Coverage and Connectivity of a Wireless
Sensor Network
CHI-FU HUANG, YU-CHEE TSENG, and HSIAO-LU WU National Chiao-Tung University
Wireless sensor networks have attracted a lot of attention recently. Such environments may consist of many inexpensive nodes, each capable of collecting, storing, and processing environmental infor-mation, and communicating with neighboring nodes through wireless links. For a sensor network to operate successfully, sensors must maintain both sensing coverage and network connectivity. This issue has been studied in Wang et al. [2003] and Zhang and Hou [2004a], both of which reach a similar conclusion that coverage can imply connectivity as long as sensors’ communication ranges are no less than twice their sensing ranges. In this article, without relying on this strong assump-tion, we investigate the issue from a different angle and develop several necessary and sufficient conditions for ensuring coverage and connectivity of a sensor network. Hence, the results signifi-cantly generalize the results in Wang et al. [2003] and Zhang and Hou [2004a]. This work is also a significant extension of our earlier work [Huang and Tseng 2003; Huang et al. 2004], which ad-dresses how to determine the level of coverage of a given sensor network but does not consider the network connectivity issue. Our work is the first work allowing an arbitrary relationship between sensing ranges and communication distances of sensor nodes. We develop decentralized solutions for determining, or even adjusting, the levels of coverage and connectivity of a given network. Ad-justing levels of coverage and connectivity is necessary when sensors are overly deployed, and we approach this problem by putting sensors to sleep mode and tuning their transmission powers. This results in prolonged network lifetime.
Categories and Subject Descriptors: C.2.2 [Computer-Communication Networks]: Network Protocols—Applications; C.3 [Special-Purpose and Application-Based Systems]—Real-time
and embedded systems
General Terms: Algorithms, Design, Performance
Additional Key Words and Phrases: Ad hoc network, coverage, connectivity, energy conservation, power control, sensor network, wireless network
Y.-C. Tsengs research is cosponsored by Taiwans MoE ATU Program, by NSC under grant numbers 93-2752-E-007-001-PAE, 95-2623-7-009-010-ET, 95-2218-E-009-020, 95-2219-E-009-007, 94-2213-E-009-004, and 94-2219-E-007-009, by MOEA under grant number 94-EC-17-A-04-S1-044, by ITRI, Taiwan, and by Intel Inc.
Author’s addresses: Department of Computer Science, National Chiao-Tung University, 1001 Ta-Hsueh Road, Hsin-Chu, Taiwan 30050, R.O.C.; email:{cfhuang, yctseng, hlwu}@csie.nctu.edu.tw. Permission to make digital or hard copies of part or all of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or direct commercial advantage and that copies show this notice on the first page or initial screen of a display along with the full citation. Copyrights for components of this work owned by others than ACM must be honored. Abstracting with credit is permitted. To copy otherwise, to republish, to post on servers, to redistribute to lists, or to use any component of this work in other works requires prior specific permission and/or a fee. Permissions may be requested from Publications Dept., ACM, Inc., 2 Penn Plaza, Suite 701, New York, NY 10121-0701 USA, fax+1 (212) 869-0481, or permissions@acm.org.
C
2007 ACM 1550-4859/2007/03-ART5 $5.00 DOI 10.1145/1210669.1210674 http://doi.acm.org/ 10.1145/1210669.1210674
ACM Reference Format:
Huang, C.-F., Tseng, Y.-C., and Wu, H.-L. 2007. Distributed protocols for ensuring both coverage and connectivity of a wireless sensor network. ACM Trans. Sens. Netw. 3, 1, Article 5 (March 2007), 24 pages. DOI= 10.1145/1210669.1210674 http://doi.acm.org/10.1145/1210669.1210674
1. INTRODUCTION
The rapid progress of wireless communication and embedded microsensing MEMS technologies has made wireless sensor networks possible. Such environ-ments may have many inexpensive wireless nodes, each capable of collecting, storing, and processing environmental information, and communicating with neighboring nodes. In the past, sensors were connected by wirelines. Today, this environment is combined with the novel ad hoc networking technology to facilitate intersensor communication [Pottie and Kaiser 2000; Sohrabi et al. 2000]. The flexibility of installing and configuring a sensor network is thus greatly improved. Recently, a lot of research activities have been dedicated to sensor networks, including the design of physical and medium-access layers [Shih et al. 2001; Woo and Culler 2001; Ye et al. 2002] and routing and trans-port protocols [Braginsky and Estrin 2002; Ganesan et al. 2001; Heinzelman et al. 2000]. Localization and positioning applications of wireless sensor net-works are discussed in Bahl and Padmanabhan [2000], Savvides et al. [2001], and Tseng et al. [2003].
Since sensors may be spread in an arbitrary manner, a fundamental issue in a wireless sensor network is to ensure coverage and connectivity. Given a sensor network, the coverage issue is concerned with how well the sensing field is monitored by sensors. In the literature, this problem has been formulated in various ways. Coverage can be regarded as a metric to evaluate the quality of service (surveillance) provided by the network. Between a given pair of points in the sensing field, some works focus on finding a path connecting these two points which is best or worst monitored by sensors when an object traverses along the path [Li et al. 2003; Meguerdichian et al. 2001; Meguerdichian et al. 2001; Veltri et al. 2003]. In Huang and Tseng [2003] and Huang et al. [2004], the coverage problem is formulated as one of determining if a 2D/3D-sensing field area is sufficiently k-covered, that is, each point in the field is within the sensing ranges of at least k sensors. The proposed approach looks at how the perimeter of each sensor’s sensing range is covered, thus leading to efficient polynomial-time algorithms. On the other hand, some works are targeted at particular applications (such as energy conservation [Abrams et al. 2004; Tian and Georganas 2003; Yan et al. 2003]), but the central idea is still related to the coverage issue.
The connectivity issue is concerned with the diversity of communication paths between sensors. This would affect network robustness and communi-cation performance. The GAF protocol [Xu et al. 2001] aims to extend the net-work lifetime by turning off redundant nodes, while keeping the same level of
routing fidelity, which is defined as uninterrupted connectivity between
com-municating nodes. GAF imposes a virtual grid on the network and nodes in the same grid coordinate with each other to determine who can sleep and for how
long. Chen et al. [2002] presents a connectivity-maintaining protocol, SPAN, which can turn off unnecessary nodes such that all active nodes are connected through a communication backbone, and all inactive nodes are directly con-nected to at least one active node. Maintaining a concon-nected network is also a basic requirement of works targeted at topology control, which involves adjust-ing sensors’ transmission power for energy efficiency and collision avoidance [Burkhart et al. 2004; Li and Hou 2004; Wattenhofer et al. 2001].
In this work, we study the relationship between sensing coverage and com-munication connectivity of a sensor network. Wang et al. [2003] proposes a cov-erage determination algorithm by looking at how intersection points between sensors’ sensing ranges are covered by their neighbors and claims that cover-age can imply connectivity as long as sensors’ communication ranges are no less than twice their sensing ranges. A Coverage Configuration Protocol (CCP) that can provide different degrees of coverage and still maintain communication connectivity is presented. If the communication ranges are less than twice the sensing ranges, Wang et al. [2003] proposes integrating CCP with SPAN [Chen et al. 2002] to provide both sensing coverage and communication connectivity. A similar result is also drawn in Zhang and Hou [2004a], and thus only the cov-erage problem is addressed. A decentralized density-control algorithm called
Optimal Geographical Density Control (OGDC) is then proposed to reduce the
number of working nodes that cover the network.
It is clear that the results in Wang et al. [2003] and Zhang and Hou [2004a] are not applicable when some sensors’ communication ranges are less than twice their sensing ranges even though others are not. Also, both Wang et al. [2003] and Zhang and Hou [2004a] assume that all sensors have the same sens-ing ranges. In this article, we relax these constraints and show necessary and/or sufficient conditions for a sensor network to be k-covered and k-connected, and to be k-covered and 1-connected. Hence, the results in Wang et al. [2003] and Zhang and Hou [2004a] can be regarded as special cases of what is proposed in this article. Based on these conditions, we then develop decentralized solutions for determining, or even adjusting, the levels of coverage and connectivity of a given network. This results in a prolonged network lifetime. As far as we know, no result has addressed the combined issues of coverage, connectivity, power management, and power control under a single framework as is done in this work. The ability to adjust the levels of coverage and connectivity makes management of the network more flexible. In emergency applications, keeping the network 1-covered and 1-connected may be sufficient. However, when an emergency occurs, higher coverage and connectivity may be needed in an on-the-fly manner. For autoconfiguration purposes, given an arbitrarily deployed sensor network, we can first calculate the coverage and connectivity levels of the network. If the coverage or connectivity level exceeds our expectation, we can make adjustments using the proposed coverage and connectivity selection protocols to prolong the network lifetime without reducing the sensing and com-municating capabilities of the network. This work is a significant extension of our earlier work [Huang and Tseng 2003; Huang et al. 2004], which addresses how to determine the level of coverage of a given sensor network but does not consider the network connectivity issue. Our work is the first work allowing an
arbitrary relationship between sensing ranges and communication distances of sensor nodes. Information about the difference of sensors’ sensing ranges is discussed in Zhang and Hou [2004b].
Some works also consider the coverage and connectivity issue but have differ-ent assumptions or applications. Shakkottai et al. [2003] considers a grid-based network consisting of sensors which may fail probabilistically and investigates the coverage, connectivity, and diameter of the network. Inanc et al. [2003] studies the problem of minimizing energy consumption by suspending sensors’ sensing and communication activities according to a Markovian stochastic pro-cess, ensuring communication connectivity and sensing coverage. However, the definitions of event coverage and path connectivity distinguish our goals from other works. Given a spatial query requesting data of interest in a geograph-ical region, the goal of Gupta et al. [2003] is to select the smallest subset of sensors which are connected and are sufficient to cover the region. The pro-posed solution is a greedy algorithm which recurrently selects a path of sensors that is connected to an already selected sensor until the given query region is completely covered.
This article is organized as follows. Section 2 gives some preliminaries. Sev-eral conditions for coverage and connectivity are presented in Section 3. De-centralized coverage and connectivity determination and adjustment protocols are developed in Section 4. Section 5 presents our simulation results. Section 6 draws our conclusions and future work.
2. PRELIMINARIES AND PROBLEM STATEMENT
We are given a set of sensors, S = {s1, s2,. . . , sn}, in a two-dimensional area
A. Each sensor si, i = 1 . . . n, is located at a known coordinate (xi, yi) inside
A and has a sensing distance of ri and a communication distance of ci. So, si
can detect an object/event located within a distance of ri from itself and talk to
another sensor within a distance of ci. Note that we make no assumption about
the relationship of ri and ci. Only bidirectional links are considered. So when
two sensors can hear each other, we say that there is a communication link, or simply a link, between them.
Definition 1. A point in A is said to be covered by siif it is within si’s sensing
range. Given an integer k, a point in A is said to be k-covered if it is covered by at least k distinct sensors. The sensor network is said to be k-covered if every point in A is k-covered.
Definition 2. The sensor network is said to be 1-connected if there is at least
one path between any two sensors. The sensor network is said to be k-connected if there are at least k disjointed paths between any two sensors.
The deployment of sensors is not of concerned in our work, and we assume the network is at least 1-covered. We formulate the general form of the coverage and connectivity problem as follows.
Definition 3. Given any two integers ks and kc, the ks-covered and
kc-connected problem, or the (ks, kc)-CC problem, is a decision problem whose
j1,L j3,L j2,L j4,L j6,L j5,L j7,L j8,L j1,R j3,R j2,R j4,R j6,R j5,R j7,R j8,R j8,L j8,R si j3,L j4,L j7,L j6,L j5,L j3,R j4,R j7,R j6,R j5,R j1,R j1,L j2,R j2,L (a) (b)
Fig. 1. Determining the perimeter coverage of a sensor si.
As far as we know, the general (ks, kc)-CC problem has not been well
ad-dressed yet. In Huang and Tseng [2003], the coverage problem has been solved in an efficient way. Following, we briefly review the result which will be used as a basis for our derivation. Given any sensor, Huang and Tseng [2003] try to look at the perimeter that bounds the sensor’s sensing range (for convenience, this may be simply referred to as the perimeter of the sensor). The algorithm essentially determines the coverage level of the sensing field A by determining the perimeter of each sensor.
Definition 4. Consider any two sensors siand sj. A point p on the perimeter
of si is perimeter-covered by sj if this point is within the sensing range of sj, that
is, the distance between p and sj is less than rj. A point p on the perimeter of si
is k-perimeter-covered if it is perimeter-covered by at least k sensors other than
si itself. Sensor 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.
THEOREM1 [HUANG ANDTSENG2003]. The whole network area A is k-covered if and only if each sensor in the network is k-perimeter-covered.
The approach in Theorem 1 looks at how the perimeter of each sensor’s sens-ing range is covered by its neighbors. For each sensor si, it tries to identify
all neighboring sensors which can contribute some coverage to si’s perimeter.
Specifically, for each neighboring sensor sj, we can determine the angle of si’s
arch, denoted by [αj, L,αj, R], which is perimeter-covered by sj. Placing all
an-gles [αj, L,αj, R] on [0, 2π] for all j’s, it is easy to determine the level of perimeter
coverage of si. For example, Figure 1(a) shows how siis covered by its neighbors
(shown in dashed circles). Mapping these covered angles in Figure 1(b), it is easy to decide that si is 1-perimeter-covered. It is shown in Huang and Tseng
[2003] that Theorem 1 can be converted into an efficient coverage determina-tion algorithm. It is to be noted that it makes no sense to consider the perimeter
of a sensor exceeding the sensing field A. So we only consider the perimeters of sensors inside A. In the extreme case that a sensor’s sensing range contains A, we simply ignore it and consider that it contributes a coverage of 1 to A.
3. CONDITIONS FOR NETWORK COVERAGE AND CONNECTIVITY
In this section, we propose theoretical foundations and necessary and/or suf-ficient conditions to solve the (ks, kc)-CC problem. We make no assumption on
the relationship between ri and ci of sensor si. We show conditions for a sensor
network to be k-covered and k-connected, and to be k-covered and 1-connected. We also show under what conditions a sensor network may provide sufficient coverage by multiple connected components.
3.1 Theoretical Fundamentals
The definition of perimeter coverage proved useful to determine the coverage level of a sensor network in Huang and Tseng [2003]. However, the network connectivity issue has not been studied. For a sensor network to operate success-fully, sensors must maintain both sensing coverage and network connectivity. Below we develop some fundamentals to achieve this goal.
Definition 5. Consider any sensor si. The neighboring set of si, denoted as
N (i), is the set of sensors each of whose sensing region intersects with si’s
sensing region.
Note that neighbors are concerned with how sensors’ coverage areas overlap, and should not be confused with communication links, which are concerned with sensors’ transmission distances.
Definition 6. Consider any sensor si. We say that si is
k-direct-neighbor-perimeter-covered, or k-DPC, if si is k-perimeter-covered and si has a link to
each node in N (i). Similarly, we say that si is
k-multihop-neighbor-perimeter-covered, or k-MPC, if siis k-perimeter-covered and sihas a (single- or multi-hop)
path to each node in N (i).
These definitions allow us to derive some coverage and connectivity proper-ties of a network.
LEMMA 1. Consider any two sensors si and sj. If each sensor in S is 1-MPC,
there must exist a communication path between si and sj.
PROOF. This proof is by construction. If si’s sensing region intersects with sj, by Definition 6, there must exist a path between si and sj, which proves this
lemma. Otherwise, draw a line segment L connecting si and sj, as illustrated
in Figure 2(a). Let L intersect si’s perimeter at point p. Since si is 1-MPC, by
Definition 6, there must exist a sensor sxin N (i) which covers p and has a path
to si. In addition, either sxmust cover sj, or sx’s perimeter must intersect L at a
point, namely q, which is closer to sjthan p is. Figure 2(b) shows several possible
combinations of sxand rx. In the former case, by Definition 6, there must exist a
path between sxand sj, and thus siand sj, which proves this lemma. In the latter
si sj sx sy sz p q r (a) (b) si s j sx p q rx si sj sx p q rx si sj sx p rx
Fig. 2. Proof of Lemma 1: (a) the path construction, and (b) possible cases of sx.
the argument until a sensor szis found which either covers sj or intersects L at
a point, say r, inside sj’s sensing range. Note that since the number of sensors
is finite, the construction must eventually terminate and the path from simust
reach sj. Otherwise, the intersection point of sj’s perimeter and L is not covered
by any sensor. As a result, there must exist a path between sz and sj which
proves this lemma.
THEOREM 2. A sensor network is k-covered and 1-connected if and only if each sensor is k-MPC.
PROOF. For the if part, we have to guarantee both the coverage and
connec-tivity. The fact that the network is k-covered has been proved by Theorem 1 because each sensor which is k-MPC is also k-perimeter-covered. In addition, Lemma 1 can guarantee that the network is 1-connected, hence proving the if part.
For the only if part, we have to show that each sensor is k-perimeter-covered and has a path to each sensor whose sensing region intersects with its region. The first concern can be ensured by Theorem 1, while the second concern can be ensured by the fact that the network is 1-connected.
THEOREM 3. A sensor network is k-covered and k-connected if each sensor is k-DPC.
PROOF. Coverage has been guaranteed by Theorem 1 since a sensor which
is k-DPC is k-perimeter-covered by definition. For the connectivity part, if we remove any k− 1 nodes from the network, it is not hard to see that each of the rest of sensors must remain 1-DPC. This implies that these sensors are also 1-MPC, and, by Lemma 1, there must exist a path between any pair of these sensors. As a result, the network is still connected after the removal of any k−1 nodes, which proves this theorem.
a
(a) (b)
Fig. 3. Observations of Theorem 2 and Theorem 3: (a) the network is 2-covered and 1-connected. The removal of sensor a will disconnect the network, and (b) the network is covered and 2-connected but no sensor is 2-DPC. Note that the sensing and communication ranges of each sensor are the same and are represented by circles.
Following we make some observations about Theorem 2 and Theorem 3. First, a major difference is that Theorem 2 can guarantee only 1 connectivity, while Theorem 3 can guarantee k connectivity. This is because, in a network where each sensor is k-MPC, the removal of any sensor may disconnect the network. For example, in the network in Figure 3(a), each sensor is 2-MPC. By Theorem 2, the network is 2-covered and 1-connected. However, if we remove sensor a, the network will be partitioned into two components. Interestingly, although the network remains 2-covered, it is no longer connected.
Second, the reverse direction of Theorem 3 may not be true. That is, if a network is k-covered and k-connected, sensors in this network may not be
k-DPC. Figure 3(b) shows an example in which the network is 2-covered and
2-connected. However, each node has a neighbor (with overlapping sensing range) to which there is no direct communication link.
Third, Theorem 3 is stronger than the results in Wang et al. [2003] and Zhang and Hou [2004a]. It is clear that when two sensors have overlapping sensing range, there is a direct communication link between these two sensors if the communication distance is at least twice the sensing distance. So what can be determined by Wang et al. [2003] and Zhang and Hou [2004a] can also be determined by Theorem 3. Furthermore, when the previous assumption does not exist, Theorem 3 may still work while Wang et al. [2003] and Zhang and Hou [2004a] do not. For example, Theorem 3 can determine that the network in Figure 4 is 1-covered and 1-connected when some sensors’ communication ranges are less than twice their sensing ranges.
3.2 Looser Connectivity Conditions
Definition 7. The direct neighboring set of si, denoted as DN(i), is the set of
sensors each of which has a communication link to siand whose sensing region
a r ca= a c cc=2rc rc b rb cb=1.5rb
Fig. 4. An example comparing Theorem 3 with results in the literature. Solid circles and dotted circles are sensors’ sensing ranges and communications ranges, respectively.
s
is
xFig. 5. An example that k-LDPC is looser than k-DPC.
denoted as MN(i), is the set of sensors each of which has a (single or multihop) path to si and whose sensing region intersects with si’s.
Definition 8. Consider any sensor si. We say that si is
k-loose-direct-neighbor-perimeter-covered, or k-LDPC, if si is k-perimeter-covered by and
only by nodes in DN(i). Similarly, we say that si is
k-loose-multihop-neighbor-perimeter-covered, or k-LMPC, if si is k-perimeter-covered by and only by nodes
in MN(i).
We comment that, for any sensor si, DN(i)⊆ MN(i) ⊆ N(i). So the definition
that si is k-LDPC is looser than that of si is k-DPC in the sense that k-DPC
guarantees that there is a link from sito each of N (i), but k-LDPC only
guaran-tees that there is a link from si to each of DN(i). For example, consider sensor
s
iFig. 6. Proof of the Lemma 2.
sensor si’s sensing range but who has no link to si, that is, sx ∈ N(i) − DN(i).
Without taking sx into account, si is 1-perimeter-covered by and only by nodes
in DN(i) and is thus 1-LDPC. However, si is not 1-DPC since it does not have
a link to each node in N (i). The definition of k-LMPC is looser than that of
k-MPC in a similar sense.
LEMMA 2. If each sensor in S is 1-LMPC, then the network can be
decom-posed into a number of connected components each of which 1-covers the sensing field A.
PROOF. This proof is by construction. For any sensor si, we try to construct a
connected component which fully covers A. (However, the proof does not guar-antee that si has a path to every sensor.) If si’s sensing region can fully cover
A, the construction is completed. Otherwise, by Definition 8, nodes in MN(i)
must perimeter-cover si’s perimeter and each has a path to si, as illustrated in
Figure 6. In addition, nodes in MN(i), together with si, form a larger coverage
region which is bounded by perimeters of nodes in MN(i). If A is already fully covered by this region, the construction is completed. Otherwise, since each sensor is 1-LMPC, we can repeat similar arguments by extending the coverage region, until the whole field A is covered.
THEOREM 4. A sensor network can be decomposed into a number of connected components each of which k-covers A if and only if each sensor is k-LMPC.
THEOREM 5. A sensor network can be decomposed into a number of k-connected components each of which k-covers A if each sensor is k-LDPC.
The proof of Theorem 4 (respectively, Theorem 5) is similar to Theorem 2 (respectively, Theorem 3) by replacing Lemma 1 with Lemma 2. We comment that, although the results of Theorem 4 and Theorem 5 do not seem to be very
Fig. 7. An example of two connected components each of which 1-covers A.
desirable if one only knows that there are multiple 1- or k-connected compo-nents in the network, this is what we have to face in practice when deploying a sensor network. An example of Theorem 4 is shown in Figure 7. Due to the rel-atively small communication ranges compared to sensing ranges, the network is partitioned into two connected components. However, each component still provides sufficient 1-coverage.
To summarize, Theorem 4 and Theorem 5 only guarantee that the network can be sufficiently covered by each connected component, while Theorem 2 and Theorem 3 can guarantee both coverage and connectivity of the whole network. When DN(i) = N(i) or MN(i) = N(i) for each sensor si, these theorems
con-verge. Also observe that Theorem 4 and Theorem 5 are more practical because each sensor only needs to collect its reachable neighbors’ information to make its decision. Most applications can be satisfied if a subset of sensors is connected and can provide sufficient coverage. The redundance caused by multiple compo-nents may be eliminated by a higher level coordinator, such as the base station, to properly schedule each component’s working time so that no two components of the network are active at the same time.
4. DISTRIBUTED COVERAGE AND CONNECTIVITY PROTOCOLS
The quality of a sensor network can be reflected by the levels of coverage and connectivity that it offers. The previous results provide us a foundation to de-termine, or even select, the quality of a sensor network by looking at how each sensor’s perimeter is covered by its neighbors. Section 4.1 shows how to trans-late there results to fully distributed coverage and connectivity determination protocols. When sensors are overly deployed, the coverage and connectivity of the network may exceed our expectation. In this case, Section 4.2 proposes a distributed quality selection protocol to automatically adjust its coverage and connectivity by putting sensors into sleep mode and tuning sensors’ transmis-sion power. In Section 4.3, we show how to integrate the previous results into one energy-saving protocol to prolong the network lifetime.
4.1 Coverage and Connectivity Determination Protocols
The goal of the protocol is to determine the levels of coverage and connectivity of the network. For a sensor to determine how its perimeter is covered, first it has to collect how its one-hop neighboring sensors’ sensing regions intersect with its sensing region and calculate the level of its perimeter coverage. Periodical
BEACON messages can be sent to carry sensors’ location and sensing range
information. On receiving such BEACON messages, a sensor can determine who its direct neighbors are and how its perimeter is covered by them. As reviewed in Section 2, determining a sensor’s perimeter coverage can be done efficiently in polynomial time [Huang and Tseng 2003]. If the level of perimeter coverage is determined to be k in this step, we can say that this sensor is k-LDPC.
If the previous level of coverage, k, is below our expectation, the sensor can flood a QUERY message to its neighbors to find out who else is having overlapping sensing regions with itself. The flooding can be a localized flooding (with a certain hop limit) to save cost. Each sensor who receives the QUERY message has to check if its sensing region intersects with the source node’s sensing region. If so, a REPLY message is sent to the source node. In so doing, the source node can calculate its level of perimeter coverage based on the received REPLY messages. If the level of perimeter coverage is determined to be kin this step (k≥ k), we can say that this sensor is k-LMPC. If this value is still below our expectation, we can take an incremental approach by flooding another QUERY with a larger hop limit until the desired level of coverage is reached or the whole network is flooded.
After these steps, each sensor can report its exploring result to the base station or a certain centralized sensor. Then the base station can determine the coverage and connectivity levels of the network. There are three possible cases. If each sensor is at least k-LDPC, the network is k-covered and k-connected. If some sensors are at least k-LMPC, while others are at least k-LDPC, the network is k-covered and 1-connected. If there exists some sensors that are neither k-LDPC nor k-LMPC, then the network must be disconnected. In this case, it is possible that the network is still sufficiently covered but is partitioned. For example, if we remove sensor a in Figure 3(a), the network is disconnected into two parts. Although these two parts together provide 2-level coverage, since sensors are unable to exchange information, such a situation can not be determined by the network.
4.2 Coverage and Connectivity Selection Protocols
When sensors are overly deployed, one may want to put some sensors into sleep mode to reduce the level of coverage. One may further reduce the transmission power of sensors to reduce the network connectivity. As far as we know, the combination of these mechanisms has not been studied in the literature. In this section, we explore these two possibilities based on the foundations developed in Section 3.
The basic idea is as follows. Suppose that we are given a sensor network that is kinit-covered and kinit-connected (this can be decided by Theorem 4
are beyond our expectation, we propose a protocol to modify the network to
ks-covered and kc-connected such that kinit≥ ks≥ kc≥ 1. First, in Section 4.2.1,
we present a sleep protocol to reduce the network to ks-LDPC (which means
ks-covered and ks-connected) by putting some sensors into sleep mode. Then,
in Section 4.2.2, a power control protocol is presented to reduce the network to
kc-LDPC. This results in a ks-covered, kc-connected network because reducing
the transmission power of a sensor will not affect its sensing range.
4.2.1 The Sleep Protocol. In this protocol, each sensor only needs to know the locations and sensing regions of its two-hop neighbors that are in the active state. Two-hop neighbor information can be obtained by attaching the direct neighbor information of each sensor in its periodical BEACON messages. The information should include a sensor’s location, sensing range, and current power setting. Since wireless sensor networks are typically considered static, the cost to exchange such information should be low. Suppose that the network is kinit-LDPC. The purpose of this protocol is to put some sensors into the sleep
mode such that the network is at least ks-LDPC, where kinit ≥ ks. For each
sensor Sx that intends to go to sleep, it will execute the following procedure.
(1) For each sy that is a direct neighbor of sxsuch that sx and sy have
overlap-ping in their sensing regions, let p(sx, sy) be the perimeter of sy’s sensing
range that is covered by sx’s sensing range. Sensor sx then calculates the
level of coverage of p(sx, sy). If the level of coverage is at least ks+1, then
sx is a candidate.
(2) If sx is a candidate for each sy that is a direct neighbor of sx, then sx is
eligible to go to the sleep mode. Then sx waits for a random backoff time
Trandand overhears if there is any other sleeping request. (One possibility
is to set Trandaccording to sx’s remaining energy.) If any sleeping request is
heard, sx will go back to Step (1), hold for another random period, and try
again. Otherwise, sx will send a SLEEP message to each of its neighbors
and wait for their responses by setting up a timer Ts.
(3) Each sy which is a neighbor of sx can reply a GRANT-SLEEP message
to sx if it has no pending grant currently. Otherwise, a REJECT-SLEEP
message is replied. Note that to avoid erroneously putting too many sensors to sleep and to maintain synchronization, a sensor can have at most one pending grant at a time. Specifically, a GRANT-SLEEP message is clear from the pending status once a CONFIRM/WITHDRAW message is received (see Step (4)).
(4) If sx can collect a GRANT-SLEEP message from each of its neighbors, sx
broadcasts a CONFIRM message to its neighbors and then goes to sleep. If any REJECT-SLEEP message is received or the timer Ts expires, sx
broadcasts a WITHDRAW message to its neighbors.
Note that in the Step (1), sx needs to know all of the direct neighbors of
sensor sy. Since sx and sy are direct neighbors, these sensors are sx’s two-hop
neighbors. Figure 8 shows an example of the protocol. If sx intends to go
Fig. 8. An example of the Sleep Protocol. Sensor sxis a candidate with respect to sensor sy. p(sx, sy) is also covered by sz and sw. If the target coverage is ks= 1, then sx is
a candidate with respect to sy. Also note that the timer T is necessary because
we assume an unreliable broadcast.
4.2.2 The Power Control Protocol. The aim of the power control protocol is to reduce the transmission power of sensors to save energy. Since this operation does not affect the sensing unit(s), the sensing capability of sensors (and thus the level of coverage of the network) is not reduced. Suppose that the network is ks-LDPC. The purpose of this protocol is to reduce some sensors’
transmission power to make the network at least kc-LDPC, where ks≥ kc. This
results in a ks-covered, kc-connected network.
This protocol assumes that each sensor knows the information of its two-hop neighbors. For sensor sx which intends to reduce its transmission powers, it
executes the following procedure.
(1) Let sy be the direct neighbor of sx that is farthest from sx. Sensor sx then
computes the perimeter coverage of the segments p(sx, sy) and p(sy, sx). If
both segments are at least (kc+ 1)-LDPC, sx is allowed to conduct power
control. Then sx waits for a random backoff time Trand and overhears if
there is any other disconnecting request. If any request is heard, sx will go
back to Step (1), hold for another random period, and try again. Otherwise,
sx will send a DISCONNECT message to sy and wait for its response by
setting up a timer Tp.
(2) On receipt of sx’s disconnecting request, if sy has no pending disconnecting
request currently, sy can reply with a GRANT-DISC message to sx.
Oth-erwise, a REJECT-DISC is replied. Note that a DISCONNECT message is clear from the pending status once a GRANT-DISC / REJECT-DISC message is received or the timer Tpexpires.
(3) If a GRANT-DISC message is received, sx can reduce its transmission
power such that only its second-farthest direct neighbor is covered and go back to Step (1) to try to further reduce its transmission power. Otherwise, a REJECT-DISC message will stop sx from reducing its transmission
Fig. 9. A power control protocol example.
Note that in the protocol just presented, sensor sy may not be able to reduce
its transmission power even if sxsuccessfully reduces its power. This is because
sy may need to maintain connectivity with other neighbors that are farther
away than sx. Another comment is that here we choose to let sx reduce its
power step-by-step. The concern is for fairness.
Figure 9 shows an example. Initially, the network is covered and 2-connected (i.e., kinit = 2). We only consider sensor sx and its two neighbors sy
and sz. We will disconnect the communication link between sxand its
farthest-direct neighbor, sy, by power control. First, sxexamines its intersection with sy.
Both segments p(sx, sy) and p(sy, sx) are 2-LDPC, so sxsends a DISCONNECT
message to sy, which will agree by replying a GRANT-DISC message. Then sx
can reduce its transmission power to the level that can reach the next farthest neighbor sz. Next, sxexamines its intersection with sz. Both segments p(sx, sz)
and p(sz, sx) are 2-LDPC, so sx sends a DISCONNECT message to sz. Suppose
that szhas a pending disconnecting request currently, it will reply a
REJECT-DISC message to sx. Then sx stops its procedure. Note that in this scenario, sy
may not necessarily reduce its transmission power even if it grants sx’s request
to reduce power. For example, sy may not be able to reduce its power because
sw wants to remain connected with sy. In order to maintain connectivity with
Fig. 10. An integrated energy-saving protocol.
(i.e., the transmission power of sxcannot reach sy, but the transmission power
of sy can reach sx). Therefore, only sxcan benefit from the transmission power.
4.3 An Integrated Energy-Saving Protocol
In Figure 10, we show how to integrate the above coverage and connectivity determination protocol, sleep protocol, and power control protocol together into one protocol. The purpose is to save energy while maintaining the quality of the network. Basically, these subprotocols are executed in this order. We assume that the goal is to achieve a ks-covered, kc-connected network, where ks ≥ kc.
wake themselves up, and one called Tcycle for sensors to recheck their local
coverage and connectivity (this is to prevent neighboring sensors from running out of batteries, thus resulting in a network weaker than ks-covered and
kc-connected). Also, a new HELP message is designed for sensors to call others’
assistance to increase the coverage and connectivity of the network (if possible) when some sensors run out of energy. Note that whenever a sensor goes to the initial state, it will use the largest transmission power to determine its local network coverage and connectivity. For example, this applies to a sensor when it receives a HELP message under a reduced transmission power status.
5. SIMULATION RESULTS
In this section, we present two sets of simulation experiments. Experiment 1 tests the network coverage and connectivity at different sensing ranges and communication ranges. Experiment 2 evaluates the performance of the proposed energy-saving protocol.
5.1 Experiment 1: Coverage and Connectivity
We have developed a simulator to compare the network coverage and con-nectivity calculated by Theorem 5 and by an exhausted search algorithm. All results in this section are from averages of at least 100 runs. The simulation environment is a 100 × 100 square area, on which sensors are randomly deployed. The sensing range and communication range of each sensor are uniformly distributed in certain ranges.
Figure 11 shows the coverage and connectivity under different communica-tion ranges. Note that Theorem 5 may not be able to find the exact coverage and connectivity levels because it only relies on local information. Our goal is to compare the results obtained by Theorem 5 (which implies coverage as well as connectivity) against the minimum of the actual coverage and actual con-nectivity obtained by an exhaustive search. So Figure 11(a) represents an ideal situation because what is found by Theorem 5 matches closely with the actual values. The gaps increase as we move to Figure 11(b), (c), and (d). This is be-cause the ratios of average communication range to average sensing range are reduced, which means that a sensor may not know of the existence of another sensor which overlaps with its own sensing range if it only examines its direct neighbors. So a certain degree of coverage and connectivity is not discovered by Theorem 5.
Next, we keep the sensing ranges fixed, but change the communication ranges variations. Figure 12 shows the coverage and connectivity in a 300-node network when we vary the mean and variation of communication ranges. Note that at each point of Figure 12(a), sensors’ communication ranges have no vari-ation, while at each point of Figure 12(b), the variation range is 20. Although in both cases Theorem 5 finds about the same values of coverage and connectivity, since the actual connectivity reduces, Theorem 5 matches more closely the ac-tual situations in the case of Figure 12(b). In Figure 13, we conduct the similar simulation by keeping the communication ranges unchanged but changing the mean and variation of the sensing ranges. The trend is similar—Theorem 5
Fig. 11. Network coverage and connectivity under different communication ranges.
Fig. 12. Network coverage and connectivity under different means and variations of communica-tion ranges.
matches the actual situations more closely when there are larger variations in sensing ranges. Also, by comparing Figure 12 and Figure 13, we observe that the gaps reduce when the ratios of the average communication range to the average sensing range increase. The reason is that as the ratio increases, a sensor is able to collect more information about its neighborhood.
Fig. 13. Network coverage and connectivity under different means and variations of sensing ranges.
5.2 Experiment 2: Network Lifetime
This section verifies our integrated energy-saving protocol for prolonging network lifetime while ensuring both coverage and communication quality. We consider three performance metrics: number of alive nodes, coverage level, and connectivity level. In these experiments, there are 300 sensors randomly deployed in a 100× 100 square area with sensing range = 15 ∼ 25 units, com-munication range= 30 ∼ 50 units, and initial energy = 8,000 ∼ 12,000 units (all in a uniform distribution). Our goal is to achieve a ks-covered and kc-connected
network. We sample the network status every 10 seconds. We assume a constant traffic rate for each sensor, and the energy cost of each transmission is proportional to a sensor’s transmission range. The energy cost for sensing is also proportional to a sensor’s sensing range [Lu et al. 2005]. Therefore, for each sensor si, the energy consumed every second is proportional to the sum of
its sensing range ri and its current communication range ci. Although this is a
simplified assumption, if the energy cost of each transmission is proportional to a sensor’s transmission distance raised to a factor of 2 or 4, our power control scheme should demonstrate even more saving in energy consumption.
Two versions of protocols are evaluated, one with the Sleep protocol only and the other with Sleep+Power Control protocol (denoted by Sleep+PC). We com-pare our results against a naive protocol where all sensors are always active, and the CCP+SPAN protocol [Wang et al. 2003]. CCP (Coverage Configuration Protocol) is a protocol that can dynamically configure a network to achieve guaranteed degrees of coverage and connectivity if sensors’ communication ranges are no less than twice their sensing ranges. If sensors’ communication ranges are less than twice their sensing ranges, Wang et al. [2003] suggests integrating CCP with SPAN which is a decentralized protocol that tries to conserve energy by turning off unnecessary nodes while maintaining a communication backbone composed of active nodes.
Figure 14(a) shows the number of alive sensors when the goal is to maintain a 2-covered and 1-connected network. In the naive protocol, because nodes are always active, the number of alive sensors drops sharply at around 150s. Sensors in CCP+SPAN protocol fail at a slower speed. Both Sleep and
Nave CCP+SPAN Sleep Sleep+PC Nave CCP+SPAN Sleep Sleep+PC
Fig. 15. Network lifetime under different communication ranges (sensing range= 15 ∼ 25). Sleep+PC protocols can significantly reduce the rate that sensors fail. Overall, Sleep+PC performs the best. This can be explained by the levels of coverage and connectivity provided by a protocol, as shown in Figure 14(b) and Fig-ure 14(c). There is too much redundancy in coverage and connectivity in both the naive and CCP+SPAN protocols. The Sleep protocol maintains the level of coverage pretty well, but the level of connectivity is still much higher than our expectation. Only Sleep+PC can maintain the best-fit coverage and connec-tivity levels. This justifies the usefulness of adopting power control to adjust the communication topology of the network. Figure 14(d) shows the network lifetime which is defined as the time before the levels of coverage and connec-tivity drop below our expectations. The lifetime of the naive protocol is around 150s. The lifetime of CCP+SPAN is around 200s. The Sleep and Sleep+PC protocols can significantly prolong network lifetime to around 340 and 410s, respectively. Figure 14(e), (f), (g), and (h) are from similar experiments where the goal is to maintain a 3-covered and 2-connected network. The trend is similar.
In the following, only the network lifetime is shown. Figure 15 shows the network lifetime under the same sensing range (15∼25) but different commu-nication ranges. In all situations, Sleep+PC performs the best. In fact, when the communication range increases, the gaps between Sleep+PC and other protocols enlarge relatively. So our power control scheme can effectively reduce network connectivity and prolong network lifetime. Basically, our schemes can perform better when sensors have larger communication ranges. This is be-cause sensors with larger transmission ranges can find more neighbors, collect more necessary information needed for making sleeping and power-controlling decisions, and thus have a better chance of going to sleep and/or shrinking their powers. Besides, it is obvious that power control can more effectively reduce network connectivity if sensors’ initial communication ranges are larger. However, how much Sleep+PC can outperform other schemes also relies on the network density and the level of coverage and connectivity to be achieved. If ksand kcare closer to kinit, our scheme is less effective. Figure 16
shows similar experiments under the same communication range (30∼50) but different sensing ranges. In Figure 17, we further test under different
Fig. 16. Network lifetime under different sensing ranges (communication range= 30 ∼ 50).
Fig. 17. Network lifetime under different coverage and connectivity requirements (sensing range = 15 ∼ 25 and communication range = 30 ∼ 50).
coverage and connectivity requirements. Around 1 to 2 times more lifetime can be demonstrated when comparing Sleep+PC to CCP+SPAN.
6. CONCLUSIONS AND FUTURE WORK
We have proposed fundamental theorems for determining the levels of coverage and connectivity of a sensor network. Earlier works are all based on stronger as-sumptions that the sensing distances and communication distances of sensors must satisfy some relations. We study this issue under an arbitrary relationship between sensing and communication ranges. Based on the proposed theorems, we have developed distributed protocols for determining the levels of coverage and connectivity of a sensor network and even for adjusting a sensor network to achieve the expected levels of coverage and connectivity. The approaches that we take are to put some sensors into the sleep mode and to reduce some sensors’ transmission power. As far as we know, the combination of these mech-anisms has not been well studied in this field, especially where coverage and connectivity issues are concerned. In our work, a deterministic model is used to
formulate sensors’ sensing and communication ranges. In reality, these values may follow a probabilistic model (such as a sensor’s ability to successfully detect at object at a distance d with a probability prob(d )). The coverage connectivity combined issue still requires further investigation in this direction.
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