ON THE OPTIMAL COVERAGE AREA FOR SOLVING ENERGY-EFFICIENT PROBLEM IN WIRELESS SENSOR NETWORK

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1 Jurnal Karya Asli Lorekan Ahli Matematik Vol. 8 No.1 (2015) Page Jurnal Karya Asli Lorekan Ahli Matematik ON THE OPTIMAL COVERAGE AREA FOR SOLVING ENERGY-EFFICIENT PROBLEM IN WIRELESS SENSOR NETWORK Kalid Abdlkader Marsal 1, Ismail Abdullah 2, Waidah Ismail 3 and Khairi Abdulrahim 4 Faculty of Science and Technology, Universiti Sains Islam Malaysia Bandar Baru Nilai, Negeri Sembilan, Malaysia. 1 kaled_992003@yahoo.com, 2 isbah@usim.edu.my, 3 waidah@usim.edu.my, 4 khairiabdulrahim@usim.edu.my Abstract : Recently, wireless sensor networks are beginning to be deployed at an accelerated pace. It is amazing to expect that the world will be covered with wireless sensor networks that are accessible via the Internet. This allows one to think of the Internet as a physical network. Wireless sensor networks are the base of wide range applications related environmental monitoring, transportation, national security, health care, surveillance, and military. In this article, recent contributions addressing energy-efficient coverage problems are surveyed through static wireless sensor networks. This article conveys the problem of maximizing networks lifetime for coverage and connectivity in wireless sensor networks. For providing sensing coverage to a set of points, target points; and for providing communication among the active sensor to transmit data at all times within the network, static sensor nodes should be randomly deployed in the region. Sensor nodes are energy-consumed and so only a minimum set of sensor nodes requires to be activated at any given time. Sensor nodes are respectively activated to achieve maximum lifetime of the network. Hence, the algorithm serves as an energy-efficient solution toward ensuring connected coverage in wireless sensor networks. It is worth mentioning that the optimal solution to the problem is NP-Complete that stimulates the need to discover efficient heuristic solutions. Keywords: coverage, energy efficiency, connectivity, wireless sensor networks, optimal 1. Introduction Wireless sensor networks represent a revolutionary stage in the world of networks. These networks are useful to various fields such as environmental monitoring, healthcare, defense, and agriculture. The main requirement for such networks is to be capable of sensing the data from the specified area of interest and to collect it at a centralized location with view of further processing and decision making. Nowadays, efficient deployment of these networks confronts multiple major challenges including connectivity, coverage and energy conservation [1]. The indispensable method for energy conservation in wireless sensor networks is to schedule intervals of sleep for external nodes, while the remaining nodes stay active to provide continuous service. In order for the sensor network to operate successfully, the active nodes must maintain both sensing coverage and network connectivity. Moreover, the network must have the ability to configure itself to any feasible degrees of coverage and connectivity in order to sustain various applications and environments with different requirements. The sensor nodes are static and are randomly deployed in the region of interest in many applications such as crisis management and military applications. Random placement of the sensor nodes is inevitable in situations, where the individual sensor placement is unviable. Event-driven reporting of the sensed data is performed by the sensor nodes wherein a sensor node, on detecting an event of interest, transmits relevant data which is collected at the sink (monitoring station). In general, more sensor nodes would be deployed than required, to make up for the lack of exact placement and to provide sufficient fault-tolerance. It is not necessary for all the sensor nodes to be active simultaneously in the network because sensor nodes collaborate to achieve a common sensing task. Sensor nodes require to actively sense or cover all 'points of interest' in the region in order to achieve 2015 Jurnal Karya Asli Lorekan Ahli Matematik Published by Pustaka Aman Press Sdn. Bhd.

2 Jurnal KALAM Vol. 8 No. 1, Page complete data collection (i.e. none of the events should go unreported in the network). The 'coverage' requirement ensures that the entire target points in the network are being covered by at least one active sensor at all times. In case of an event has been detected by one of the sensor nodes, the information should be transmitted to the central monitoring station. The 'connectivity' requirement guarantees that any active sensor in the network can transmit or communicate to the monitoring station at all times, using relay sensor nodes, if required. The sensor nodes have limited battery (energy) resources and it may not be adequate to continuously recharge their energy levels. This is another reason why the sensor nodes need to be deployed amply or in large numbers in expected to operate for longer time periods with limited resources. Therefore, it is necessary to develop an energy-efficient solution towards maintaining coverage and connectivity in the network. At the same time, other sensors could 'sleep' or deactivate themselves, saving their energy for future use. Additionally, activating all the sensor nodes simultaneously leads to excessive energy consumption in the system limiting the network lifetime, such as activation of schedule is also not advisable because abundant data will be collected at the sink if multiple active sensors are covering any particular target point. Moreover in such case multiple sensor nodes would sense a particular event in the region and would try to transmit simultaneously, causing unnecessary packet collisions owing to excessive crowding for the wireless channel. Therefore activating a minimal set of sensor nodes at any time causes better resource benefit in the network and also the network lifetime is prolonged. Sufficiently high density deployment of sensors in the region leads the initial decision making task to identify a feasible set of sensors, which needs to be activated in order to ensure the coverage requirement in the network. Finding a convenient set which includes the minimum number of sensors is referred to as the energy-efficient coverage problem in this article. Such set of sensors is adequate to support the network for a while. However maximizing the life time of the network requires finding the maximum number of such disjoint feasible sets that is referred to as the lifetime coverage problem. if all the feasible sets are also necessary for providing connectivity, the emerging problem is referred to as the lifetime coverage and connectivity problem. In this article, the aim is to maximize the network lifetime that is defined as the time until both coverage and connectivity requirements are met in the network, and beyond which connected coverage cannot be ensured. Sensors would not be subject to recharge as in crisis management. The sensor nodes active set would eventually get consumed of their energy content, and would vanish. Simultaneously, another set of sensors would need to be activated. All sensors of the active set, which were active during the same time interval, would die together in the presence of significant temporal correlation in the events occurring at different target points. This would result in activating a fresh set of sensor nodes, causing the newly formed active set to be disjoint from the previous one. In this manner, mutually exclusive sets of sensors would get activated in sequence. Increasing the network lifetime is tantamount to maximizing the number of such sequences. Consequently, determining the maximum number of irrelevant sets of sensors enables each set to individually ensure coverage and connectivity (i.e. lifetime coverage and connectivity problem), serving as an optimal solution towards maximizing network lifetime. If the sensor nodes are rechargeable, maximizing network lifetime would result in the minimum energy consumption at the nodes. A minimum recharge rate for the sensor nodes could then be computed that would be adequate to persistently support the network. The main contribution of this article is to develop an algorithm to compute an energy-efficient solution for the life time coverage and connectivity problem in wireless sensor networks. The approach here is different from other relevant approaches in the following aspects: It formulates a simpler problem of energy-efficient coverage using linear programming techniques and benefits the structure of the formulations to develop efficient approaches for the coverage phase. It considers the visible set of sensor nodes which can provide both coverage and connectivity. Each of the sensors can sense as well as propagate information. Sensing and communication ranges of the sensors are independent. 120

3 Kalid Abdlkader Marsal et. al. 2. Literature Review Coverage has been extensively handled in many studies. Area coverage, where the aim is to monitor a specified region, has been conveyed in [2], [3], [4].Target coverage, which is considered here, has also been studied from the perspective of maximal support path in [5], [6]. The notions of classifying the sensor nodes into disjoint sets, where each set can independently ensure coverage and thus could be activated respectively, has been investigated in [7], [2]. Cardei and Du in [7] point out that the disjoint set cover problem is NP-complete and propose an efficient heuristic for set cover computations using a mixed integer programming formulation. As a whole, the sensing and communication range of a sensor node may be different and these ranges also vary significantly across sensor nodes. For example, in high density regions of target points, deploying sensor nodes with large sensing range might be helpful to achieve data collection benefits and better resource utilization. However, wide-range communication at sensor nodes would cause more channel contention and collisions in the wireless network. On the other hand, in a remote region with a few target points, it would be helpful to deploy sensors with a small sensing range and wide-range communication. 3. Energy-Efficient Coverage This article first develops some combinational results which are used to formulate the energyefficient coverage problem as a linear integer problem Notations and definitions Consider a network of N sensors intended to provide sensing coverage to a set of m target points in the region. The target points and the sensors are randomly deployed in the region. Sensors are deployed abundantly to provide maximum coverage lifetime and may have independent sensing radii. Let S be the set of all sensors and M be the set of all target points. Definition 1: Utility of a sensor s S, u(s), is the total number of target points t sensing range of s. M which lie in the Definition 2: Coverage of a target point t ϵ M with respect to a set S of sensors, C S (t), is the total number of sensors s S which cover the target point t. Definition 3: A Feasible set S f is a set of sensors s ϵ S such that for all target points t sensor s S f which covers t. M, there is a Definition 4: A Cover Set A is a set of sensors s ϵ S such that, A is a feasible set. is not a feasible set. Hence a cover set does not contain any abundant sensors and activating one such cover set is adequate to provide coverage in the network. Definition 5: Utility of a cover set A is defined as U (A) = (1) Definition 6: Coverage of the set M of target points with respect to a set S of sensors is defined as C(M) = (2) 121

4 Jurnal KALAM Vol. 8 No. 1, Page Properties Next, some results related to the utility and coverage provided by a cover set are developed. Lemma 1: For a cover set A, C A (M) = U (A). Proof: Consider a target point t M. let X denote the set of sensors which cover t such that X A. we have c A (t) = Now for all s X, t adds one to the utility of sensor s, u(s) thus, t adds (= c A (t)) to the utility of the cover set A. Also t adds c A (t) (= to the coverage of the set M. Summing over all target points t M, we have the desired result. Note that lemma 1 is also valid for any feasible set. Lemma 2: For a cover set A of size k (i.e. = k), the utility of any sensor s A is upper bounded by (m+1-k). Proof: Since A is a non-redundant feasible set, for all sensors s A there must be a target point t which is not covered any other sensor s' A (otherwise the set A - is a cover set and hence the set A is not a cover set) thus, we have 122 and c A (t) = 1. Since = k, there are at least k target points in M which have their coverage, with respect to the set A, equal to one, Now let s max A be the sensor node with maximum utility in A. since the total number of target points is m and there must be at least (k-1) target points which are not covered by s max, u (max) (m+1-k) From Lemma 2and the fact that ( = k ), we have the following corollary. Corollary 1: the utility of a cover set A of size k is upper bounded as, U (A) (m+1-k)k. Next we generalize the bounds to the utility of a cover set of any size. Lemma 3: The utility of a cover set A is bounded as, m U (A) ( ) 2, 0. Proof: Since A is a cover set, each target point t M must have coverage c A (t) 1. This together with the fact that, leads to C A (M) m. Now fromlemma 1, the lower band follows. From Corollary 1, we have U (A) (m+1-k)k, where k =. Let f (k) = (m+1-k) k. then from simple calculus, f (k) is maximized at k =. Hence, U (A) ( 2. From lemmas 1 and 3, we observe that the coverage provided to the set of target points by any cover set is bounded only in terms of the total number of target points and these bounds are independent of the size of the cover set Linear Program Next we address the issue of energy-efficient coverage in sensor networks. The objective is to find a feasible set A (A S), which comprises of a minimum number of sensors. It is easy to observe that the set A must be a cover set, as defined in Section II-A. Definition 7: Excess Utility of a sensor s A, u'(s), is defined as, u'(s) = u (s) 1. Similarly, the excess utility of a cover set A is defined as U' (A) = Note that U' (A) = U (A) - Let S E denote a set of sensors which provides coverage to the network utilizing minimum number of sensors. Then S E is a cover set with minimum cardinality. That is, S, such that S' is a cover set, Since set S E is a cover set, it also classifies the following constraints:

5 Kalid Abdlkader Marsal et. al. S S E : u' (s) 0 (3) t M : cs E (t) 1 (4) C SE (M) = + U' (S E ) (5) (3) is satisfied since otherwise the set S E - would be a feasible set, and hence the set S E would not be a cover set. (4) holds, since the set S E is a feasible set. (5) follows from lemma 1. From (5), we have, = C SE (M) U' (S E ). In other words, the set S E minimizes over all sets of sensors A, the objective function Y (A) defined as, Subject to the following constraints Y (A) = C A (M) U' (A) (6) S A : u'(s) 0 (7) t M : c A (t) 1 (8) To pose the above optimization problem as a linear program, let us introduce the following variables. For each sensor node, si, where i, let xi denote whether s i A. That is, I : 1 i N, the variable x i takes on value 0, if s i A. 1, if s i A. For each target point t j M, where j, let y i j denote whether the target point t j is covered by sensor s i. Precisely, i, j, the variable y ij takes on value 1, if sensor s i covers target point t j. 0, otherwise. Note that the variables y ij could be precomputed once the network configuration is known, and are effectively constant thereafter. Now the energy-efficient coverage problem can be expressed as the following linear integer program [8], with N variables and N + m constraints. Minimize i subject to ( i [1..N], j [1..m] (9) and (10) correspond to (7) and (8) respectively. From the structure of the above optimization problem, We observe that minimizing the objective function Y(A) (Equation 6), is the same as minimizing the sum of the coverage of all target points with respect to A and maximizing the sum of utilities of all sensors s A, simultaneously. 4. Lifetime Coverage and Connectivity Problem This problem differs from the energy-efficient coverage problem posed in section II in the following aspects: Each set of sensors needs to ensure both coverage and connectivity in the network. The aim is to find multiple such disjoint (or mutually exclusive) sets, and in particular the maximum of such sets. In this section the problem is formulated more precisely and there will be comments on its optimal solution. 123

6 Jurnal KALAM Vol. 8 No. 1, Page Problem Formulation Consider a region of interest represented by a Manhattan grid, with m target points distributed randomly in the grid [1]. N static sensor nodes are deployed randomly in the region. Sensors sense information and transmit it to the monitoring station, which is located at the center of the grid. Every sensor has a sensing capability covering a disk of radius R c. A sensor s whose distance from another sensor S' (or the monitoring station) is less than its transmission radius (R c ), can transmit directly to s' (or the monitoring station). All the sensors have a constant charge level C at the time of deploying. An active sensor has the ability to sense as well as transmit information to its surrounding sensors, which lie within its transmission radius (R c ). All active sensors can also act as relay nodes. A Passive sensor is unable to sense or relay information. An available sensor is one which has a nonzero energy level and is currently inactive or passive. Let S be the set of all sensors and M be the set of all target points. Let A ᶹ denote the set of available sensors (which is initially the same as S). Definition 8: An active sensor s has a path to monitoring station MS in the set S iff s can transmit to MS or there are relay sensors r 1,, r k S such that Sensor s can transmit to sensor r 1. I : 1 i k, sensor r i can transmit to sensor r i+1 Sensor r k can transit to MS. Definition 9: An active set, A, is the set of sensors s S such that A is a feasible set. s A : there is a path from s to MS in A. An active set is adequate to maintain coverage and connectivity requirements in the network for a while. Let such a set of sensors, A, be activated at time T 1. At some time T 2 T1, one or more sensors in the set A would get consumed of their energy completely. At this time a new set of sensors s A ᶹ needs to be computed, which shall be activated to satisfy the above requirements. Assuming all the sensors in the set A die at the time, the new set should be mutually exclusive to all the active sets computed earlier. Thus the computed active sets could be activated in succession, where each one of them contributes towards one round of sensor nodes activation in the network. Given a particular network configuration, let R be the number of rounds thus computed (i.e. R = number of disjoint active sets computed fro S). Assuming the sensors spend trivial amount of energy in the sleep or inactive state, the number of rounds, R, represents the network lifetime. That is, if duration of a round is T, the network lifetime equals RT. Hence, a large computed R would imply longer network lifetime. On the contrary, a trivial activation decision could be set the initial active set A to be the same as S, which results in minimum R( = 1) or minimum network lifetime. The objective of the lifetime coverage and connectivity problem is to compute the maximum R and to compute the active sets (A 1,., A R ) corresponding to the round (1,., R), or equivalently, Maximize R such that I, j : A i is an active set. A i A j =, I j Through multiple solutions, the one with least energy consumption is desired. For example, consider two solutions have the same value of R but different set of disjoint active sets (A 1,., A R ) and (A' 1,.., A' R ) respectively. The solution consists of those active sets for which the total number of sensors activated is the least (i.e. the former set is a solution if ). 124

7 Kalid Abdlkader Marsal et. al NP Completeness of the Optimal solution Finding the maximum number of disjoint active sets of sensors in S, where each of the active sets can independently provide coverage and connectivity in the network is the optimal solution to the lifetime coverage and connectivity problem. Cardei and Du [1] have shown that computing an optimal set cover (or the lifetime coverage problem) is NP-complete. The problem that is faced also needs to compute an optimal set cover. Moreover, the computed sets must also guarantee connectivity in the network. Hence, this problem has a solution represented in the following section. 5. Conclusion Sensor nodes are energy-constrained and hence only a minimal set of sensor nodes needs to be activated at any given time. Node activation schedules ought to be in succession in order to fulfill maximum lifetime of the network. Eventually, the optimal solution to the lifetime coverage and connectivity problem is NP-Complete. It is possible to compute the optimal solution by extensive search for small network sizes (small values of N and m). However, the complexity of computing the optimal solution grows very rapidly with the size of the network. Hence, an efficient heuristic solution approach requires to be developed. References [1] M. Cardei and J. Wu, "Coverage in Wireless Sensor Networks", Hand-book of sensor Networks, M. Ilyas and I. Magboub (eds.), CRC Press, [2] Slijepcevic.S and M. Potkonjak, Power Efficient Organization of Wireless Sensor Networks, Proc. of IEEE International Conference on Communications 2 (2001) Accessed 2001, from [3] Honghai Zhang and Jennifer C. Hou, Maintaining Sensing Coverage and Connectivity in Large Sensor Networks, NSF International Workshop on Theoretical and algorithmic Aspects of Sensor, Ad Hoc Wireless and Peer-to-peer Networks, Feb,2004. [4] Wang,X. G. Xing, Y. Zhang, C. Lu, R. Pless, and C. D. Gill, Integrated Coverage and Connectivity Con_guration in Wireless Sensor Networks, accepted to the First ACM Conference on Embedded Networked Sensor Systems (2003). Accessed November 05-0, 2003, from xing, yfzhang, lu, pless, cdgill@cse.wustl.edu [5] K. Dasgupta, M. Kukreja and K. Kalpakis, "Topology- Aware Placement and Role Assignment for Energy-Efficient Information Gathering in Sensor Networks" Proc. Of Eighth IEEE International Symposium Computers and Communication, 2003, Vol. 1, pp [6] Meguerdichian.S, F. Koushanfar, M. Potkonjak, and M. Srivastava, Coverage Problems in Wireless Ad-Hoc Sensor Networks, IEEE Infocom Accessed 2001, from [7] S. Shakkottai, R. Srikant and N. Shroof, Unreliable Sensor Grids: Coverage, Connectivity and Diameter, Proc. Of IEEE INFOCOM, [8] Christos H. Papadimitriou, Keneth Steiglitz, "Combinatorial Optimization: Algorithms and Complexity", Dover Publication Inc.,