a study of k-coverage and measures of connectivity in 3d wireless sensor networks-o94

A Study of k-Coverage and Measures ofConnectivity in 3D Wireless Sensor Networks

Habib M. Ammari, Member, IEEE, and Sajal K. Das, Senior Member, IEEE

AbstractIn a wireless sensor network (WSN), connectivity enables the sensors to communicate with each other, while sensing

coverage reflects the quality of surveillance. Although the majority of studies on coverage and connectivity in WSNs consider

2D space, 3D settings represent more accurately the network design for real-world applications. As an example, underwater sensor

networks require design in 3D rather than 2D space. In this paper, we focus on the connectivity and k-coverage issues in 3D WSNs,

where each point is covered by at least k sensors (the maximum value of k is called the coverage degree). Precisely, we propose the

Reuleaux tetrahedron model to characterize k-coverage of a 3D field and investigate the corresponding minimum sensor spatial

density. We prove that a 3D field is guaranteed to be k-covered if any Reuleaux tetrahedron region of the field contains at least

k sensors. We also compute the connectivity of 3D k-coveredWSNs. Based on the concepts of conditional connectivity and forbidden

faulty sensor set, which cannot include all the neighbors of a sensor, we prove that 3D k-coveredWSNs can sustain a large number of

sensor failures. Precisely, we prove that 3D k-covered WSNs have connectivity higher than their coverage degree k. Then, we relax

some widely used assumptions in coverage and connectivity in WSNs, such as sensor homogeneity and unit sensing and

communication model, so as to promote the practicality of our results in real-world scenarios. Also, we propose a placement strategy of

sensors to achieve full k-coverage of a 3D field. This strategy can be used in the design of energy-efficient scheduling protocols for

3D k-covered WSNs to extend the network lifetime.

Index Terms3D k-covered wireless sensor networks, Reuleaux tetrahedron, coverage, connectivity.

1 INTRODUCTION

A wireless sensor network (WSN) consists of a largenumber of resource-limited (such as CPU, storage,battery power, and communication bandwidth), tiny de-vices, which are called sensors. These sensor nodes can sensetask-specific environmental phenomenon, perform in-net-work processing on the sensed, and communicate wirelesslyto other sensor nodes or to a sink (also, called data gatheringnode), usually via multihop communications. WSNs can beused for a variety of applications dealing with monitoring,control, and surveillance.

In the literature, most of the works on WSNs dealt with2D settings, where sensors are deployed on a planar field.However, there exist applications that cannot be effectivelymodeled in the 2D space. For instance, WSNs deployed onthe trees of different heights in a forest, or in a buildingwith multiple floors, or underwater applications [2], [3]require the design in the 3D space. Oceanographic datacollection, pollution monitoring, offshore exploration, dis-aster prevention, and assisted navigation are typicalapplications of underwater sensor networks [2]. In [27],different deployment strategies have been proposed for 2D

and 3D communication architectures in underwater acous-tic sensor networks, where sensors are anchored at thebottom of the ocean for the 2D design and float at differentdepths of the ocean to cover the entire 3D region. As shownin [31], both coverage of space and energy-efficient datarouting for a telepresence application require 3D design.

In this paper, we study the coverage and connectivityissues in 3D k-covered WSNs, where sensors are deployed ina 3D field, such that every location is covered by at leastk sensors (a property known as k-coverage, where themaximum value of k is called the coverage degree). Thelimited battery power of sensors and the difficulty ofreplacing and/or recharging batteries in hostile environ-ments require that the sensors be deployed with highdensity (up to 20 sensors per cube meter [30]) in order toextend the network lifetime. Moreover, to cope with theproblem of faulty sensors due to low battery power andachieve high data accuracy, redundant coverage (in parti-cular, k-coverage) of the same region is necessary. Asmentioned, the essential function of a WSN is to monitor afield of interest and report the sensed data to the sink forfurther processing. However, sensing coverage would bemeaningless if the sensed data cannot reach the sink due tothe absence of communication paths between it and thesource sensors (or data generators). In other words, networkconnectivity must be guaranteed. A WSN is said to be faulttolerant if it remains functionally connected in spite of somesensor failures. That is why each source sensor has to beconnected to the sink by multiple communication paths.This is why we focus on connectivity. Hence, a WSN will beable to function properly if both coverage and connectivityare maintained simultaneously. This implies that all sourcesensors and the sink should belong to the same connectedcomponent.

IEEE TRANSACTIONS ON COMPUTERS, VOL. 59, NO. 2, FEBRUARY 2010 243

. H.M. Ammari is with the Wireless Sensor and Mobile Ad-hoc Networks(WiSeMAN) Research Lab, Department of Computer Science, HofstraUniversity, 205 Adams Hall, Hempstead, NY 11549.E-mail: [email protected].

. S.K. Das is with the Center for Research in Wireless Mobility andNetworking (CReWMaN), Department of Computer Science and En-gineering, University of Texas at Arlington, Room 249B, Nedderman Hall,Arlington, TX 76019-0015. E-mail: [email protected].

Manuscript received 9 June 2008; revised 31 Jan. 2009; accepted 9 June 2009;published online 28 Oct. 2009.Recommended for acceptance by R. Marculescu.For information on obtaining reprints of this article, please send e-mail to:[email protected], and reference IEEECS Log Number TC-2008-06-0282.Digital Object Identifier no. 10.1109/TC.2009.166.

0018-9340/10/$26.00 2010 IEEE Published by the IEEE Computer Society

Authorized licensed use limited to: IEEE Xplore. Downloaded on May 13,2010 at 11:46:58 UTC from IEEE Xplore. Restrictions apply.

Traditional (or unconditional) connectivity assumes thatany subset of nodes can potentially fail at the same time,including all the neighbors of a given node. While suchconnectivity can measure the fault tolerance of small-scalenetworks, it is not effective for large-scale dense networks,such as highly dense, 3D k-covered WSNs. In such networkswith thousands of sensors, it is highly unlikely that all theneighbors of a given sensor node fail simultaneously. This isdue to high-density deployment and heterogeneity of thesensors. Assuming a uniform sensor distribution, the ratioof the size of the neighbor set of a given sensor to the totalnumber of sensors in a WSN covering a field of volume V is4R3=3V , where R V 1=3 is the radius of the communica-tion range of the sensors. The probability of failure of all theneighbors of a given sensor could be identified with thisratio, and hence, is very low. Furthermore, in real-worldsensing applications [19], WSNs have heterogeneoussensors with unequal energy levels and different sensing,processing, and communication capabilities, thus increasingthe network reliability and lifetime [36]. This also impliesthat the probability that the entire neighbor set of a sensorfails at the same time is very low.

In this study, we use a more general (and realistic)concept of connectivity, called conditional connectivity [15]with respect to some property P, defined as follows: Theconditional connectivity of a connected graph G is thesmallest number of nodes of G whose removal disconnectsG into components each of which has property P. Anothergeneralization of connectivity, called restricted connectivity[12], is based on the concept of forbidden faulty set, where allneighbors of a node cannot simultaneously fail.

1.1 Contributions

In this paper, we address the following questions concern-ing coverage and connectivity in 3D k-covered WSNs:

1. Given a 3D field, what is the minimum sensor spatialdensity to guarantee full k-coverage of the field?

2. What is the connectivity and conditional connectiv-ity of 3D k-covered WSNs?

To this end, our main contributions are summarized asfollows:

1. We characterize k-coverage of 3D WSNs by usingthe fundamental result of the intersection of convexsets stated in Hellys Theorem. Based on thischaracterization and the concept of Reuleaux tetra-hedron, we compute the minimum sensor spatialdensity required to guarantee full k-coverage of a3D field. We find that this density is k=0:422r30 forhomogeneous 3D k-covered WSNs, where r0 r=1:066and r is the sensing radius of sensors. For hetero-geneous 3D k-covered WSNs, r is replaced with rmin,the minimum sensing radius of all sensors. Oursimulation results match well with the analyticalresults.

2. Based on the minimum sensor spatial density, weprove that the connectivity of 3D k-covered WSNs ismuch higher than the degree k of sensing coverageprovided by the network. More precisely, thisconnectivity is 9:926R=r03k for homogeneousk-covered WSNs, where r0 r=1:066 and R is the

radius of the communication range of sensors. Forheterogeneous 3D k-coveredWSNs, r is replacedwithrmin and R with Rmin, where Rmin is the minimumradius of the communication range of all sensors.

3. We show that the traditional connectivity metricused to capture network fault-tolerance underesti-mates the resilience of 3D k-covered WSNs. Inparticular, we find that the conditional connectivityof 3D k-covered WSNs is equal to r0 2R3 r30=r30k for homogeneous k-covered WSNs, wherer0 r=1:066. For heterogeneous 3D k-coveredWSNs,r andR are replaced with rmin andRmin, respectively.

The remainder of this paper is organized as follows:Section 2 reviews related work on coverage and connectiv-ity. Section 3 introduces the basic assumptions anddefinitions. Section 4 discusses coverage and connectivityof 3D k-covered WSNs, while Section 5 focuses on theirconditional connectivity. Section 6 shows how to relax someof the assumptions used in our analysis. Section 7 concludesthe paper.

2 RELATED WORK

In this section, we review a sample of approaches forcoverage and connectivity in 2D and 3D WSNs. For morecomprehensive review on coverage and connectivity inWSNs, the interested reader is referred to [8], [13].

2.1 Coverage and Connectivity of 2D WSNs

In [1], a directional sensors-based approach is proposed forWSN coverage, where the coverage region of a directionalsensor depends on the location and the orientation ofsensors. A differentiated coverage algorithm is presented in[9] for heterogeneous WSNs, where different network areasdo not have the same coverage degree. In [14], the authorsproposed centralized and distributed algorithms for con-nected sensor cover, so the network can self-organize itstopology in response to a query and activate the necessarysensors to process the query. The relationship betweensensing coverage and communication connectivity of WSNsis studied and distributed protocols to guarantee that bothcoverage and connectivity are proposed in [18]. Based onthe Voronoi diagram and graph search algorithms, anoptimal polynomial time (worst and average case) algo-rithm for coverage calculation is proposed in [26].

Studies on k-coverage to maintain connectivity in 2DWSNs have been the focus of several works. The first tostudy coverage and connectivity in WSNs in an integratedfashion is due to [34], where the authors proved that if thecommunication radius of sensors is double their sensingradius, a network is connected given the coverage isprovided. They also proposed a coverage configurationprotocol based on the degree of coverage of the application.In [17], polynomial-time algorithms are presented for thecoverage problem to check whether every point in a field isat least k-covered, depending on whether sensors have thesame or different sensing ranges. Efficient distributedalgorithms are proposed in [24] to optimally solve the bestcoverage problem with the least energy consumption. In[37], Zhang and Hou proposed a distributed algorithm inorder to keep a small number of active sensors in a WSNregardless of the relationship between sensing and com-munication ranges. The results in [34] and [37] are

244 IEEE TRANSACTIONS ON COMPUTERS, VOL. 59, NO. 2, FEBRUARY 2010


improved in [32] by proving that if the original network isconnected and the identified active nodes can cover thesame region as all the original nodes, then the networkformed by the active nodes is connected when thecommunication range is at least twice the sensing range.An optimal deployment strategy is described in [5] toachieve both full coverage and 2-connectivity regardless ofthe relationship between communication and sensing radiiof sensors. A distributed and localized algorithm using theconcept of the kth-order Voronoi diagram is proposed in[40] to provide fault tolerance and extend the networklifetime, while maintaining a required degree of coverage.

Necessary and sufficient conditions for 1-covered,1-connected wireless sensor grid networks are presented in[29]. A variety of algorithms have been proposed tomaintainconnectivity and coverage in large WSNs. In [20], Kumaret al. proposed k-barrier coverage for intrusion detection.The authors established an optimal deployment pattern forachieving k-barrier coverage and developed efficient globalalgorithms for checking such coverage. As shown in [21], theminimum number of sensors needed to achieve k-coveragewith high probability is approximately the same regardlessof whether sensors are deployed deterministically orrandomly, if sensors fail or sleep independently with equalprobability. The changes of the probability of k-coveragewith the radius of the sensing range of sensors or the numberof sensors are studied in [33]. The k-coverage set and thek-connected coverage set problems are formalized in termsof linear programming with two nonglobal solutions in [35].The coverage problem in heterogeneous planar WSNs as aset intersection problem is formulated in [22] and [23] whichalso derived analytical expressions to which quantify thestochastic coverage. It is proved in [38] that the requireddensity to k-cover a square region depends on both the sidelength of the square field and the coverage degree k.2.2 Coverage and Connectivity of 3D WSNs

Coverage and connectivity in 3D WSNs have gainedrelatively less attention in the literature. A placementstrategy based on Voronoi tessellation of 3D space isproposed in [4], which creates truncated octahedral cells.Several fundamental characteristics of randomly deployed3D WSNs for connectivity and coverage are investigated in[28], which compute the required sensing range to guaranteecertain degree of coverage of a region, the minimum andmaximum network degrees for a given communicationrange as well as the hop-diameter of the network. Related toour work is the novel result discussed in [43], which provedthat the breadth of the Reuleaux tetrahedron is not constant.This shows that the properties of 2D space cannot be directlyextended to 3D space. Indeed, the Reuleaux triangle [44](counterpart of Reuleaux tetrahedron in 2D space) has aconstant width. Note that the Reuleuax tetrahedron is thesymmetric intersection of four congruent spheres such thateach sphere passes through the centers of the other threespheres. However, the Reuleaux triangle [44] corresponds tothe symmetric intersection of three congruent disks such thateach disk passes through the centers of the other two disks.Thus, its constant width is equal to the radius of these disks.

Our work can be viewed as an extension of [4] byconsidering k-coverage in 3D WSNs. Moreover, existingworks on coverage and connectivity in WSNs assumed thenotion of traditional connectivity, while our work considersamore realisticmeasure, namely, conditional connectivity [18],

which is based on the concept of forbidden faulty set [15].Also, our work exploits the result given in [43] with regardto the breadth of the Reuleaux tetrahedron discussed earlier.This helps us provide correct measures of connectivity andfault tolerance of 3D WSNs based on an accurate character-ization of k-coverage of 3D fields.

3 ASSUMPTIONS AND DEFINITIONS

In this section, we present the assumptions made in ouranalysis and define some key concepts. Relaxation of someof the assumptions will be discussed in Section 6 andAppendix B.

We consider staticWSNs,which represent the sensing andcommunication ranges of sensors as spheres. From now on,the sensing and communication ranges of a sensor si arecalled sensing and communication spheres of radii ri and Ri,respectively, centered at iwhich represents the location of si.In a homogeneous (heterogeneous)WSNs, the sensors have (donot have) the same sensing ranges and the same communica-tion ranges. The communication links between sensors aresupposed to be perfectly reliable, while sensor nodes can failor die independently due to low battery power. We considera cubic field of volume V in 3D euclidean space. Moreover,we assume that the volumes of the sensing and communica-tion spheres of sensors are negligible compared to V suchthat ri, Ri V 1=3.

Let us define the following terms. A location of afield is said to be 1-covered (or sensed) if it belongs to thesensing sphere of at least one sensor. A 3D convexregion C is said to be k-covered if each location in C iscovered by at least k sensors.

The neighbor set of a sensor si is defined as Nsi fsj : ji jj Rig, where j is the location of the neighbor sjof si and ji jj is the euclidean distance between i and j.

The breadth of 3D convex region C is the maximumdistance between tangential planes on opposing faces oredges of C.

A communication graph of a WSN is a graph G S; L,where S is a set of sensors and L is a set of communicationlinks between them such that for all si, sj 2 S, si; sj 2 L ifji 0jj Ri.

The connectivity of a communication graph G of a WSNis equal to k if G can be disconnected by removing at leastk sensors.

4 UNCONDITIONAL CONNECTIVITY OF 3Dk-COVERED WSNS

In this section, we show how to guarantee k-coverage of a3D field, derive the corresponding minimum sensor spatialdensity, and compute measures of unconditional (or tradi-tional) connectivity for homogeneous 3D k-covered WSNs,where any subset of sensors can fail. The results forheterogeneous WSNs are summarized in Appendix A.

4.1 Minimum Sensor Density for k-Coverage

Lemma 1 characterizes the breadth of a 3D k-coveredconvex region.

Lemma 1. A 3D convex region C is guaranteed to be k-coveredwith exactly k sensors if its breadth is less than or equal to r,the radius of the sensing spheres of the sensors.

AMMARI AND DAS: A STUDY OF k-COVERAGE AND MEASURES OF CONNECTIVITY IN 3D WIRELESS SENSOR NETWORKS 245


Proof (By contradiction). Assume that the breadth of a 3Dconvex region C does not exceed r and C is not k-coveredwhen exactly k sensors are deployed in it. Notice thateach of these k sensors is located on the boundary orinside of C. Thus, there must be at least one location 2 C that is not k-covered. In other words, there is atleast one sensor si located at

0 such that j 0j > r,which contradicts our hypothesis that the breadth of Cdoes not exceed r. tuIt is true that the deployment of k sensors in a 3D convex

region, say C, whose breadth is larger than r cannotguarantee its k-coverage, where r is the radius of the sensingrange of the sensors. Let pi and pj be two points inC such thatone sensor si is located at pi and pi; pj b > r, where b isthe breadth of C and pi; pj is the euclidean distancebetween pi and pj. Given that b > r, it is impossible for si tosense any event that occurs at pj. Thus, there is at least onesensor (i.e., si) among those k sensors, which cannot cover pj,and hence, C cannot be k-covered.

Next, we compute the minimum sensor spatial densityrequired for guaranteeing k-coverage of a 3D field. To thisend, we compute the maximum volume of a 3D convexregion C that is guaranteed to be k-covered when exactlyk sensors are deployed in it. First, we state Hellys Theorem[6], a fundamental result that characterizes the intersectionof m convex sets in n-dimensional space, where m n 1.Hellys Theorem (Intersection of convex sets) [6]. Let be a

family of convex sets in IRn such that for m n 1, anym members of have a nonempty intersection. Then, theintersection of all members of is nonempty.

From Hellys Theorem [6], we infer that givenk 4 sensors, a 3D convex region C is k-covered by thosek sensors if and only if C is 4-covered by any four of those ksensors. Given that the breadth of C is r, the networkinduced by sensors located in C is guaranteed to beconnected if R r, where r and R are, respectively, theradii of the sensing and communication spheres of sensors.Now, let us address the first question: What is the minimumsensor spatial density necessary to guarantee full k-coverage of a3D field? Theorem 1 computes the minimum sensor density.

Theorem 1. Let r be the radius of the sensing spheres of sensorsand k 4. The minimum sensor spatial density required tofully k-cover a 3D field is computed as

r; k k0:422r30

; 1

where r0 r=1:066.Proof. Let Ck be the intersection of k sensing spheres and

assume that their centers do not coincide in a 3D field. FromLemma 1, it follows that Ck is guaranteed to be k-coveredby exactly k sensors if its breadth does not exceed theradius r of the sensing spheres of sensors. Thus, themaximum volume of Ck is obtained when its breadth isequal to r. From Hellys Theorem [6], it follows that theintersection of k sensing spheres is not empty if theintersection of any four of these k spheres is not empty. Onthe other hand, the intersection set operator requires thatthe maximum intersection volume of these k sensingspheres be equal to that of four spheres provided that themaximum distance between any pair of these k sensors

does not exceed r. Let us focus on the analysis of foursensing spheres. The maximum overlap volume of foursensing spheres such that every point in this overlapvolume is 4-covered corresponds to the configuration: thecenter of each sensing sphere is at distance r from thecenters of all other three sensing spheres. Precisely, thesensing sphere of each of the four sensors passes throughthe centers of the other three sensing spheres, as shown inFig. 1. The edges between the centers of these four spheresform a regular tetrahedron and the shape of the intersectionvolume of these four spheres is known as the Reuleauxtetrahedron [43] and denoted by RT r. Unfortunately, itwas proved that the Reuleaux tetrahedron does not have aconstant breadth [43]. Indeed, while the distance betweensome pairs of points on the boundary of the Reuleauxtetrahedron RT r is equal to r, the maximum distancebetween other pairs of points on the boundary ofRT r isequal to 1:066r [43], i.e., slightly larger than r. This impliesthat the Reuleaux tetrahedron RT r cannot be k-coveredwith exactly k sensors given that the distance betweensome pairs of points on the boundary of RT r is largerthan r, where r is the radius of the sensing spheres of thesensors. Therefore, the Reuleaux tetrahedron that isguaranteed to be k-covered with exactly k sensors shouldhave a side length equal to r0 r=1:066. The volume of theReuleaux tetrahedron RT r0 is given by [43]

volr0 3ffiffiffi2

p 49 162 tan1

ffiffiffi2

pr30=12 0:422r30:

Thus, RT r0 is the maximum volume that can bek-covered by exactly k sensors, where k 4. Weconclude that the maximum volume of Ck, denoted by

volmaxCk, is equal to volmaxCk 0:422r30. Given thatvolmaxCk has to contain k sensors to k-cover Ck, weconclude that the minimum sensor spatial density per

unit volume required for full k-coverage of a 3D field is

computed as

r; k k=volmaxCk k=0:422r30:ut

We should mention that the notion of the arc lengthdiscussed in [43] corresponds to the (maximum) breadth ofthe Reuleaux tetrahedron. Indeed, it is possible to find twoparallel plans that bound the Reuleaux tetrahedron suchthat the maximum distance between these two plans isequal to r only.


Fig. 1. (a) Intersection of four symmetric spheres and (b) theircorresponding Reuleaux tetrahedron.


It is worth emphasizing that the value of r; k is tight inthe sense that it is minimum given that k sensors should belocated within volmaxCk, such that Ck is guaranteed to bek-covered with exactly these k sensors. Also, notice thatr; k depends only on the coverage degree k dictated by asensing application and the radius r of the sensing range ofsensors. Moreover, r; k increases with k. Indeed, highcoverage degree k requires more sensors to be deployed, andhence, a denserWSN is necessary. Also, r; k decreases as rincreases. When the sensing range gets larger, a fewernumber of sensors is needed to fully k-cover a 3D field. Bothof these observations reflect the real behavior of the sensors.Thus, r; k does not depend on the size of the field asopposed to the result of the stochastic approach in [38].

Lemma 2 uses Theorem 1 and states a sufficient conditionfor k-coverage of a 3D field.

Lemma 2. A 3D field is guaranteed to be k-covered if anyReuleaux tetrahedron region of side r0 in the field contains atleast k sensors, where r0 r=1:066 and k 4.

It is worth noting that there may be some differencesbetween analytical and simulation/implementation resultsdue to the boundary effects. In fact, it is impossible todecompose a 3D field into complete Reuleaux tetrahedra suchthat the Reuleaux tetrahedra close to the border of a 3D fieldlie entirely inside it. Hence, more than necessary number ofsensors is used to k-cover the border of the 3D field.

Next, based on the minimum sensor spatial density(Theorem 1) and another criterion to be specified in thefollowing section, we compute the network connectivity of3D k-covered WSNs.

4.2 Computing Network Connectivity

Data accuracy depends on the size of the connectedcomponent that contains the sink. It reaches the highestvalue when the sink belongs to the largest connectedcomponent of the network. Thus, high-quality coveragerequires all source sensors be connected to the sink. That iswhy we focus on the sink to compute the connectivity of3D k-covered WSNs. In other words, connectivity of WSNsshould be so defined as to take into consideration theinherent structure of this type of network. Indeed, sensorshave neither the same role nor the same impact on thenetwork performance. Thus, measuring the connectivity ofWSNs should account for their specific morphology, wherethe sink is the most critical node in the network. Hence, wecompute the connectivity of 3D k-covered WSNs based onthe size of the connected component that includes the sink.Theorem 2 deals with homogeneous WSNs.

Theorem 2. Let G be a communication graph of a homogeneous3D k-covered WSN deployed in a cubic field of volume V . Theconnectivity of G, which is denoted by G, is given by

1G G 3G; 2where

1G 12:0243k;

3G RV2=3k

0:422r30;

in which r0 r=1:066, R=r, and k 4.

Proof. The optimum position of the sink in terms of energy-efficient data gathering is the center of the cubic field[25]. Let 0 be the location of the sink s0. We consider thefollowing three cases depending on the size of theconnected component that includes the sink. Also, giventhat sensor failure is due to low battery power, weassume that the sink has infinite source of energy, thusexcluding the possibility of a faulty sink.

Case 1: Single-node connected component. In thiscase, there are at least two components, one of thembeing the single-node component containing the sink.Finding the minimum number of nodes to disconnect thenetwork requires that the disconnected network has onlytwo components. In order to isolate the sink, all itsneighbors must fail. Hence, at least the communicationsphere of the sink should contain no sensor but the sink.

Let N be a random variable that counts the number ofsensor failures to isolate the sink s0. Given that sensorsare randomly and uniformly deployed in a volume Vwith density r; k per unit volume, where R V 1=3,the expected number of neighbors of the sink is given by

EN r; kkB0; Rk; 2awhere jB0; Rj 4R3=3 is themeasure of the volume ofthe communication sphere B0; R of the sink s0 locatedat 0. Thus, the expected minimum number of sensor

failures to isolate s0 is equal to EN. Substituting (1) in(2a), we find that the network connectivity is given by

1G EN 12:0243k; 2bwhere R=r. Figs. 2 and 3 plot the function 1G in(2b). Clearly, 1G increases with the ratio and thedegree of coverage k. More importantly, 1G is muchhigher than k.

Case 2: Nontrivial connected components. Twoconfigurations of the disconnected network are ofparticular interest where the two connected componentsof the disconnected network are separated by a vacantregion (or gap). Furthermore, any pair of sensors, onefrom each component, are separated by a distance atleast equal to R in order to prohibit any communicationbetween the two components. In the first configuration(Fig. 4a), the component including the sink is reduced to


Fig. 2. Plot of the function 1G (fix k and vary ).


its communication sphere. Thus, the volume of thevacant region, denoted by gap0; R, and surroundingthe component of the sink, is given by

kgap0; Rk 42R3=3 4R3=3 9:333R3: 2cThus, the expected minimum number of sensor failures

to isolate the component of the sink is given by

EN r; kkgap0; Rk: 2dSubstituting (1) in (2d), we find that the networkconnectivity is equal to

2G EN 84:1643k; 2e

where R=r.In the second configuration, the original network is

split into two components such that the vacant regionforms a cuboid, denoted by cubR, and whose sides areR; V 1=3 and V 1=3, as shown in Fig. 4b. Now, thisconfiguration corresponds to the smallest connectedcomponent containing the sink if the field has to bedivided into two regions such that none of themsurrounds the other. Thus, the expected minimumnumber of sensor failures to isolate the connectedcomponent containing the sink is given by

EN kcubRk; 2fwhere

kcubRk RV 2=3: 2gSubstituting (1) and (2g) in (2f), it follows that network

connectivity is equal to

3G EN RV2=3k

0:422r30: 2h

It is easy to check that 3G > 2G since R V 1=3.Case 3: Largest connected component. This config-

uration is totally opposite to the one given in Case 1 andhas only one isolated sensor. Since we are interested ink-coverage of the entire field, such a network isconsidered as disconnected. The network connectivityis the same as in Case 1. Thus,

1G G 3G:It is easy to check that G > k. tuAll these bounds and those to be derived in next sections

are based on our fundamental result stated in Theorem 1.Since these bounds are based on the minimum sensorspatial density for full k-coverage of a 3D field, they arelower bounds.

4.3 Boundary Effects

Network connectivity is defined as the minimum number ofsensors whose failure (or removal) disconnects the network.Given the cubic geometry of a 3D field we considered,sensors located close to the border of the field are affectedby the boundary effects. Indeed, the communication rangesof these sensors cover areas outside of the deployment area,and hence, have fewer number of communication neighborscompared to all other sensors (especially the ones whosecommunication ranges lay entirely inside the field). How-ever, the proposed network connectivity measures considerthe sink as the most critical node in the network and whoseisolation would definitely kill the entire network. Becausethe optimum position of the sink in terms of energy-efficientdata gathering is the center of the cubic field [25], theboundary effects do not exist at all, and thus, our derivedbounds on the connectivity are correct. Even when our


Fig. 3. Plot of the function 1G (fix and vary k).

Fig. 4. 2D projection of nontrivial-connected components of thedisconnected network.


approach for computing network connectivity measuresconsiders all sensors as critical and peer-to-peer (seeAppendix C for detailed discussion), the boundary effectsdo not have any impact on the derived bounds on networkconnectivity. Indeed, we are interested in the minimumnumber of sensor failures to disconnect the network.

As far as sensor deployment to achieve 3D k-coverage isconcerned, we should mention that the boundary effectshave an impact on the performance of the network. ByLemma 2, a 3D field is guaranteed to be fully k-covered if andonly if each Reuleaux tetrahedron region in the field containsat least k sensors. However, it is impossible to randomlydecompose a 3D field into an integer number of Reuleauxtetrahedron regions because of the boundary of the field.Indeed,most of the Reuleaux tetrahedron regions close to theborder of a 3D field do not entirely lay inside the deploymentarea. Therefore, more than necessary sensors would beneeded to achieve k-coverage of these Reuleaux tetrahedronregions on the border of the field. Simulation results reportedin Section 4.1.2 show that the sensor spatial density necessaryto fully k-cover a cubic field is a bit higher than the boundgiven in Theorem 1, mainly due to the boundary effects.

Next, we introduce new measures of connectivity of3D k-covered WSNs by placing a specific constraint on asubset of sensors that would fail.

5 CONDITIONAL CONNECTIVITY OF 3D k-COVEREDWSNS

In this section, we use the concepts of conditional connectivity[15] and forbidden faulty set [12], as a remedy to the aboveshortcomings of the traditional (unconditional) connectivitymetric.Ourapproach isbasedon forbidden faulty sensor sets.

Let G S;E be a communication graph representing a3D k-covered WSN. Define a forbidden faulty sensor set of Gas a set of faulty sensors that includes the entire neighborset of a given sensor. Consider the property P: A faultysensor set cannot include the entire neighbor set of a givensensor. A faulty sensor set satisfying property P is denotedby FP, where FP S and defined by

FP fU Sj8si 2 S : Nsi 6 Ug:The conditional connectivity of G with respect to P, denotedby G : P, is the minimum size of FP such that thegraph Gd S FP; ESFP is disconnected, where ESFPis a set of remaining communication edges between thenonfaulty sensors.

As we will see, our results prove that 3D k-coveredWSNscan sustain a larger number of sensor failures under therestriction imposed on the faulty sensor set. According toTheorem 1, the minimum number of sensors necessary tok-cover a cubic field of volume V is given by jSminj V k=0:422r3, where r is the radius of the sensing spheres ofsensors. The probability of the failure of the entire neighborset of a given sensor can be identified with the ratio of thesize of the neighbor set of a given sensor to the total numberof sensors jSminj. This ratio is equal to 4R3=3V , which is verylow given that R V 1=3. This shows that the traditionalconnectivity, which does not impose any restriction on thefaulty sensor set, is not a useful metric for 3D k-coveredWSNs, which are highly dense networks.

5.1 Homogeneous 3D k-Covered WSNs

Theorem 3 computes the conditional connectivity of

homogeneous 3D k-covered WSNs.

Theorem 3. The conditional connectivity of a homogeneous 3D

k-covered WSN k 4 is given by

G : P r0 2R3 r30k

r30; 3

where r0 r=1:066.Proof. We consider the following two cases based on the

type of connected component that contains the sink.Case 1: Smallest connected component. According to

our conditional connectivity model, no sensor can beisolated, and hence, no trivial component can be part ofthe disconnected network. Under the assumption offorbidden faulty sensor set, the smallest connectedcomponent that is disconnected from the rest of thenetwork and contains the sink can be determined asfollows: In order to achieve k-coverage of the cubic field,every location must be k-covered, including the location0 of the sink s0. Otherwise, the k-coverage property willnot be satisfied. Therefore, the smallest connectedcomponent that includes the sink consists of k sensorsdeployed in the Reuleaux tetrahedron of side r0 andcentered at 0. In order to disconnect the sink under theforbidden faulty sensor set constraint, the Reuleauxtetrahedron should be surrounded by an empty annulusof width equal to R (sensors located in the annulus havefailed), as can be seen in Figs. 5 and 6. The Reuleauxtetrahedron together with this annulus forms a largerReuleaux tetrahedron of side r0 2R. The volume of theannulus, denoted by A0; R, is equal to

kA0; Rk 0:422r0 2R3 0:422r30:Therefore, the expected conditional minimum number of

sensor failures to disconnect the smallest component

including the sink can be computed as

EN : P r; kkA0; Rk: 3a


Fig. 5. Two nested concentric Reuleaux tetrahedra.


Substituting (1) in (3a), we find that the conditionalconnectivity is given by

1G : P EN : P r0 2R3 r30k

r30: 3b

It is easy to check that the forbidden faulty setconstraint is satisfied for both the faulty sensors (locatedinside the annulus and which have failed) and nonfaultysensors (located outside the annulus). Indeed, any sensorin the inner Reuleaux tetrahedron still has nonfaultyneighbors in the inner Reuleaux tetrahedron. Besides, anysensor outside the outer Reuleaux tetrahedron still hasnonfaulty neighbors outside the outer Reuleaux tetrahe-dron. Also, any faulty sensor within the annulusAR hasnonfaulty neighbors located in the inner Reuleauxtetrahedron and outside the outer Reuleaux tetrahedron.

Case 2: Largest connected component. This case issimilar to the previous one. However, the sink belongs tothe largest connected component. Hence, the discon-nected network consists of two components: the oneincluding the sink and a second component associatedwith sensors located in a Reuleaux tetrahedron of side r.Using the same analysis as in Case 1, we obtain the sameconditional network connectivity:

2G : P 1G : P : 3cFrom both cases 1 and 2, the conditional connectivity of

homogeneous 3D k-covered WSNs is computed as

G : P 1G : P :ut

5.2 Heterogeneous 3D k-Covered WSNs

We observe that computing the conditional connectivity ofheterogeneous 3D k-covered WSNs is not a straightforwardgeneralization of the approach used previously for homo-geneous 3D k-covered WSNs. If we use the same process aspreviously, we either violate the forbidden faulty sensor setconstraint or maintain network connectivity. Precisely, ifthe width of the annulus containing the faulty sensors isRmax, then sensors whose communication radii are lessthan or equal to Rmax=2 may be located in the annulus.

Thus, the entire neighbor set of this type of sensors wouldfail at the same time, and hence, the property P would beviolated (Fig. 7a). Now, if the width of the annuluscontaining the faulty sensors is less than Rmax, then thenonfaulty sensors of one connected component will be ableto communicate with the nonfaulty sensors of the otherconnected component of the disconnected network. Hence,the network is still connected (Fig. 7b). In this case, it isimpossible to find an exact value of conditional connectiv-ity for heterogeneous 3D k-covered WSNs. Next, wecompute their lower and upper bounds based on the typesof sensors in and around the annulus.

Lemma 3. The conditional connectivity of the heterogeneous 3D

k-covered WSNs is given by

1G : FP G : P 2G : FP ; 4where

1G : P r0min 2Rmin3 r0min3k

r0min3

;

2G : P rmax 2Rmax3 r3maxk

r0min3

;


Fig. 6. 2D projection of an annulus of width R surrounding a Reuleauxtetrahedron of side r.

Fig. 7. 2D projection: (a) forbidden fault sensor set constraint violated

and (b) connectivity maintained.


in which

k 4; r0min minfrj=1:066 : sj 2 Sg;rmax maxfrj=1:066 : sj 2 Sg;Rmin minfRj : sj 2 Sg; andRmax maxfRj : sj 2 Sg:

Proof. As above, we consider the following two casesdepending on the size of the connected component thatincludes the sink:

Case 1: Smallest connected component. In order tocompute a lower bound on the conditional connectivityof heterogeneous 3D k-covered WSNs, we consider theReuleaux tetrahedron centered at location 0 of the sinks0, which will be disconnected from the network. First,we assume that the annulus containing the faulty sensorsas well as the volume surrounding it contain only leastpowerful sensors, and hence, the width of this annulus isequal to Rmin. Also, to guarantee that the sink will not beisolated, which would violate the forbidden faulty sensorset constraint, the Reuleaux tetrahedron centered at 0should have a side equal to rmin. These two conditionshelp disconnect the network while satisfying the for-bidden faulty sensor set constraint. The volume of theannulus A0; Rmin is given by

kA0; Rmink 0:422rmin 2Rmin3 0:422r3min: 4aHence, the expected conditional minimum number ofsensor failures to disconnect the connected componentincluding the sink (or the inner Reuleaux tetrahedron)from the rest of the network is computed as

EN : P rmin; kkA0; Rmink; 4bwhere

rmin; k k0:422r3min

and

rmin minfrj=1:066 : sj 2 Sg:Thus, the conditional network connectivity is given by

1G : P EN : P rmin 2Rmin3 r3mink

r3min; 4c

where k 4 and Rmin minfRj : sj 2 Sg.In order to compute an upper bound on the condi-

tional connectivity of heterogeneous 3D k-coveredWSNs, we assume that sensors inside the annulus arethe most powerful ones. Thus, the side of the innerReuleaux tetrahedron should be equal to rmax, while thewidth of the annulus surrounding it should be equal toRmax. It is easy to check that this setup will disconnectthe network, while satisfying the forbidden faulty setconstraint. The upper bound on the conditional con-nectivity is given by

2G : P EN : P rmin; kkA0; Rmaxk

rmax 2Rmax3 r3maxk

r3min:

4d

where

rmin; k k0:422r3min

; k 4;

rmin minfrj=1:066 : sj 2 Sg;rmax maxfrj=1:066 : sj 2 Sg; andRmax maxfRj : sj 2 Sg:

Case 2: Largest connected component. In this case,the sink belongs to the largest connected component ofthe disconnected network. Hence, the previous analysisapplies to any sensor in the network instead of the sink.Thus, the conditional connectivity of heterogeneous 3Dk-covered WSNs satisfies

1G;P G;P 2G;P :ut

Next, we relax the assumptions used in our previousanalysis to enhance the practicality of our results.

6 DISCUSSIONS

The analysis of the minimum sensor spatial densitynecessary for k-coverage of a 3D field and networkconnectivity of 3D k-covered WSNs is based on the unitsphere model, where the sensing and communication rangesof sensors are modeled by spheres. In other words, sensorsare supposed to be typically isotropic. Although thisassumption is the basis for most of the protocols forcoverage and connectivity in WSNs, it may not holduniversally, and thus, may not be valid in practice. InAppendix B, we show how to relax this assumption in orderto promote the applicability of our results to real-world3D WSN scenarios, and summarize our results for theconvex model, where the sensing and communicationsranges of sensors are convex but not necessarily spherical.Moreover, we assumed that our results for networkconnectivity hold for a degree of sensing coverage k, wherek 4. In this section, we show how to relax severalassumptions to account for more realistic scenarios.

6.1 Relaxing the Assumption of k 4The analysis of k-coverage and connectivity for 3Dk-covered WSNs are valid for all k 4. Since the breadthof the Reuleaux tetrahedron is equal to r, our results canalso be used for k 3. That is, a 3D field can be k-coveredby deploying k sensors in the Reuleaux tetrahedron, wherek 3. However, the network would be denser thannecessary (especially for k 1) and the coverage degreewould be higher than that dictated by the application.

6.2 Sensor Placement Strategy

Notice that under the assumption of spherical model, it isimpossible to achieve a degree of coverage exactly equal to kin all the locations of the cube. Thus, a sensor placementstrategy to achieve k-coverage should be devised in such away that every location in the cube is k0-covered, where k0 isvery close to k. This placement strategy should benefit fromthe geometry of the Reuleaux tetrahedron. The sensorplacement problem can be transformed into a problem ofcovering a cube with overlapping sets of congruent



Reuleaux tetrahedra. An optimal covering consists in usinga minimum number of Reuleaux tetrahedra by minimizingthe overlap volume between them. As described inSection 4.1.2, two adjacent Reuleaux tetrahedra overlapsuch that the faces of their corresponding regular tetrahedraare entirely coinciding with each other. Thus, the curvededges of the Reuleaux tetrahedra should overlap, so thesame subset of sensors deployed on these curved edgescould participate in covering the space associated with bothReuleaux tetrahedra, thus minimizing the number ofsensors required to k-cover the cube. Notice that sensorslocated in a 3D lens, which corresponds to the overlapvolume of two adjacent Reuleaux tetrahedra of side length r,are at distance r from any point in their volumes, and hence,participate in k-covering both tetrahedra. The design ofduty-cycling protocols to k-cover a 3D field with a minimumnumber of sensors should select sensors based on thisobservation.

6.3 Sink-Independent Connectivity Measures

Although in centralized algorithms, the concept of sink iswell defined, it is likely that distributed algorithms, such asthe consensus-based algorithm, will be implemented forWSNs. In this case, the concept of (fixed) sink would losevalue. It would be interesting to revise the definition ofconnectivity to take this concept into account. We suggestthat all the nodes be considered as peer-to-peer. Thus, wedefine connectivity with respect to all sensors in thenetwork. Given the geometry of the deployment field thatwe consider (cube), the boundary sensors, i.e., sensors locatedat the boundary of the cube, have small neighbor sets. Inparticular, the sensors located at the eight corners of thecube sb1, . . . , sb8 as shown in Fig. 8 are highly likely to havethe smallest neighbor set. Our measures of connectivity willbe based on one of these boundary sensors to find theminimum number of sensors of its neighbor set that shouldfail in order to disconnect the network. It is easy to checkthat compared to the sink, the size of the neighbor set of aboundary sensor is equal to a quarter of that of the neighborset of the sink. Thus, the previous connectivity measurescomputed in Sections 4 and 5 with respect to the sinkremain the same for a boundary sensor but are weighted bya coefficient equal to 1

4. For more details, the interested

reader is referred to Appendix C.

6.4 Stochastic Sensing and Communication Models

In the deterministic sensing model, which has been consideredso far, a point in a field is covered by a sensor si based onthe euclidean distance ; si between and si. In thispaper, we use coverage of a point and detection of an

event interchangeably. Formally, the coverage Cov; si ofa point by a sensor si is equal to 1, if ; si r, and 0,else. As can be seen, this sensing model considers thesensing range of a sensor as a sphere, and hence, all sensorreadings are precise and have no uncertainty. However,given the signal attenuation and the presence of noiseassociated with sensor readings, it is necessary to consider amore realistic sensing model by defining Cov; si usingsome probability function. That is, the sensing capability ofa sensor must be modeled as the probability of successfuldetection of an event, and hence, should depend on thedistance between it and the event as well as the type ofpropagation model being used (free-space versus multi-path). Indeed, it has been showed that the probability thatan event in a distributed detection application can bedetected by an acoustic sensor depends on the distancebetween the event and the sensor [10]. A realistic sensingmodel for passive infrared (PIR) sensors that reflect theirnonisotropic range was presented in [7]. This sensingirregularity of PIR sensors was verified by simulations [7].Thus, in our stochastic sensing model, the coverage Cov; siis defined as the probability of detection p; si of an event atpoint by sensor si as follows:

p; si e;si ; if ; si r;0; otherwise;

5

where represents the physical characteristic of the sensorssensing units and 2 4 is the path-loss exponent.Precisely, we have 2 for the free-space model and 2 < 4 for the multipath model. Our stochastic sensing modelis motivated by Elfes one [11], where the sensing capabilityof a sonar sensor is modeled by a Gaussian probabilitydensity function. A probabilistic sensing model for coverageand target localization in WSNs was proposed in [42]. Thissensing model considers ; si r re instead of ; si,where r is the detection range of the sensors and re < r is ameasure of detection uncertainty.

Under the stochastic sensing model, a point in a field issaid to be probabilistically k-covered if the detection prob-ability of an event occurring at by at least k sensors is atleast equal to some threshold probability 0 < pth < 1.

In this section, we exploit the results of Section 4.1 tocharacterize probabilistic k-coverage in 3D WSNs based onour stochastic sensing model. Theorem 4 computes theminimum k-coverage probability pk;min such that every point ina field is probabilistically k-covered.

Theorem 4. Let r be the radius of the nominal sensing range ofthe sensors, r0 r=1:066, and k 4. The minimumk-coverage probability so that each point in a 3D field isprobabilistically k-covered by at least k sensors under ourstochastic sensing model defined in (5) is computed as

pk;min 1 1 er0 k

: 6

Proof. First, we identify the least k-covered point in a 3Dfield, so we can compute pk;min. By Lemma 2, k sensorsshould be deployed in a Reuleaux tetrahedron region ofside r0 r=1:066 in the field to achieve k-coverage of a3D field with a minimum number of sensors. It is easy tocheck that the least k-covered point lc is the one thatcorresponds to the configuration where all deployed


Fig. 8. Eight boundary sensors located on the corners of a cube.


k sensors are located at a distance r0 r=1:066 from lc.In other words, lc is the farthest point from all thosek sensors. Hence, the distance between lc and each ofthese k sensors is equal to r0 r=1:066. Thus, theminimum k-coverage probability for the least k-coveredpoint lc by k sensors under the stochastic sensing modelin (5) is given by

pk;min 1Yki11 p; si 1 1 er

k:

utThe stochastic k-coverage problem is to select a mini-

mum subset Smin S of sensors such that each point in a3D field is k-covered by at least k sensors and the minimumk-coverage probability of each point is at least equal to somegiven threshold probability pth, where 0 < pth < 1. Thishelps us compute the stochastic sensing range rs, whichprovides probabilistic k-coverage of a field with a prob-ability no less than pth. Lemma 4 computes the value of rs.

Lemma 4. Let k 3 and 2 4. The stochastic sensingrange rs of the sensors that is necessary to probabilisticallyk-cover a 3D field with a minimum number of sensors andwith a probability no lower than 0 < pth < 1 is given by

rs 1log1 1 pth1=k1=

; 7

where represents the physical characteristic of the sensorssensing units.

Proof. From (6), we deduce that pk;min pth ) rs 1 log1 1 pth1=k1=. Since we are interested inthe minimum number of sensors to probabilisticallyk-cover a 3D field, we should consider the maximumvalue of rs, i.e., the maximum stochastic sensing range ofthe sensors. This will allow the sensors to probabilisti-cally k-cover as much space of the 3D deployment field aspossible. Thus, rs 1 log1 1 pth1=k1=. tuThe upper bound on the stochastic sensing range rs of the

sensors computed in (7) will be used to compute ourmeasures of connectivity and fault tolerance of 3D k-coveredWSNs under the assumption of more realistic, stochasticsensing, and communication models. Fig. 9 shows rs fordifferent values of pth and kwhile considering the free-spacemodel ( 2) (Fig. 9a) and the multipath model ( 4)(Fig. 9b). Note that rs decreases as a function of pth, k, and .This is due to the fact that the minimum probability pk;min ofk-coverage of the same location bymultiple sensors decreasesas pth, k, and increase.

Lemma 5 states a sufficient condition for probabilistick-coverage of a 3D field based on our stochastic sensingmodel in (5), the threshold probability pth, and the degree kof coverage.

Lemma 5. Let k 4. A 3D field is probabilistically k-coveredwith a probability no lower than 0 < pth < 1 if any Reuleauxtetrahedron of maximum breadth rs=1:066 in the field containsat least k sensors.

Lemma 6 states a sufficient condition for connectivitybetween sensors under our stochastic sensing model.

Lemma 6. Let k 4. The sensors that are selected to k-cover a3D field with a probability no less than 0 pth 1 under thestochastic sensing model defined in (5) are connected if the

radius of their stochastic communication range Rs is at leastequal to their stochastic sensing range rs., Rs rs.Given that network connectivity is defined as theminimum

number of sensors of a neighbor set which need to fail inorder to disconnect the network, we should consider theminimum stochastic communication range of the sensors,i.e., Rs rs, to minimize the size of the neighbor set of eachsensor. Therefore, to find our unconditional and conditionalmeasures of network connectivity and fault tolerance for3D k-coveredWSNs usingmore realistic scenarios, includingstochastic models for sensing and communications, we needto replace r by rs and R by rs in Sections 4 and 5, andAppendices A, B, and C.

6.4.1 Simulation Results

In this section, we present the simulation results using ahigh-level simulator written in the C language. We considera cubic field of side length 1,000 m. All simulations arerepeated 200 times and the results are averaged.

Fig. 10 plots the sensor spatial density as a function ofthe degree of coverage k for different values of thethreshold probability pth and for a path-loss exponent 2. As expected, the density increases with pth. Indeed,as we increase pth, more sensors would be needed toachieve the same degree of coverage k. Recall that thebreadth of the Reuleaux tetrahedron that is guaranteed tobe covered with exactly k sensors decreases as pth and increase. Precisely, this breadth is equal to rs=1:066.However, for pth 0:8, the sensor density tends to decreasewhen k goes from 4 to 5 and increases afterward. Thisbehavior is clearly noticeable for pth 0:9. This is mainly


Fig. 9. Upper bound of rs versus k.


due to the stochastic nature of the sensing range of thesensors, which depends on the logarithm of pth, thethreshold probability, k, and .

Fig. 11 plots the achieved degree of coverage k versus thetotal number of deployed sensors. Moreover, we vary bothpth and fix 2. Definitely, higher number of deployedsensors would yield higher coverage degree. Here, also, anyincrease in pth would require a larger number of deployedsensors to provide the same degree of coverage. Notice thatthe same observation holds for pth 0:8 and pth 0:8 as inthe previous experiment.

6.5 3D Sensing Applications

In the case of WSNs deployed on the trees of differentheights in a forest, the sensors could be seen almosteverywhere in the space. Pompili et al. [27] proposeddifferent deployment strategies for 2D and 3D communica-tion architectures for underwater acoustic sensor networks,where the sensors are anchored at the bottom of the oceanfor the 2D design and float at different depths of the oceanto cover the entire 3D region. Indeed, oceanographic datacollection, pollution monitoring, offshore exploration, dis-aster prevention, and assisted navigation are all typicalapplications of underwater sensor networks [2], [3]. ForWSNs deployed in buildings with multiple floors, sensorsare placed on the ground and/or the wall, but the networksseldom contain sensors floating in the middle of the room.The first examples show that our proposed 3D model isvalid and can be applied to choose the sensor density inpractical problems. The last example, however, shows thelimited validity of our model due to the restriction imposedon the placement of sensors inside buildings or rooms.

7 CONCLUSION

In this paper, we investigated coverage and connectivity in3D k-covered WSNs. Indeed, emerging applications, such asunderwater acoustic sensor networks, require 3D design.We proposed the Reuleaux tetrahedron model to guaranteek-coverage of a 3D field. Based on the geometric propertiesof Reuleaux tetrahedron, we derived the minimum sensorspatial density to ensure k-coverage of a 3D space. Also, wecomputed the connectivity of homogeneous and hetero-geneous 3D k-covered WSNs. Our approach takes intoaccount an inherent characteristic of WSNs in that the sinkhas a critical role in terms of data processing and decision

making, compared to the rest of the network. Thus, wecomputed the connectivity of 3D k-covered WSNs based onthe size of the connected component that includes the sink.We conclude that the connectivity of 3D k-covered WSNs ismuch higher than the degree of sensing coverage kprovided by the network. The traditional connectivitymetric, however, is defined in an abstract way. Moreprecisely, it does not consider the inherent properties ofWSNs because it assumes that any subset of nodes can failsimultaneously. This assumption is not valid for hetero-geneous 3D k-covered WSNs. To compensate for theseshortcomings, we proposed more realistic measures ofconnectivity based on the concept of forbidden faulty set.Using this concept, we found that 3D k-covered WSNs cansustain a large number of sensor failures.

We believe that our results have practical significance forsensor network designers to develop 3D applications withprescribed degrees of coverage and connectivity. Theseconnectivity measures can be exploited in the design offault-tolerant topology control protocols for 3D k-coveredWSNs. Our future work will focus on the design of efficientsensor deployment strategies for 3D k-covered WSNs. Also,we are interested in the design of data routing protocols onduty-cycled 3D k-covered WSNs. Indeed, joint k-coverageand routing poses major challenges due to the time-varyingconnectivity of the network. This is mainly due to the factthat the sensors are turned on or off to save energy andextend the network lifetime.

APPENDIX A

HETEROGENEOUS 3D k-COVERED WSNS

A.1 Minimum Sensor Density for k-Coverage

Achieving k-coverage of a 3D field by heterogeneoussensors would depend on the least powerful ones in termsof their sensing capabilities. Lemmas 7 and 8 correspond toLemma 1 and Theorem 1, respectively.

Lemma 7. If the breadth of a 3D convex region C is at most equalto the minimum radius rmin of the sensing spheres of sensors,then C is guaranteed to be k-covered if k (k 4) sensors aredeployed in it, where rmin minfrj=1:066 : sj 2 Sg.From Lemma 7, it follows that connectivity between

sensors located in the Reuleaux tetrahedron of side rminrequires that Rmin be at least equal to rmin, i.e., Rmin rmin.


Fig. 10. Sensor spatial density versus k. Fig. 11. k versus nd.


Lemma 8. Let rmin be the minimum radius of the sensing spheresof sensors and k 4. The minimum sensor spatial densityneeded for k-coverage of a 3D field by heterogeneous WSNs isgiven by

rmin; k k0:422r3min

; 8

where rmin minfrj=1:066 : sj 2 Sg.A.2 Network Connectivity

Lemma 9 computes the connectivity measures for hetero-geneous 3D k-covered WSNs.

Lemma 9. Let G be a communication graph of a heterogeneous3D k-covered WSN with Rmin rmin and k 4. Theconnectivity of the graph G is given by

4G G 3G; 9where

3G RmaxV2=3k

0:422r3min; 4G 12:02432k;

2 Rmin=rmin; k 4; rmin minfrj=1:066 : sj 2 Sg;Rmin minfRj : sj 2 Sg; and Rmax maxfRj : sj 2 Sg:

APPENDIX B

CONVEX SENSING AND COMMUNICATION MODEL

The assumption of spherical sensing and communicationranges of sensors given in Section 3 may not hold in real-world WSN platforms. It has been observed in [39] that thecommunication range of MICA motes [16] is asymmetricand depends on the environments. It is also found in [41]that the communication range of radios is highly probabil-istic and irregular.

In this Appendix, for problem tractability, we consider aconvex model, where the sensing and communication rangesof sensors are convex but not necessarily spherical.

First, we define the notion of largest enclosed sphere of a3D convex region C as a sphere that lies entirely inside Cand whose diameter is equal to the minimum distancebetween any pair of points on the boundary of the region C.

B.1 Homogeneous 3D k-Covered WSNs

We consider homogeneous sensors that have the sameconvex sensing ranges and communication ranges. Lem-mas 10 and 11 correspond to Lemma 1 and Theorem 1,respectively. Their proof is similar to that in Section 4 byusing the largest enclosed sphere instead of the sensingsphere.

Lemma 10. If k 4 homogeneous sensors are deployed in a3D convex region C, then the region C is k-covered; if itsbreadth does not exceed rled, the radius of the largest enclosedsphere of the sensing range.

Lemma 11. The minimum sensor spatial density required toguarantee k-coverage of a 3D field is given by

rled; k k0:422rled0

3; 10a

where rled stands for the radius of the largest enclosed sphere ofthe sensing range, r0led rled=1:066, and k 4.

Now, we discuss how those results can be derived usinga convex model, where the sensing and communicationranges of the sensors may not necessarily be spherical.

B.2 Heterogeneous 3D k-Covered WSNs

Lemmas 12 and 13 correspond to Lemma 1 and Theorem 1,respectively.Theirproof is also the sameas that inSection4byusing the largest enclosed sphere instead of sensing sphere.

Lemma 12. A 3D convex region C is guaranteed to be k-coveredwhen exactly k heterogeneous sensors are deployed in it, ifthe breadth of C does not exceed rminled , where r

minled

minfrled=1:066 : sj 2 Sg and k 4.Lemma 13. The minimum sensor spatial density required to

k-cover a 3D field is given by

rminled ; k

k0:422rminled 2

; 10b

where rminled minfrled=1:066 : sj 2 Sg and k 4.

The measures of network connectivity can be derivedusing the same approach as in Section 4.1.2 and Lemma 13.Thus, the assumption of unit sphere model for sensing andcommunication ranges of sensors can be relaxed with theaid of the largest enclosed sphere of their sensing range.

APPENDIX C

CASE OF UNDERWATER WSNS

The results in Sections 4 and 5 and in Appendices A and Bare only applicable to the connectivity of sink node.Although the connectivity of sink is critical, in somescenarios, such as underwater WSNs [2], [3], any sensormay be critical due to the high cost, for instance.

In the following, we extend our network connectivitymeasures for 3D k-covered WSNs to the case where anysensor in the network is critical. Specifically, we consider aboundary sensor, i.e., a sensor located at one corner of thecubic field. Such a boundary sensor has the minimumnumber of communication neighbors given that all sensorsare located within the deployment regionthe cube, andhence, the actual communication range of a boundarysensor is only a quarter of its communication sphere. In [34],a boundary sensor is considered to compute the connectiv-ity of 2D k-covered WSNs.

Theorem 4 summarizes the connectivity measures withrespect to a boundary sensor for homogeneous 3D k-coveredWSNs. The case of heterogeneous WSNs and the case ofsensors with convex sensing and communication ranges canbe treated similar to Sections 4 and 5 aswell as Appendices Aand B. We omit the proof of Theorem 4 as we use the sameanalysis in Sections 4 and 5, and Appendices A and B.

Theorem 4. Let G be a communication graph of a homogeneous3D k-covered WSN deployed in a cubic field, where the radii ofthe sensing and communication spheres of sensors are r and R,respectively. The connectivity of G is computed as

G 3:023k; 11



whereas the conditional connectivity of G is given by

G : P 3:02r0 R3 r30k

r30; 12

where r0 r=1:066, R=r, and k 4.

ACKNOWLEDGMENTS

The authors gratefully acknowledge the insightful com-ments of the anonymous reviewers which helped improvethe quality and presentation of the paper significantly. Thework of H.M. Ammari is partially supported by theUS National Science Foundation (NSF) grant 0917089 and aNew Faculty Start-Up Research Grant from Hofstra Collegeof Liberal Arts and Sciences Deans Office. The work of S.K.Das is partially supported by the AFOSR grant A9550-08-1-0260 and NSF grants IIS-0326505 and CNS-0721951. Hiswork is also supported by (while serving at) the NSF. Anyopinion, findings, and conclusions or recommendationsexpressed in this material are those of the authors and do notnecessarily reflect the views of the NSF.

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Habib M. Ammari received the Diploma ofEngineering and Doctorat de Specialite degreesin computer science from the Faculty ofSciences of Tunis, Tunisia, in 1992 and 1996,respectively, the MS degree in computer sciencefrom Southern Methodist University in Decem-ber 2004, and the PhD degree in computerscience and engineering from the University ofTexas at Arlington (UTA) in May 2008. He is anassistant professor of computer science in the

Department of Computer Science at Hofstra University, where he is thefounding director of the Wireless Sensor and Mobile Ad-Hoc Networks(WiSeMAN) Research Laboratory. He was on the faculty of the SuperiorSchool of Communications of Tunis (SupCom Tunis), from 1992 to2005 (an engineer of computer science, 1992-1993; a lecturer ofcomputer science, 1993-1997; an assistant professor of computerscience, 1997-2005; received tenure in 1999). His main researchinterests lie in the areas of wireless sensor and mobile ad hocnetworking, and multihop mobile wireless Internet architectures andprotocols. In particular, he is interested in coverage, connectivity,energy-efficient data routing and information dissemination, faulttolerance, security in wireless sensor networks, and the interconnectionbetween wireless sensor networks, mobile ad hoc networks, and theglobal IP Internet. He received the US National Science Foundation(NSF) Research Grant Award and the Faculty Research and Develop-ment Grant Award from Hofstra College of Liberal Arts and Sciences,both in 2009. He published his first book Challenges and Opportunitiesof Connected k-Covered Wireless Sensor Networks: From SensorDeployment to Data Gathering (Springer, 2009). He received the JohnSteven Schuchman Award for 2006-2007 Outstanding Research by aPhD Student and the Nortel Outstanding CSE Doctoral DissertationAward, both from UTA in 2008 and 2009, respectively. He was arecipient of the TPC Best Paper Award from EWSN 08 and the BestContribution Paper Award from IEEE PerCom 08-Google PhD forum.He also was an ACM Student Research Competition (ACM SRC)nominee at ACM MobiCom 05. He was selected for inclusion in the2006 edition of Whos Who in America and the 2008-2009 HonorsEdition of Madison Whos Who Among Executives and Professionals.He serves as an associate editor of the International Journal ofCommunication Systems and the International Journal of NetworkProtocols and Algorithms. He is on the Editorial Board of theInternational Journal of Mobile Communications and the InternationalJournal on Advances in Networks and Services. He also is on theEditorial Review Board of the International Journal of DistributedSystems and Technologies. He is the cofounder and a coeditor, withDr. Sylvia Silberger, Associate Professor and Chair of the Department ofMathematics at Hofstra University, of the Sciences Undergraduate AndGraduate REsearch Experiences (Sciences U-AGREE) Journal, whichis published at Hofstra University. He served as the program cochair/workshop cochair of WiMAN 10, IWCMC 10, IQ2S 09, and WiMAN 09.He has served as a reviewer for several international journals, includingthe IEEE Transactions on Mobile Computing, the IEEE Transactions onParallel and Distributed Systems, the ACM Transactions on SensorNetworks, the IEEE Transactions on Vehicular Technology, the IEEETransactions on Wireless Communications, the Mobile and NetworkApplications, the Wireless Networks, the Ad Hoc Networks, theComputer Networks, the Ad Hoc & Sensor Wireless Networks, theInternational Journal of Sensor Networks, the Information ProcessingLetters, the Computer Communications, the Journal of Parallel andDistributed Computing, the Information Sciences, the InternationalJournal of Computer and Applications, and the Data and KnowledgeEngineering Journal, and as a technical program committee member ofnumerous IEEE and ACM conferences and symposia, including theIEEE Infocom, IEEE ICDCS, IEEE PerCom, SSS, IEEE MASS, IEEEMSN, IEEE LCN, and EWSN. He is a member of the IEEE.

Sajal K. Das is a university distinguishedscholar professor of computer science andengineering and the founding director of theCenter for Research in Wireless Mobility andNetworking (CReWMaN) at the University ofTexas at Arlington (UTA). He is currently aprogram director at the US National ScienceFoundation (NSF) in the Division of ComputerNetworks and Systems. He is also an E.T.S.Walton professor of Science Foundation of

Ireland; a visiting professor at the Indian Institute of Technology (IIT)at Kanpur and IIT Guwahati; an honorary professor of Fudan Universityin Shanghai and international advisory professor of Beijing JiaotongUniversity, China; and a visiting scientist at the Institute of InfocommResearch (I2R), Singapore. His current research interests includewireless and sensor networks, mobile and pervasive computing, smartenvironments and smart heath care, pervasive security, resource andmobility management in wireless networks, mobile grid computing,biological networking, applied graph theory, and game theory. He haspublished more than 400 papers and more than 35 invited book chaptersin these areas. He holds five US patents in wireless networks and mobileInternet, and coauthored the books Smart Environments: Technology,Protocols, and Applications (Wiley, 2005) and Mobile Agents inDistributed Computing and Networking (Wiley, 2009). He is a recipientof the IEEE Computer Society Technical Achievement Award (2009),the IEEE Engineer of the Year Award (2007), and several Best PaperAwards in various conferences such as EWSN 08, IEEE PerCom 06,and ACM MobiCom 99. At UTA, he is a recipient of the Lockheed MartinTeaching Excellence Award (2009), UTA Academy of DistinguishedScholars Award (2006), University Award for Distinguished Record ofResearch (2005), College of Engineering Research Excellence Award(2003), and Outstanding Faculty Research Award in Computer Science(2001 and 2003). He is frequently invited as keynote speaker atinternational conferences and symposia. He serves as the foundingeditor-in-chief of Elseviers Pervasive and Mobile Computing (PMC)Journal, and also as an associate editor of the IEEE Transactions onMobile Computing, the ACM/Springer Wireless Networks, the IEEETransactions on Parallel and Distributed Systems, and the Journal ofPeer-to-Peer Networking. He is the founder of the IEEE WoWMoMSymposium and the cofounder of the IEEE PerCom Conference. He hasserved as the general and technical program chair as well as TPCmember of numerous IEEE and ACM conferences. He is a seniormember of the IEEE.

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