Sensor Coverage in Wireless Sensor Networks Deying Li1, Hai Liu2 1 School of Information, Renmin University of China 2 Department of Computer Science, Hong Kong Baptist University, Hong Kong Abstract -- Coverage problem is an important and fundamental issue in sensor networks, which reflects how well a sensor network is monitored or tracked by sensor. In this chapter, we survey the current works on coverage problem in sensor networks. Two types of sensor coverage are investigated: area coverage and target coverage. Combining with sensor development mechanism (deterministic, statistical) and wireless sensor network properties (e.g. network connectivity, energy efficient and fault tolerant for connectivity and sensing etc), various coverage problems have been introduced and discussed in details. We focus on the most representative problems in each domain and present a comprehensive review and analysis of various existed algorithms, techniques.

1. Introduction Wireless sensor networks (WSNs) have received significant attention of researchers in recent years due to its wide range of applications such as military surveillance, environmental monitoring, forest fire detection, healthcare and other areas [Akyildiz02, Chong2003]. A wireless sensor network composes of a large scale of sensor devices (called sensor nodes) equipped with sensor unit, a wireless communication unit, a battery power unit and a programmable embedded processor. The sensor nodes are capable of sensing, data processing, and communicating with each other via radio transceivers. They coordinate with each other to establish a network to remotely interact with the physical world, such as to monitor a geographical region or a set of targets spread across a geographical region, and to report sensed data to the monitoring center, which is connected to the base station. Wireless networks can be random or deterministic deployed in physical environments to collect information from an area of interest in a robust and autonomous manner. An important research issue in wireless sensor network is the coverage problem which reflects how well the deployed sensor nodes can monitor a set of targets. Sensor activation scheduling under constraint on covering of targets is called the coverage problem in the literature. There are two types of targets: area [Cardei02, Carle04, SP01, Tian02, Wang03, Zhang05] and point [CardeiDu05, Kar03, CTW05, Cheng05, Li07a, Li07b]. In area coverage problem, a set of sensors is given and distributed over a geographical region to monitor a given area. But in point coverage problem, a set of sensors is given and distributed over a geographical region to monitor a set of points (or targets). The sensing range of a sensor is typically model as disk in the 2D space or as a sphere in the 3D space, with the sensor located in the center. The communication range of a sensor is modeled in the same way. A sensor can monitor all targets that fall in its sensing range. The data sent by the sensor can be received by all sensors that fall in its communication range. A sensor’s transmission range is typically larger than its sensing range. 1

Technology limitations of sensors could seriously affect the quality of service. Stringent power supply of wireless sensor nodes is the most critical limitation, because those nodes are usually powered by batteries that may not be possible to be recharged or replaced after they are deployed in hostile or hazardous environments [Yan 03]. Recently researchers have found that the significant energy saving can be achieved by elaborate managing the duty cycle of nodes in WSN with high node density and it can prolong the network lifetime. In this approach, some nodes are scheduled to sleep (or enter a power saving mode) while the remaining active nodes keep working. However, the excessive-number of sleep nodes will lead to a WSN to be disconnected, i.e. the set of working nodes will be isolated. In addition, the over-loading of working nodes will cause these nodes to be easily exhausted and failed, which also cause a WSN to be disconnected and, consequently, invalidate the data collection and transmission. It is therefore crucial to determine a small number of sensors that still cover the given area (or a set of targets) or divide all sensors into maximum number of subsets such that each subset still cover the given area or a set of targets. This is a lifetime requirement. These selected active sensors are connected so that the sensor can report the detected data to the monitoring center. This is a connectivity requirement. . Sensors are prone to be failure. The over-loading of working sensor nodes will cause easily exhausted and failed. In addition to possible hardware or software malfunctions, sensors may fail because of severe weather conditions or other hash physical environment in the sensor filed. It is therefore crucial to construct a fault-tolerant WSN that will continuously provide needed services despite sensor failures. This is the fault-tolerance requirement. The fault-tolerance requirement includes two types: coverage fault-tolerance and connectivity fault-tolerance. The sensor coverage problem can be further divided into single coverage and multiple coverage. In single coverage, each target or point in the area must be monitored by at least one working sensor. In multiple coverage, each target or point in the area needs to be monitored by at least k different working sensors, which is called as the flat k-area-coverage problem for area coverage [Gallais07]. There is another definition on k-area-coverage. An area is k-covered if there exist k distinct sets of sensors so that each one can provide fully coverage of the sensing area, which is called layered k-area-coverage problem [Gallais07]. For connectivity fault-tolerance requirement, the coverage problem can be further divided into 1-connectivity and k-connectivity coverage problem. In this chapter, we survey recent contributions which address the coverage problem. In section 2, we survey area coverage problem including multiple area coverage problem, and connected coverage problem. Section 3 investigates point coverage problem, which includes connected point coverage problem, multiple connected multiple point coverage problem, and breach coverage problem.

2. Area Coverage The area coverage problem has been well studied and a review of existing solutions for area coverage problem is described in [SSW05, Cardei06, ChenK07]. 2

Most of works in the survey [Cardei06] and [ChenK07] were published before 2003 and 2005, respectively. The survey work in [SSW05] focuses on only constructing area-dominating sets for sensor area coverage. We present in the chapter a comprehensive and updated survey on sensor area coverage. There are various coverage problems including area coverage, k-coverage, m-connected k-coverage problems. The area coverage problem is defined as follows: Definition 1(Area Coverage problem) A set of sensors are given and distributed over a geographical region to monitor a given area, an area coverage problem is to find a minimum number of sensors to work such that each physical point in the area is monitored by at least a working sensor. Definition 2(k-coverage) An area is k-coverage if each physical point in the area is covered by at least k ( k ≥ 1 ) working (or active) sensors. Definition 3 (m-connected) The communication graph of a given set of sensors M is m-connected if for any two vertices in M, there are m vertex-disjoint paths between the two vertices. A equivalent definition is, after the removal of any k-1 vertices in M, the resulted graph is still connected. Definition 4 (m-connected k-coverage problem) A set of sensors are given and distributed over a geographical region to monitor a given area, an m-connected k-coverage problem is to find a minimum number of sensors to work such that each physical point in the area is monitored by at least k active sensors and the active sensors form a m-connected graph. Most of algorithms or protocols for coverage problem guarantee full coverage, that is, each physical point must be covered. There are some algorithms or protocols for coverage problem which does not guarantee 100% coverage, such as PEAS [YZLZ03] and the approach in [Cardei02]. There is another objective except selecting a minimal set of working nodes in the area coverage: to divide all sensors into a maximum number of disjoint sets of sensors (or non-disjoint sets) such that each set fully covers the area. Selecting a minimal a set of working nodes reduce power consumption and prolongs network lifetime. In the same way, dividing all sensors into a maximum number of disjoint(or non-disjoint) sets which activate successively prolongs network lifetime. We now give a comprehensive literature review of existing solutions and their contributions which address various area coverage problems. In the following subsections, we introduce detail algorithms and solutions.

2.1 Area coverage without connectivity guarantee 2.1.1 Maximize the number of disjoint sets For energy efficient area coverage, the works in [SP01, Cardei02] consider a large population of sensors, deployed randomly for area monitoring. Slijepcevic and Potkonjak [SP01] proposed an energy conservation technique for area coverage in wireless sensor networks. It selects and successively activates mutually exclusive sets of sensor nodes, such that each set completely monitors the entire monitored area. The authors propose a heuristic to this problem. It first divides the monitored area into fields, which is a set of points. Two points belong to same 3

filed if and only if they are covered by the same set of sensors. After the fields are established, for each sensor, a list of all fields covered by that sensor is created. The set of all fields in the area is denoted as A, and set of sensors as C. The authors transform the coverage problem as the set k-cover problem: Does C contain k disjoint covers for A. Then present a heuristic solution for the set k-cover to get a heuristic solution for the coverage problem. Their method achieves energy saving by increasing the number of disjoin covers. The results on the set k-cover problem [SP01] solve a fair version where the objective is to maximize k such that every cover contains all the physical points. In many environments, requiring that a cover contain all the physical points may be too strict. For instance, there is a single area that is monitored by only one sensor but all other areas are monitored by hundreds of sensors. Except for that single area, all other areas could be covered for many times by dividing the sensors into covers. But in the fair version, the sensors can not be partitioned at all because only all sensors to monitor all areas. Figure 1 shows this case. Fig. 1 (b) can cover the whole monitoring area. But in Figure 1(c), (d), the small area A is not be covered. All areas except area A can be covered by at three different sensors while small area A is only covered by one sensor.

Figure 1. The single area A is covered by only one sensor. Abrams et al [Abrams04] study a variation of the set k-cover problem: to find a partition of the subsets into k covers so that the number of times that areas(fields) are covered by the partition, is maximized. Three approximation algorithms are presented: randomized, distributed greedy, and centralized greedy. In the randomized algorithm, each sensor simply assigns itself to a cover chosen uniformly at random from the set of all possible covers. In the distributed greedy algorithm, each sensor assigns itself, in turn, to the cover with the minimum intersection between the areas the sensor monitors and the areas monitored by the cover thus far. The centralized greedy algorithm is similar to the distributed greedy except that an area in the intersection is 4

weighted. Cardei et al [Cardei02] propose another efficient method to achieve energy saving by organizing the sensors nodes into a maximum number of disjoint dominating sets which are activated successively. Only the sensors from the active set are responsible for monitoring the monitored area and all other nodes are in a sleep mode. The authors prove that the maximum disjoint dominating sets problem is NP-complete, and any polynomial-time approximation algorithm has a lower bound of 1.5. Based on the sequential coloring algorithm, the authors propose a heuristic to compute maximum number of disjoint dominating sets in an undirected graph. Compared to the work in [SP01], the maximum number of disjoint dominating sets is greater or equal than the maximum number of covers. This is valid because the sensors in one cover also form a dominating set. Therefore, approach in [Cardei02] potentially achieves better energy saving than approach in [SP01]. However, the approach [SP01] can achieve the full area coverage constraint, but there are small coverage lapses in the monitored area for approach in [Cardei02]. For example, as in Figure 2, there is only one set cover {S1, S2}, but there are two disjoint dominating sets {S1} and {S2}. Considering disjoint dominating sets compared [Cardei02] with disjoint covers method [SP01], in this example the longevity of the network is double from the point of view of energy resources. However, there are some uncovered parts of the target area in [Cardei02].

Figure 2. two sensors are deployed on monotoried area. 2.1.2 Minimize the number of active nodes Approaches in [SP01] and [Cardei2002] are to divide sensors to maximum number of disjoint sets (or dominating sets), each set can monitor the sensed area. These sets are activated successively in order to prolong lifetime. Tian and Georganas [Tian03, Tian02] propose another energy-efficient node scheduling scheme for area coverage in synchronous networks where sensing range is equal to the transmission range. The main objective of the algorithm is to minimize number of working nodes, as well as maintain the original sensing coverage. It requires every node to be aware of its own and its neighbor location information. At the beginning of each round, each node selects a time-out interval. At the end of the interval, if a node sees that neighbors together cover its monitoring area, the node transmits a retreat message to all its neighbors and goes into the sleep mode. Otherwise, the node remains active, but does not transmit any message. The process repeats periodically to allow for changes 5

in monitoring status. In this scheme, each node must know its neighbor location information and has to do accurate geometrical calculation to determine whether or take an off-duty status. Tian and Georgannas [Tian04] propose three different alternative node scheduling schemes for area coverage, which are location and calculation-free. In the Nearest-neighbor based scheme, after each node collects distance information to its all neighboring nodes, it determine its working status by examining if its distance to the nearest neighbor is not more than the threshold. If affirmative, the node can take off-duty status. In the neighbor number-based scheme, each node collects its all neighbors’ information, and determines its working status by examining if its neighbors’ number exceeds a given threshold. If affirmative, the node will take off-duty status. In probability-based scheme, each node generates a random number from [0,1) and checks if the number is less than the off-duty probability, if it is, the node takes off-duty status, otherwise, it sets its status as on-duty.

2.2 Connected Area Coverage In all schemes introduced in above sections, the working sensors may not be connected, and thus reporting to a monitoring center can not be proceeded. In order to collect information from the sensor nodes to monitoring center, the active sensors are desired be connected. A frequently addressed objective is to determine a minimal number of active sensors to maintain monitoring the given area as well as connectivity. Next we will introduce several connected coverage mechanisms. 2.2.1 Transmission range equals to the sensing range Ye et al [YZCLZ02, YZCLZ03] present PEAS, a distributed, probing-based density control algorithm for robust sensing coverage. In this work, a subset of sensor nodes operative mode maintains coverage while others are put into sleep. Each sensor node has the same probing range Rp and may vary its transmission power and choose a power level to cover a circular area given a radius. In PEAS, each node has three operation modes: sleeping, probing and working. Initially all sensor nodes are in the sleeping mode. Each node sleeps for an exponentially distributed duration generated according to a probability density function (PDF). When sleeping time expires, the sensor enters the probing mode. The probing node uses an appropriate transmission power to broadcast a PROBE message within its local probing range Rp. Any working node(s) within that range should respond with a REPLY message, also sent within the range of Rp. If the probing node hears a REPLY, it goes back to the sleeping mode for another random period of time. If the probing node does not hear any REPLY, it enters the working mode and starts monitoring until it fails or consumes all its energy. The probing range can be adjusted to achieve different levels of coverage redundancy. The choice of probing range also affects network connectivity. The authors also study the asymptotic connectivity of PEAS. With this protocol, the probability having full coverage of a monitored area is close to 1 if the sensing range Rt ≥ (1 + 5) R p . There is a problem that PEAS does not ensure that the coverage area of a sleeping node is completely covered by other active nodes, i.e. it does not guarantee complete coverage. Figure 3 illustrates PEAS, with the black nodes being active and the white 6

nodes being in sleep mode, because each white node is contained within Rp to one of the active nodes. But there is a coverage area of the white node which is not completely covered by the active nodes. This protocol has limited usefulness because it is probabilistic and does not ensure full area coverage.

Figure 3. PEAS for area coverage. Carle and Simplot [Carle04] propose another mechanism for energy-efficient connected area coverage for the case when all sensor nodes have the same range and the communication range equals the sensing range. The goal of the algorithm is to select the minimum number of active nodes to cover the given area. The authors modify one of existing protocol for connected dominating-set protocol (e.g. Dai and Wu’s algorithm in [Dai 03] to find area coverage rather than node coverage. In the modification protocol, each node computes its timeout function based on its priority and listens to messages from other nodes before deciding its dominating status at the end of a timeout interval. A node choosing gateway status always transmits a message (positive advertising) to all its neighbors. A node choosing not to monitor its area has the option of transmitting this information to its neighbors (negative advertising) or not. The protocol runs: using a simple perimeter coverage scheme [Tian02], a node computes the area covered by each node that transmits either positive or negative advertising and includes the transmitting node in a subset; at the end of its timeout interval, the node computes a subgraph of its one-hop neighbors that sent advertisements(these are its neighbors with higher priority); If this subgraph is connected and the nodes in subgraph fully cover the node’s area, the node opts for sleeping status; Otherwise, the node chooses active status. The distributed dominant-pruning algorithm can prove that the set of active nodes is connected. Figure 4 shows an example that how a node does decision for its status. (a) Node A decides to be active because its active neighbors do not fully cover its monitoring area. (b) Node A decides to be inactive because its monitoring area is covered by its active neighbors that are connected. (c) Node A decides to be active because its active neighbors are not connected.

7

Figure 4. Example of configurations for area-coverage decision.

2.2.2 Transmission range is at least twice of the sensing range Wang et al [Wang03] and Zhang et al [Zhang 03] first discuss how to combine consideration of coverage and connectivity maintenance in a single activity scheduling. An important, but intuitive result for maintaining sensing coverage and connectivity by keeping a minimal number of sensor nodes in the active mode has been proved by Zhang and Hou [Zhang03]. The authors first investigate the relationship between coverage and connectivity, and prove that if the transmission range is at least twice of the sensing range, a complete coverage of a convex area implies connectivity among the working nodes in the active mode. Second, the authors derive, under the ideal case in which node density is sufficiently high, a set of optimality conditions under which a subset of working sensor nodes can be chosen for full coverage. Based on the optimality conditions, the authors propose a decentralized and localized density control algorithm, called optimal geographical density control(OGDC). OGDC is under assumptions: the transmission range is at least twice of the sensing range, each node is aware of its own position, and all nodes are time synchronized. At any time, a node is in one of the three states: UNFECIDED, ON, OFF. Time is divided into rounds. At the beginning of each round, all the nodes wake up, set their states to “UNDECIDED”, and carry out the operation of selecting working nodes. By the end of the execution, all the nodes change their states to either “ON” or “OFF” and remain in that state until the beginning of the next round. This decision is based on the power-on messages. Every node keeps a list with neighbor information. When a node receives a power-on message, it checks whether its neighbors cover its sensing area, and if so, it will change to OFF state. A node decides to change into the ON state if it is the closest node to the optimal location of an ideal working node. The process of selecting working nodes (in a decentralized manner) in each round commences by randomly selecting a sensor node A to be the starting node (Figure 5). Then one of its neighbors with an approximate distance of 3r , B, is selected to be a working node. To cover the crossing point of disk A and B, the node, Q, whose position is closest to the optimal position C is then selected to become a working node. The process continues until all the nodes change their states 8

to either “ON” or “OFF”, and the set of modes with “ON” states forms the working set.

Figure 5. The process of selecting working nodes. Wang et al [Wang03] also prove that the transmission range is at least twice of the sensing range, and the area to be covered is convex, then the area coverage also implies connectivity among the covering sensors. Wang [Wang 03] and Zhang et al [Zhang 03, 05] provide a sufficient condition for safe scheduling integration in those fully covered networks. However, random node deployment often makes initial sensing holes inside the deployed area inevitable even in an extremely high-density network. Tian and Georgnnas [Tian05] enhance their work to support general wireless sensor networks by proving another conclusion: “the communication range is twice of the sensing range” is the sufficient condition and the tight lower bound to ensure that complete coverage preservation implies connectivity among active nodes if the original network topology (consisting of all the deployed nodes) is connected. That is, the authors prove that if active nodes form a completely coverage, and the original topology is connected, when the transmission range is twice of the sensing range, then the induced subgraph by active nodes is connected. When the transmission range is less than twice of the sensing range, then the induced subgraph by active nodes may be disconnected. Wu and Yang[Wu04] extend a result from [Zhang 03] where only uniform sensing range among all sensors is used. Wu and Yang consider cases where each sensor is able to select one of two or three adjustable ranges and the transmission range is at least twice of the sensing range, with the goal of minimizing the overlapped sensing area. They present two new energy-efficient models of different sensing ranges. Jiang and Dou[Jiang04] describe several improvements to algorithm in [Tian02]. The authors present a distributed and localized density control algorithm for wireless sensor networks, which all nodes have the same sensing range and the transmission range is at least twice of the sensing range. The authors apply the perimeter criterion that a circle is covered completely if perimeters of other circle covering it are fully covered by other covering circles. In the algorithm, a sensor is in one of the two states: “ACTIVE” and “NON-ACTIVE”. At the beginning, all nodes are in ACTIVE state. 9

Network lifetime is divided into rounds, and each round has a scheduling phase followed by a sensing. The scheduling phase is further divided into two sub-phases: neighbor discovery phase and evaluating phase. At the beginning of the neighbor discovery phase, node broadcast a hello message to its one-hop neighbors and sets a timer to wait for neighbors’ hello message. Upon this timer expires, node has obtained knowledge about one-hop neighbors and construct its neighbor set and effective neighbor set. Then entering the evaluating phase, sensor begins to evaluate the density control algorithm to decide which state it should go. In each time round, the ACTIVE nodes work for the sensing task and the NON-ACTIVE nodes will turn off their sensing and communication units to save energy.

2.2.3 Arbitrary Ratio of transmission range to sensing range Gallais et al [Gallais08] generalize the approach in [Carle04] for an arbitrary ratio of sensing range and transmission range. The approach are based on a time-out scheme, in addition to being fully localized, has a very small communication overhead. When a round starts, each node selects a time out and listens to messages sent by other nodes before the time-out expires. Sensor nodes whose sensing area is not fully covered when the deadline expires decide to remain active for the considered round and transmit an activity message announcing it. There are four variants in the approach, depending on whether or not withdrawal and retreat messages are transmitted. Covered nodes decide to sleep, with or without transmitting a withdrawal message to inform neighbors about the status. After hearing from more neighbors, active sensors may observe that they became covered and may decide to alter their original decision and transmit a retreat message. In this approach, the covering criterion which has been already applied in [Jiang04], [Xing05] and [Zhang05] is applied on the borders of the sensing area of each sensor[Gallais 06], the node using it verifies whether or not its sensing area is fully covered. The details of the protocol include how the time-out is decided, and how the area coverage and connectivity tests are performed. The test for connectivity of covering circles must be performed when the transmission range is less than twice of the sensing range, that is, when the transmission range is less than twice of the sensing range, a node can decide to turn off if and only if its neighbors fully cover and are also connected. Sheu et al [Sheu 07] study query execution over a specific geographical region. And propose an efficient distributed protocol to find minimum number of connected active sensor nodes to cover the queried region. Assumptions: Transmission ranges and sensing ranges differ between sensors, and the sensing range of a sensor node may differ from its transmission. The proposed protocol consists of two phases-self-pruning phase and sensing nodes discovery phase. In the beginning of the protocol, each sensor node is assumed to have the information of its 1-hop-cover neighbors. In the self-pruning phase, each node checks whether or not its sensing area is completely covered by its higher priority neighbors by using the perimeter covering criterion in [Huang 03]. If no, it becomes a sensing node. The authors prove that the 10

sensing nodes selected by self-pruning can fully cover the queried region when the deployed sensor nodes cover the queried region. In the sensing nodes discovery phase, each of the considered perimeters is subdivided into sub-perimeters, based on the intersections with other considered circles. For each such sub-perimeter, the sensor with the highest priority, among nodes covering this sub-perimeter, is active. After the two phases, the selected sensing nodes are connected and can cover the queried region. Gupta et al [Gupta 03] study the connected sensor coverage problem: Given a query over a sensor network, select a minimum set of sensors, called connected sensor cover, such that a) the sensing regions of the selected sensors cover the entire geographical region of the query, and b) the selected sensors form a connected communication graph. The authors first prove that the connected sensor coverage problem is NP-complete and then propose a centralized greedy algorithm. The proposed algorithm is as follows: Let M be the set of sensors already selected for inclusion in the connected sensor cover by the greedy algorithm at any stage. Initially, M is an empty set. The algorithm starts with including in M an arbitrary sensor that lies within the query’s region. At each stage, the greedy algorithm selects a sensor C along with a path of sensors P that forms a communication path between C and some sensor in M with maximum benefit of P, add selected path P to M, till query’s region is covered by sensors in M. In the algorithm, the benefit of P is defined as the number of uncovered valid subelements covered by P per sensor. At any stage of the algorithm, the communication subgraph induced by M is connected. A straightforward distributed version of the same algorithm is also given. Zhou et al [Zhou04a] address Variable Radii Connected Sensor Cover problem which generate the problem in [Gupta03]: Given a query region in the network, each node has vary its sensing range and transmission range where they can not exceed the maximum sensing range and the maximum transmission, selecting a subset of sensors which forms connected sensor cover such that the total energy cost (including sensing cost and transmission cost) is minimized. The authors design various centralized and distributed algorithms-Voronoi based algorithm, Greedy algorithm and Steiner tree based algorithm. One of the designed centralized algorithms (called CGA) is shown as O(logn)-approximation. CGA works as follows. CGA maintain a set of selected sensors M along with their assigned transmission and sensing range, and increases the covered region while keeping connectivity of M. At each stage, either adds to M a “path” of sensors or increases the sensing range of a sensor in M, whichever gives the maximum “benefit”. CGA terminates when the given query region is completely covered by the assigned sensing regions of the sensors in M.

2. 3 k-area coverage Sensor nodes usually are deployed into remote and inhospitable area to monitor targets. Because severe weather conditions or other hash physical environment in the sensor filed or the over-loading of working sensor nodes, sensors are prone to fail. It is therefore crucial to construct a k-coverage problem ( k ≥ 1 ), in which each physical point is covered at least k different sensor nodes. There are many existing works to 11

address k-coverage problem. Next, we give a survey on k-coverage problem. Sensor networks are often desired to prolong the lifetime of operation. This is usually achieved by putting sensors to sleep for most of their lifetime. On the other hand, the intrusion detection applications require guaranteed k-coverage off protected region at all times. To determine the appropriate number of sensors to deploy that achieves both goals simultaneously becomes a challenging problem. Kumar et al [Kumar 04] study this problem: Given an area to be protected, how many sensors should be deployed so that every point in the region is covered by at least k sensors, and given that the network must last for a specified length of times? The authors consider three kinds of deployments for a sensor network on a unit squarea n×

n grid, random uniform (for all n points), and Poisson (with density n). In

all three deployments, each sensor is active with probability p. A critical condition for three deployments is derived. And the authors show that the conditions for deterministic deployments are similar to the conditions for random deployments. 2.3.1 k-area coverage without connectivity guarantee The k-area coverage problem addressed in [Gallais06a] consists in building k distinct subsets of active nodes (layers) so that each layer covers the area. The authors propose a decentralized protocol. Sensors are randomly deployed over a square area and activity is imagined in a rounded fashion. At each round, every node decides its status between either monitoring for the entire round or getting passive until the next decision phase. Every sensor is aware of required coverage degree, denoted as k. A node A can find smallest i so that ith layer of the area covered by that node is not fully covered by its neighbors. Then, if i ≤ k , A decides to be active at layer i and sends a positive acknowledgement announcing its activity layer i and its geographical position. Otherwise, it decides to be passive and no message is sent. Figure 6 shows that that sensor A first evaluates the coverage provided by neighbors of layer 1(black nodes on Figure 6(b) before deciding to evaluate the coverage at layer 2(Figure 6(c)). Finally, Figure 6(d) shows that A is covered at all 2 layers. A takes its activity decision depending on its required coverage degree k. If k>2, then A gets active at layer 3 and sends a position acknowledgment. If k=2, then A gets passive without sending any message.

Figure 6. Evaluation of coverage. Cai et al [Cai07] propose a precise and energy-aware coverage control protocol, named Area-based Collaborative Sleeping (ACOS). Based on the net sensing area of 12

a sensor, which is covered only by the sensor and not covered by other active sensors, the ACOS controls the mode of sensors to maximize the coverage degree, minimizing the energy consumption. Each sensor node has four states: Sleep, PreWakeUP, Awake, and Overdue. Initially, each sensor is Sleep with timer, when node s wake up, its state changes from Sleep to PreWakeUP, node u sends a broadcast message, to its neighbors within radius 2r and waits for T seconds. When any neighboring sensor v with Active receives this message, node v sends back reply message including its location. After u receives reply messages, u computes the net area ratio, if the net area ratio is less than the threshold, u return back to Sleep state. If the net area ration is more than the threshold, u changes to Awake state, and initialize its wake timer and broadcast a Wake-Notification message. When node u is still in the Awake state and its wake time expires, it changes from Awake to Overdue state. When a node which is in Awake or Overdue state hears a Wake_Notification message, it re-calculates the net area ratio to repeat the process. The state transition diagram is in Figure 7.

Figure 7. State transition diagram of ACOS. Hefeeda and Baghen [Heffeda07] study –coverage problem: Given n already-deployed sensors in a target area, and a desired coverage degree k ≥ 1 , select a minimal subset of sensors to cover all sensor locations such that every location is within the sensing range of at least k different sensors. The authors model the k-coverage problem as a set system (X, R) where X is the set of sensor locations and RC is a the collection of subsets of X created by intersecting disks of radius r with points of X, for which an optimal hitting set corresponds to an optimal solution for k-coverage. And propose an approximation algorithm with a logarithmic ratio for computing near-optimal hitting sets [BG95]. A fully distributed version of the proposed algorithm is designed and implemented. There are various theoretical works on area coverage problem in wireless sensor networks. Xing et al [Xing2004] presents a theoretical analysis of greedy geographic routing protocols on wireless sensor networks that must provide sensing coverage over a geographic area. The authors prove that the Greedy Geographic Forwarding[Karp00, Stoj01] and their new greedy protocol always succeed in any sensing covered network when the communication range is at least twice the sensing range. Liu and Towsley [LiuB04] approach the coverage problem from a theoretical perspective and explored the fundamental limits of the coverage of a large-scale sensor network. The authors study three fundamental coverage measures of 13

large-scale sensor networks: Area coverage, node coverage, and detectability. These measures are determined by basic network parameters and have important implications on network planning and protocol performance of sensor networks. Ke et al [Ke07] proves that deploying sensors on grid points to construct a wireless sensor network that fully covers critical grids using minimum sensors (Critical-Grid Coverage problem) and that fully covers a maximum total weight of grids using a given number of sensors(Weighted-Grid Coverage problem) are each NP-Complete.

k-area coverage with the transmission range being at least twice sensing range The network connectivity is rarely treated in existing works on k-area coverage. Wang et al [Wang03] prove that when the transmission range is at least twice the sensing range, a set of working nodes that forms k-coverage a convex region forms a k-connected communication graph. Tian et al [Tian05] enhance the result in [wang03] for general random deployment network to prove that when the transmission range is at least twice of the sensing range, and the system sensing coverage is completely k-degree preserved after node scheduling, if a network graph is originally k-connected, the induced subgraph by the active nodes must be k-connected. Most of existing results on k-area coverage rely on this theorem to focus on area coverage only without addressing the problem of the connectivity preservation. Wang et al [Wang 03] generate the result in [Zhang 03]. And propose the coverage configuration protocol (CCP) that is a decentralized protocol that only depends on local states of sensing neighbors and can provide different degrees of coverage requested by applications. In CCP, each node determines its eligibility using the k-coverage eligibility algorithm based on the information about its sensing neighbors, and may switch state dynamically when its eligibility. Given a requested coverage degree k, a node is ineligible if every location within its coverage is already k-covered by other active nodes in its neighborhood. The authors prove that a convex region is k-cover if it contains intersection points between sensors or between sensors’ and region boundary and all these intersection points are k-covered. Based on this, a sensor is ineligible to turn active if all the intersection points inside its sensing circle are at least k-covered. Every node maintains a table of known sensing neighbors based on the beacons (hello messages) that it receives from its communication neighbors. A node can be in one of three states: SLEEP, ACTIVE and LISTEN. All nodes start in the SLEEP state for a random time. When the sleep timer expires, a node in the sleep state enters LISTEN state. When a beacon (HELLO, WITHDRAW or JOIN message) is received, a node in the listen state evaluate its eligibility. If it is eligible, it starts a join timer, otherwise it returns to the SLEEP state. If it becomes ineligible after the join timer is stated, it cancels the join timer. If the join timer expires, the node broadcast a JOIN beacon and enters the active state. If the listen timer expires, it starts a sleep timer and returns to the SLEEP state. Once a node is in the active state, it re-evaluate the coverage eligibility every time it receives HELLO message and decide whether to go into the SLEEP state or remain in the ACTIVE state. If the ratio of the communication range to the sensing range is more than 2, CPP 2.3.2

14

can guarantee connectivity. But CPP does not guarantee connectivity when the ratio of the communication range to the sensing range is less than 2. The authors also present a simple approach for integrating CCP with an existing connectivity maintenance protocol, SPAN [ChenJ01] to provide sensing coverage and communication connectivity. The proposed protocol in [Sheu 07] can be extended to solve k-coverage problem, which can find a set of sensing nodes satisfy the k-coverage request. The protocol is as follows: Assume that a set SN1 of sensing nodes is got in the self-pruning phase. If a non-sensing node is aware of its neighboring nodes in SN1, it can delete these sensing nodes from its 1-hop-cover neighboring set and execute the self-pruning again to determine whether it can be a sensing node. After the second iteration, all the non-sensing nodes can determine their roles-sensing nodes or non-sensing nodes, then get the second coverage set SN2 to fully cover the queried region if the remaining sensor nodes can fully cover the queried region. SN1 and SN2 form a 2-coverage. Applying the above procedures, k-coverage can be got. Lu et al [Lu06] address the k-coverage Maintenance Problem: Given a sensor group S deployed in region R and a natural number k, find subset S ' with the minimum number of sensors such that S ' is able to maintain k-coverage. That is, for any position v in R, if v can be k-covered by S, it must be k-covered by S ' ; Otherwise, the coverage degree of v in S ' is same as in S. It assumes that the transmission range is at least twice the sensing range. The authors propose a scalable coverage maintenance scheme (called as SCOM). SCOM assume that each node knows its location and can acquire the location of neighbors through one-hop communication. Time is slotted into rounds. At the beginning of each round, SCOM runs in two phases: Decision phase and optimization phase. In the decision phase, each sensor is initially in BOOTSTRAP state and has an empty active neighbor list. Before making the decision of turning on or off, each sensor sets a back-off timer depending on its residual energy. When a sensor’s timer expires, the sensor checks whether its sensing region is k-covered by the sensors in the active neighbor list using the redundancy eligibility rule for homogenous or heterogeneous, and switches to ACTIVE or INACTIVE state accordingly. If a sensor decides to turn into ACTIVE state, it broadcast a TURNON beacon with its coordinates to it’s the neighbors. Upon receiving the TURNON beacon, a neighbor adds the sender into the active neighbor list. In the optimization phase, sensors optimize the coverage by turning off redundant active sensors while still guaranteeing the required coverage. The Sensor Scheduling for k-Coverage(SSC) problem is investigated in [Gao06]. Which requires to efficiently schedule the sensors, such that the monitored region can be k-covered throughout the whole network lifetime with maximizing network lifetime. All the sensors have uniform transmission range and sensing range. And the transmission range is at least twice the sensing range. The authors model the SSC problem to find maximum number of disjoint k-cover sets. In [Huang03], the authors prove that the entire monitored region is k-covered if and only if each sensor in the monitored region is k-perimeter-covered. Consider any two sensors si and sj. A point on the perimeter of si is perimeter-covered by sj if this point is within the sensing 15

range of sj. si is k-perimeter-covered if all points on the perimeter of si are perimeter-covered by at least k sensors other than si itself. A segment of si’s perimeter is k-perimeter-covered if all points on the segment are perimeter-covered by at least k sensors other than si itself. Figure 8 shows an example: the perimeter of si between two arrows is covered by sensor sj. Based on this result, Gao et al propose a greedy algorithm, PCL-Greedy-Selection(GS). The main idea of GS is to iteratively construct subset by choosing sensors from the area with the lowest sensor density. When construct an individual subset, the sensor with a small PCL value is added to the subset. In addition, the authors develop a guideline for designing a sensor deployment by employing density control.

Sj

Si

Figure 8. An example of perimeter-coverage. 2.3.3 Connected k-area coverage Area coverage protocols aim at turning off redundant sensor nodes while ensuring full coverage of the area by the remaining active nodes. Providing k-area coverage means that every physical point of the monitored area is sensed by at least k sensors. Connectivity of the active nodes subset must also be provided so that monitoring reports can reach the sink stations. Existing solution hardly address these two issues as a unified one. The works in [Zhou04b, Zhou05, Gallias07] address coverage and connectivity as a unified one. Next, we review them. Zhou et al [Zhou04b, Zhou05] study the k-area coverage problem and the connectivity preservation problem. Zhou et al consider the problem of selecting a minimum size connected K-cover, which is defined as a set of sensors M such that each point in the sensor network is “covered” by at least K different sensor in M, and the communication graph induced by M is connected. The authors design a centralized O(logn)-approximation algorithm. The greedy algorithm is a generalization of the centralized approximation algorithm in [Gupta 03] for the connected 1-coverage problem. The Greedy Algorithm maintains a set of M of selected sensors and at each stage, select a candidate sensor without belong to M and a candidate path of sensor with maximum “K-Benefit” with respect M, add the selected path to M. This is repeated until the query region is k-covered by M. The distributed version of the Greedy algorithm is also given. Zhou et al [Zhou 05] address a more general, variable radii sensor model, choosing a subset of sensors such that they maintain a k1-connectivity and k2-cover, wherein every sensor can adjust both its sensing and transmission ranges, and the overall energy consumption is minimized. The energy consumption includes sensing energy consumption and transmission energy consumption. The authors propose a distributed and localized Voronoi-based algorithm. The Voronoi-based algorithm 16

works as follows. Initially, each sensor node in the sensor network is active, and gathers locations of all the nodes in the l-hop active neighborhood. Each active sensor node computes its k2 th –order local Voronoi cell, and the neighbors in the k1-RNG over active nodes. It uses the V-R assignment method to assign itself sensing and transmission radius. Each node computes its sleep benefit, based on the sleep benefit, choose a sensor with the most sleeping benefit among all its local voronoi neighbors to become inactive. A sensor node is chosen to become inactive only if the remaining active sensors are capable of k1-covering the query region and maintaining k1-connectivity of their communication graph. Repeat above processes. The algorithm terminates when no more sensors can be made inactive. Gallais and Carle [Gallais07] consider connected k-coverage problem. And consider two definitions for the k-area coverage problem: the flat k-area coverage problem and the layered k-area coverage problem. The authors propose a localized algorithm that can be applied to time-synchronized networks. Each node selects a time-out, which depends on the remaining energy, and has some random number, while listening to messages from neighboring nodes. Once the timeout ends, u takes its activity decision based on known neighboring nodes. It so evaluates its coverage according to the appropriate coverage evaluation scheme. If completely k-covered according to the flat k-area coverage issue, if u decides to be passive and turns into sleep mode. Otherwise, u remains active and sends a positive acknowledgment message which contains the values of its communicating and sensing range with its position. Any node with a longer timeout that receives this message adds u to its neighbor table. For the layered k-area coverage issue, Nodes still listen for messages during a given timeout before making their activity decision and choosing an activity layer whose number is included in the messages. A node u sorts its neighbors according to a number of layers. Then, u evaluates if at least k virtual activity layers fully cover its area S(u). If no, u remains active and chooses the uncovered activity layer which has the lowest number, and sends an activity message to announce its status. About connectivity, when CR ≥ 2 SR , connectivity is ensured. When CR0, and an undirected graph G=(V, E) find a subset of nodes C ⊆ V such that each node in V is dominated by at least k different nodes in C, and the number of nodes in C is minimized. The k-CCS is to add another constraint that 23

subgraph induced by C is connected. The k-CS is formulated as an integer linear programming and then a centralized LP-based algorithm for k-CS is proposed. LP-based algorithm includes two steps: the first step is to compute the optimal solution of relax LP problem, the second step is to round the optimal solution to solution of ILP: if the solution of optimal solution of LP is greater than some value, set this variable to 1, otherwise, zero. Non-global solutions for k-CS/k-CSS are proposed: cluster based algorithm and pruning-based algorithm. Cluster base algorithm runs as follows: sequentially apply a traditional clustering algorithm k time to get k sets of clusterheads, find gateways to connect the first set, then add other nodes to all clusterheads and gateways to form the k-CS/k-CSS. In the pruning-based algorithm, all nodes are initially assumed as active. Each node using 2-hop neighborhood to determines its status. Initially, all nodes are marked. Each node u is given a unique priority, L(u). Each node broadcast its neighbor set N(u), and build a subset C (u ) which is formed by u’s all neighbors with higher priorities than u. Node u is umnmarked if C(u) is connected(this constraint is removed for k-CS) and for any neighbor w of u, there are k distinct nodes in C(u), such that w is a neighbor of all the k nodes. All marked nodes form k-CS/k-CSS. Li et al [Li07b] address k-connected m-coverage problem which is different from the coverage problem in [Yang06]. The coverage problem is : Given a set of sensors and a set of targets, and a coverage mapping from sensors to targets, and constants k and m, k ≥ 1, m ≥ 1 , find a minimum number of sensors such that each target is covered at least m sensors and the selected sensors is k-connected. The k-connected coverage problem and k-connected m-coverage problem are NP-hard. In [Li07b], the authors first study m-coverage problem, which is formulated as ILP, then propose an approximation algorithm based on LP. Based on solution of m-coverage problem and algorithms for k-connected augmentation [Li07a], two heuristics (kmTS algorithm and kmReverse algorithm) are proposed for k-connected m-coverage problem. Two algorithms include two steps: the first step is to construct a m-coverage of targets; The second step is to increase small size nodes to this m-coverage such that the subgraph by these increased nodes and nodes of m-coverage is k-connected using the algorithms [Li07a].

3.3.3. Breach Coverage Network lifetime has been recognized as an important factor in sensor network design. To extend sensor network lifetime, one potential approach is to divide sensors into disjoint subsets, each of which can cover all targets. Each subset is switched to active mode and sleep mode alternatively, so that at any time there is only one set of sensors active to prolong network lifetime. The size of sensor cover sets is not put any constraint. However, the number of deployed sensors is usually very large and the base station may not provide a bandwidth large enough for receiving data from all sensors in the cover sets. In this situation, a complete coverage is sometime not 24

available. Maybe there exists some targets can not be monitored by any sensor. A target is in breach if it is not monitored by any sensor. There are some coverage breach problems studied in the literature [Slijepcevi01, Chengxi05, 07, WangC07, Thai05]. Cheng et al [Chengx05, Chengx07] study three coverage breach problems: Minimum Breach problem, Minimum Individual Breach Time problem and Minimum Maximal Breach problem. The Minimum Breach problem is : Given a set A of fixed points and a set S of sensors, organize sensors into disjoin subsets Ci, i=1,2…, K, where each subset | Ci |≤ W and the overall breach is minimized. The authors prove the three problems are NP-hard. The three coverage breach problem are formulated as 0-1 linear integer programming problems. The minimum breach problem is formulated as a 0-1 integer programming problem as following: K

M

min{∑∑ (1 − ykj )} k =1 j =1

N

∑a x i =1

ij k ,i

≥ yk , j

K

∑x N

∑x i =1

= 1,

∀i = 1,...N ;

=W,

∀k = 1..., K ;

k ,i

k =1

k ,i

∀j = 1,..., M , k = 1,...K ;

yk , j ∈ {0,1}

∀k = 1...K , j = 1,...M ;

xk ,i ∈ {0,1}

∀k = 1...K , i = 1...N .

A Greedy approximation algorithm and a heuristic based on the LP-relaxation are proposed. In a greedy strategy, iteratively pick the most coverage-effective sensor and put it in its fit position until all sensors are put into subsets. Each subset can have at most W sensors. LP-based heuristic(called Relaxation) includes three steps; In the first step, the integer programming(IP) problem is relaxed to a linear programming (LP) problem, and compute an optimal solution for LP. In the second step, using greedy strategy to find an integer solution based on the optimal solution of LP. In the third step, the solution from (IP) problem is used to construct the subsets. In [Thai05], Thai et al present two new linear programming based models, Minimum Coverage Breach under Bandwidth constraints (MCBB) and Maximum Network Lifetime under bandwidth constraints (MNLB) to solve the joint optimization on energy and bandwidth utilization. MCBB problem is: Given a collection C of subsets of a finite set R, find a family of p order pairs (Sj, tj) such that the total coverage breach is minimized. Where Sj is a set cover and tj is the time duration between 0 and 1 for Sj to be active. MNLB problem is to find a family of p

25

order pairs (Sj, tj) such that

∑

p

t

j =1 j

is maximized. In the two models, sensors are

organized into non-disjoint set cover. The MCBB problem and the MNLB are NP-hard, and can be formulated as mix integer programming. The authors propose two approximation algorithms based on the optimal solution of relax linear programming to solve them.

4.

Conclusion

In the chapter, we investigate the current works on coverage problem in sensor networks, and classify them into two categories: sensor area coverage and target coverage. We focus on the most representative problems in each domain and present a comprehensive review and analysis of various existed algorithms and techniques.

Acknowledgement This research is partially supported by the National Natural Science Foundation of China under grant 10671208, and Key Laboratory of Data Engineering and Knowledge Engineering (Renmin University of China), MOE.

Reference [Abrams04] Z. Abramsm A. Goel, and S. Plotkln, “Set K-cover Algorithms for Energy Efficient Monitoring in Wireless Sensor Networks”, Proc. Of International Symposium on Information Processing in Sensor Networks, (ISPN),2004, pp. 424-432. [Akyildiz02] I.F. Akyildiz, W. Su, Y. Sankarasubrmaniam and E. Cayirci, “A Survey on Sensor Networks”, IEEE Communication Magazine, (Aug, 2002) 102-114. [BG95] H. Bronnimann and M. Goodrich, “Almost Optimal Set Covers in Finite VC-dimension,” Discrete and Computational Geometry, 4(14), 1995. [Cai07] Y. Cai, M. Li, W. Shu, and M.Y. Wu, “ACOS: An Area-based Collaborative Sleeping Protocol for Wireless Sensor Networks,” International Journal of Ad Hoc & Sensor Wireless Networks, Vol.3, No.1, pp.77--97, 2007. [Cardei02] Cardei, D. MacCallum, X. Cheng, M. Min, X. Jia, D.Y. Li, and D.-Z. Du: “Wireless Sensor Networks with Energy Efficient Organization,” Journal of Interconnection Networks, 3(2002) 213-229. [Cardei06] M. Cardei, Jie Wu: “Energy-efficient Coverage Problems in Wireless Ad-hoc Sensor Networks”, Computer Communications 29(4): 413-420 (2006). [CardeiDu05] M. Cardei and D.Z. Du, “Improving Wireless Sensor Network Lifetime through Power Aware Organization”, ACM Wireless Networks, 11( 2005), pp. 333-340. [CardeiW05] M. Caredi, J. Wu, M. Lu, M.O. Pervaiz, “Maximum Network Lifetime in Wireless Sensor Networks with Adjustable Sensing Ranges”, WIMob, Proc. of the IEEE International Conference on Wireless and Mobile Computing, Networking and Communications, 2005, pp.438-445. 26

[Carle04] J. Carle and D. Simplot, “Energy Efficient Area Monitoring for Sensor Networks”, IEEE Computer, vol. 37, No. 2(2004), pp. 40-46. [Chakrabarty02] K. Chakrabarty, S. S. Iyengar, H. Qi and E. Cho, “Grid Coverage for. Surveillance and Target Location in Distributed Sensor Networks”, IEEE Transactions on Computer, 2002,51(12):1448-1453. [ChenJ01] B. Chen, K. Jamieson, H. Balakrishnan, and R. Morris, “Span: An Energy-Efficient Coordination Algorithm for Topology Maintenance in Ad Hoc Wireless Networks,” MobiCom 2001, pp. 16-21. [Chengx05] X. Cheng, L. Ruan, and W. Wu, “Achieving Minimum Coverage Breach under Bandwidth Constraints in Wireless Sensor Networks”, INFOCOM05. [Chengx07] X. Cheng, L. Ruan, and W. Wu, “Coverage Breach Problems in Bandwidth-Constrained Sensor Networks”, ACM Transactions on Sensor Networks, 2(3), 2007. [ChenK07] Jie Chen and Xenofon Koutsoukos, "Survey on Coverage Problems in Wireless Ad Hoc Sensor Networks", IEEE SouthEastCon 2007. Richmond, VA, March 22-25, 2007. [Cheng05] M.X. Cheng, L. Ruan, and Weili Wu, “Achieving Minimum Coverage Breach under Bandwidth Constraints in Wireless Sensor Networks”, Infocom’05. [Chong03] C. Y. Chong and S. P. Kumar, “Sensor Networks: Evolution, Opportunities, and Challenges”, Proc. IEEE, pp. 1247-1256, vol.91., no. 8, 2003. [CTW05]M. Cardei, M. Thai, Y. Li, and Weili Wu, “Energy-efficient Target Coverage in Wireless Sensor Networks”, Infocom’05. [Dai03] F. Dai and J. Wu, “Distributed Dominant Pruning in Ad Hoc Networks”, ICC2003, pp.353-357. [Gallias06a] A. Gallais, J. Carle, D, Simplot-Ryl, and Ivan Stojmenovic, “ Localized Sensor Area Coverage with Low Communication Overhead”, Proc. Fouth IEEE int’l Conf. Pervasive Computing and Comm. (PerCom) 2006. [Gallais06b] A. Gallais, J. Carle, and D. Simplot-Ryl, “Ensuring k-coverage in Wireless Sensor Networks with Realistic Physical Layers”, IEEE SENSORS 2006, pp.22-25. [Gallais07] A. Gallais and J. Carle, “An Adaptive Localized Algorithm for Multiple Sensor Area Coverage”, in Proc. of the 21st International Conference on Advanced Networking and Applications (AIAN’07). [Gallais08] A. Gallais, J. Carle, D, Simplot-Ryl, and Ivan Stojmenovic, “ Localized Sensor Area Coverage with Low Communication Overhead”, IEEE Transactions on Mobile Computing, vol.7, No. 5, 2008, pp. 1-12. [Gupta03] H. Gupta, S. R. Das, and Q. Gu, “Connected Sensor Cover: Self-organization of Sensor Networks for Efficient Query Execution”, Proceedings of the 4th ACM International Symposium on Mobile Ad Hoc Networking and Computing (MobiHoc), 2003. [Hefeeda07] M. Hefeeda and M. Bagheri, “Randomized k-Coverage Algorithms For Dense Sensor Networks,” In Proc. of IEEE INFOCOM 2007, pp. 2376--2380, Anchorage, AK, May 2007. [Huang03] C. F. Huang and Y. C. Tseng, “The Coverage Problem in a Wireless 27

Sensor Networks”, Proc. Of the 2nd ACM int. Conf. on Wireless Sensor Networks and Application, 2003. PP.115-121. [Jian04] J. Jiang and W. Dou, “ A Coverage Preserving Density Control Algorithm for Wireless Sensor Networks,” Proc. Third int’l Conf. Ad-Hoc Networks and Wireless (ADHOC-NOW),2004. [Jaggi06] N. Jaggi, and A. A. Abouzeid, “Energy-Efficient Connected Coverage in Wireless Sensor Networks”, Proceedings of 4th Asian International Mobile Computing Conference (AMOC 2006), Kolkata, India, January 4-7, 2006. [Kar03] K. Kar and S. Banerjee, “Node Placement for Connected Coverage in Sensor Networks”, Proc. of WiOpt 2003: Modeling and Optimization in Mobile, Ad Hoc, and Wireless Networks(2003). [Karp00] Brad Karp and H.T. Kung, “GPSR: Greedy Perimeter Stateless Routing for Wireless Networks”, Mobile Computing and Networking, 2000, pp. 243-254. [Ke07] W. Ke, B. Liu, and M. Tsai, “ Constructing a Wireless Sensor Network to Fully Cover Critical Grids by Deploying Minimum Sensors on Grid Points is NP-Complete”, IEEE Transactions on Computers, Vol.56, No. 5, 2007, pp. 710-715. [Li07a] D. Li, J Cao, M Liu, Y. Zheng, “K-connected Coverage Problem in Wireless Sensor Networks,” COCOA2007. [Li07b] D. Li, J Cao, D. Liu, “Algorithms for the m-coverage Problem and k-connected m-Coverage Problem in Wireless Sensor Networks,” NPC2007. [LiuB04] B. Liu and D. Towsley, “A Study of the Coverage of Large-scale Sensor Networks”. Proceedings of the 1st IEEE International Conference on Mobile Ad Hoc and Sensor Systems (MASS2004). Pp. 475-483. Fort Lauderdale, Florida, October 2004. [Liu06]H. Liu, P. Wan, and X. Jia, "Maximal Lifetime Scheduling for Sensor Surveillance Systems with K Sensors to 1 Target," IEEE Transactions on Parallel and Distributed Systems, vol. 17, no. 12, pp. 1526-1536, Dec. 2006. [Liu07] H. Liu, X. Jia, P. Wan, C. Yi, S. Kami Makki, and Niki Pissinou, "Maximizing Lifetime of Sensor Surveillance Systems," IEEE Transactions on Networking, vol. 15, no. 2, pp. 334-345, Apr. 2007. [Liuz06] Zheng Liu, “Maximizing Network Lifetime for Target Coverage Problem in Heterogeneous Wireless Sensor Networks”, LNCS4864, 2007, pp. 457-468. [LuW05] M. Lu, J. Wu, M. Cardei, and M. Li, “Energy-Efficient Connected Coverage of Discrete Targets in Wireless Sensor Networks”, ICCNMC 2005, LNCS3619, pp.43-52. [Lu06] J. Lu, J. Wang, and T. Suda, “Scalable Coverage Maintenance for Dense Wireless Sensor Networks”, IEEE SECON2006. [Yan03] T.Yan, T. He, and J.A. Stankovic, “Differentiated Surveilance for Sensor Networks”, Proc. First Int’l Conf. Embedded networked Sensor Systems, pp.51-62, 2003. [Schrijver1986] A. Schrijver, Theory of Linear and Integer Programming, Wiley, Chichester 1986. [Sheu07] J. Sheu, S. Tu, and C. Yu, “A Distributed Query Protocol in Wireless Sensor Networks”, Wireless Personal Communication, 41(2007), pp. 449-464. 28

[SSW05] D. Simplot-Ryl, I. Stojmenovie, and J. Wu, “Energy-efficient Backbone Construction, Broadcasting, and Area Coverage in Sensor Networks”, Handbook of Sensor Networks: Algorithms and Architectures (I. Stojmenovie ed.), Wiley, pp.343-380, 2005. [SP01] S. Slijepcevic, M. Potkonjak, “Power Efficient Organization of Wireless Sensor Networks”, IEEE International Conference on Communications 2001, 472-476. [Stojmenovic01] Ivan Stojmenovic and Xu Lin, “LOOP-free Hybrid Sing-path/flooding Routing Algorithms with Guaranteed Delivery for Wireless Networks,” IEEE Transactions on Parallel and Distributed Systems, vol.12(10),, pp. 1023-1032, 2001. [Tian03] D. Tian and N. D. Georganas, “ A Node Scheduling Scheme for Energy Conservation in Large Wireless Sensor Networks”, Journal of Wireless communations and Mobile Computing, 3(2003) 271-290. [Tian02] D. Tian and N. D. Georganas, “ A Coverage-preserving Node Scheduling Scheme for Large Wireless Sensor Networks”, Proc. of the 1st ACM Workshop on Wireless Sensor Networks and Applications(2002). [Tian04] D. Tian and N. D. Georganas, “Location and Calculation-free Node Scheduling Scheme in Large Wireless Sensor Networks,” Ad Hoc Networks, 2(2004), 65-85. [Tian05] D. Tian and N. D. Georganas, “Connectivity Maintenance and Coverage Preservation in Wireless Sensor Networks”, Ad Hoc Networks, 3(2005) pp. 744-761. [Thai05] T. Thai, F. Wang, and D. Du, “Coverage Problems in Wireless Sensor Networks: Designs and Analysis,” International Journal of Sensor Networks, special issue on Coverage Problems in Sensor Networks, 2005. [Wang03] X. Wang, G. Xing, Y. Zhang, C. Lu, R. Pless, C. D. Gill, “Interated Coverage and Connectivity Configuration in Wireless Sensor Networks”, in: Proceedings of the first ACM Conference on Embedded Networked Sensor Systems(SenSys 2003), Los Angles, November 2003. [Wu04] J. Wu and S. Yang, “Coverage and connectivity in sensor networks with adjustable ranges”, proc. Of the 2004 International Workshop on Mobile and Wireless Networks, 2004. [Wang06] J. Wang and N. Zhong, “Efficient Point Coverage in Wireless Sensor Networks”, Journal of Combinatorial Optimization, 3(11)2006 , pp. 291-304. [WangC07] C. Wang, M. Thai, Y. Li F. Wang W.i Wu, “Minimum Coverage Breach and Maximum Network Lifetime in Wireless Sensor Networks”, in Proc. of Global Telecommunications Conference 2007. GLOBECOM '07. IEEE. [Xing04] G. Xing, C. Lu, R. Pless, and Q. Huang, “Greedy Geographic Routing is Good Enough in Sensing Covered Networks”, INFOCOM2004. [Xing05] G. Xing, X. Wang, Y. Zhang, C. Lu, R. Pless , and C. Gill, “Integrated Coverage and Connectivity Configuration for Energy Conservation in Sensor Networks,” ACM Trans. Sensor Networks, vol.1 no.1. pp. 36-72, 2005. [Xu06] X. Xu and S. Sahni’ “Approximation Algorithms for Wireless Sensor Deployment,” IEEE Transactions on Computer, 12(56), 2007, pp. 1681-1695. 29

[Yang06] S. Yang, F. Dai, M. Cardei, J. Wu, F. Patterson, “On Connected Multiple Point Coverage in Wireless Sensor Networks” International Journal of Wireless Information Networks, 4(13), 2006 , pp. 289-301. [YZCLZ02] F. Ye, G. Zhong, S. Lu,, and L. Zhang, “Energy Efficient Robust Sensing Coverage in Large Sensor Networks”, Technical Report, UCLA, 2002. [YZLCZ03] F. Ye, G. Zhong, J. Cheng, S. Lu, and L. Zhang, “PEAS: A Robust Energy Conserving Protocol for Long-lived Sensor Networks”, In Proc. of the 23nd International Conference on Distributed Computing Systems (ICDCS), 2003. [Zhang 03] H. Zhang, J. C. Hou, “Maintaining Sensing Coverage and Connectivity in large sensor networks, Technical Report UIUCDCS-R2003-2351, June 2003. [Zhang05] H. Zhang, J. C. Hou, “Maintaining Sensing Coverage and Connectivity in Large Sensor Networks”, Ad Hoc ＆ Sensor Wireless Networks, 1(1), pp.89-124, 2005. [Zhou04a] Z. Zhou, A. Das, and H. Gupta, “Variable Raddi Connected Sensor Cover in Sensor Networks,” Proc. of the IEEE International Conference on Sensor and Ad Hoc Communications and Networks (SECON), 2004. [Zhou04b] Zongheng Zhou, Samir Das, and Himanshu, Gupta, “Connected K-Coverage Problem in Sensor Networks,” Proc. of 13th International Conference on Computer Communications and Networks (ICCCN), 2004. [Zhou05] Z. Zhou, Samir Das, and H. Gupta, “Fault Tolerant Connected Sensor Cover with Variable Sensing and Transmission Ranges,” IEEE SECON05.

30

1. Introduction Wireless sensor networks (WSNs) have received significant attention of researchers in recent years due to its wide range of applications such as military surveillance, environmental monitoring, forest fire detection, healthcare and other areas [Akyildiz02, Chong2003]. A wireless sensor network composes of a large scale of sensor devices (called sensor nodes) equipped with sensor unit, a wireless communication unit, a battery power unit and a programmable embedded processor. The sensor nodes are capable of sensing, data processing, and communicating with each other via radio transceivers. They coordinate with each other to establish a network to remotely interact with the physical world, such as to monitor a geographical region or a set of targets spread across a geographical region, and to report sensed data to the monitoring center, which is connected to the base station. Wireless networks can be random or deterministic deployed in physical environments to collect information from an area of interest in a robust and autonomous manner. An important research issue in wireless sensor network is the coverage problem which reflects how well the deployed sensor nodes can monitor a set of targets. Sensor activation scheduling under constraint on covering of targets is called the coverage problem in the literature. There are two types of targets: area [Cardei02, Carle04, SP01, Tian02, Wang03, Zhang05] and point [CardeiDu05, Kar03, CTW05, Cheng05, Li07a, Li07b]. In area coverage problem, a set of sensors is given and distributed over a geographical region to monitor a given area. But in point coverage problem, a set of sensors is given and distributed over a geographical region to monitor a set of points (or targets). The sensing range of a sensor is typically model as disk in the 2D space or as a sphere in the 3D space, with the sensor located in the center. The communication range of a sensor is modeled in the same way. A sensor can monitor all targets that fall in its sensing range. The data sent by the sensor can be received by all sensors that fall in its communication range. A sensor’s transmission range is typically larger than its sensing range. 1

Technology limitations of sensors could seriously affect the quality of service. Stringent power supply of wireless sensor nodes is the most critical limitation, because those nodes are usually powered by batteries that may not be possible to be recharged or replaced after they are deployed in hostile or hazardous environments [Yan 03]. Recently researchers have found that the significant energy saving can be achieved by elaborate managing the duty cycle of nodes in WSN with high node density and it can prolong the network lifetime. In this approach, some nodes are scheduled to sleep (or enter a power saving mode) while the remaining active nodes keep working. However, the excessive-number of sleep nodes will lead to a WSN to be disconnected, i.e. the set of working nodes will be isolated. In addition, the over-loading of working nodes will cause these nodes to be easily exhausted and failed, which also cause a WSN to be disconnected and, consequently, invalidate the data collection and transmission. It is therefore crucial to determine a small number of sensors that still cover the given area (or a set of targets) or divide all sensors into maximum number of subsets such that each subset still cover the given area or a set of targets. This is a lifetime requirement. These selected active sensors are connected so that the sensor can report the detected data to the monitoring center. This is a connectivity requirement. . Sensors are prone to be failure. The over-loading of working sensor nodes will cause easily exhausted and failed. In addition to possible hardware or software malfunctions, sensors may fail because of severe weather conditions or other hash physical environment in the sensor filed. It is therefore crucial to construct a fault-tolerant WSN that will continuously provide needed services despite sensor failures. This is the fault-tolerance requirement. The fault-tolerance requirement includes two types: coverage fault-tolerance and connectivity fault-tolerance. The sensor coverage problem can be further divided into single coverage and multiple coverage. In single coverage, each target or point in the area must be monitored by at least one working sensor. In multiple coverage, each target or point in the area needs to be monitored by at least k different working sensors, which is called as the flat k-area-coverage problem for area coverage [Gallais07]. There is another definition on k-area-coverage. An area is k-covered if there exist k distinct sets of sensors so that each one can provide fully coverage of the sensing area, which is called layered k-area-coverage problem [Gallais07]. For connectivity fault-tolerance requirement, the coverage problem can be further divided into 1-connectivity and k-connectivity coverage problem. In this chapter, we survey recent contributions which address the coverage problem. In section 2, we survey area coverage problem including multiple area coverage problem, and connected coverage problem. Section 3 investigates point coverage problem, which includes connected point coverage problem, multiple connected multiple point coverage problem, and breach coverage problem.

2. Area Coverage The area coverage problem has been well studied and a review of existing solutions for area coverage problem is described in [SSW05, Cardei06, ChenK07]. 2

Most of works in the survey [Cardei06] and [ChenK07] were published before 2003 and 2005, respectively. The survey work in [SSW05] focuses on only constructing area-dominating sets for sensor area coverage. We present in the chapter a comprehensive and updated survey on sensor area coverage. There are various coverage problems including area coverage, k-coverage, m-connected k-coverage problems. The area coverage problem is defined as follows: Definition 1(Area Coverage problem) A set of sensors are given and distributed over a geographical region to monitor a given area, an area coverage problem is to find a minimum number of sensors to work such that each physical point in the area is monitored by at least a working sensor. Definition 2(k-coverage) An area is k-coverage if each physical point in the area is covered by at least k ( k ≥ 1 ) working (or active) sensors. Definition 3 (m-connected) The communication graph of a given set of sensors M is m-connected if for any two vertices in M, there are m vertex-disjoint paths between the two vertices. A equivalent definition is, after the removal of any k-1 vertices in M, the resulted graph is still connected. Definition 4 (m-connected k-coverage problem) A set of sensors are given and distributed over a geographical region to monitor a given area, an m-connected k-coverage problem is to find a minimum number of sensors to work such that each physical point in the area is monitored by at least k active sensors and the active sensors form a m-connected graph. Most of algorithms or protocols for coverage problem guarantee full coverage, that is, each physical point must be covered. There are some algorithms or protocols for coverage problem which does not guarantee 100% coverage, such as PEAS [YZLZ03] and the approach in [Cardei02]. There is another objective except selecting a minimal set of working nodes in the area coverage: to divide all sensors into a maximum number of disjoint sets of sensors (or non-disjoint sets) such that each set fully covers the area. Selecting a minimal a set of working nodes reduce power consumption and prolongs network lifetime. In the same way, dividing all sensors into a maximum number of disjoint(or non-disjoint) sets which activate successively prolongs network lifetime. We now give a comprehensive literature review of existing solutions and their contributions which address various area coverage problems. In the following subsections, we introduce detail algorithms and solutions.

2.1 Area coverage without connectivity guarantee 2.1.1 Maximize the number of disjoint sets For energy efficient area coverage, the works in [SP01, Cardei02] consider a large population of sensors, deployed randomly for area monitoring. Slijepcevic and Potkonjak [SP01] proposed an energy conservation technique for area coverage in wireless sensor networks. It selects and successively activates mutually exclusive sets of sensor nodes, such that each set completely monitors the entire monitored area. The authors propose a heuristic to this problem. It first divides the monitored area into fields, which is a set of points. Two points belong to same 3

filed if and only if they are covered by the same set of sensors. After the fields are established, for each sensor, a list of all fields covered by that sensor is created. The set of all fields in the area is denoted as A, and set of sensors as C. The authors transform the coverage problem as the set k-cover problem: Does C contain k disjoint covers for A. Then present a heuristic solution for the set k-cover to get a heuristic solution for the coverage problem. Their method achieves energy saving by increasing the number of disjoin covers. The results on the set k-cover problem [SP01] solve a fair version where the objective is to maximize k such that every cover contains all the physical points. In many environments, requiring that a cover contain all the physical points may be too strict. For instance, there is a single area that is monitored by only one sensor but all other areas are monitored by hundreds of sensors. Except for that single area, all other areas could be covered for many times by dividing the sensors into covers. But in the fair version, the sensors can not be partitioned at all because only all sensors to monitor all areas. Figure 1 shows this case. Fig. 1 (b) can cover the whole monitoring area. But in Figure 1(c), (d), the small area A is not be covered. All areas except area A can be covered by at three different sensors while small area A is only covered by one sensor.

Figure 1. The single area A is covered by only one sensor. Abrams et al [Abrams04] study a variation of the set k-cover problem: to find a partition of the subsets into k covers so that the number of times that areas(fields) are covered by the partition, is maximized. Three approximation algorithms are presented: randomized, distributed greedy, and centralized greedy. In the randomized algorithm, each sensor simply assigns itself to a cover chosen uniformly at random from the set of all possible covers. In the distributed greedy algorithm, each sensor assigns itself, in turn, to the cover with the minimum intersection between the areas the sensor monitors and the areas monitored by the cover thus far. The centralized greedy algorithm is similar to the distributed greedy except that an area in the intersection is 4

weighted. Cardei et al [Cardei02] propose another efficient method to achieve energy saving by organizing the sensors nodes into a maximum number of disjoint dominating sets which are activated successively. Only the sensors from the active set are responsible for monitoring the monitored area and all other nodes are in a sleep mode. The authors prove that the maximum disjoint dominating sets problem is NP-complete, and any polynomial-time approximation algorithm has a lower bound of 1.5. Based on the sequential coloring algorithm, the authors propose a heuristic to compute maximum number of disjoint dominating sets in an undirected graph. Compared to the work in [SP01], the maximum number of disjoint dominating sets is greater or equal than the maximum number of covers. This is valid because the sensors in one cover also form a dominating set. Therefore, approach in [Cardei02] potentially achieves better energy saving than approach in [SP01]. However, the approach [SP01] can achieve the full area coverage constraint, but there are small coverage lapses in the monitored area for approach in [Cardei02]. For example, as in Figure 2, there is only one set cover {S1, S2}, but there are two disjoint dominating sets {S1} and {S2}. Considering disjoint dominating sets compared [Cardei02] with disjoint covers method [SP01], in this example the longevity of the network is double from the point of view of energy resources. However, there are some uncovered parts of the target area in [Cardei02].

Figure 2. two sensors are deployed on monotoried area. 2.1.2 Minimize the number of active nodes Approaches in [SP01] and [Cardei2002] are to divide sensors to maximum number of disjoint sets (or dominating sets), each set can monitor the sensed area. These sets are activated successively in order to prolong lifetime. Tian and Georganas [Tian03, Tian02] propose another energy-efficient node scheduling scheme for area coverage in synchronous networks where sensing range is equal to the transmission range. The main objective of the algorithm is to minimize number of working nodes, as well as maintain the original sensing coverage. It requires every node to be aware of its own and its neighbor location information. At the beginning of each round, each node selects a time-out interval. At the end of the interval, if a node sees that neighbors together cover its monitoring area, the node transmits a retreat message to all its neighbors and goes into the sleep mode. Otherwise, the node remains active, but does not transmit any message. The process repeats periodically to allow for changes 5

in monitoring status. In this scheme, each node must know its neighbor location information and has to do accurate geometrical calculation to determine whether or take an off-duty status. Tian and Georgannas [Tian04] propose three different alternative node scheduling schemes for area coverage, which are location and calculation-free. In the Nearest-neighbor based scheme, after each node collects distance information to its all neighboring nodes, it determine its working status by examining if its distance to the nearest neighbor is not more than the threshold. If affirmative, the node can take off-duty status. In the neighbor number-based scheme, each node collects its all neighbors’ information, and determines its working status by examining if its neighbors’ number exceeds a given threshold. If affirmative, the node will take off-duty status. In probability-based scheme, each node generates a random number from [0,1) and checks if the number is less than the off-duty probability, if it is, the node takes off-duty status, otherwise, it sets its status as on-duty.

2.2 Connected Area Coverage In all schemes introduced in above sections, the working sensors may not be connected, and thus reporting to a monitoring center can not be proceeded. In order to collect information from the sensor nodes to monitoring center, the active sensors are desired be connected. A frequently addressed objective is to determine a minimal number of active sensors to maintain monitoring the given area as well as connectivity. Next we will introduce several connected coverage mechanisms. 2.2.1 Transmission range equals to the sensing range Ye et al [YZCLZ02, YZCLZ03] present PEAS, a distributed, probing-based density control algorithm for robust sensing coverage. In this work, a subset of sensor nodes operative mode maintains coverage while others are put into sleep. Each sensor node has the same probing range Rp and may vary its transmission power and choose a power level to cover a circular area given a radius. In PEAS, each node has three operation modes: sleeping, probing and working. Initially all sensor nodes are in the sleeping mode. Each node sleeps for an exponentially distributed duration generated according to a probability density function (PDF). When sleeping time expires, the sensor enters the probing mode. The probing node uses an appropriate transmission power to broadcast a PROBE message within its local probing range Rp. Any working node(s) within that range should respond with a REPLY message, also sent within the range of Rp. If the probing node hears a REPLY, it goes back to the sleeping mode for another random period of time. If the probing node does not hear any REPLY, it enters the working mode and starts monitoring until it fails or consumes all its energy. The probing range can be adjusted to achieve different levels of coverage redundancy. The choice of probing range also affects network connectivity. The authors also study the asymptotic connectivity of PEAS. With this protocol, the probability having full coverage of a monitored area is close to 1 if the sensing range Rt ≥ (1 + 5) R p . There is a problem that PEAS does not ensure that the coverage area of a sleeping node is completely covered by other active nodes, i.e. it does not guarantee complete coverage. Figure 3 illustrates PEAS, with the black nodes being active and the white 6

nodes being in sleep mode, because each white node is contained within Rp to one of the active nodes. But there is a coverage area of the white node which is not completely covered by the active nodes. This protocol has limited usefulness because it is probabilistic and does not ensure full area coverage.

Figure 3. PEAS for area coverage. Carle and Simplot [Carle04] propose another mechanism for energy-efficient connected area coverage for the case when all sensor nodes have the same range and the communication range equals the sensing range. The goal of the algorithm is to select the minimum number of active nodes to cover the given area. The authors modify one of existing protocol for connected dominating-set protocol (e.g. Dai and Wu’s algorithm in [Dai 03] to find area coverage rather than node coverage. In the modification protocol, each node computes its timeout function based on its priority and listens to messages from other nodes before deciding its dominating status at the end of a timeout interval. A node choosing gateway status always transmits a message (positive advertising) to all its neighbors. A node choosing not to monitor its area has the option of transmitting this information to its neighbors (negative advertising) or not. The protocol runs: using a simple perimeter coverage scheme [Tian02], a node computes the area covered by each node that transmits either positive or negative advertising and includes the transmitting node in a subset; at the end of its timeout interval, the node computes a subgraph of its one-hop neighbors that sent advertisements(these are its neighbors with higher priority); If this subgraph is connected and the nodes in subgraph fully cover the node’s area, the node opts for sleeping status; Otherwise, the node chooses active status. The distributed dominant-pruning algorithm can prove that the set of active nodes is connected. Figure 4 shows an example that how a node does decision for its status. (a) Node A decides to be active because its active neighbors do not fully cover its monitoring area. (b) Node A decides to be inactive because its monitoring area is covered by its active neighbors that are connected. (c) Node A decides to be active because its active neighbors are not connected.

7

Figure 4. Example of configurations for area-coverage decision.

2.2.2 Transmission range is at least twice of the sensing range Wang et al [Wang03] and Zhang et al [Zhang 03] first discuss how to combine consideration of coverage and connectivity maintenance in a single activity scheduling. An important, but intuitive result for maintaining sensing coverage and connectivity by keeping a minimal number of sensor nodes in the active mode has been proved by Zhang and Hou [Zhang03]. The authors first investigate the relationship between coverage and connectivity, and prove that if the transmission range is at least twice of the sensing range, a complete coverage of a convex area implies connectivity among the working nodes in the active mode. Second, the authors derive, under the ideal case in which node density is sufficiently high, a set of optimality conditions under which a subset of working sensor nodes can be chosen for full coverage. Based on the optimality conditions, the authors propose a decentralized and localized density control algorithm, called optimal geographical density control(OGDC). OGDC is under assumptions: the transmission range is at least twice of the sensing range, each node is aware of its own position, and all nodes are time synchronized. At any time, a node is in one of the three states: UNFECIDED, ON, OFF. Time is divided into rounds. At the beginning of each round, all the nodes wake up, set their states to “UNDECIDED”, and carry out the operation of selecting working nodes. By the end of the execution, all the nodes change their states to either “ON” or “OFF” and remain in that state until the beginning of the next round. This decision is based on the power-on messages. Every node keeps a list with neighbor information. When a node receives a power-on message, it checks whether its neighbors cover its sensing area, and if so, it will change to OFF state. A node decides to change into the ON state if it is the closest node to the optimal location of an ideal working node. The process of selecting working nodes (in a decentralized manner) in each round commences by randomly selecting a sensor node A to be the starting node (Figure 5). Then one of its neighbors with an approximate distance of 3r , B, is selected to be a working node. To cover the crossing point of disk A and B, the node, Q, whose position is closest to the optimal position C is then selected to become a working node. The process continues until all the nodes change their states 8

to either “ON” or “OFF”, and the set of modes with “ON” states forms the working set.

Figure 5. The process of selecting working nodes. Wang et al [Wang03] also prove that the transmission range is at least twice of the sensing range, and the area to be covered is convex, then the area coverage also implies connectivity among the covering sensors. Wang [Wang 03] and Zhang et al [Zhang 03, 05] provide a sufficient condition for safe scheduling integration in those fully covered networks. However, random node deployment often makes initial sensing holes inside the deployed area inevitable even in an extremely high-density network. Tian and Georgnnas [Tian05] enhance their work to support general wireless sensor networks by proving another conclusion: “the communication range is twice of the sensing range” is the sufficient condition and the tight lower bound to ensure that complete coverage preservation implies connectivity among active nodes if the original network topology (consisting of all the deployed nodes) is connected. That is, the authors prove that if active nodes form a completely coverage, and the original topology is connected, when the transmission range is twice of the sensing range, then the induced subgraph by active nodes is connected. When the transmission range is less than twice of the sensing range, then the induced subgraph by active nodes may be disconnected. Wu and Yang[Wu04] extend a result from [Zhang 03] where only uniform sensing range among all sensors is used. Wu and Yang consider cases where each sensor is able to select one of two or three adjustable ranges and the transmission range is at least twice of the sensing range, with the goal of minimizing the overlapped sensing area. They present two new energy-efficient models of different sensing ranges. Jiang and Dou[Jiang04] describe several improvements to algorithm in [Tian02]. The authors present a distributed and localized density control algorithm for wireless sensor networks, which all nodes have the same sensing range and the transmission range is at least twice of the sensing range. The authors apply the perimeter criterion that a circle is covered completely if perimeters of other circle covering it are fully covered by other covering circles. In the algorithm, a sensor is in one of the two states: “ACTIVE” and “NON-ACTIVE”. At the beginning, all nodes are in ACTIVE state. 9

Network lifetime is divided into rounds, and each round has a scheduling phase followed by a sensing. The scheduling phase is further divided into two sub-phases: neighbor discovery phase and evaluating phase. At the beginning of the neighbor discovery phase, node broadcast a hello message to its one-hop neighbors and sets a timer to wait for neighbors’ hello message. Upon this timer expires, node has obtained knowledge about one-hop neighbors and construct its neighbor set and effective neighbor set. Then entering the evaluating phase, sensor begins to evaluate the density control algorithm to decide which state it should go. In each time round, the ACTIVE nodes work for the sensing task and the NON-ACTIVE nodes will turn off their sensing and communication units to save energy.

2.2.3 Arbitrary Ratio of transmission range to sensing range Gallais et al [Gallais08] generalize the approach in [Carle04] for an arbitrary ratio of sensing range and transmission range. The approach are based on a time-out scheme, in addition to being fully localized, has a very small communication overhead. When a round starts, each node selects a time out and listens to messages sent by other nodes before the time-out expires. Sensor nodes whose sensing area is not fully covered when the deadline expires decide to remain active for the considered round and transmit an activity message announcing it. There are four variants in the approach, depending on whether or not withdrawal and retreat messages are transmitted. Covered nodes decide to sleep, with or without transmitting a withdrawal message to inform neighbors about the status. After hearing from more neighbors, active sensors may observe that they became covered and may decide to alter their original decision and transmit a retreat message. In this approach, the covering criterion which has been already applied in [Jiang04], [Xing05] and [Zhang05] is applied on the borders of the sensing area of each sensor[Gallais 06], the node using it verifies whether or not its sensing area is fully covered. The details of the protocol include how the time-out is decided, and how the area coverage and connectivity tests are performed. The test for connectivity of covering circles must be performed when the transmission range is less than twice of the sensing range, that is, when the transmission range is less than twice of the sensing range, a node can decide to turn off if and only if its neighbors fully cover and are also connected. Sheu et al [Sheu 07] study query execution over a specific geographical region. And propose an efficient distributed protocol to find minimum number of connected active sensor nodes to cover the queried region. Assumptions: Transmission ranges and sensing ranges differ between sensors, and the sensing range of a sensor node may differ from its transmission. The proposed protocol consists of two phases-self-pruning phase and sensing nodes discovery phase. In the beginning of the protocol, each sensor node is assumed to have the information of its 1-hop-cover neighbors. In the self-pruning phase, each node checks whether or not its sensing area is completely covered by its higher priority neighbors by using the perimeter covering criterion in [Huang 03]. If no, it becomes a sensing node. The authors prove that the 10

sensing nodes selected by self-pruning can fully cover the queried region when the deployed sensor nodes cover the queried region. In the sensing nodes discovery phase, each of the considered perimeters is subdivided into sub-perimeters, based on the intersections with other considered circles. For each such sub-perimeter, the sensor with the highest priority, among nodes covering this sub-perimeter, is active. After the two phases, the selected sensing nodes are connected and can cover the queried region. Gupta et al [Gupta 03] study the connected sensor coverage problem: Given a query over a sensor network, select a minimum set of sensors, called connected sensor cover, such that a) the sensing regions of the selected sensors cover the entire geographical region of the query, and b) the selected sensors form a connected communication graph. The authors first prove that the connected sensor coverage problem is NP-complete and then propose a centralized greedy algorithm. The proposed algorithm is as follows: Let M be the set of sensors already selected for inclusion in the connected sensor cover by the greedy algorithm at any stage. Initially, M is an empty set. The algorithm starts with including in M an arbitrary sensor that lies within the query’s region. At each stage, the greedy algorithm selects a sensor C along with a path of sensors P that forms a communication path between C and some sensor in M with maximum benefit of P, add selected path P to M, till query’s region is covered by sensors in M. In the algorithm, the benefit of P is defined as the number of uncovered valid subelements covered by P per sensor. At any stage of the algorithm, the communication subgraph induced by M is connected. A straightforward distributed version of the same algorithm is also given. Zhou et al [Zhou04a] address Variable Radii Connected Sensor Cover problem which generate the problem in [Gupta03]: Given a query region in the network, each node has vary its sensing range and transmission range where they can not exceed the maximum sensing range and the maximum transmission, selecting a subset of sensors which forms connected sensor cover such that the total energy cost (including sensing cost and transmission cost) is minimized. The authors design various centralized and distributed algorithms-Voronoi based algorithm, Greedy algorithm and Steiner tree based algorithm. One of the designed centralized algorithms (called CGA) is shown as O(logn)-approximation. CGA works as follows. CGA maintain a set of selected sensors M along with their assigned transmission and sensing range, and increases the covered region while keeping connectivity of M. At each stage, either adds to M a “path” of sensors or increases the sensing range of a sensor in M, whichever gives the maximum “benefit”. CGA terminates when the given query region is completely covered by the assigned sensing regions of the sensors in M.

2. 3 k-area coverage Sensor nodes usually are deployed into remote and inhospitable area to monitor targets. Because severe weather conditions or other hash physical environment in the sensor filed or the over-loading of working sensor nodes, sensors are prone to fail. It is therefore crucial to construct a k-coverage problem ( k ≥ 1 ), in which each physical point is covered at least k different sensor nodes. There are many existing works to 11

address k-coverage problem. Next, we give a survey on k-coverage problem. Sensor networks are often desired to prolong the lifetime of operation. This is usually achieved by putting sensors to sleep for most of their lifetime. On the other hand, the intrusion detection applications require guaranteed k-coverage off protected region at all times. To determine the appropriate number of sensors to deploy that achieves both goals simultaneously becomes a challenging problem. Kumar et al [Kumar 04] study this problem: Given an area to be protected, how many sensors should be deployed so that every point in the region is covered by at least k sensors, and given that the network must last for a specified length of times? The authors consider three kinds of deployments for a sensor network on a unit squarea n×

n grid, random uniform (for all n points), and Poisson (with density n). In

all three deployments, each sensor is active with probability p. A critical condition for three deployments is derived. And the authors show that the conditions for deterministic deployments are similar to the conditions for random deployments. 2.3.1 k-area coverage without connectivity guarantee The k-area coverage problem addressed in [Gallais06a] consists in building k distinct subsets of active nodes (layers) so that each layer covers the area. The authors propose a decentralized protocol. Sensors are randomly deployed over a square area and activity is imagined in a rounded fashion. At each round, every node decides its status between either monitoring for the entire round or getting passive until the next decision phase. Every sensor is aware of required coverage degree, denoted as k. A node A can find smallest i so that ith layer of the area covered by that node is not fully covered by its neighbors. Then, if i ≤ k , A decides to be active at layer i and sends a positive acknowledgement announcing its activity layer i and its geographical position. Otherwise, it decides to be passive and no message is sent. Figure 6 shows that that sensor A first evaluates the coverage provided by neighbors of layer 1(black nodes on Figure 6(b) before deciding to evaluate the coverage at layer 2(Figure 6(c)). Finally, Figure 6(d) shows that A is covered at all 2 layers. A takes its activity decision depending on its required coverage degree k. If k>2, then A gets active at layer 3 and sends a position acknowledgment. If k=2, then A gets passive without sending any message.

Figure 6. Evaluation of coverage. Cai et al [Cai07] propose a precise and energy-aware coverage control protocol, named Area-based Collaborative Sleeping (ACOS). Based on the net sensing area of 12

a sensor, which is covered only by the sensor and not covered by other active sensors, the ACOS controls the mode of sensors to maximize the coverage degree, minimizing the energy consumption. Each sensor node has four states: Sleep, PreWakeUP, Awake, and Overdue. Initially, each sensor is Sleep with timer, when node s wake up, its state changes from Sleep to PreWakeUP, node u sends a broadcast message, to its neighbors within radius 2r and waits for T seconds. When any neighboring sensor v with Active receives this message, node v sends back reply message including its location. After u receives reply messages, u computes the net area ratio, if the net area ratio is less than the threshold, u return back to Sleep state. If the net area ration is more than the threshold, u changes to Awake state, and initialize its wake timer and broadcast a Wake-Notification message. When node u is still in the Awake state and its wake time expires, it changes from Awake to Overdue state. When a node which is in Awake or Overdue state hears a Wake_Notification message, it re-calculates the net area ratio to repeat the process. The state transition diagram is in Figure 7.

Figure 7. State transition diagram of ACOS. Hefeeda and Baghen [Heffeda07] study –coverage problem: Given n already-deployed sensors in a target area, and a desired coverage degree k ≥ 1 , select a minimal subset of sensors to cover all sensor locations such that every location is within the sensing range of at least k different sensors. The authors model the k-coverage problem as a set system (X, R) where X is the set of sensor locations and RC is a the collection of subsets of X created by intersecting disks of radius r with points of X, for which an optimal hitting set corresponds to an optimal solution for k-coverage. And propose an approximation algorithm with a logarithmic ratio for computing near-optimal hitting sets [BG95]. A fully distributed version of the proposed algorithm is designed and implemented. There are various theoretical works on area coverage problem in wireless sensor networks. Xing et al [Xing2004] presents a theoretical analysis of greedy geographic routing protocols on wireless sensor networks that must provide sensing coverage over a geographic area. The authors prove that the Greedy Geographic Forwarding[Karp00, Stoj01] and their new greedy protocol always succeed in any sensing covered network when the communication range is at least twice the sensing range. Liu and Towsley [LiuB04] approach the coverage problem from a theoretical perspective and explored the fundamental limits of the coverage of a large-scale sensor network. The authors study three fundamental coverage measures of 13

large-scale sensor networks: Area coverage, node coverage, and detectability. These measures are determined by basic network parameters and have important implications on network planning and protocol performance of sensor networks. Ke et al [Ke07] proves that deploying sensors on grid points to construct a wireless sensor network that fully covers critical grids using minimum sensors (Critical-Grid Coverage problem) and that fully covers a maximum total weight of grids using a given number of sensors(Weighted-Grid Coverage problem) are each NP-Complete.

k-area coverage with the transmission range being at least twice sensing range The network connectivity is rarely treated in existing works on k-area coverage. Wang et al [Wang03] prove that when the transmission range is at least twice the sensing range, a set of working nodes that forms k-coverage a convex region forms a k-connected communication graph. Tian et al [Tian05] enhance the result in [wang03] for general random deployment network to prove that when the transmission range is at least twice of the sensing range, and the system sensing coverage is completely k-degree preserved after node scheduling, if a network graph is originally k-connected, the induced subgraph by the active nodes must be k-connected. Most of existing results on k-area coverage rely on this theorem to focus on area coverage only without addressing the problem of the connectivity preservation. Wang et al [Wang 03] generate the result in [Zhang 03]. And propose the coverage configuration protocol (CCP) that is a decentralized protocol that only depends on local states of sensing neighbors and can provide different degrees of coverage requested by applications. In CCP, each node determines its eligibility using the k-coverage eligibility algorithm based on the information about its sensing neighbors, and may switch state dynamically when its eligibility. Given a requested coverage degree k, a node is ineligible if every location within its coverage is already k-covered by other active nodes in its neighborhood. The authors prove that a convex region is k-cover if it contains intersection points between sensors or between sensors’ and region boundary and all these intersection points are k-covered. Based on this, a sensor is ineligible to turn active if all the intersection points inside its sensing circle are at least k-covered. Every node maintains a table of known sensing neighbors based on the beacons (hello messages) that it receives from its communication neighbors. A node can be in one of three states: SLEEP, ACTIVE and LISTEN. All nodes start in the SLEEP state for a random time. When the sleep timer expires, a node in the sleep state enters LISTEN state. When a beacon (HELLO, WITHDRAW or JOIN message) is received, a node in the listen state evaluate its eligibility. If it is eligible, it starts a join timer, otherwise it returns to the SLEEP state. If it becomes ineligible after the join timer is stated, it cancels the join timer. If the join timer expires, the node broadcast a JOIN beacon and enters the active state. If the listen timer expires, it starts a sleep timer and returns to the SLEEP state. Once a node is in the active state, it re-evaluate the coverage eligibility every time it receives HELLO message and decide whether to go into the SLEEP state or remain in the ACTIVE state. If the ratio of the communication range to the sensing range is more than 2, CPP 2.3.2

14

can guarantee connectivity. But CPP does not guarantee connectivity when the ratio of the communication range to the sensing range is less than 2. The authors also present a simple approach for integrating CCP with an existing connectivity maintenance protocol, SPAN [ChenJ01] to provide sensing coverage and communication connectivity. The proposed protocol in [Sheu 07] can be extended to solve k-coverage problem, which can find a set of sensing nodes satisfy the k-coverage request. The protocol is as follows: Assume that a set SN1 of sensing nodes is got in the self-pruning phase. If a non-sensing node is aware of its neighboring nodes in SN1, it can delete these sensing nodes from its 1-hop-cover neighboring set and execute the self-pruning again to determine whether it can be a sensing node. After the second iteration, all the non-sensing nodes can determine their roles-sensing nodes or non-sensing nodes, then get the second coverage set SN2 to fully cover the queried region if the remaining sensor nodes can fully cover the queried region. SN1 and SN2 form a 2-coverage. Applying the above procedures, k-coverage can be got. Lu et al [Lu06] address the k-coverage Maintenance Problem: Given a sensor group S deployed in region R and a natural number k, find subset S ' with the minimum number of sensors such that S ' is able to maintain k-coverage. That is, for any position v in R, if v can be k-covered by S, it must be k-covered by S ' ; Otherwise, the coverage degree of v in S ' is same as in S. It assumes that the transmission range is at least twice the sensing range. The authors propose a scalable coverage maintenance scheme (called as SCOM). SCOM assume that each node knows its location and can acquire the location of neighbors through one-hop communication. Time is slotted into rounds. At the beginning of each round, SCOM runs in two phases: Decision phase and optimization phase. In the decision phase, each sensor is initially in BOOTSTRAP state and has an empty active neighbor list. Before making the decision of turning on or off, each sensor sets a back-off timer depending on its residual energy. When a sensor’s timer expires, the sensor checks whether its sensing region is k-covered by the sensors in the active neighbor list using the redundancy eligibility rule for homogenous or heterogeneous, and switches to ACTIVE or INACTIVE state accordingly. If a sensor decides to turn into ACTIVE state, it broadcast a TURNON beacon with its coordinates to it’s the neighbors. Upon receiving the TURNON beacon, a neighbor adds the sender into the active neighbor list. In the optimization phase, sensors optimize the coverage by turning off redundant active sensors while still guaranteeing the required coverage. The Sensor Scheduling for k-Coverage(SSC) problem is investigated in [Gao06]. Which requires to efficiently schedule the sensors, such that the monitored region can be k-covered throughout the whole network lifetime with maximizing network lifetime. All the sensors have uniform transmission range and sensing range. And the transmission range is at least twice the sensing range. The authors model the SSC problem to find maximum number of disjoint k-cover sets. In [Huang03], the authors prove that the entire monitored region is k-covered if and only if each sensor in the monitored region is k-perimeter-covered. Consider any two sensors si and sj. A point on the perimeter of si is perimeter-covered by sj if this point is within the sensing 15

range of sj. si is k-perimeter-covered if all points on the perimeter of si are perimeter-covered by at least k sensors other than si itself. A segment of si’s perimeter is k-perimeter-covered if all points on the segment are perimeter-covered by at least k sensors other than si itself. Figure 8 shows an example: the perimeter of si between two arrows is covered by sensor sj. Based on this result, Gao et al propose a greedy algorithm, PCL-Greedy-Selection(GS). The main idea of GS is to iteratively construct subset by choosing sensors from the area with the lowest sensor density. When construct an individual subset, the sensor with a small PCL value is added to the subset. In addition, the authors develop a guideline for designing a sensor deployment by employing density control.

Sj

Si

Figure 8. An example of perimeter-coverage. 2.3.3 Connected k-area coverage Area coverage protocols aim at turning off redundant sensor nodes while ensuring full coverage of the area by the remaining active nodes. Providing k-area coverage means that every physical point of the monitored area is sensed by at least k sensors. Connectivity of the active nodes subset must also be provided so that monitoring reports can reach the sink stations. Existing solution hardly address these two issues as a unified one. The works in [Zhou04b, Zhou05, Gallias07] address coverage and connectivity as a unified one. Next, we review them. Zhou et al [Zhou04b, Zhou05] study the k-area coverage problem and the connectivity preservation problem. Zhou et al consider the problem of selecting a minimum size connected K-cover, which is defined as a set of sensors M such that each point in the sensor network is “covered” by at least K different sensor in M, and the communication graph induced by M is connected. The authors design a centralized O(logn)-approximation algorithm. The greedy algorithm is a generalization of the centralized approximation algorithm in [Gupta 03] for the connected 1-coverage problem. The Greedy Algorithm maintains a set of M of selected sensors and at each stage, select a candidate sensor without belong to M and a candidate path of sensor with maximum “K-Benefit” with respect M, add the selected path to M. This is repeated until the query region is k-covered by M. The distributed version of the Greedy algorithm is also given. Zhou et al [Zhou 05] address a more general, variable radii sensor model, choosing a subset of sensors such that they maintain a k1-connectivity and k2-cover, wherein every sensor can adjust both its sensing and transmission ranges, and the overall energy consumption is minimized. The energy consumption includes sensing energy consumption and transmission energy consumption. The authors propose a distributed and localized Voronoi-based algorithm. The Voronoi-based algorithm 16

works as follows. Initially, each sensor node in the sensor network is active, and gathers locations of all the nodes in the l-hop active neighborhood. Each active sensor node computes its k2 th –order local Voronoi cell, and the neighbors in the k1-RNG over active nodes. It uses the V-R assignment method to assign itself sensing and transmission radius. Each node computes its sleep benefit, based on the sleep benefit, choose a sensor with the most sleeping benefit among all its local voronoi neighbors to become inactive. A sensor node is chosen to become inactive only if the remaining active sensors are capable of k1-covering the query region and maintaining k1-connectivity of their communication graph. Repeat above processes. The algorithm terminates when no more sensors can be made inactive. Gallais and Carle [Gallais07] consider connected k-coverage problem. And consider two definitions for the k-area coverage problem: the flat k-area coverage problem and the layered k-area coverage problem. The authors propose a localized algorithm that can be applied to time-synchronized networks. Each node selects a time-out, which depends on the remaining energy, and has some random number, while listening to messages from neighboring nodes. Once the timeout ends, u takes its activity decision based on known neighboring nodes. It so evaluates its coverage according to the appropriate coverage evaluation scheme. If completely k-covered according to the flat k-area coverage issue, if u decides to be passive and turns into sleep mode. Otherwise, u remains active and sends a positive acknowledgment message which contains the values of its communicating and sensing range with its position. Any node with a longer timeout that receives this message adds u to its neighbor table. For the layered k-area coverage issue, Nodes still listen for messages during a given timeout before making their activity decision and choosing an activity layer whose number is included in the messages. A node u sorts its neighbors according to a number of layers. Then, u evaluates if at least k virtual activity layers fully cover its area S(u). If no, u remains active and chooses the uncovered activity layer which has the lowest number, and sends an activity message to announce its status. About connectivity, when CR ≥ 2 SR , connectivity is ensured. When CR0, and an undirected graph G=(V, E) find a subset of nodes C ⊆ V such that each node in V is dominated by at least k different nodes in C, and the number of nodes in C is minimized. The k-CCS is to add another constraint that 23

subgraph induced by C is connected. The k-CS is formulated as an integer linear programming and then a centralized LP-based algorithm for k-CS is proposed. LP-based algorithm includes two steps: the first step is to compute the optimal solution of relax LP problem, the second step is to round the optimal solution to solution of ILP: if the solution of optimal solution of LP is greater than some value, set this variable to 1, otherwise, zero. Non-global solutions for k-CS/k-CSS are proposed: cluster based algorithm and pruning-based algorithm. Cluster base algorithm runs as follows: sequentially apply a traditional clustering algorithm k time to get k sets of clusterheads, find gateways to connect the first set, then add other nodes to all clusterheads and gateways to form the k-CS/k-CSS. In the pruning-based algorithm, all nodes are initially assumed as active. Each node using 2-hop neighborhood to determines its status. Initially, all nodes are marked. Each node u is given a unique priority, L(u). Each node broadcast its neighbor set N(u), and build a subset C (u ) which is formed by u’s all neighbors with higher priorities than u. Node u is umnmarked if C(u) is connected(this constraint is removed for k-CS) and for any neighbor w of u, there are k distinct nodes in C(u), such that w is a neighbor of all the k nodes. All marked nodes form k-CS/k-CSS. Li et al [Li07b] address k-connected m-coverage problem which is different from the coverage problem in [Yang06]. The coverage problem is : Given a set of sensors and a set of targets, and a coverage mapping from sensors to targets, and constants k and m, k ≥ 1, m ≥ 1 , find a minimum number of sensors such that each target is covered at least m sensors and the selected sensors is k-connected. The k-connected coverage problem and k-connected m-coverage problem are NP-hard. In [Li07b], the authors first study m-coverage problem, which is formulated as ILP, then propose an approximation algorithm based on LP. Based on solution of m-coverage problem and algorithms for k-connected augmentation [Li07a], two heuristics (kmTS algorithm and kmReverse algorithm) are proposed for k-connected m-coverage problem. Two algorithms include two steps: the first step is to construct a m-coverage of targets; The second step is to increase small size nodes to this m-coverage such that the subgraph by these increased nodes and nodes of m-coverage is k-connected using the algorithms [Li07a].

3.3.3. Breach Coverage Network lifetime has been recognized as an important factor in sensor network design. To extend sensor network lifetime, one potential approach is to divide sensors into disjoint subsets, each of which can cover all targets. Each subset is switched to active mode and sleep mode alternatively, so that at any time there is only one set of sensors active to prolong network lifetime. The size of sensor cover sets is not put any constraint. However, the number of deployed sensors is usually very large and the base station may not provide a bandwidth large enough for receiving data from all sensors in the cover sets. In this situation, a complete coverage is sometime not 24

available. Maybe there exists some targets can not be monitored by any sensor. A target is in breach if it is not monitored by any sensor. There are some coverage breach problems studied in the literature [Slijepcevi01, Chengxi05, 07, WangC07, Thai05]. Cheng et al [Chengx05, Chengx07] study three coverage breach problems: Minimum Breach problem, Minimum Individual Breach Time problem and Minimum Maximal Breach problem. The Minimum Breach problem is : Given a set A of fixed points and a set S of sensors, organize sensors into disjoin subsets Ci, i=1,2…, K, where each subset | Ci |≤ W and the overall breach is minimized. The authors prove the three problems are NP-hard. The three coverage breach problem are formulated as 0-1 linear integer programming problems. The minimum breach problem is formulated as a 0-1 integer programming problem as following: K

M

min{∑∑ (1 − ykj )} k =1 j =1

N

∑a x i =1

ij k ,i

≥ yk , j

K

∑x N

∑x i =1

= 1,

∀i = 1,...N ;

=W,

∀k = 1..., K ;

k ,i

k =1

k ,i

∀j = 1,..., M , k = 1,...K ;

yk , j ∈ {0,1}

∀k = 1...K , j = 1,...M ;

xk ,i ∈ {0,1}

∀k = 1...K , i = 1...N .

A Greedy approximation algorithm and a heuristic based on the LP-relaxation are proposed. In a greedy strategy, iteratively pick the most coverage-effective sensor and put it in its fit position until all sensors are put into subsets. Each subset can have at most W sensors. LP-based heuristic(called Relaxation) includes three steps; In the first step, the integer programming(IP) problem is relaxed to a linear programming (LP) problem, and compute an optimal solution for LP. In the second step, using greedy strategy to find an integer solution based on the optimal solution of LP. In the third step, the solution from (IP) problem is used to construct the subsets. In [Thai05], Thai et al present two new linear programming based models, Minimum Coverage Breach under Bandwidth constraints (MCBB) and Maximum Network Lifetime under bandwidth constraints (MNLB) to solve the joint optimization on energy and bandwidth utilization. MCBB problem is: Given a collection C of subsets of a finite set R, find a family of p order pairs (Sj, tj) such that the total coverage breach is minimized. Where Sj is a set cover and tj is the time duration between 0 and 1 for Sj to be active. MNLB problem is to find a family of p

25

order pairs (Sj, tj) such that

∑

p

t

j =1 j

is maximized. In the two models, sensors are

organized into non-disjoint set cover. The MCBB problem and the MNLB are NP-hard, and can be formulated as mix integer programming. The authors propose two approximation algorithms based on the optimal solution of relax linear programming to solve them.

4.

Conclusion

In the chapter, we investigate the current works on coverage problem in sensor networks, and classify them into two categories: sensor area coverage and target coverage. We focus on the most representative problems in each domain and present a comprehensive review and analysis of various existed algorithms and techniques.

Acknowledgement This research is partially supported by the National Natural Science Foundation of China under grant 10671208, and Key Laboratory of Data Engineering and Knowledge Engineering (Renmin University of China), MOE.

Reference [Abrams04] Z. Abramsm A. Goel, and S. Plotkln, “Set K-cover Algorithms for Energy Efficient Monitoring in Wireless Sensor Networks”, Proc. Of International Symposium on Information Processing in Sensor Networks, (ISPN),2004, pp. 424-432. [Akyildiz02] I.F. Akyildiz, W. Su, Y. Sankarasubrmaniam and E. Cayirci, “A Survey on Sensor Networks”, IEEE Communication Magazine, (Aug, 2002) 102-114. [BG95] H. Bronnimann and M. Goodrich, “Almost Optimal Set Covers in Finite VC-dimension,” Discrete and Computational Geometry, 4(14), 1995. [Cai07] Y. Cai, M. Li, W. Shu, and M.Y. Wu, “ACOS: An Area-based Collaborative Sleeping Protocol for Wireless Sensor Networks,” International Journal of Ad Hoc & Sensor Wireless Networks, Vol.3, No.1, pp.77--97, 2007. [Cardei02] Cardei, D. MacCallum, X. Cheng, M. Min, X. Jia, D.Y. Li, and D.-Z. Du: “Wireless Sensor Networks with Energy Efficient Organization,” Journal of Interconnection Networks, 3(2002) 213-229. [Cardei06] M. Cardei, Jie Wu: “Energy-efficient Coverage Problems in Wireless Ad-hoc Sensor Networks”, Computer Communications 29(4): 413-420 (2006). [CardeiDu05] M. Cardei and D.Z. Du, “Improving Wireless Sensor Network Lifetime through Power Aware Organization”, ACM Wireless Networks, 11( 2005), pp. 333-340. [CardeiW05] M. Caredi, J. Wu, M. Lu, M.O. Pervaiz, “Maximum Network Lifetime in Wireless Sensor Networks with Adjustable Sensing Ranges”, WIMob, Proc. of the IEEE International Conference on Wireless and Mobile Computing, Networking and Communications, 2005, pp.438-445. 26

[Carle04] J. Carle and D. Simplot, “Energy Efficient Area Monitoring for Sensor Networks”, IEEE Computer, vol. 37, No. 2(2004), pp. 40-46. [Chakrabarty02] K. Chakrabarty, S. S. Iyengar, H. Qi and E. Cho, “Grid Coverage for. Surveillance and Target Location in Distributed Sensor Networks”, IEEE Transactions on Computer, 2002,51(12):1448-1453. [ChenJ01] B. Chen, K. Jamieson, H. Balakrishnan, and R. Morris, “Span: An Energy-Efficient Coordination Algorithm for Topology Maintenance in Ad Hoc Wireless Networks,” MobiCom 2001, pp. 16-21. [Chengx05] X. Cheng, L. Ruan, and W. Wu, “Achieving Minimum Coverage Breach under Bandwidth Constraints in Wireless Sensor Networks”, INFOCOM05. [Chengx07] X. Cheng, L. Ruan, and W. Wu, “Coverage Breach Problems in Bandwidth-Constrained Sensor Networks”, ACM Transactions on Sensor Networks, 2(3), 2007. [ChenK07] Jie Chen and Xenofon Koutsoukos, "Survey on Coverage Problems in Wireless Ad Hoc Sensor Networks", IEEE SouthEastCon 2007. Richmond, VA, March 22-25, 2007. [Cheng05] M.X. Cheng, L. Ruan, and Weili Wu, “Achieving Minimum Coverage Breach under Bandwidth Constraints in Wireless Sensor Networks”, Infocom’05. [Chong03] C. Y. Chong and S. P. Kumar, “Sensor Networks: Evolution, Opportunities, and Challenges”, Proc. IEEE, pp. 1247-1256, vol.91., no. 8, 2003. [CTW05]M. Cardei, M. Thai, Y. Li, and Weili Wu, “Energy-efficient Target Coverage in Wireless Sensor Networks”, Infocom’05. [Dai03] F. Dai and J. Wu, “Distributed Dominant Pruning in Ad Hoc Networks”, ICC2003, pp.353-357. [Gallias06a] A. Gallais, J. Carle, D, Simplot-Ryl, and Ivan Stojmenovic, “ Localized Sensor Area Coverage with Low Communication Overhead”, Proc. Fouth IEEE int’l Conf. Pervasive Computing and Comm. (PerCom) 2006. [Gallais06b] A. Gallais, J. Carle, and D. Simplot-Ryl, “Ensuring k-coverage in Wireless Sensor Networks with Realistic Physical Layers”, IEEE SENSORS 2006, pp.22-25. [Gallais07] A. Gallais and J. Carle, “An Adaptive Localized Algorithm for Multiple Sensor Area Coverage”, in Proc. of the 21st International Conference on Advanced Networking and Applications (AIAN’07). [Gallais08] A. Gallais, J. Carle, D, Simplot-Ryl, and Ivan Stojmenovic, “ Localized Sensor Area Coverage with Low Communication Overhead”, IEEE Transactions on Mobile Computing, vol.7, No. 5, 2008, pp. 1-12. [Gupta03] H. Gupta, S. R. Das, and Q. Gu, “Connected Sensor Cover: Self-organization of Sensor Networks for Efficient Query Execution”, Proceedings of the 4th ACM International Symposium on Mobile Ad Hoc Networking and Computing (MobiHoc), 2003. [Hefeeda07] M. Hefeeda and M. Bagheri, “Randomized k-Coverage Algorithms For Dense Sensor Networks,” In Proc. of IEEE INFOCOM 2007, pp. 2376--2380, Anchorage, AK, May 2007. [Huang03] C. F. Huang and Y. C. Tseng, “The Coverage Problem in a Wireless 27

Sensor Networks”, Proc. Of the 2nd ACM int. Conf. on Wireless Sensor Networks and Application, 2003. PP.115-121. [Jian04] J. Jiang and W. Dou, “ A Coverage Preserving Density Control Algorithm for Wireless Sensor Networks,” Proc. Third int’l Conf. Ad-Hoc Networks and Wireless (ADHOC-NOW),2004. [Jaggi06] N. Jaggi, and A. A. Abouzeid, “Energy-Efficient Connected Coverage in Wireless Sensor Networks”, Proceedings of 4th Asian International Mobile Computing Conference (AMOC 2006), Kolkata, India, January 4-7, 2006. [Kar03] K. Kar and S. Banerjee, “Node Placement for Connected Coverage in Sensor Networks”, Proc. of WiOpt 2003: Modeling and Optimization in Mobile, Ad Hoc, and Wireless Networks(2003). [Karp00] Brad Karp and H.T. Kung, “GPSR: Greedy Perimeter Stateless Routing for Wireless Networks”, Mobile Computing and Networking, 2000, pp. 243-254. [Ke07] W. Ke, B. Liu, and M. Tsai, “ Constructing a Wireless Sensor Network to Fully Cover Critical Grids by Deploying Minimum Sensors on Grid Points is NP-Complete”, IEEE Transactions on Computers, Vol.56, No. 5, 2007, pp. 710-715. [Li07a] D. Li, J Cao, M Liu, Y. Zheng, “K-connected Coverage Problem in Wireless Sensor Networks,” COCOA2007. [Li07b] D. Li, J Cao, D. Liu, “Algorithms for the m-coverage Problem and k-connected m-Coverage Problem in Wireless Sensor Networks,” NPC2007. [LiuB04] B. Liu and D. Towsley, “A Study of the Coverage of Large-scale Sensor Networks”. Proceedings of the 1st IEEE International Conference on Mobile Ad Hoc and Sensor Systems (MASS2004). Pp. 475-483. Fort Lauderdale, Florida, October 2004. [Liu06]H. Liu, P. Wan, and X. Jia, "Maximal Lifetime Scheduling for Sensor Surveillance Systems with K Sensors to 1 Target," IEEE Transactions on Parallel and Distributed Systems, vol. 17, no. 12, pp. 1526-1536, Dec. 2006. [Liu07] H. Liu, X. Jia, P. Wan, C. Yi, S. Kami Makki, and Niki Pissinou, "Maximizing Lifetime of Sensor Surveillance Systems," IEEE Transactions on Networking, vol. 15, no. 2, pp. 334-345, Apr. 2007. [Liuz06] Zheng Liu, “Maximizing Network Lifetime for Target Coverage Problem in Heterogeneous Wireless Sensor Networks”, LNCS4864, 2007, pp. 457-468. [LuW05] M. Lu, J. Wu, M. Cardei, and M. Li, “Energy-Efficient Connected Coverage of Discrete Targets in Wireless Sensor Networks”, ICCNMC 2005, LNCS3619, pp.43-52. [Lu06] J. Lu, J. Wang, and T. Suda, “Scalable Coverage Maintenance for Dense Wireless Sensor Networks”, IEEE SECON2006. [Yan03] T.Yan, T. He, and J.A. Stankovic, “Differentiated Surveilance for Sensor Networks”, Proc. First Int’l Conf. Embedded networked Sensor Systems, pp.51-62, 2003. [Schrijver1986] A. Schrijver, Theory of Linear and Integer Programming, Wiley, Chichester 1986. [Sheu07] J. Sheu, S. Tu, and C. Yu, “A Distributed Query Protocol in Wireless Sensor Networks”, Wireless Personal Communication, 41(2007), pp. 449-464. 28

[SSW05] D. Simplot-Ryl, I. Stojmenovie, and J. Wu, “Energy-efficient Backbone Construction, Broadcasting, and Area Coverage in Sensor Networks”, Handbook of Sensor Networks: Algorithms and Architectures (I. Stojmenovie ed.), Wiley, pp.343-380, 2005. [SP01] S. Slijepcevic, M. Potkonjak, “Power Efficient Organization of Wireless Sensor Networks”, IEEE International Conference on Communications 2001, 472-476. [Stojmenovic01] Ivan Stojmenovic and Xu Lin, “LOOP-free Hybrid Sing-path/flooding Routing Algorithms with Guaranteed Delivery for Wireless Networks,” IEEE Transactions on Parallel and Distributed Systems, vol.12(10),, pp. 1023-1032, 2001. [Tian03] D. Tian and N. D. Georganas, “ A Node Scheduling Scheme for Energy Conservation in Large Wireless Sensor Networks”, Journal of Wireless communations and Mobile Computing, 3(2003) 271-290. [Tian02] D. Tian and N. D. Georganas, “ A Coverage-preserving Node Scheduling Scheme for Large Wireless Sensor Networks”, Proc. of the 1st ACM Workshop on Wireless Sensor Networks and Applications(2002). [Tian04] D. Tian and N. D. Georganas, “Location and Calculation-free Node Scheduling Scheme in Large Wireless Sensor Networks,” Ad Hoc Networks, 2(2004), 65-85. [Tian05] D. Tian and N. D. Georganas, “Connectivity Maintenance and Coverage Preservation in Wireless Sensor Networks”, Ad Hoc Networks, 3(2005) pp. 744-761. [Thai05] T. Thai, F. Wang, and D. Du, “Coverage Problems in Wireless Sensor Networks: Designs and Analysis,” International Journal of Sensor Networks, special issue on Coverage Problems in Sensor Networks, 2005. [Wang03] X. Wang, G. Xing, Y. Zhang, C. Lu, R. Pless, C. D. Gill, “Interated Coverage and Connectivity Configuration in Wireless Sensor Networks”, in: Proceedings of the first ACM Conference on Embedded Networked Sensor Systems(SenSys 2003), Los Angles, November 2003. [Wu04] J. Wu and S. Yang, “Coverage and connectivity in sensor networks with adjustable ranges”, proc. Of the 2004 International Workshop on Mobile and Wireless Networks, 2004. [Wang06] J. Wang and N. Zhong, “Efficient Point Coverage in Wireless Sensor Networks”, Journal of Combinatorial Optimization, 3(11)2006 , pp. 291-304. [WangC07] C. Wang, M. Thai, Y. Li F. Wang W.i Wu, “Minimum Coverage Breach and Maximum Network Lifetime in Wireless Sensor Networks”, in Proc. of Global Telecommunications Conference 2007. GLOBECOM '07. IEEE. [Xing04] G. Xing, C. Lu, R. Pless, and Q. Huang, “Greedy Geographic Routing is Good Enough in Sensing Covered Networks”, INFOCOM2004. [Xing05] G. Xing, X. Wang, Y. Zhang, C. Lu, R. Pless , and C. Gill, “Integrated Coverage and Connectivity Configuration for Energy Conservation in Sensor Networks,” ACM Trans. Sensor Networks, vol.1 no.1. pp. 36-72, 2005. [Xu06] X. Xu and S. Sahni’ “Approximation Algorithms for Wireless Sensor Deployment,” IEEE Transactions on Computer, 12(56), 2007, pp. 1681-1695. 29

[Yang06] S. Yang, F. Dai, M. Cardei, J. Wu, F. Patterson, “On Connected Multiple Point Coverage in Wireless Sensor Networks” International Journal of Wireless Information Networks, 4(13), 2006 , pp. 289-301. [YZCLZ02] F. Ye, G. Zhong, S. Lu,, and L. Zhang, “Energy Efficient Robust Sensing Coverage in Large Sensor Networks”, Technical Report, UCLA, 2002. [YZLCZ03] F. Ye, G. Zhong, J. Cheng, S. Lu, and L. Zhang, “PEAS: A Robust Energy Conserving Protocol for Long-lived Sensor Networks”, In Proc. of the 23nd International Conference on Distributed Computing Systems (ICDCS), 2003. [Zhang 03] H. Zhang, J. C. Hou, “Maintaining Sensing Coverage and Connectivity in large sensor networks, Technical Report UIUCDCS-R2003-2351, June 2003. [Zhang05] H. Zhang, J. C. Hou, “Maintaining Sensing Coverage and Connectivity in Large Sensor Networks”, Ad Hoc ＆ Sensor Wireless Networks, 1(1), pp.89-124, 2005. [Zhou04a] Z. Zhou, A. Das, and H. Gupta, “Variable Raddi Connected Sensor Cover in Sensor Networks,” Proc. of the IEEE International Conference on Sensor and Ad Hoc Communications and Networks (SECON), 2004. [Zhou04b] Zongheng Zhou, Samir Das, and Himanshu, Gupta, “Connected K-Coverage Problem in Sensor Networks,” Proc. of 13th International Conference on Computer Communications and Networks (ICCCN), 2004. [Zhou05] Z. Zhou, Samir Das, and H. Gupta, “Fault Tolerant Connected Sensor Cover with Variable Sensing and Transmission Ranges,” IEEE SECON05.

30