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- Coverage and Connectivity Issues in Wireless Sensor Networks
- Convex position estimation in wireless sensor networks
- A node scheduling scheme for energy conservation in large wireless sensor networks
- FLOW An efficient forwarding scheme to mobile sink in wireless sensor networks
- A coverage-preserving node scheduling scheme for large wireless sensor networks
- Coverage and Connectivity Issues in Wireless Sensor Networks
- Integrated coverage and connectivity configuration in wireless sensor networks
- Connectivity maintenance and coverage preservation in wireless sensor networks. Ad Hoc Netw
- Coverage problems in wireless ad-hoc sensor networks
- Connected K-target coverage problem in wireless sensor networks with different observation scenarios
- Rendezvous Planning in Wireless Sensor Networks with Mobile Elements
- Coverage in Wireless Ad-hoc Sensor Networks
- SECURITY TOPOLOGY IN WIRELESS SENSOR NETWORKS WITH ROUTING OPTIMISATION
- Spatial-Temporal Coverage Optimization in Wireless Sensor Networks
- Mostly-Sleeping Wireless Sensor Networks Connectivity, k-Coverage, and α-Lifetime

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Coverage with Connectivity in Wireless Sensor Networks

(Invited Paper) Xin Liu Department of Computer Science University of California Davis, CA 95616 email:liu@cs.ucdavis.edu

Abstract— In this paper, we study coverage with connectivity properties in large wireless sensor networks. We consider three classes: full coverage with connectivity, partial coverage with connectivity, and constrained coverage with connectivity. We outline two simple network topologies to satisfy the constrained coverage with connectivity criterion. We compare the surveillance performance and deployment cost for networks with different coverage with connectivity criteria. Together, they cover a whole spectrum of surveillance quality in wireless sensor networks at different cost. We outline potential research topics in the area.

I. I NTRODUCTION Recent advancements in microelectronics, digital signal processing, and low-power RF techniques have enabled the deployment of large wireless sensor networks. Wireless sensor networks can be deployed in areas without infrastructure support, in hostile ?elds, and under harsh environments. Applications of such sensor networks include spatially and temporally dense environmental monitoring, battle?eld monitoring, seismic structure response study, precision farming, traf?c monitoring, etc. The development of wireless sensor networks will have signi?cant impact on both scienti?c adventures and our daily life. We consider a wireless sensor network where sensor nodes have both sensing ability and communication ability. Coverage and connectivity are basic requirements in a wireless sensor network. The objective of such a network is to detect events of interests or collect data and then report the obtained information to a fusion center. Therefore, connectivity, i.e., the ability to report information to the fusion center, is as critical as sensing coverage. Thus, we consider the coverage with connectivity property in sensor networks. We focus on large sensor networks. Because it is often either impossible or undesirable to deploy sensor nodes precisely, we speci?cally consider the case where sensor nodes are randomly deployed in a large ?eld. Different applications may have different degrees of coverage and connectivity (CC) requirements. We classify the CC requirements as follows: full CC, partial CC, and constrained CC. We discuss the performance and cost of these different criteria in this paper.

The work was supported by NSF through CAREER Award #0448613 and Grant #0520126

Full coverage with connectivity means that every location in the ?eld is covered by at least one node and information at this location can be reported to the fusion center. In other words, the fusion center can obtain information at any location in the whole surveillance ?eld. Applications with such full CC requirements include battle ?eld monitoring, intrusion detection, etc. To provide higher degrees of information/decision accuracy and fault tolerance, K-coverage and K-connectivity may be desirable [1]. To be more speci?c, Kcoverage(connectivity) allows at least K ? 1 node failures while maintaining coverage(connectivity). Such applications may include distributed detections, mobility tracking, event monitoring in high security areas. Such sensor networks are discussed in Section III. There are applications that requires less stringent coverage and connectivity guarantee. For example, instead of knowing the temperature of every location in the ?eld, knowing the temperatures in 80% of the area might provide suf?cient information for the temperature pro?le of the ?eld. We call this partial coverage with connectivity. Figure 1 shows such a case. Compared with full CC, a network with partial coverage and connectivity requirements needs much less sensor nodes. Results from percolation theories apply to such networks, as discussed in Section IV. We also introduce a new class, named constrained coverage with connectivity. In constrained CC, the maximum size of an area that an event can occur without being reported to the fusion center is bounded. Consider a sensor network that is deployed to monitor forest wild?re. Constrained CC implies that it is required that a wild?re must be detected and reported before it propagates to a ?eld of a certain size. Another type of applications of constrained CC is to collect data that are spatial correlated, such as temperature and humidity. Thus, to limit the maximum size of isolated area provides a con?dence level of the information in the whole ?eld. Constrained CC has great ?exibility to balance the tradeoff between surveillance quality and deployment cost, and is discussed in more detail in Section V. In this paper, we discuss the above mentioned CC criteria. Based on the results in the literature on different coverage and connectivity models, we compare their surveillance performance and deployment cost, and identify possible research problems.

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Fig. 1. An illustration of coverage with connectivity. Nodes 1 and 2 are not covered and connected.

II. S YSTEM M ODEL A location in the ?eld is covered if an event happened at this location can be detected by one or more sensor nodes. Let Rs be the sensing range of a sensor node. A location is covered if its distance to at least one sensor node is no larger than Rs . Two sensor nodes are directly connected if the distance of the two nodes is smaller than the communication range, Rc . A sensor node is connected to the fusion center if there exists a route from the sensor to the fusion center. The route consists of a sequence of sensor nodes where no hop in the route is longer than the communication range. In a sensor network, both coverage and connectivity are important. After all, the purpose of a wireless sensor network is to monitor and report the information to the fusion center. Therefore, connectivity, i.e., the ability to report information to the fusion center, is also critical, and thus we consider the CC property in sensor networks. Figure 1 illustrates the idea of CC. Sensor nodes are randomly deployed in the ?eld. We draw a circle of radius r = Rc /2 centered at each sensor node. If the circle of two nodes overlap, then they are directly connected and thus can communicate with each other. If there exists a connected component that includes both the sensor and the fusion center, then the sensor node can communicate with the fusion center. For the ease of illustration, we assume that the sensing range, Rs , equals Rc /2. In other words, the circles illustrate both sensing and communication capabilities of a sensor node. Note that an event happened inside the circles of sensor nodes 1 and 2 can be detected by at least one sensor nodes. However, this information cannot be reported to the fusion center and thus is useless. From the data-collection viewpoint, there is no difference whether sensors 1 and 2 exist or not. In contrast, an event can be both detected and reported if it happens inside other circles. We assume the area to be considered in signi?cantly larger than the sensing and communication ranges, and thus ignore boundary effects. In each CC case, we start with the assumption Rs = Rc /2 and then discuss more general settings. We note that the ratio between Rs and Rc is a critical factor in the CC property and further study is needed to fully understand the impact of the ratio on the CC property. We say that an area is isolated if an event happened in the area cannot be reported to the fusion center. To elaborate,

a location is isolated if 1) it is not covered by any sensor nodes, or 2) if it is covered by one or more sensor nodes but these sensor nodes are not connected to the fusion center. For example, the areas covered by sensors 1 and 2 are considered as isolated areas. As mentioned earlier, we study random deployed sensor networks. There are two widely used models. The ?rst model assumes that nodes are distributed according to a Poisson point process with a density λ, and the second one assumes that n nodes are uniformly distributed in the network of size A. When n = λA and A goes to in?nite, these two models have the same properties because of the law of large numbers. Thus, in this paper we use these two models alternatively. We note that consider a homogeneous network with a rangebased sensing and communication model in this paper. More general models can be considered. For instance, coverage model based on estimation theory and signal processing has been studied in [2], [3], distance-based sensing intensity model developed in [4], [5], [6]. In [7], the authors study both pure ad-hoc and hybrid networks and show that the introduction of a sparse network of base stations can signi?cantly improve the connectivity. More general communication models, such as SINR (Signal to Inference and Noise Ratio) model and information theoretic model, are of research interest. Related work include [8], [9] and references therein. In [8], the authors study the case where other transmissions cause interference, and show that the interference coef?cient is an important factor in connectivity. In [9], the authors study the effects of irregular shapes of communication models and shows that connectivity properties in the literature using the geometric disk abstraction hold for more general irregular shapes. Generalization to such (more realistic) network models are certainly desired, but beyond the scope of this paper. Coverage and connectivity is an important component considered in many operations of sensor networks, including, clustering, synchronization, query and information discovery, deployment and redeployment. In particular, CC is often studied jointly with sleep-awake scheduling [10], [11], [12] for energy conservation and surveillance quality provisioning. Coverage and connectivity has been studied extensively in the literature and there are good survey papers in the area [13], [10]. We do not plan to survey all works in the area, and refer readers to these references and references therein for a detailed discussion on survey. Coverage is classi?ed into three types, namely area coverage, point/targe coverage, and barrier coverage, in [14]. Following their terminology, our focus in the paper is on the issue of area coverage and the corresponding connectivity. III. F ULL C OVERAGE WITH C ONNECTIVITY Full coverage with connectivity means that every location in the ?eld is covered by at least one node and the information regarding this location can be routed to the fusion center. The basic question is the number of nodes needed to satisfy such a requirement under random deployment. In [15], the authors study both the coverage and connectivity properties of a random network where sensor nodes are

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distributed in a ?eld of size A according to a homogeneous Poisson process. Recall that Rs and Rc are sensing and communication ranges respectively. It is shown that 1) if 2 πRs λ(A) ≤ (1??) ln A, then there exists uncovered locations in the ?eld with probability one for any ? > 0; 2) if 2 πRs λ(A) ≥ (1 + ?) ln A, then every location in the ?eld is covered by at least one sensor nodes with probability one for 2 any ? > 0. It is also shown that if πRc λ(A) ≤ (1 ? ?) ln A, then there exists unconnected sensor nodes with probability one. On the other hand, using the techniques in [16], it can 2 be shown that if πRc λ(A) ≥ (1 + ?) ln A, then every node is connected with probability one. Similar results have been obtained using different network models and communication models. For example, the authors of [16] consider a unit area with n randomly located nodes. Suppose that each node can communicate with nodes within a circle of area πr2 (n) = (log n + c(n))/n. Then the network is asymptotic connected if and only if c(n) → ∞. It is shown in [17] that each node should be connected to Θ(log n) neighbors in order to guarantee asymptotic connectivity for a random adhoc network. In [18], the authors study the connectivity and coverage of a regular sensor grid with unreliable nodes in a unit square region. They show that the necessary and suf?cient conditions for the network to cover the region while all active nodes are connected are of the form p(n)r2 (n) ? log(n)/n, where two nodes within distance r(n) can communicate with each other and p(n) is the probability that a node is active. Other applications, e.g., distributed detections, may required higher degrees of coverage to improve information accuracy and fault tolerance. Thus, K-coverage and Kconnectivity have been proposed and studied, where Kcoverage(connectivity) allows at least K ? 1 node failures while maintaining coverage(connectivity) [1]. Differentiated coverage and connectivity have been studied [1], [19] for ?eld with different surveillance requirements. Fault tolerant deployment that provides K-connectivity is studied in [20]. In particular, the relationship between K-coverage and Kconnectivity under various ratio between Rc and Rs are studied in [21], [22], [23]. In summary, the required sensor node density grows logarithmically with the size of the ?eld to guarantee full coverage with connectivity; i.e., λ(A) = Θ(ln A) and the average number of node is n = Θ(A ln A). It is clear that more sensor nodes are needed to guarantee K-coverage and connectivity [20]. When A is large, to deploy a fully-covered and connected sensor network can be costly. Among all three classes, full CC provides the best surveillance quality. It can be considered as the extreme case for the partial CC where the coverage requirement is 100%. It can also be considered as a constrained coverage with connectivity where the size of the maximum isolated area is zero. On the other hand, to guarantee full CC, the number of nodes required is large, on the order of A ln A, which is also the highest among the three. IV. PARTIAL C OVERAGE WITH C ONNECTIVITY As discussed in Section III, the number of nodes required to guarantee full CC in a wireless sensor network is on the order

of A ln A where A is the size of the ?eld. Such a network can be costly, which is avoidable for certain applications with less stringent requirements. Consider a ?eld of area A and nodes are randomly deployed in the network with density λ. Recall that the sensing range is 2 Rs . On average e?πRs λ percent of the whole area will not be covered. Thus, for the whole ?eld to be covered, λ has to go to in?nite. What happens if λ is bounded and both coverage and connectivity are required? Percolation theory studies such cases [24]. It states that there exists a critical density λc (as a function of Rc ). If the density of nodes is lower than λc , then the number of nodes in each connected component is bounded almost surely. On the other hand, if the density of nodes in the ?eld is higher than λc , then there exists a unique “giant” connected component. This giant connected component includes an in?nite number of nodes almost surely as the size of the ?eld goes to in?nite. Note that it is only guaranteed that there exists a giant connected component. There is no guarantee on the size of the ?eld that is covered by the giant component or the percentage of the nodes that are connected in the giant component. There is no guarantee that the fusion center is a part of the giant component, either. The case where λ < λc is less interesting because almost surely the network is disconnected; i.e., a ?nite number of sensor nodes is connected to the fusion center among an in?nite number of sensor nodes. We focus on the case where λ > λc . It has been shown numerically in [25] that the critical density is λc = 1.4365 . 2 Rc

In Figure 2, we show the result in several different densities in a square with L = 100, and Rc = 1. The size of the ?eld is A = L2 = 10000. The number of nodes in each ?gure equals (1 + ?)λc L2 , where ? = ?0.1, 0, 0.1, 0.3. In the ?gure, the largest connected component is marked black, and different levels of gray indicate other connected components. The case where ? = ?0.1 corresponds to the under-critical case, where the size of each connected component is bounded with probability 1 as L goes to in?nite. We can observe that there are a lot of small clusters, which are not connected to each other. The case where ? = 0 corresponds to the critical case. The cases where ? = 0.1, 0.3 correspond to the supercritical cases, where there exists an unbounded components as L goes to in?nite with probability 1. We observe that there exists one giant component that include most of the nodes. Of course, the larger the value of ?, the larger the number of nodes included in the giant component. For ? = 0.3, we can see that most area is covered by the giant connected component. Compare with full coverage with connectivity, the number of sensor nodes deployed is relatively small, (1 + ?)λc A vs. (1 + 2 ?)A ln A/(πRc ). There are a few questions involved in the partial coverage with connectivity problem. First, there is no guarantee that the origin or the fusion center is a part of the giant component. Suppose the fusion center locate at the origin in Figure 2. When ? = 0.1, although there exists a giant component, the fusion center is only connected to a small portion of nodes (a

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Fig. 2.

Percolation property at different values of ?, ? = ?0.1, 0, 0.1, 0.3.

gray component). How can we deploy networks such that the fusion center is connected to most nodes? Heuristically, more sensor nodes should be deployed in areas closer to the fusion center to guarantee more nodes are connected to the fusion center. To achieve such a goal systematically is an interesting research issue involving both theoretical study and protocol design. Second, we are interested in the size of the area that the fusion center can obtain information. A natural question to ask is as follows: what is the percentage of the ?eld that is covered with connectivity, i.e., what is the ratio of the size of the giant component to the whole area? A randomly deployed network with density λ covers Qs = 2 1 ? e?λπRs portion of the whole area. In other words, Qs is the area covered by all levels of gray (including black) over the whole area in Figure 2. However, it is more interesting to know Qc , the percentage of the ?eld that is covered with connectivity, i.e., the area covered by black. We make the optimistic assumption that the fusion center is connected to the largest connected component. Note that Qs is only an upperbound of Qc because areas covered by different levels of

gray (not including black) are covered, but not connected with the largest component. Thus, they should be excluded when we count the area of CC. Next, we de?ne Qu , which a tighter bound for Qc . Let P1 be the probability that a single sensor node is not connected to any other nodes. We have Qu =

2 = Qs ? πRs P1

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2 2 1 ? e?λπRs ? π 2 Rs Rc λe?λπRc .

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In other words, Qu exclude areas covered by nodes that is not connected with any other nodes. Table I compares Qs , the sensing percentage, Qu , an upperbound for sensing with connectivity, Qc , the sensing with connectivity percentage, which is obtained by simulation, and pn , the percentage of the number of nodes belong to the largest cluster over the total number of nodes. We note that Qu is a slightly better bound on Qc than Qs . It is obvious that when ? increases, more and more sensor nodes are connected and they cover a larger and larger area. For example, when ? = 2, all nodes are connected and they cover “most” of the area. An important problem is to quantify the required density given a coverage requirement.

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? 0.1 0.3 0.5 1 2

Qs 0.7109 0.7693 0.8159 0.8953 0.9661

Qu 0.7022 0.7651 0.8140 0.8950 0.9661

Qc 0.5928 0.7435 0.8053 0.8926 0.9653

pn 0.8521 0.977 0.993 0.999 1

TABLE I P ERCENTAGE OF AN AREA BEING COVERED AND CONNECTED , AND ITS BOUNDS .

We also want to understand the characteristics of isolated areas. Metrics of practical interests include the average and maximum size of isolated areas, and the percentage of isolated areas. Some applications may require such information. For example, if the sensor network is deployed to detect forest ?re, then the largest isolated area is the maximum size of possible damage of a wild?re before being detected. We may be able to apply results from coverage theories here [26], [12]. There are interesting properties between the sensing range and the communication range. Because we do not understand exactly the coverage property even when Rc = Rs /2, it is more challenging to understand the CC property for different ratio of Rc and Rs . Alternatively, we may want to answer the following question: what should the sensing range be such that α portion of the ?eld is covered and connected, say α = 99% given Rc ? In summary, to guarantee full CC, the number of sensor nodes is on the order of A ln A. To guarantee the existence of a giant component, the number of required nodes is on the order of A. The giant component covers a large portion of the whole ?eld. The important issues are to quantify the density needed to satisfy the predetermined CC requirements and to guarantee that the fusion center is connected in the ”giant” component. V. C ONSTRAINED C OVERAGE WITH C ONNECTIVITY From the previous discussion, we know that to guarantee the existence of a giant component, the number of sensor nodes needed is lower-bounded by λc A. Even if we only need coarse information of the surveillance ?eld, we cannot decrease the number of sensor nodes needed. Furthermore, there is no guarantee of the size of isolated areas. This motivates the constrained coverage with connectivity criterion that provides speci?c guarantees on isolated areas. There are applications that have speci?c constraints on isolated areas. One type of applications is to monitor events that propagate, such as wild?re, plant disease, and soil contamination. It is required to detect an event before its affected area grows larger than a certain threshold. Another type of applications is to monitor properties with high spatial correlation, such as temperature and humidity. Limiting the size of isolated area provides a con?dence level of the information gathered. There are different types of coverage constraints based on applications. Some of the typical constraints are described as follows:

The maximum size of a disk that can be placed in an isolated area is bounded. Such a constraint is useful for applications that detect events propagating omnidirectionally, e.g, wild?re in forest and soil contamination. ? The maximum length of a straight line that can be placed in an isolated area is bounded. An example of such applications is target tracking, where the target moves along a straight trajectory. ? The maximum size of an isolated area is bounded. Such a constraint can bound the maximum size of an event, e.g., a plant disease propagates randomly (instead of omnidirectionally) in a forest/orchard. For such applications, we propose two types of network topologies, namely the comb topology and the grid topology. In these topologies, we assume that sensor nodes can be deployed randomly along straight lines. For example, sensor nodes can be deployed by an airplane ?ying at a low attitude or a vehicular driving along a road. Figure 3 shows the comb topology and Figure 4 the grid topology. In the ?gure, √ the ?eld being monitored is a square of size A, and L = A is the perimeter. The lines in the ?gures are the trajectories where sensor nodes are placed on. If sensor nodes along the lines are connected and the lines are covered by sensor nodes, then the placed network can be con?gured to satisfy the constrained CC requirements. The comb model can satisfy the ?rst requirement. In the ?gure, V is the distance between two consecutive lines. Thus, the maximum diameter of a disk is bounded by V . (We assume that Rs = Rc /2 and V >> Rs for simplicity. ) The grid model can satisfy all three requirements and provide better fault tolerance, but requires more nodes to satisfy the ?rst requirement compared to the line model. For the ?rst requirement, the maximum diameter of the disk is bounded by V . For the second requirement, the maximum √ length of any straight line is bounded by 2V . For the third requirement, the maximum size of an isolated area is bounded by V 2 . In order to estimate the number of nodes needed to deploy such a network, we ?rst present Theorem 5.1, which states a necessary and a suf?cient condition for a line to be connected and covered. Assume n nodes are uniformly distributed along a line of length L. Theorem 5.1: If L L log , n = (1 ? ?) Rc Rc then the network of length L will be disconnected with probability 1 as L → ∞ for any ? > 0. On the other hand, a suf?cient condition for the linear network to be connected is L L log n = (1 + ?) Rc Rc for any ? > 0. The above theorem states a necessary and a suf?cient condition for a linear network to be connected. To provide full coverage of a linear network is similar. We simply need to replace Rc in the above theorem with 2Rs . Thus, based on the CC requirements, we can calculate the number of nodes needed. The proposed topologies provides

?

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Fig. 3. Comb model for constrainted coverage with connectivity.

Fig. 4.

Grid model for constrainted coverage with connectivity.

the ?exibility to build a surveillance network based on the sensing/monitoring quality requirement. For example, consider the ?rst requirement. Assume that the maximum diameter of an isolated disk is V and Rs = Rc /2. Then, to satisfy such a requirement, we need √ √ √ A A A A n = (1 + ?) ln ? ln A. +2 V Rc Rc V Based on the value of V , we obtain a full spectrum of surveillance quality. In other words, the number of nodes needed for a task is adjusted according to the coarseness of the requirement. √ Note that 0 ≤ V ≤ A. While full CC and partial CC (when the node density is high) provide ?ne surveillance quality, the proposed topologies can provide coarse surveillance quality at low cost, i.e., when V is relatively large. We assume Rs = Rc /2 so that full coverage and full connectivity on the line are equivalent. If Rs < Rc /2, we can replace Rc in the previous discussion with 2Rs to guarantee full coverage and full connectivity on the line. This is important for constraints 2 and 3, but not for constraint 1. If Rs > Rc /2, full connectivity guarantees full coverage on the line and all three constraints can be satis?ed. It is desirable to further study the case of Rs < Rc /2 to understand the coverage property assuming full connectivity. Results from full coverage case may be extended here. Sensor nodes may scatter along a predetermined trajectory instead of being placed precisely on the trajectory when a linear sensor network is deployed. The objective is to maintain coverage with connectivity along the trajectory. The number of nodes required will depend on the scattering pattern. We note that a linear deployment can be fragile and prone to node failures. Therefore, in practice, the sensors can be deployed in a narrow and long trajectory instead of a line to improve reliability. Furthermore, the grid topology provides better resilience to node failures than the comb topology. Research topics include reliability and cost analysis in a narrow and long trajectory and necessary and suf?cient conditions for CC in a linear network given a scattering pattern. Such results are important for linear network deployments in general.

VI. C ONCLUSION AND F UTURE R ESEARCH T OPICS In this paper, we study coverage and connectivity properties in wireless sensor networks. We note that coverage without connectivity is meaningless in wireless networks. We thus present and discuss three types of CC criteria. Together, they cover a whole spectrum of surveillance quality in wireless sensor networks. Table II compares the performance and cost of these different criteria. Full CC provides the best surveillance quality at the highest cost. The number of nodes required is on the order of A ln A, where A is the size of the surveillance ?eld. To provide K-coverage and K-connectivity requires even more nodes. When some sensor nodes run out of energy or fail to function, a full CC network may degrade to a partial CC network. In a partial coverage network, “most” of the area is covered with connectivity. A fusion center can detect events in “most” cases. The density of sensor nodes needed is lower-bounded by the critical density. When the density of sensor nodes falls below the critical density, “most” of the network is disconnected although the fusion center might be connected to a small number of nodes. We notice that the connectivity property is very sensitivity to the density when it is close to the critical density. To provide better faulttolerance performance, we could over-provision the network, e.g., ? ≥ 0.1. The constrained CC is slightly different from the previous two because we have some degree of control on the sensor placement. We place sensor nodes (randomly) along a trajectory. We propose two simple network topologies. The major advantage of constrained CC is its ?exibility: the number of sensor nodes required depends on the desired surveillance precision. In other words, when we have a lot of sensor nodes, the network can be deployed to provide high quality surveillance information. When the number of available sensor nodes is limited, a coarse information collecting network can be deployed. Depending on applications, the sensing range and the communication range of a sensor node can be very different. As we emphasized in this paper, connectivity is critical to a wireless sensor network. The relationship between coverage

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Classi?cation Full Coverage Partial Coverage Constrained Coverage

Performance Best Average good Depends on precision TABLE II

Cost Highest (O(A ln A)) medium (O(A)) Depends on precision (O(A/V ln A))

C OMPARE DIFFERENT CLASSES

and connectivity is an interesting research topic. When full CC is considered, if Rc ≥ 2Rs , the coverage implies connectivity [1]. If Rc < 2Rs , then full connectivity implies full coverage. The problem is more interesting when partial CC is considered. We note that the density of the nodes should be larger than the critical density to guarantee the existence of a giant connected component. Under such conditions, the area of the ?elds that is covered with connectivity depends on the sensing range. In the case of constrained CC, similar issues exist. In particular, the challenge is to guarantee under coverage requirements two and three under various ratios between Rc and Rs . In summary, given a value of Rc , we can determine the number of nodes and deployment pattern needed to maintain the required connectivity. Given required connectivity, to determine the property of the CC area is challenging under various ratios of Rs and Rc . In this paper, we focus on a homogeneous wireless sensor network with simple range-based isotropic sensing and communication models. Generalization to more realistic network models are certainly desired. There are a few promising directions, including signal-processing-based sensing model, SINR-based or information-theoretic communication model [8], [27], [28], irregular shape of communication model [9], and hybrid network architecture [7]. For instance, coverage model based on estimation theory and signal processing has been studied in [2], [3], distance-based sensing intensity model developed in [6]. CC properties for hybrid sensor networks has been explored in [29]. Such coverage models provide quanti?cations of coverage quality and thus will signi?cantly change the CC property. Another important issue to be further studied is fault tolerance in large sensor networks [20]. We note that K-coverage and K-connectivity belongs to this category. However, most current work consider homogeneous network with homogeneous and independent failures. In practice, network failures can be temporally and spatially correlated. For instance, a node, depleted of battery, will impose heavier load on its neighboring nodes, which causes neighboring nodes to die quickly. Another example of correlated failure is a catastrophic event happened in a certain geographic area, which affects a large portion of sensor nodes in the given area. Potential remedies include heterogenous deployment and network redeployment. In particular, redeployment is a critical and practical issue to guarantee CC property in a long-lived sensor network. Theoretical study and protocol development are both desired in this research ?eld. While this paper focuses on random deployment of sensor networks, we note the importance of carefully designed deployment, e.g., [22], [30]. Carefully designed deployment

can improve surveillance quality, provide fault tolerance, and reduce the device cost, at the expense of high initial deployment cost. For example, in hybrid networks, it is very likely that more capable and critical nodes, such as base stations or gateways, can be deployed at pre-designed locations at a reasonable cost. Therefore, designed deployment needs to be further studies. In summary, we study CC properties in large wireless sensor networks. We classify and compare three types of CC criteria that cover a whole spectrum of surveillance quality, and discuss future research topics. R EFERENCES

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