Constructing Low-Connectivity and Full-Coverage Three

IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL. 28, NO. 7, SEPTEMBER, 2010 1

Constructing Low-Connectivity and Full-CoverageThree Dimensional Sensor Networks

Chuanlin Zhang, Xiaole Bai, Jin Teng, Dong Xuan, Weijia Jia†

Abstract—Low-connectivity and full-coverage three dimen-sional Wireless Sensor Networks (WSNs) have many real-worldapplications. By low connectivity, we mean there are at least kdisjoint paths between any two sensor nodes in a WSN, wherek ≤ 4. In this paper, we design a set of patterns to achieve 1-, 2-,3- and 4-connectivity and full-coverage, and prove their optimal-ity under any value of the ratio of communication range rc oversensing range rs, among regular lattice deployment patterns. Wefurther investigate the evolutions among all the proposed low-connectivity patterns. Finally, we study the proposed patternsunder several practical settings.

Index Terms—Wireless sensor network topology, Three di-mensional networks, Optimal deployment pattern, Coverage,Connectivity

I. INTRODUCTION

In this paper, we study the problem of how to construct athree dimensional (3D) Wireless Sensor Network (WSN) thatachieves low-connectivity and full-coverage by using the leastnumber of sensors. By low connectivity, we mean there are atleast k disjoint paths between any two sensors, where k ≤ 4.

A. Motivation

Our research on the above problem is motivated by thefollowing two important facts.− Low-connectivity and full-coverage three dimensional

WSNs have many real-world applications: WSNs deployedin 3D aerial space can be used to support intelligent 3D vision

†Corresponding authorChuanlin Zhang is with the Department of Mathematics, Jinan University,

Guangzhou, 510632, P. R. China. E-mail: [email protected]. Xiaole Baiis with the Department of Computer Science of University of MassachusettsDartmouth, MA, 02747, U.S.A. E-mail: [email protected]. Jin Teng andDong Xuan are with the Department of Computer Science and Engineering,The Ohio State University, Columbus, OH 43210, U.S.A. E-mail: tengj,[email protected]. Weijia Jia is with the Department of ComputerScience, City University of Hong Kong, Hong Kong SAR, China. E-mail:[email protected]. An earlier version was published with the same namein the 9th ACM International Symposium on Mobile Ad Hoc Networking andComputing (MobiHoc), May 2009.

This work was supported in part by High Technology Development Programand Natural Science Foundation of Guangdong & Natural Science Foundationof P. R. China under Grant No. 2009B01080030, No. 32207013 and No.07005930; the US National Science Foundation (NSF) under Grant No.CNS0916584, CAREER Award CCF-0546668 and the Army Research Office(ARO) under grant No. AMSRD-ACC-R 50521-CI; SAR Hong Kong RGCGeneral Research Fund (GRF) No. (CityU 114908), (CityU 114609), CityUApplied R & D Centre (ARD(Ctr)) No. 9681001 and ShenZhen BasicResearch Fund No. JC200903170456A . The first author was supportedpartially by HK SAR GRF No. (CityU 114908) while working on this paper.Any opinions, findings, conclusions, and recommendations in this paper arethose of the authors and do not necessarily reflect the views of the fundingagencies.

Manuscript received June 4, 2009; revised February 8, 2010.

systems [1], human paropsia [2], constructing aerial defensesystem [3], and aerosphere pollution monitoring [4] etc. 3Dunderwater WSNs also have various important applications[5], [6]. Sensor nodes have sensing capability, and in manyapplications, need to connect/communicate with each otherfor data routing and aggregation. Low-connectivity WSNs arepaticularly popular in real-world 3D WSN applications [5],[6], [7]. One of the important reasons is that 3D sensors aremuch more expensive than their 2D counterparts.

− The optimal deployment pattern problem is fundamentalin 3D WSNs and other wireless networks: Research towardssolving this problem has both practical and theoretical signif-icance. First, the most obvious benefit of optimal deploymentis fewer sensor nodes and higher cost efficiency. We noticethat sometimes it is possible to maintain the exact position of3D sensors, as monitoring applications in a highrise buildingor watery areas [5], [6]. Meanwhile, even if it is not easyto deploy 3D sensors at desirable locations, e.g., in space,the optimal patterns can still be reference value and provideinsights for a sub-optimal deployment. Second, a good de-ployment pattern can be helpful to the engineering of thewhole WSN. The design of sensor node functionality can begreatly influenced by deployment patterns. For example, wecan strike a good balance between the total number of nodesand the complexity of a single node, if we know how differentcombinations of sensing and transmission power levels workout. Deployment patterns also provide guidelines for topologyplanning and routing protocols.

Put into a larger picture, the 3D deployment problem inWSN brings to the front the fundamental optimization of ser-vice provision and intra-net connectedness, which is importantto any kind of wireless networks, such as cellular networksor Wireless Mesh Networks. Optimal deployment patternsfor WSNs can be easily adapted to help the deployment ofthese networks. Finally, it is simply interesting to look at aclassic problem of space covering in the modern context ofnetworking. The general methodology employed here to solvethe problem may provide insights for more generic geometricand topological problems.

B. Related Work

In general, optimal deployment pattern in WSNs is relatedto the covering problem in computational geometry. Coveringpoints using a minimum number of given geometric bodies hasbeen extensively studied for disks on a large 2D area [8], [9],disks on a bounded square [10], [11], orthogonal rectangles[12], fat convex bodies [13], etc. For 2D WSNs satisfying


both coverage and connectivity requirements, several possiblepatterns and some theoretical guidelines are presented andanalyzed in [14], [15], [16], [17], [18].

However, the results derived in 2D WSNs can hardly beextended to 3D WSNs. As the application prospective of 3DWSNs is promising, there is an urgent need and a great interestto derive optimal patterns to guide real-world deployment.So far, there are two sets of work related to the optimaldeployment pattern problem in 3D WSNs.

− Sphere Covering and Packing in Discrete ComputationalGeometry One closely related problem in discrete computa-tional geometry is covering problem, especially sphere cover-ing in 3D Euclidean space. In 1887, Lord Kelvin provided aconjectured answer to the problem of “What is the optimal wayto fill a three dimensional space with cells of equal volume, sothat the surface area (interface area) is minimum?”. His answerstates that the Voronoi polyhedrons in the optimal coveringstrategy are 14-sided truncated octahedrons [20]. To date, theproof for Kelvin’s conjecture remains open. However, thereare valuable efforts on the covering problem under certainconditions. One important condition is the spheres are placedfollowing certain regularity. In [21], R. Bambah first provedthat the least covering density of a 3D space by identicalspheres is 5

√5π/24 (the definition of covering density will be

given later). E. Barnes in [22] and L. Few in [23] proved thesame result in different ways. Another closely related problemin discrete computational geometry is sphere packing in 3DEuclidean space. Sphere packing considers arrangements ofnon-overlapping identical spheres filling a space. There havebeen several works on the packing problem [26]. One of themost famous results is known as Kepler’s conjecture. In 1611,Johannes Kepler conjectured the maximum possible densityfor sphere packing is π/

√18. There is no rigorous proof until

2005. T. C. Hales in [27] accomplished the proof showingthat no packing of identical spheres in 3D Euclidean spacecan have density greater than π/

√18, which is the density of

the face-centered cubic packing [27], [28], [29]. None of theabove efforts considers connectivity in 3D Euclidean space.− Connectivity and Coverage in WSN Deployment S.

Alam and Z. Haas in [30] suggested the sensor deploymentpattern that creates the Voronoi tessellation of truncated oc-tahedral cells in 3D space. The suggestion is directly fromKelvin’s conjecture. The numerical data in [30] illustratestruncated octahedron tessellation is better than the tessellationsof cube, hexagonal prism, and rhombic dodecahedron. How-ever, the optimality proof for truncated octahedron tessellationis untouched. Besides the efforts focused on the optimaldeployment strategy, there are some works in 3D sensordeployment addressing other issues related to coverage. In[31], a deployment algorithm is proposed to “repair” coverageholes once they are discovered in a 3D volume. In [32],some sufficient conditions are presented for sensors in a givendeployment to check if every point in a 3D volume is coveredby at least k sensors. In [33], Zhao et al. presented severalschemes and corresponding theoretical analysis for surfacecovering on complicated 3D surfaces. None of these studiesprovide any lead towards the optimality of deployment patterns

0.5 1 1.5 20

1

2

3

4

5

6x 10

5

rc/r

s

Nu

mb

er

of

Se

nso

r N

od

es

Ne

ed

ed

full−coverage and 2−(1−)connectivity

full−coverage and 4−(3−)connectivity

full−coverage and 14−connectivity

Fig. 1. Sensor needed for various rc/rs to achieve 2- (1-), 4- (3-) and14-connectivity by optimal lattice patterns, respectively. The 3D deploymentspace is 1, 0003m3. Sensing range rs is 30m and communication range rcvaries from 15m to 60m.

in 3D WSNs. We notice there are many valuable efforts on3D routing and localization, e.g., [35], [36], [7], [37], etc.Our research can act as a complement to them by providingnew network topologies as a carrier for their algorithms andprotocols. In 2D WSNs, X. Bai et al. in [16], [17], and[18] studied several optimal deployment patterns to achievemultiple connectivity and full coverage in 2D WSNs. Theoptimality of the proposed patterns in [18] is proved under theconstraints of regularity. X. Bai et al. have made an attemptto further the exploration of optimal lattice patterns into threedimensional space, and some results on high connectivity arepresented in [42]. Another approach to optimal deployment isadopted by Ammari et al. [34] through calculating the criticaldensity for a fully-covered and connected 3D WSN.

C. Our Contributions

In this paper, we study the low-connectivity and full-coverage deployment problem in the domain of lattice. Ourwork is inspired by the efforts both in discrete computationalgeometry and WSNs. The research interest for optimal pat-terns under certain regularity constraints has arisen because,besides pure theoretical intentions, it has been noticed thatmany important natural constructs show strong regularityin their constructing components. In 3D space, one of themost universal and important structures with the propertiesof periodicity and homogeneity is lattice. Typical examplesinclude structures of numerous crystalline solids in our dailylife, like coal, salt, ice, etc.

We highlight our contributions as follows.− New deployment patterns with optimality: We have de-

signed a set of patterns to construct 3D WSNs that achievefull coverage and k-connectivity where k ≤ 4. We have provedthe optimality of 1-connectivity, 2-connectivity, 3-connectivityand 4-connectivity patterns under any value of rc/rs amongregular lattice deployment patterns (defined in Section II).Compared with the pattern proposed in [30] that achieves 14-connectivity, our patterns can save a large amount of nodes,as shown in Fig. 1. We observe when rc/rs = 1, the numberneeded to achieve 14-connectivity is around 2.5 times that to


achieve 3- or 4-connectivity, and 3.5 times that to achieve1- or 2-connectivity. This number difference will increase asrc/rs decreases. When rc/rs = 0.5, the number needed toachieve 14-connectivity is almost 6 times that to achieve 3-or 4-connectivity, and around 18 times that to achieve 1- or2-connectivity.− Pattern evolutions: We have investigated the evolutions

among all the proposed low-connectivity patterns. This studyprovides insights on exploration for optimal patterns.− Practical setting considerations: Above optimal patterns

are based on the sphere sensing and communication models.As we know, to build up theoretical foundation, abstractionis inevitable. However, it is also important to study thepatterns under more practical models. In the paper, we studythe proposed patterns when practical models are considered.Besides, we also investigate an inherited problem among lowconnectivity networks, i.e. the long path problem [17], for theproposed deployment patterns.

The rest of paper is organized as follows. After presentingdefinition and notations in Section II, we introduce 1- and 2-connectivity patterns and prove their optimality in Section III.In Section IV, we present 3- and 4-connectivity patterns, provetheir optimality, and further discuss pattern evolutions amongall the proposed patterns. Practical considerations are studiedin Section V, followed by the conclusion of this paper.

II. DEFINITIONS AND NOTATIONS

Similar to the context for optimal deployment pattern re-search in 2D, we consider that all sensors are of the same typeand have sphere-shaped communication domain with radiusrc and sensing field with radius rs. The deployment region isconsidered vast enough such that its boundary can be ignored.We discuss in Section V practical considerations which arebeyond these mathematical abstractions. In the following, weintroduce some important definitions.

Definition 1: Right Parallelepiped, Axle Set, F-diagonal, B-diagonal: A hexahedron is called a right parallelepiped if itsbases are parallelograms aligned one directly above the otherand has lateral faces that are rectangles. Any three edges of aparallelepiped are called an axle set if any two of them are notparallel. The diagonals of the parallelepiped faces are calledF-diagonals. The body diagonals of a parallelepiped are calledB-diagonals.

Definition 2: Basic Lattice, Seed Parallelepiped: Given aright parallelepiped α, the set Λ is called a basic latticegenerated by right parallelepiped α if Λ is composed of all thevertices generated by shifting α to its three edges’ directionswith shift distance being integer times the correspondingedge length. This right parallelepiped α is called the seedparallelepiped for Λ.

Definition 3: Body-Centered Lattice: Given a basic latticeΛ′ generated by seed parallelepiped α, point set Λ is called abody-centered lattice if it is composed of all points in Λ′ andall the center points of α in the process of generating Λ′.

A body-centered lattice is called body-centered cubic lattice(bcc lattice in short) when its seed parallelepiped is a cube.

Fig. 2. Lattice patterns that achieves 1- or 2-connectivity and full coverage.The dashed lines illustrate how lattices are constructed. The connectionlinks are shown in bold solid lines. (a1) Λ2−1 pattern (rc/rs < 4/3)and Λ2−2 pattern (4/3 ≤ rc/rs < 12/

√9 + 32

√3). Λ2−1 and Λ2−2

share the same structure, but have different edge lengths (a2) Λ2−3 pattern(12/

√9 + 32

√3 ≤ rc/rs < 2

√3/

√5) and Λ2−4 pattern (rc/rs ≥

2√3/

√5). Λ2−3 and Λ2−4 share the same structure, but have different edge

lengths

Meanwhile a body-centered cuboid lattice will be generatedif its seed parallelepiped is a cuboid. In this paper, we studyregular lattices. A regular lattice Λ is either a basic latticegenerated by its seed right parallelepiped α or a body-centeredcuboid lattice generated by its seed cuboid α.

Definition 4: Coverage Lattice with Radius r: Given latticeΛ and spheres with radius r centering at each point in Λ, Λ iscalled a coverage lattice with radius r if every point in a 3Dvolume can be covered by at least one sphere.

Definition 5: Lattice Λ Pattern: Given sensors with sensingrange rs and a lattice Λ, a sensor deployment scheme is calledlattice Λ pattern if sensors are deployed at each point in Λ andΛ is a coverage lattice with rs.

From Definition 5, when “lattice pattern” is used in thispaper, full coverage is always implied.

Definition 6: Covering Density: If Λ is a coverage latticewith radius r and generated by seed parallelepiped α, then theratio of the total volume of the spheres with radius r coveringα to the volume of α is called covering density of Λ withradius r, denoted by σ(Λ, r).

Given a fixed rs and two lattices Λ and Λ′, if σ(Λ, rs) <σ(Λ′, rs), then lattice Λ pattern is better than lattice Λ′ patternsince less sensor nodes are needed in lattices Λ pattern toachieve full coverage.

Definition 7: Optimal Lattice Pattern: Given sensing rangers, a lattice Λ pattern is called the optimal lattice pattern ifσ(Λ, rs) is minimum among all regular lattice patterns.

III. LATTICE PATTERN FOR 1- AND 2-CONNECTIVITY

Due to their symmetry, lattice patterns with exactly oddconnectivity do not exist. We only need to consider thosethat achieve even connectivity. Naturally, the optimal latticepatterns that achieve 1-connectivity are optimal ones thatachieve 2-connectivity.

A. Pattern Description

The proposed lattice patterns for 1- or 2-connectivity in 3Dspace are shown in Fig. 2.


− When rc/rs < 4/3, the pattern follows a body-centeredlattice, denoted by Λ2−1, which is generated by a cuboid α

with upper and bottom faces each with edge length e1 =√

12√

(3r2s − r2c + rs√9r2s − 2r2c ), e2 = (3rs +

√9r2s − 2r2c )/2

and its center. The height of α is rc. This seed cuboid α andits center are illustrated by ABCDEFGH and I in Fig. 2(a1).Any sensor is able to connect with its two neighbors along thedirection of height, as illustrated by sensor A in Fig. 2(a1).− When 4/3 ≤ rc/rs < 12/

√9 + 32

√3, the pattern

follows a body-centered lattice, denoted by Λ2−2, which isgenerated by a cuboid α with upper and bottom faces eachwith edge length e3 = e4 =

√4r2s − r2c/4 and its center.

The height of α is rc. This seed cuboid α and its center areillustrated by ABCDEFGH and I in Fig. 2(a1). Any sensorcan connect with its two neighbors as illustrated by sensor A.− When 12/

√9 + 32

√3 ≤ rc/rs < 2

√3/√5, the pattern

follows a body-centered lattice, denoted by Λ2−3, which isgenerated by a cube α with edge length e5 = 2rc/

√3 and

its center. This seed cube and its center is illustrated byABCDEFGH and I in Fig. 2(a2). Any sensor is able to connectwith its two neighbors along the direction of B-diagonal, asillustrated by sensor I in Fig. 2(a2).− When 2

√3/√5 ≤ rc/rs, the pattern also follows a body-

centered lattice, denoted by Λ2−4, which is generated by acube α with edge length e6 = 4rs/

√5 and its center. This

seed cube and its center is illustrated by ABCDEFGH andI in Fig. 2(a2). Any sensor is able to connect with its twoneighbors along the direction of B-diagonal, as illustrated bysensor I in Fig. 2(a2).

We note that some extra nodes are needed at the boundariesof 3D deployment volume for global connectivity when latticeΛ2−1 or Λ2−2 patterns are used. More discussions on this issueare provided in Section V.

Theorem 1: Among regular lattice patterns that achieve 1-or 2-connectivity and full coverage in 3D space, the latticeΛ2−1 pattern is an optimal pattern when rc/rs < 4/3, thelattice Λ2−2 pattern is an optimal pattern when 4/3 ≤ rc/rs <

12/√

9 + 32√3, the lattice Λ2−3 pattern is an optimal pattern

when 12/√9 + 32

√3 ≤ rc/rs < 2

√3/√5, and the lattice

Λ2−4 pattern is an optimal pattern when 2√3/

√5 ≤ rc/rs.

B. Optimality Proof

In this section, we first present the proof road map forTheorem 1, followed by proof details.− Road map From Definition 7, to prove optimality is

equivalent to find the lattice Λ pattern with the least coveringdensity σ(Λ, r), which is denoted by σMIN . To get σMIN ,we need to consider all regular basic lattice patterns as wellas regular body-centered lattice patterns, obtain σ′

MIN ’s forall cases, and then compare them.

Covering density is 4πr3s/(3V ) for basic lattice patterns,and is 8πr3s/(3V ) for body-centered lattice patterns, where Vis the volume of the seed parallelepiped. To obtain σ′

MIN foreach case, we are to obtain the maximum volume, which isdenoted by V ′

MAX , for each case, as is shown in equation (1),

V ′MAX = max f(x, y, z, γ) = xyz sin γ , (1)

Fig. 3. (a)The parallelogram at the middle of height z is denoted by Ωz/2.(b) The most efficient way to cover the middle parallelogram at z/2.

where x, y and z are the lengths of three non-parallel edgesof the right parallelepiped and γ is the included angle of thebottom parallelogram.

V ′MAX can be obtained by solving a nonlinear optimization

problem (1) under constraints generated from full coverageand desirable connectivity. In the following, we present theproof details.

− Proof Both coverage constraints and connectivity con-straints are considered explicitly for (1). Coverage constraintsare derived by first properly choosing a certain face or a certaingeometry point or both of the seed parallelepiped and thenletting them be covered.

Since constraints are different for basic lattice pattern caseand for body-centered lattice pattern case. In the followingpart, we show how to obtain these constraints for basic latticesand body-centered lattice, respectively.

Case A. Basic Lattice We consider coverage constraintsfirst, then connectivity constraints.

We denote the constraints for satisfying full coverage inthis case by Cov-BL. As shown in Fig. 3 (a), we denotethe parallelogram at z/2 in a seed parallelepiped by Ωz/2.Covering Ωz/2 is a necessary and sufficient condition for fullcoverage.

The most efficient way to cover Ωz/2 is to let the overlappedarea of any three disks (intersections of sensing spheres bythe plane) be zero, as illustrated in Fig. 3(b). Let E be theex-center of the triangle ABD (with 3 acute angles, implying∠DAB = γ ≤ π/2). Then distance between the ex-centerto any of the three vertices is the smallest disk radius wecan have. With a little calculation, we can get the coverageconstraint as x2 + y2 − 2xy cos γ ≤ (4r2s − z2) sin2 γ.

Now we consider the constraints for connectivity. The con-nection edge can be either one edge of the seed parallelepiped,or a F-diagonal, or a B-diagonal. Note that in basic lattice, if aF-diagonal or a B-diagonal is the connection edge, then at leasttwo edges of the base are also the connection edges. Hence,we only need to consider two cases here, namely, x ≤ rc orz ≤ rc. Denote these two connectivity constraint by Con-BL-1and Con-BL-2.

The constraints of Cov-BL AND Con-BL-1 can be writtenas

x2 + y2 − 2xy cos γ ≤ (4r2s − z2) sin2 γ and x ≤ rc. (2)

The constraints of Cov-BL AND Con-BL-2 then can be written


Fig. 4. (a) The area in base ABCD that is not covered by four sensors atbase vertices is indicated by shaded area. (b) The uncovered area should becovered by the sensor G that is at the center. |GN | = z/2.

as

x2 + y2 − 2xy cos γ ≤ (4r2s − z2) sin2 γ and z ≤ rc (3)

We solve these two optimization problems and register everyσ for comparison with cases of body-centered lattice andchoose the smallest to be the optimal pattern. Note that it isnot necessary to consider the range rc/rs ≥ 4/

√5 since the

pattern proposed in [30] from Kevin’s conjecture can achieve14-connectivity in this range.

Case B. Body-Centered Lattice We consider coverageconstraints first, then connectivity constraints. Though wemainly study body-centered cuboid lattice, yet we’ll deriveconstraints for more general lattices, i.e. body-centered latticesgenerated by right parallelepiped.

Denote the coverage constraints from this case by Cov-BCL.Among three edges, x, y and z, at most one of them can belarger than 2rs. Otherwise, the face with edges that are bothlarger than 2rs can never be covered. Hence, we have threecases to consider as follows. First, no edge is longer than2rs. Denote the coverage constraints from this case by Cov-BCL-1. Second, one edge of base (at plan xoy) is longer than2rs. Without loss of generality, we can assume x ≥ 2rs. Thecorresponding coverage constraints are denoted by Cov-BCL-2. Third, the height of seed parallelepiped, z, is longer than2rs. The constraints are denoted by Cov-BCL-3.

We now consider the first case where there is no edge islonger than 2rs. A sufficient and necessary condition for fullcoverage of body-centered lattice is that all faces of the seedparallelepiped are covered. We start from the base. On the xOyplane, as illustrated in Fig. 4(a), we have |AE| = |DE| =|AF | = |BF | = rs, ∠DAB = γ. Both |EN | and |FN |should be smaller than

√r2s − z2/4 to ensure the uncovered

area should be covered by the sensor G at the center, asillustrated in Fig. 4(b). We have the following constraints (4).

y2 + z2 − x2 ≤ 2y sin γ√4r2s − x2

x2 + z2 − y2 ≤ 2x sin γ√4r2s − y2

.

(4)Now we study the face on plane xoz, as illustrated in Fig.

5, we have |AE| = |JE| = |JF | = |KF | = rs and sensorG locates at the center, and M is its projection on the sideface ABJK. To ensure the area that is not covered by the foursensors at face vertices should be covered by the sensor G,

Fig. 5. (a) The area in a side face ABKJ that is not covered by the foursensors at base vertices is indicated by shaded area. (b) The uncovered areashould be covered by the sensor G that is at the center. M is the projectionof G at the side face.

we have following condition

|ME| ≤

√r2s −

y2 sin2 γ

4, |MF | ≤

√r2s −

y2 sin2 γ

4.

That is,x+ y cos γ −√

4r2s − z2 −√4r2s − y2 sin2 γ ≤ 0

y2 + z2 − x2 − 2z√4r2s − x2 ≤ 0

.

(5)By exchange x with y, we can have the constraints for theface at plan yoz as follows. y + x cos γ −

√4r2s − z2 −

√4r2s − x2 sin2 γ ≤ 0

x2 + z2 − y2 − 2z√4r2s − y2 ≤ 0

.

(6)Combine (4), (5) and (6), we obtain constraints Cov-BCL-1.Similarly, we can obtain the constraints Cov-BCL-2 and Cov-BCL-3 for the second case and third case respectively.

Now we consider the constraints for connectivity require-ments. We’ll focus on the regular lattice patterns as mentionedin Section II. With body-centered cuboid lattices, we need toconsider three cases. First, one edge, x or y, is the connectionedge. We denote the constraints for this case by Con-BCL-1.Second, the height, z, is the connection edge. The constraintsfor this case is denoted by Con-BCL-2. Third, the half ofB-diagonal is the connection edge. We have the constraintsx2 + y2 + z2 − 2xy cos γ ≤ 4r2c . The constraints for this caseis denoted by Con-BCL-3.

To obtain the optimal deployment pattern for body centeredlattice, we are to solve (1) under following constraints cases.

Case B.1. Constraints can be expressed as (Cov-BCL-1 ORCov-BCL-2 OR Cov-BCL-3)AND Con-BCL-1.



As with basic lattice, we solve above optimization prob-lems, register their σ′

MIN ’s. After comparing all σ′MIN ’s and

choosing the smallest, we can get the optimal pattern, whichwas presented in Theorem 1.

− Remarks Solving above nonlinear optimization problemsis not trivial. However, when general lattice patterns areconsidered, coverage constraints are even more difficult toget and they have more complicated expressions. To solve


Fig. 6. Lattice patterns that achieves 3- or 4-connectivity and full coverage.The dashed lines illustrate how lattices are constructed. The connectionlinks are shown in bold solid lines. (b1) Λ4−1 pattern(rc/rs < 4/3)and Λ4−2 pattern (4/3 ≤ rc/rs < 2 3

√2/

√3). Λ4−1 and Λ4−2 share

the same structure, but have different edge lengths. (b2) Λ4−3 pattern(2 3√2/

√3 ≤ rc/rs ≤ 2

√3/

√5) and Λ4−4 pattern (rc/rs ≥ 2

√3/

√5).

Λ4−3 and Λ4−4 share the same structure, but have different edge lengths

nonlinear optimization problems for such cases is hard. Weconjecture that our proposed patterns here are also optimalamong general lattice patterns. Its proof is our on-going work.

It is worth mentioning that the authors have tackled a similaroptimization problem to achieve full coverage and 6- and14- coverage in [42]. Though the problems and solutionsare seemingly similar in form. However, due to the low-connectivity and full coverage context studied in this paper,there exist at least two major differences in the mathemat-ical core. First, as the connectivity gets low, the possiblechoice of connection edges is given much more lattitude. Sobesides difficulty in classification and case merge, we alsoneed to develop totally different coverage contraints for ’tall’and ’flat’ parallelepiped, i.e., Cov-BCL-2 and Cov-BCL-3,which cannot happen in high-connectivity cases. The difficultyof problem construction is almost tripled. Second, becauseof less connecitivity constraints, the feasible region of theoptimization problem is much larger, making the solutionmuch more difficult. With high-connectivity cases, the symme-try among different connectivity constraints often makes thesolution easy. However, we have little symmetry with low-connectivity cases. So it is often necessary to seek recourse tomore mathematical techniques, like advanced inequalities andnumerically assisted analysis. So the computation and analysiscomplexity is much higher in this paper.

IV. LATTICE PATTERN FOR 3- AND 4-CONNECTIVITY

In this section, we present lattice patterns to achieve 3-and 4-connectivity. Similar to the 2- and 1-connectivity case,optimal 4-connectivity lattice patterns are also optimal 3-connectivity lattice patterns.

A. Pattern Description

The optimal lattice pattern that achieves 3- or 4-connectivityis shown in Fig. 6.− When rc/rs < 4/3, the pattern follows a body-centered

lattice, denoted by Λ4−1, which is generated by a cuboid αwith upper and bottom faces each with edge length e7 = rcand e8 = 2rs +

√4r2s − 2r2c . The height of α is rc. This

seed cuboid and its center is illustrated by ABCDEFGH andI in Fig. 6(b1). Any sensor is able to connect with its four

neighbors in one plane. For example, in Fig. 6(b1) sensor Eis connected with A, N, M and H.

− When 4/3 ≤ rc/rs < 2 3√2/√3, the pattern also follows

a body-centered lattice, denoted by Λ4−2, which is generatedby a cuboid α with upper and bottom faces each with edgelength e9 = 4rs/3 and e10 = 8rs/3. The height of α is 4rs/3.This seed cuboid and its center is illustrated by ABCDEFGHand I in Fig. 6(b1). Any sensor is able to connect with its fourneighbors in one plane. For example, in Fig. 6(b1) sensor Eis connected with A, N, M and H.

− When 2 3√2/√3 ≤ rc/rs ≤ 2

√3/√5, the pattern follows

a body-centered lattice, denoted by Λ4−3, which is generatedby a cube α with edge length e11 = 2rc/

√3 and its center.

This seed cube and its center is illustrated by ABCDEFGHand I in Fig. 6(b2). Any sensor is able to connect with its twoneighbors along the direction of B-diagonals. For example, inFig. 6(b2) sensor I is connected with A, B, G and H.

− When 2√3/√5 ≤ rc/rs, the pattern also follows a body-

centered lattice, denoted by Λ4−4, which is generated by acube α with edge length e12 = 4rs/

√5 and its center. This

seed cube and its center is illustrated by ABCDEFGH andI in Fig. 6(b2). Any sensor is able to connect with its twoneighbors along the direction of B-diagonals. For example, inFig. 6(b2) sensor I is connected with A, B, G and H.

We notice that some nodes are needed to add at theboundaries of 3D deployment volume to achieve global con-nectivity when lattice Λ4−1 or Λ4−2 patterns are adopted.More discussion on this issue are provided in Section V.

Theorem 2: Among regular lattice patterns that achieve 3-or 4-connectivity and full coverage in 3D space, the latticeΛ4−1 pattern is an optimal pattern when rc/rs < 4/3, thelattice Λ4−2 pattern is an optimal pattern when 4/3 ≤ rc/rs <2 3√2/√3, the lattice Λ4−3 pattern is an optimal pattern when

2 3√2/√3 ≤ rc/rs < 2

√3/

√5, and the lattice Λ4−4 pattern is

an optimal pattern when 2√3/√5 ≤ rc/rs.

B. Optimality Proof

In this section, we present the proof of Theorem 2. Ingeneral, the proof road map of 3- and 4-connectivity latticeis similar to that of 1- and 2-connectivity, which has beenelaborated in Section III-B. The basic steps to formulate theoptimization problems are the same: first, this problem can berepresented as a non-linear constrained optimization problem,then we classify many different cases into several typical onesand calculate the minimum covering density for each, andfinally we compare all results to reach the final conclusion. Thecoverage constraints are the same, i.e. Cov-BL, Cov-BCL-1,Cov-BCL-2 and Cov-BCL-3, but the connectivity constraintsare different.

With the help of several particular properties of regularlattices, e.g., centrosymmetry, we are able to make the fol-lowing classification. For basic lattices, there are two cases toconsider. The first case is two non-parallel edges of the bottomface in a lattice cell. The second is the two non-parallel edgesof the side face in a lattice cell. For body centered cuboidlattices, there are also two cases. The first case is two non-parallel edges of the side face. The second is one half the bodydiagonal of a lattice cell.


With calculations and comparisons, we obtain that whenrc/rs > 2 3

√2/√3, one of the Body Centered Lattice patterns

is the optimal pattern and when rc/rs ≤ 2 3√2/√3 another

one of the Body Centered Lattice patterns is the optimal case.And this concludes the proof of Theorem 2.

C. Pattern Evolution

It is interesting and important to study how different patternsare related and how they evolve.

We note that from symmetry, the lattice Λ2−2 pattern shownin Fig. 2(a1) actually achieves 4-connectivity, the lattice Λ2−3

pattern shown in Fig. 2(a2) actually achieves 8-connectivity(sensor I can connect with sensors A, B, C, D, E, F, G, andH), and the lattice Λ2−4 pattern shown in Fig. 2(a2) actuallyachieve 14-connectivity (sensor I can connect with sensors A,B, C, D, E, F, G, H as well as the sensors at centers of 6neighboring cubes). We also note that, the lattice Λ4−3 patternshown in Fig. 2(b2) actually achieves 8-connectivity (sensorI can connect with sensors A, B, C, D, E, F, G, and H), andthe lattice Λ4−4 pattern shown in Fig. 2(b2) actually achieves14-connectivity (sensor I can connect with sensors A, B, C, D,E, F, G, H as well as the sensors at centers of 6 neighboringcubes). It is also interesting to note that, given a lattice Λpattern is an optimal lattice pattern to achieve k1-connectivityand it actually achieves k2-connectivity (k2 > k1), it will alsobe an optimal lattice pattern to achieve k2-connectivity, since anetwork with high connectivity also implies low connectivity.So in their own effective intervals, the lattice Λ2−2 pattern isoptimal for networks up to 4-connectivity, Λ2−3 and Λ4−3 areoptimal for up to 8-connectivity, while Λ2−4 and Λ4−4 areoptimal for up to 14-connectivity.

A close look at lattice pattern Λ2−1, Λ2−2, Λ4−2,Λ2−3(Λ4−3) and Λ2−4(Λ4−4) reveals how patterns evolveas rc/rs increases. In lattice Λ2−1, Λ2−2 or Λ4−2 patterns,connectivity follows “lines” or “planes”. The “lines” and“planes” are separated from each other, hence extra sensorsare needed to add at volume boundaries to assure globalconnectivity. In lattice Λ2−3 and Λ2−4 pattern, where onesensor can connect to 8 or 14 neighbors, connectivity follows“bodies” and no extra sensors are needed for global connecti-vity. Transmission of basic connectivity structure from “lines”or “planes” to “bodies” reflects transmission of constraints.When rc is small, connectivity constraints dominates. Thestructure follows the most efficient way to achieve requiredconnectivity, e.g., “lines” in lattice Λ2−1 pattern. When rc islarge, coverage constraints dominates. The structure followsthe most efficient way to achieve full coverage while connec-tivity is the byproduct. It is witnessed by the final convergenceof patterns to lattice Λ2−4(Λ4−4) pattern for full coverage.Lattice Λ2−2 and Λ2−3 patterns illustrates the gradual shiftingof constraints dominance from connectivity to coverage.

V. PRACTICAL CONSIDERATIONS

In this section, we discuss some practical issues beyond ourmathematical abstraction.− On Long Path Problem: As we discussed above, some

extra nodes are needed at the volume boundaries for global

Fig. 7. ABCDEFGH is the volume where sensors are deployed. (a) LatticeΛ2−1 pattern is used. In the boundary face, JK is to connect all the “lines”in one plane, and NM is to connect all planes. (b) the long path problem canbe overcome by adding such connection faces inside the target volume.

connectivity in lattice Λ2−1, Λ2−2, Λ4−1 and Λ4−2 patterns.As shown in Fig. 7, ABCDEFGH is the volume where sensorsare deployed. In Fig. 7(a), we show the case when lattice Λ2−1

pattern is deployed. In this case, to achieve global connectivity,extra sensors need to be deployed at the boundary faces ADHEand BCGF. For each boundary faces, we first add sensorsto connect “lines” in the lattice Λ2−1 pattern, as illustratedby JK in Fig. 7(a), such that 2-connectivity is achieved inplanes, for which we illustrate one by IJKL. Then we connectthese parallel planes by adding sensors as illustrated by MN.The number of extra nodes needed in one boundary face isrelatively small. For instance, when rc = rs = 30m, weneed around 17,200 nodes to achieve 2-connectivity in a cubewith edge length 1000m. while in each boundary face, weneed around 1.7% (300) extra nodes. The possible long pathproblem can be overcome by adding these faces inside thedeployment volume, as shown in Fig. 7(b). For Λ4−1 andΛ4−2, we have separately connected planes already rather thanlines. So even smaller number of nodes are needed to constructa connection face.

− On Practical Sensing and Communication Models: Fora practical sensing model, S. Megerian et al. in [38] proposethat the quality of sensing gradually attenuates with increasingdistance. Y. Zhou et al. in [39] propose a probabilistic sensingmodel where the detection probability changes for differenttarget distance. When the above models are used, sensingsphere can still be obtained by setting a sensing range thresh-old that is decided by desirable sensing quality or detectionprobability. For some sensor types, the sensing capability mayvary along different directions. One typical model obtainedfrom real device experiments by Cao et al. in [40] suggeststhe sensing capability roughly follows Gaussian distributionover different directions. Denote the average sensing radiusover all the directions by µ and the standard deviation by σ2.In a particular direction, the probability for sensing range X

being x is given by PX = x = exp(− (x−µ)2

2σ2 )/(σ√2π).

As σ increases, the sensing field is more similar to a sphere.Though it is possible for the sensing range x to be negative, weconsider this probability to be negligible, because, accordingto statistics approximation, the possibility for x to lie out of the3σ region on µ’s both side can be regarded as zero. We studyby simulation the impact from such sensing irregularity oncoverage in lattice Λ2 pattern. The results are shown in Fig. 8.We notice that higher sensing irregularity will result in loweroverall coverage. This is an expected phenomenon, because


0.5 1 1.5 20.97

0.975

0.98

0.985

0.99

0.995

1

rc/r

s

Co

ve

rag

e P

erc

en

tag

e

σ2

=5

σ2

=10

σ2

=20

Fig. 8. Sensors each with µ = 30m are deployed in a 10003m3 cubefollowing the lattice Λ2 pattern. The coverage in percentage is obtained byfirst generating 106 points within the cube, and then checking how manypercent of them are covered. Every value presented is the average over 100times simulation.

with optimal patterns, we often push the sensing spheres toextremities, so that many points in the space lie precariouslyclose to the covering boundary of any covering sphere. Witha little variance in the sphere, these points are likely tobecome not covered. The larger the variance σ, the more theuncovered areas are. When rc/rs ≥ 2

√3/5, the coverage

will not change since the deployment pattern keeps the same.One interesting observation is that there exists incrementon coverage percentage from rc/rs = 0.5 to 2 3

√3/

√3. It

illustrates that smaller rc, which implies more overlapped area,does not always suggest better tolerance of sensing irregularity.Tolerance also depends on the pattern architecture. In latticepattern Λ2−1 and Λ2−2, sensors are connected in “lines”.Sensing irregularity along lines can be tolerated well. LatticeΛ2−3 patterns can tolerate irregularity along all directions.But when rc further increases, we have discovered that thepercentage of overlapped sensing area, which is essentially abuffer for variance on sensing range, decreases within a certaininterval. The decreasing of the overlapped area dictates lesstolerance capability, which explains the rapid decrement ofcoverage percentage. The simulation results for 4-connectivitypattern generate the same observation. We do not present themdue to the space limitations.

In reality, the communication wireless signal undergoesattenuation and various disruptive physical phenomena in the3D space. We consider a widely used model suggested byZuniga and Krishnamachari in [41]. This model establishedthe function of the distance between the transmitter and the re-ceiver and the communication link quality measured by packetreception rate (PRR). PRR at distance d can be expressed asPRR(d) = (1 − 1

2e−Pt−PL(d)−Pn

2 )8ℓ, where Pt is the outputpower of the transmitter, PL(d) is the path loss at distanced, Pn is the noise floor and ℓ is the frame length. (Interestedreader may refer to [41] for detailed derivation.) Under thismodel, we consider a connection established between twonodes only if the PRR from each other is above a certainthreshold. By simulation, we investigate the effect from theabove model on the probability for one sensor in lattice Λ2

pattern to connect all 2 neighbors. The results are shown inFig. 9. (The results for 4-connectivity lattice pattern are similarand not presented here due to the limitations of space.) Wenotice from Fig. 9 that that probability transition from 1 to

00.5

11.5

2

0

20

40

600

0.2

0.4

0.6

0.8

1

rc/r

sTransmission Prower Pt (dbm)

Pro

ba

bili

ty o

f co

nn

ect

ing

2 s

pci

fic

ne

igh

bo

rs w

ith

PR

R >

= 0

.95

Fig. 9. A connection is considered established when PRR ≥ 0.95. rc/rs =0.3 ∼ 1.8. For each combination of Pt and an optimal deployment pattern,we run simulation 10, 000 times. The probability is then the ratio of numberof times over 10, 000 when a sensor can connect with 2 neighbors. Otherparameters are from empirical data [41].

0 is sharp. This implies the connectivity will deteriorate fastwhen transmission power decreases in lattice Λ2 pattern.

It is also worth noting that interference might be a substan-tial problem for WSN, if deployed improperly. Interferenceamong different nodes may severely compromise the commu-nication quality and lead to less or even no connectivity at all.However, this is unlikely to happen in our optimal patterns. Weendeavor to cover a space with the smallest number of sensors.That is to say, our scheme deploys sensors as sparsely aspossible. In almost all cases, the overlapping communicationrange of different sensors are minimized under our scheme. Soit can be expected that the possibility of interference, whichonly happens in overlapped areas, will be low in our optimalpatterns. Moreover, we can make use of such MAC layerscheduling algorithms as TRAMA, B-MAC and CC-MAC,which are surveyed in [43], to counter interference, if it is veryserious. By scheduling the activities of antennas based on theknowledge of the optimal patterns, we can greatly diminish thechances that two interfering nodes might request the channelat the same time.

− On Node Location Imprecision: Individual sensors some-times will not be exactly deployed at the precalculated lo-cations due to deployment errors or boundary and locationconstraints. Denote the maximum random error by ε. In aconservative way, to ensure the coverage and connectivity, wecan use r′s = rs − ε, and r′c = rc − ε to decide sensing andcommunication sphere and then the optimal pattern accord-ingly.

However, not all errors or boundary constraints are knownor predictable before actual deployment. So node locationimprecision can be regarded under these scenarios as randomsingle point error or random misalignment of a series of nodes.As was pointed out in [44], random errors and deploymentmisalignments may greatly increase the number of nodesnecessary to cover an area in a lattice-based deployment. Soit is also important to see how our designed patterns cantolerate such imprecision. We consider the following threetypical cases:

The first is totally random error of every single node. Weadopt the model described in [44], that a node may lie around


0.5 1 1.5 20.96

0.965

0.97

0.975

0.98

0.985

0.99

0.995

1

rc/r

s

Co

ve

rag

e P

erc

en

tag

e

R=0.1rs

R=0.25rs

R=0.4rs

Fig. 10. Sensors each with µ = 30m are deployed in a 10003m3 cubefollowing the lattice Λ2 pattern. We add a horizontal random shift to all nodesin the same horizontal plane. The coverage in percentage is obtained by firstgenerating 105 points within the cube, and then checking how many percentof them are covered. Every value presented is the average over 1000 timessimulation.

the target location, bounded by a maximum error spherewith radius R. By drawing the line of the sensing coveragepercentage against R, we find that the figure is extremelysimilar to Fig. 8. The σ2 = 5, σ2 = 10 and σ2 = 20 casesin Fig. 8 are roughly equivalent to R = 0.07rs, R = 0.1rsand R = 0.2rs cases. The reason of similarity behind is notvery hard to see: the random error of sensor node locationcan be translated to variance of sensing range under certainrules, especially when the coverage of each generated pointis decided independently. The same phenomenon also occurswhen evaluating the connectivity with random errors. Thelocation errors can be counted uniquely into noise Pn. So wedo not present the results in detail due to space limitation.

The second and third cases are random imprecision in ahorizontal plane and progressive misalignment of horizontalplanes. Both cases are inspired by a generic observation madein [45], that in 3D space, especially in underwater scenarios,nodes on the same plane may undergo similar displacementafter deployment and we assume nodes normally do notfluctuate in the vertical direction.

In the second case, nodes remain in their horizontal planebut may disperse around their designated point with uniformprobability in an error circle of Radius R. The results areshown in Fig. 10. From the figure, we can see that for the inter-vals corresponding to Λ2−1 and Λ2−2, the coverage percentageincreases slowly with the increase of rc/rs. However, Λ2−3

tolerates this kind of location imprecision especially well. Itcan be accounted for by the decrease of the percentage ofoverlapped sensing area with the increase of rc/rs. If rc/rsfurther increases, the optimal patterns remain the same, so thecovering percentage stays almost at the same level.

The third case deals with progressive misalignment ofhorizontal lattice plane. That is to say each plane shifted moretowards a direction compared to the lower one. An intuitiveexplanation is that the whole area covered by the WSN isskewed to a certain angle. This is very typical in aerospaceor water areas, because currents may push the lattice to anoblique angle. The gradual shift on every horizontal latticeplane comply with a uniform distribution of [0, R] in anydirection, i.e., bounded by an error circle of radius R. Asis shown in Fig. 11, the coverage percentage is very high

0.5 1 1.5 20.99

0.992

0.994

0.996

0.998

1

rc/r

s

Co

ve

rag

e P

erc

en

tag

e

R=0.1rs

R=0.25rs

R=0.4rs

Fig. 11. Sensors each with µ = 30m are deployed in a 10003m3 cubefollowing the lattice Λ2 pattern. We add a progressive random shift to allnodes in the same horizontal plane. The coverage in percentage is obtainedin the same way as Fig. 10.

over 99.2%, even when the shift reaches 0.4rs. It means thatthe skewed position of the lattice can hardly compromise theoptimality of our designed patterns.

VI. CONCLUSION

In this paper, we have designed a set of patterns for low-connectivity (k ≤ 4) and full-coverage WSNs. We haveproved the optimality of 1-, 2-, 3- and 4-connectivity patternsunder any value of the ratio of rc/rs among the right latticedeployment patterns. We have investigated the evolutionsamong all the proposed low-connectivity patterns. The pro-posed patterns can save a significant number of sensor nodes,and provide insights for further pattern optimality explorationin 3D WSNs.

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Chuanlin Zhang received the BSc and MSc de-gree from Southwest Normal University, Chongqing,China, in 1984 and 1987, and the PhD degreefrom Jilin University, Changchun, China, in 1995.Currently, he is a professor in Faculty of Infor-mation Science and Technology, Jnan University,Guangzhou, China. His research interests includesensor networks, algorithm and complexity, QoSfor internet and communications, network securityand mobile computing, numerical computing and itsapplication, computational economics and finance.

Xiaole Bai is now an assistant professor at Uni-versity of Massachusetts Dartmouth. He receivedhis Ph.D. degree from the Department of ComputerScience and Engineering at the Ohio State Univer-sity, his M.S. degree in the area of networking atHelsinki University of Technology, Finland, in 2003and his B.S degree at Southeast University, China, in1999. His research interests include network scienceand engineering, cyber space security, distributedcomputing and theory.

Jin Teng is now in the first year of his PhDprogram with the Department of Computer Scienceand Engineering at the Ohio State University. Hereceived his B.S and M.S degrees in ElectronicEngineering at Shanghai Jiao Tong University, Chinain 2006 and 2009. His research interests mainlyinclude wireless communication architecture, QoSof wireless networks, network coverage in WSN andcyber space security.

Dong Xuan is an associate professor in the De-partment of Computer Science and Engineering, theOhio State University. He received his B.S. and M.S.degrees in Electronic Engineering from ShanghaiJiao Tong University (SJTU), China, in 1990 and1993, and Ph.D degree in Computer Engineeringfrom Texas A&M University in 2001. His researchinterests include real-time computing and commu-nications, network security, sensor networks anddistributed systems. He is a recipient of the NSFCAREER award.

Weijia Jia is a Professor in the Department ofComputer Science of City University of Hong Kong.He received BSc and MSc from Center South Uni-versity, China, in 1982 and 1984 and Master ofApplied Science and PhD from Polytechnic Facultyof Mons, Belgium in 1992 and 1993 respectively, allin Computer Science. His research interests includenext generation wireless communication, protocolsand heterogeneous networks; distributed systems,multicast and anycast QoS routing protocols.

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