with fixed sensing range).

14 140 J Comb Optim (2008) 15: Effect of number of sensors on network lifetime This set of simulations evaluates how the number of sensors affects the network lifetime. For GS and the ideal case, a fixed sensing rage of 50 for each sensor is used; for GSA, the maximum possible sensing range of a sensor is 50 and each sensor can adjust its sensing range. The number of sensors ranges from 50 to 200. Figure 3 shows how the network lifetime is affected by the number of sensors and k. Our algorithms have stable performances. The network lifetime increases linearly with respect to the network size. The network lifetime decreases as k increases since more sensors are required for a subset. The network lifetime for GS is at least 90% of the network lifetime for the ideal case. GSA has a longer network lifetime than both of them since each sensor is allowed to adjust its sensing range to further extend network lifetime. It is shown in Fig. 4 that the number of sensors in each subset keeps constant and this also consolidates with Fig. 3. Fig. 3 Network lifetime Fig. 4 Number of sensors/subset

15 J Comb Optim (2008) 15: Fig. 5 Remaining energy Fig. 6 Network lifetime Figure 5 shows the efficiency of the energy usage, in terms of the amount of the remaining energy when network lifetime ends. The present of remaining energy is due to the unused sensors which cannot find other cooperators to form a cover set. Less remaining energy indicates more efficient energy usage. As shown in Fig. 5, GSA is better than GS. In the worst case where there are 200 sensors and k = 4, the remaining energy is at most the total initial energy of 70 sensors. Because of the random sensor deployment, those sensors cannot find their cooperators such that they have no chance to work. This problem can be partly solved by applying the density control scheme which is shown in Sect Effect of sensing range on network lifetime The purpose of this set of simulations is to evaluate how the sensing range of a sensor affects the network lifetime. The number of sensors is set to 50. For GS and the ideal

16 142 J Comb Optim (2008) 15: Fig. 7 Number of sensors/subset Fig. 8 Remaining energy case, the fixed sensing rage ranges from 30 to 80; for GSA, the maximum possible sensing range of a sensor ranges from 30 to 80 and each sensor can adjust its sensing range. As shown in Fig. 6, GSA provides much longer network lifetime than both GS and the ideal case. With larger sensing ranges, all of them have longer network lifetimes. One reason is that with larger sensing ranges, the number of sensors in each subset decreases (Fig. 7) such that more subsets can be constructed. GSA outperforms the other two algorithms because it can adjust the sensing range of each sensor and less energy is wasted in overlapped covered regions. In Fig. 7, it is shown that GS and GSA have the same number of sensors in each subset. This shows that GS does not allocate unnecessary sensors into a subset and GSA reduces sensors sensing ranges efficiently.

17 J Comb Optim (2008) 15: Fig. 9 Network lifetime vs. number of sensors Fig. 10 Remaining energy vs. number of sensors Figure 8 shows that the efficiency of the energy usage improves as the sensing range increases. It drops from (about 30 sensors total initial energy) to (about 10 sensors total initial energy). The higher the k is, the more energy remains. 7.2 Simulations with density control The random sensor deployment may make it difficult for some sensors to find cooperators such that they cannot make any contribution on prolonging the network lifetime. In this set of simulations, we apply the density control scheme discussed in Sect. 6 on sensor deployment and evaluate how this scheme affects the performances of GS and GSA.

18 144 J Comb Optim (2008) 15: Fig. 11 Network lifetime vs. sensing range Fig. 12 Remaining energy vs. sensing range Figure 9 shows that the density control scheme improves the network lifetime remarkably compared with Fig. 3. On average, there is a 25% improvement on network lifetime. Without the density control, there is a lot of remaining energy as shown in Fig. 5. Figure 10 shows that the remaining energy is limited to the total initial energy of about 20 sensors in the worst case where 200 sensors are deployed and k = 4, which is a 29% improvement. We also investigate how the sensing range affects network performance when the density control is applied and the results are shown in Figs. 11 and 12 where 50 sensors are deployed for each network. Figure 11 shows that network lifetime is stable when the sensing range varies from 30 to 80. This is because sensors are deployed in a uniform manner, and larger sensing ranges will not help sensors find more cooperators. Figure 12 shows that larger sensing ranges lead to less remaining energy.

19 J Comb Optim (2008) 15: Conclusion and future work In this paper, we investigate a new SSC problem of scheduling sensors to provide k-coverage for a monitored area with the purpose of maximizing the network lifetime. We propose two heuristic algorithms, GS and GSA, to solve the SSC problem. GS deals with the case where sensors have fixed sensing ranges and sensors are divided into disjoint cover sets. GSA deals with the case where sensors can adjust their sensing ranges and sensors are divided into non-disjoint cover sets. In addition, we develop a guideline for users to better design a sensor deployment plan by employing density control. Simulation results are presented to evaluate our proposed algorithms. We will further investigate the k-coverage scheduling problem with more constraints, such as connectivity and communication range, bandwidth limitation, transmission delay requirement and etc. In addition, other non-greedy heuristics as well as distributed algorithms are also of our interest. Acknowledgements This work is partly supported by NSF CAREER Award under Grant No. CCF and the National Grand Fundamental Research 973 Program of China under Grant No. 2006CB References Abrams Z, Goel A, Plotkin S (2004) Set K-cover algorithms for energy efficient monitoring in wireless sensor networks. In: Proceedings of third international symposium on information Akyildiz IF, Su W, Sankarasubramaniam Y, Cayirci E (2002) Wireless sensor networks: a survey. Comput Netw 38: Cardei M, Du D-Z (2005) Improving wireless sensor network lifetime through power aware organization. ACM Wirel Netw 11(3): Cardei M, Wu J (2004) Energy-efficient coverage problems in wireless ad hoc sensor networks. J Comput Commun Sensor Netw Cardei M, Thai M, Li Y, Wu W (2005a) Energy-efficient target coverage in wireless sensor networks. In: IEEE INFOCOM 2005, Miami, USA, March 2005 Cardei M, Wu J, Lu N, Pervaiz MO (2005b) Maximum network lifetime with adjustable range. In: IEEE international conference on wireless and mobile computing, networking and communications (WiMob 05), August 2005 Cheng MX, Ruan L, Wu W (2005) Achieving minimum coverage breach under bandwidth constraints in wireless sensor networks. In: Proceedings of the 24th conference of the IEEE communications society (INFOCOM) Gupta H, Das S, Gu Q (2003) Connected sensor cover: self-organization of sensor networks for efficient query execution. In: MobiHoc 03, Annapolis, MD, June 1 3, 2003 Huang Q (2003) Solving an open sensor exposure problem using variational calculus. Technical Report WUCS-03-1, Washington University, Department of Computer Science and Engineering, St. Louis, MO Huang C, Tseng Y (2003) The coverage problem in a wireless sensor network. In: WSNA 03, San Diego, CA, September 2003 Jiang J, Dou W (2004) A coverage-preserving density control algorithm for wireless sensor. In: Proceedings of 3rd international conference, ADHOC-NOW 2004, Vancouver, Canada, July 22 24, 2004 Jie Wu, Yang S (2004) Coverage issue in sensor networks with adjustable ranges. In: Proceedings of the 2004 international conference on parallel processing workshops Kumar S, Lai TH, Balogh J (2004) On k-coverage in a mostly sleeping sensor network. In: Proceeding of the 10th international conference on mobile computing and networking, Philadelphia, PA, pp Li X, Wan P, Frieder O (2003) Coverage in wireless ad hoc sensor networks. IEEE Trans Comput 52(6): Meguerdichian S, Koushanfar E, Potkonjak M, Snvastava M (2001) Coverage problems in wireless ad-hoc sensor networks. In: IEEE Infocom, pp

20 146 J Comb Optim (2008) 15: Mehta DP, Lopez MA, Lin L (2003) Optimal coverage paths in ad-hoc sensor networks. IEEE Int Conf Commun 1: Wang X, Xing G, Zhang Y, Lu C, Pless R, Gill C (2003) Integrated coverage and connectivity configuration in wireless sensor networks. SenSys 03, Los Angeles, CA, November 2003 Yan T, He T, Stankovic JA (2003) Differentiated surveillance for sensor networks. In: Proceedings of ACM SenSys03, November 2003, pp Zhou Z, Das S, Gupta H (2004) Connected k-coverage problem in sensor networks. In: Proceedings of the international conference on computer communications and networks (ICCCN)