Sensor Scheduling for k-coverage in Wireless Sensor Networks

Transcription

1 Sensor Scheduling for k-coverage in Wireless Sensor Networks Shan Gao, Chinh T. Vu, and Yingshu Li Department of Computer Science Georgia State University Atlanta, GA 333, USA {sgao, chinhvtr, 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 region 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 a heuristic algorithm for it. In addition, we develop a guideline for users to better design a sensor deployment plan to save energy by employing density control. Simulation results are presented to evaluate our proposed algorithm. 1 Introduction Sensor networks which usually consist of a large number of sensors are attracting people s attentions. They can sense and collect information from all kinds of objects in the monitored area. Furthermore, they can process the gathered information and send it back to users. Therefore, they are being widely employed for military fields, national security, environmental monitoring, traffic control, health, industry, disaster prevention and recovery [1]. However, current sensor networks still have some limitations that prevent them from better serving the people. The limitations are as following: limited power at each sensor, limited communication ability, limited computation ability, limited wireless bandwidth, large number of nodes in a network, huge deployment area and infinite sensing data streams. These limitations bring a lot of challenging problems. In this paper, we address the k-coverage problem which requires that every point of the whole monitored area can be covered by at least k sensors at any time. To deploy a sensor network, an aircraft may be used to spread the sensors into an area when ground access is not possible. This causes the lack of accurate placement of sensors, which will be compensated by deploying more redundant sensors. Another reason for deploying redundant sensors is to provide faulttolerance, since sensors are prone to failures [1]. If any point in the monitored area is monitored by at least k sensors, proper operation of the network can still be ensured, even if some sensors fail. The required coverage level k may J. Cao et al. (Eds.): MSN 26, LNCS 432, pp , 26. c Springer-Verlag Berlin Heidelberg 26

2 Sensor Scheduling for k-coverage in Wireless Sensor Networks 269 be different for different applications. In friendly environment such as home monitoring, k can be set to a small value, while in hostile environment such as battle fields, k should be set to a large value. Even for a single sensor network, k may be different. For example, for forest fire detection, k may be low in the rainy season, but high in the dry season. One may say that since much redundant sensors are deployed, the k-coverage problem can be easily solved. However, considering the power limitation of sensor networks, to make all sensors remain active greatly shortens the network lifetime. It is shown in [2] that each sensor spends.34w to.7w power when it is in the transmit, receive and idle states, however, only.3w in the sleep state. In addition, the lifetime of a battery discharging in short bursts with significant off-time is approximately twice as much as that in a continuous operation mode [3]. These facts indicate that a good active/sleep scheduling mechanism can dramatically extend network lifetime. Therefore, while maintaining the coverage level k, only a subset of the sensors is needed to be active at any time. In this paper, our contributions are as following: i) we define the problem of Sensor Scheduling for k-coverage (SSC) which is NP-hard; ii) we design a heuristic algorithm for the SSC problem which divides the sensors into subsets, such that a schedule can be worked out by activate these subsets successively to extend network lifetime; and iii) we propose a density control scheme for sensor deployment to reduce the number of unallocated sensors such that the network efficiency is improved. 2 Related Work Recently, the coverage problem, a fundamental problem in sensor networks about how well an area is monitored by sensors, has attracted people s attentions. Basically, there are three kinds of coverage problems [6] which are target coverage problem, area coverage problem and breach coverage problem. The work in [7] [8] addressed the target coverage problem where the purpose is to cover all the targets. The work in [4], [], [9] [11] addressed the area coverage problem where the purpose is to cover the whole monitored area. The breach coverage problem is addressed in [12] where the purpose is to minimize the number of uncovered targets. Some other work, [14] and [1], tried to find a path which is best or worst monitored by sensors and connects two given points inside or outside the surveillance area and this path can indicate the sensing ability of the sensor network in the best or the worst situation. None of the above work considers the k-coverage requirement for the purpose of quality of surveillance. To the best of our knowledge, not much work address the k-coverage problem. Wang et al. [9] first studies 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 (subset of sensors) that can provide k-coverage. However, the size of the obtained subset cannot be guaranteed to be small. In [], the authors designed a greedy heuristic for the k-coverage problem and the size of their constructed

3 27 S. Gao, C.T. Vu, and Y. Li cover set is within O(logn) factor of the optimal. The main idea is to select a candidate path which has the maximum K-Benefit value. Both of these two work only consider constructing one cover set instead of dividing sensors into subsets such that each subset can provide k-coverage. In [13], the authors study the sensor deployment problem so that k-coverage can be guaranteed. In [4], the coverage problem is formulated as a decision problem, whose goal is to determine whether each point in the monitored region is covered by at least k sensors. The main idea is to check the perimeter coverage level 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-perimeter-covered. Based on this work, we design in this paper a heuristic algorithm to divide the sensors into subsets and each of the subset can provide k-coverage, such that the network lifetime can be maximized. The differences between these algorithms and ours are: 1) our algorithms provide solutions to k-cover the monitored area; 2) in our algorithms, k-coverage is 1% guaranteed; 3) there is no limitation on sensor s sensing range which could 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 region and no two sensors are located at the same location. Every point in the region needs to be continuously monitored (covered) by at least k sensors. The network lifetime is defined as the total duration during which the whole region 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 region. To extend the network lifetime, instead of making all the sensors to be active throughout the whole network lifetime, a subset of the sensorscanbeturnedontoprovidekcoverage at any time, while the rest sensors are in 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 [9]. All the sensors have uniform transmission range and sensing range. 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 region, schedule the activities of the sensors such that at any time, the whole region can be k-covered and the network lifetime is maximized. In [], the authors consider the problem of constructing a single connected k- coverage set and this problem (CCP) is proved to be NP-hard. Therefore, the SSC problem is NP-hard since CCP is a special case of the SSC problem when the number of the constructed k-coverage sets is one. 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

4 Sensor Scheduling for k-coverage in Wireless Sensor Networks 271 the region. To solve the SSC problem, we can divide the sensors into disjoint subsets. Each subset can k-cover the whole region, where k-cover indicates for every point in the monitored region, at least k sensors can cover this point. These subsets can be scheduled to be active successively. For each subset, its lifetime is decided by the sensor which has the least power. Therefore, the lifetime of the entire network highly depends on the number of subsets. The following notations are used to formulate the SSC problem and to describe our algorithm. K: If all the sensors are active, any point in the monitored region 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. S: The set of all the sensors. m: All the sensors can be divided into at most m subsets and each subset can k-cover the monitored region. C i :Theith subset, 1 i m. cov i : The coverage level of the C i, which means any point in the monitored region is covered by at least cov i sensors which belong to C i. Our goal is to construct as many subsets as possible such that i) each subset can k-cover the whole monitored region; ii) the network lifetime is maximized. Then the SSC problem is formulated as Objective: Max m Subject to: 1 i m C i S C i C j =, 1 i, j m, i j cov i k, 1 i m 4 Disjoint Cover Sets with Fixed Sensing Range In this section, we present a greedy heuristic for the SSC problem. In [4], the authors proved that the entire monitored region is k-covered if and only if each sensor in the monitored region is k-perimeter-covered. k-perimeter-cover requires that any point on the perimeter of a sensor i be covered by at least k sensors other than sensor i. Based on this fact, we propose a greedy algorithm, PCL-Greedy- Selection (GS). We define the Perimeter Coverage Level (PCL) of a sensor a as the number of the sensors in the same set that cover any point on a s perimeter of the sensing area. The lower the PCL is, the smaller the node density (the number of nodes per unit area) is. The main idea of GS is to iteratively construct subsets C i by choosing sensors from the area with the lowest sensor density. When construct an individual C i, the sensor with a smaller PCLvalue will be added to C i at each step. In this way, we can include as less sensors as possible in C i and these sensors are distributed in the area as widely as possible, such that more sensors can be left to join

5 272 S. Gao, C.T. Vu, and Y. Li Algorithm 1. PCL-Greedy-Selection(k, S) 1: Sort S in non-decreasing order based on their PCLvalues 2: while S is not empty do 3: cov i getcoveragelevel(c i) 4: if cov i <kthen : node the first sensor in S 6: Add node to C i 7: Remove node from S 8: else 9: PruneGreedySelection(k, S, C i) 1: Add C i to C 11: i ++ 12: end if 13: end while 14: output C other subsequent subsets and the overlapped sensing regions in each subset are reduced as much as possible. This also indicates when construct a subset C i,the area with smaller node density is taken care of with higher priority. The GS is shown in Algorithm 1.. The input includes k, a user-specified coverage level, and S, the set of all the sensors. The output is a collection of subsets C, and each subset can k-cover the whole monitored region. To justify if a subset C i can k-cover the entire monitored region, we can use the method proposed in [4] and we call it getcoveragelevel(c i ). Firstly, all the sensors in S are sorted in non-decreasing order based on their PCL values. Then sensors are added to 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 be constructed in the same manner. GS stops when we can no longer construct a subset that can k-cover the whole monitored region. Since each subset is constructed in a greedy manner, it is possible that there exist some redundant sensors in a subset. Therefore, after 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 to conduct this operation is P runegreedyselection which is described in Algorithm 2.. In this algorithm, given a subset C i,wecheckforeachsensorinc i to see if the removal of it will make cov i smaller than k. If a sensor is redundant (after the removal of this sensor, cov i is still no less than k), it will be added back to S. In [4], the authors have shown the fact that if no two sensors are located at the same location, the whole monitored region is k-covered if and only if each sensor is k-perimeter-covered. Based on this fact, the correctness of our algorithm is guaranteed. Theorem 1. The time complexity of GS is O(n 2 dlog(d)). Here, n is the number of the sensors, and d = max(d 1,..., d i,..., d n ) where d i is the number of neighbors of sensor i.

6 Sensor Scheduling for k-coverage in Wireless Sensor Networks 273 Algorithm 2. PruneGreedySelection(k, S, C i ) 1: for j =1to 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) : if cov i k then 6: Add s j to S 7: else 8: Add s j back to C i 9: end if 1: end for Proof. The time for sorting S is O(nlogn). There are n iterations in the while loop. At each iteration, the main part that dominate the time complexity is getareacoveragelevel or P runegreedyselection. The function, getareacoveragelevel, is proposed in [4]. Its time complexity is O( C i dlog(d)), where C i is the size of a subset C i. The time complexity for P runegreedyselection is O( C i 2 dlog(d)). Therefore, the time complexity of GS is O(n 2 dlog(d)). The number of the subsets constructed by GS decides the network lifetime. The following theorem gives the bound of the number of the constructed subsets in the ideal cases. Theorem 2. Given some sensors K-covering an area, if the sensors sensing range is fixed and the constructed subsets are disjoint, the maximum number of subsets m is K k,wherek (K k) is the minimum coverage level that the sensor network can provide if all the sensors are activate. 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, thatis,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 too. To prove this, without loss of generality, we assume m = K k α. 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 cases. Because (K K k k), the remaining sensors could construct α more subset(s). This leads to a contradiction. Thus, the lower bound of m is K k too. Based on the upper bound and the lower bound of m, we conclude that m = K k. Density Control of the Sensor Deployment From Theorem 2, we can see there is a linear relationship between K and the number of the constructed subsets. This is also validated through the simulation

7 274 S. Gao, C.T. Vu, and Y. Li results in Section 6. As the network lifetime is decided by the number of the constructed subsets, 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 areas in a monitored region have different sensor densities. From the simulation results, we found that there always exist some sensors that were not allocated to any subset which is a waste of resource. The waste is due to the difference between the sensor density of the area near the border of the monitored region and the sensor density of the area at the center of the monitored region. The unallocated sensors are usually the ones at the center of the monitored region. The sensors near the borders have smaller PCL values and the sensors at the center have larger PCL values. GS adds sensors to a subset beginning from the sensors with smaller PCL values. Thus, it is possible after all the sensors near the border have been added to some subsets, there still exist some sensors at the center and no more subsets can be constructed to provide k-coverage for the whole monitored region. Therefore, to extend the network lifetime, the PCL values of the sensors need to be balanced, so that the closer to the border the area is, the more sensors this area should have. To guarantee balancing the PCL values, the number of the neighbors of the sensors close to the borders should be equal to the number of the neighbors of the sensors at the center. We derive a relationship between the sensor density of the area near the border and the sensor density of the area at the center in Theorem 3. We define a disk centered at c as D c. The sensor density of D c is denoted as ρ c,and ρ c = number of sensors in D c, D c where D c is the area of D c. Theorem 3. Assume the sensor density at the center of the monitored region A is ρ c. To guarantee the number of the neighbors of the sensors close to the borders be equal to the number of the neighbors of the sensors at 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. 1, D c is the disk centered at c (center of the monitored region) with radius of 2R s and D p is the disk centered at p with radius of 2R s minus area A. Sincewe 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

8 Sensor Scheduling for k-coverage in Wireless Sensor Networks 27 Fig. 1. Density computation condition: ρ p = Dc D ρ p c.weknow D c = π(2r s ) 2 and D p = A 1 +2A 2 = π(2r s ) 2 2π 2α 2π +2( 1 2 r (2R s ) 2 r 2 ) = 4(π α)rs 2 + r 4Rs 2 r 2,whereA 1 is the area filled with dashed lines, A 2 is the area filled with dotted lines and α = arccos r 4πR 2R s. Hence, ρ p = 2 s r 4(π arccos +r ρ c. 2Rs )R2 s 4R 2 s r2 Based on Theorem 3, users can develop a plan for deploying sensors such that any point in the monitored region may be covered by almost the same number of sensors. This scheme can reduce the number of unallocated sensors. In other words, the amount of wasted recoursecan be minimized and the network lifetime can be further extended. 6 Simulation Results In this section, we evaluate GS s performance by conducting simulations to measure the network lifetime in terms of evaluating the constructed subsets, the number of unallocated sensors, and the effect of density control mentioned in Theorem 3. Networks are randomly generated in a fixed region of 1 1. 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 times experiments. 6.1 Performance of Greedy Selection In this section, we study the performance of GS algorithm. The results of our algorithm are compared with the ideal case which is proved in Theorem 2. Fig. 2(a) shows the comparisons between GS s results and the ideal results when the number of sensors varies from to 2 and Density Control is applied. We can see the actual numbers of subsets are close to the ideal results. In fact, the ratio (actual/ideal) are between 8% and 9% stably. When the sensing range of the sensors varies from 3 to 8, Fig. 2(b) shows that the results of our algorithm are still very close to the ideal numbers. The ratio tends to be stabilized between 8% and 9%. Fig. 2(c) shows that our results are almost the same as the ideal results when Density Control is not applied. Actually the ratios are all above 9%, whereas it is not good enough. The reason is that the percentage

9 276 S. Gao, C.T. Vu, and Y. Li The number of subsets k=1(ideal) k=1(actual) k=2(ideal) k=2(actual) k=4(ideal) k=4(actual) The number of subsets k=1(ideal) k=1(actual) k=2(ideal) k=2(actual) k=4(ideal) k=4(actual) The number of Sensors (a) The number of sensors varies from to 2 with DC Sensors Sensing Range (b) The sensing range varies from 3 to 8 with DC The number of Subsets k=1(ideal) k=1(actual) k=2(ideal) k=2(actual) k=4(ideal) k=4(actual) The number of Subsets k=1(ideal) k=1(actual) k=2(ideal) k=2(actual) k=4(ideal) k=4(actual) The number of Sensors (c) The number of sensors varies from to 2 without DC Sensing Range (d) The sensing range varies from 3 to 8 without DC Fig. 2. Compare GS s results with the ideal results of the used sensors is quite low, only 61.34% on average. Due to low usage percentage, there are enough redundant sensors for constructing more subsets. By applying Density Control, the usage percentage is improved to 83.17% on average. There are less redundant sensors (potential unallocated sensors) left. The comparison when increasing sensors sensing range without Density Control is shown in Fig. 2(d). All these results show that GS s results are stable and very close to the ideal results whatever changing the number of sensors or the sensors sensing range. 6.2 Effect of Total Number of Sensors on Network Lifetime The purpose of this set of simulations is to evaluate how the total number of deployed sensors affects the network lifetime. The sensing range of a sensor is set to. Fig. 3(a) shows how many subsets can be constructed when the number of sensors ranges from to 2. As shown in Fig. 3(a), the number of the constructed subsets increases linearly with respect to the network size. This fact is also validated by Theorem 2. It is shown in Fig. 3(b) that the number of nodes per subset keeps constant and this also consolidates with Fig. 3(a). As the coverage level k increases, the number of the constructed subsets decreases since more sensors are required for a subset. Fig. 3(c) illustrates the number of the unallocated sensors. Around 33% 4% sensors are not allocated on average. After studying the experiment data, we

10 Sensor Scheduling for k-coverage in Wireless Sensor Networks 277 The number of Subsets k=1 (no dc) k=2 (no dc) k=4 (no dc) The number of Sensors per Subset k=1 (no dc) k=2 (no dc) k=4 (no dc) The number of Sensors (a) The number of subsets The number of Sensors (b) The number of sensors per subset 8 7 k=1 (no dc) k=2 (no dc) k=4 (no dc) 3 2 k=1 (no dc) k=2 (no dc) k=4 (no dc) The number of Unallocated Sensors The number of Subsets The number of Sensors (c) The number of unallocated sensors Sensing Range (d) The number of subsets The number of Sensors per Subset k=1 (no dc) k=2 (no dc) k=4 (no dc) The number of Unallocated Sensors k=1 (no dc) k=2 (no dc) k=4 (no dc) Sensing Range (e) The number of sensors per subset Sensing Range (f) The number of unallocated sensors Fig. 3. Effect of total number of sensors and the sensing range on the network lifetime. Density Control is NOT applied. found that only a few small areas close to the corners of the region are not covered by these unallocated sensors. The reason is that the density of sensors close to the center of the region is larger than the one close to the corners and borders. Therefore, there are not enough sensors near the borders to form some subsets with the sensors at the center of the region. To solve this problem, we can reduce the number of unallocated sensors through density control which will be evaluated in section 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. sensors are deployed in the region.

11 278 S. Gao, C.T. Vu, and Y. Li sensor Deploy sensors with DC Deploy sensors without DC Fig. 4. Deploy sensors with and without DC The sensing range ranges from 3 to 8. Fig. 3(d) shows that more subsets are constructed as the sensing range increases. Even one sensor can cover the whole area when the sensing range becomes very large. However, only those placed at the center of the region can solely k-covertheentireregionexcept that the sensors have very large sensing range. Those sensors close to the corners and borders need other sensors cooperation to k-cover the whole region. Thus, the curves of the number of subsets do not keep increasing. There is the maximum number of subsets which is shown in Theorem 2. Fig. 3(e) indicates that larger sensing range leads to fewer sensors in each subset. Larger sensing range also makes more sensors at the center of the region be used. In the previous simulation, they are unallocated sensors generally. With larger sensing range, they can provide the required coverage level without the help from the sensors close to the corners and borders. Fig. 3(f) validates this fact. On average, 39.27% sensors are not allocated into any subset. 6.4 Effect of Density Control on Network Lifetime In this set of experiments, we apply Density Control (DC) for sensor deployment and evaluate its effectiveness. When deploy the sensors, we apply Theorem 3 to control the density of sensors in the monitored region. By employing DC, we can deploy more sensors in the areas close to the corners and borders such that all the sensors have almost the same number of neighbors. In other words, the monitored region is covered uniformly. The effect of DC is presented in Fig. 4. Due to the space limitation, only the numerical results are shown as following. In the simulations on increasing the number of sensors from to 2, we observed that compared with the cases where DC is not applied, by employing DC, up to 74.6% (averagely 39.8%) more subsets can be constructed and there are up to 66% (averagely 6.6%) less unallocated sensors. In the simulations on increasing the sensing range from 3 to 8, compared with the cases where DC is not applied, by employing DC, we can obtain 39.68% more subsets on average and the number of the unallocated sensors is reduced 18.91% on average. Therefore, by deploying sensors more rationally, sensors are used more effectively.

12 Sensor Scheduling for k-coverage in Wireless Sensor Networks Conclusion and Future Work In this paper, we investigate a new SSC problem of scheduling sensors to provide k-coverage for a monitored region with the purpose of maximizing the network lifetime. We propose a heuristic algorithm to solve the SSC problem. In addition, we develop a guideline for users to better design a sensor deployment plan by employing density control. Theoretical analyses as well as simulation results are presented to evaluate our proposed algorithm. We will further investigate the k-coverage scheduling problem with more constraints, such as connectivity, adjustable sensing range 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. References 1. I. F. Akyildiz, W. Su, Y. Sankarasubramaniam and E. Cayirci, Wireless sensor networks: a survey, Computer Networks, 38: , V. Raghunathan, C Schurgers, S. Park, and M. B. Srivastava, Energy-aware wireless microsensor networks, IEEE Signal Processing Magazine 19:4, L. Benini, G. Castelli, A. Macii, E. Macii, M. Poncino and R. Scarsi, A Discrete- Time Battery Model for High-Level Power Estimation, Proceedings of DATE, pp.3 39, C. Huang and Y. Tseng, The coverage problem in a wireless sensor network, WSNA 3, San Diego, CA, Sep Z. Zhou, S. Das, and H. Gupta, Connected k-coverage problem in sensor networks, in Proceedings of the International Conference on Computer Communications and Networks, M. Cardei and J. Wu, Energy-Efficient Coverage Problems in Wireless Ad Hoc Sensor Networks, Journal of Computer Communications on Sensor Networks, M. Cardei, M. Thai, Y. Li and W. Wu, Energy-Efficient Target Coverage in Wireless Sensor Networks, IEEE INFOCOM 2, Miami, FL, Mar M. Cardei, D.-Z. Du, Improving Wireless Sensor Network Lifetime through Power Aware Organization, ACM Wireless Networks, 11(3):333 34, X. Wang, G. Xing, Y. Zhang, C. Lu, R. Pless, C. Gill, Integrated coverage and connectivity configuration in wireless sensor networks, SenSys 3, Los Angeles, CA, Nov H. Gupta, S. Das, Q. Gu, Connected Sensor Cover: Self-Organization of Sensor Networks for Efficient Query Execution, MobiHoc 3, Annapolis, MA, Jun Z. Abrams, A. Goel, S. Plotkin, Set K-Cover Algorithms for Energy Efficient Monitoring in Wireless Sensor Networks, Proc. of Third International Symposium on Information, M. X. Cheng, L. Ruan, and W. Wu, Achieving Minimum Coverage Breach under Bandwidth Constraints in Wireless Sensor Networks, IEEE INFOCOM 2, Miami, FL, Mar. 2.

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