Arbitrary Obstacles Constrained Full Coveragein Wireless Sensor Networks ?

Haisheng Tan† Yuexuan Wang‡ Xiaohong Hao‡

Qiang-Sheng Hua† Francis C.M. Lau†

† Department of Computer Science, The University of Hong Kong,Pokfulam, Hong Kong, China

‡ Institute for Theoretical Computer Science, Tsinghua University,Beijing, 100084, China

Abstract. Coverage is critical for wireless sensor networks to monitor aregion of interest and to provide a good quality of service. In most caseswe need to achieve full coverage which means every point inside theregion (excluding the obstacles) must be covered by at least one sensor.The problem of placing the minimum number of sensors to achieve fullcoverage for a region with obstacles is NP-hard. Most existing coveragemethods, such as contour-based ones, simply place sensors along theboundaries to cover the holes near obstacles and the region boundary.These methods are inefficient especially when obstacles or the regionbecome irregular. In this paper, based on computational geometry, wedesign a full coverage method, which accurately finds the uncovered holesand places sensors efficiently for both the regular and irregular obstaclesand regions. Specifically, we show that the more irregular of the obstaclesand the region, the more sensors our method saves.

Key words: Wireless sensor networks, Coverage, Obstacles, Computa-tional Geometry

1 Introduction

Wireless sensor networks (WSNs) can be deployed in a region of interest forarea monitoring and event detection. It has been applied extensively in military,civilian and health care, such as environmental monitoring, intrusion detection,cancer monitoring, and smart agriculture [1]. Coverage is one of the fundamentalissues in WSNs. It’s used to determine how well the region is monitored and oftenthe service can be provided.

In the literature, paper [3] studies how to place disks to fully cover a plane.The authors prove it’s asymptotically optimal, in terms of the number of disks

? The work presented in this paper was supported in part by Hong Kong RGC-GRFgrants (7136/07E and 714009E), the National Basic Research Program of ChinaGrant Nos. 2007CB807900, 2007CB807901, the National Natural Science Foundationof China Grant Nos. 60604033,60553001, and the Hi-Tech research and DevelopmentProgram of China Grant No. 2006AA10Z216.

2 H. Tan et al.

used, to place disks on the vertices of equilateral triangles (Figure 1). In addi-tion, several other optimal deployment patterns have been proposed to achievefull coverage and k-connectivity (k ≤ 6) on a plane in [4]. The Art Gallery prob-lem [14] studies how to guard a gallery with the minimum number of cameras,in which a location is guarded as long as line-of-sight exits from a camera. Com-pared with the above related works, the coverage of a sensor s in our problemis not only constrained by a limited sensing radius, rs, but also the existing ofobstacles and the region boundary (Figure 2). The problem of placing the min-

Fig. 1: The optimal placement pat-tern on a plane

Fig. 2: Coverage of a sensor in a finiterectangle with an obstacle (the grayarea is covered)

imum number of sensors to achieve full coverage for a region with obstacles hasbeen proved to be NP-hard [5]. Paper [2] categorizes the approaches for cover-age into three groups: force based [6, 7], grid based [4, 8, 13] and computationalgeometry based [10–12]. Authors in [7] propose a virtual force algorithm (VFA)to enhance the coverage after an initial random placement over a region withobstacles. Their method extremely depends on the sensors’ mobility and energy.In [9], an efficient algorithm is designed for a robot to place sensors when ob-stacles exist. It achieves coverage by carefully designing the movements of therobot. [12] firstly deploys sensors with distance

√2rs along the boundaries of

obstacles and the region. Then delaunay triangulation is applied to determinethe positions for adding sensors for full coverage. The whole region is dividedinto single-row and multi-row regions in [13]. Besides full coverage, the authorsin [13] also guarantee the network connectivity. In order to cover the holes, theydeploy sensors along the boundaries of obstacles and regions with a constantdistance based on the relationship between rs and the sensor’s communicationradius rc. To handle areas near obstacles and the region boundary, the exist-ing methods for static sensors [9, 12, 13] actually simply place sensors along theboundaries with constant distances. So we call them contour-based methods. Toachieve full coverage, these methods seem suitable for the simple regular regionsand obstacles, such as the boundaries are long straight lines. However, whenobstacles and the region become arbitrarily irregular, such as containing comb-shaped boundaries, these methods become inefficient. As without considering thespecific shapes of boundaries, they may not ensure full coverage unless placingsensors at each turning points on the boundaries. In fact, even the best resultsfor contour-based methods that place sensors with optimal dynamic distancescan be inefficient (Figure 3).

In this paper, inspired by the inefficiency of existing deployment methodswhen handling irregular boundaries, we consider the shapes of boundaries ex-

Arbitrary Obstacles Constrained Full Coverage in WSNs 3

plicitly. Based on computational geometry, we design algorithms to find the holesaccurately and cover them efficiently. Our method performs excellently for boththe regular and irregular obstacles and regions, including the extremely irregularones. The rest of the paper is organized as follows: In Section 2, we define themodels and the problem. In Section 3, our placement method is described. Sec-tion 4 contains the analysis and experiments. Section 5 gives more discussion.Section 6 concludes the whole paper and points out the future work.

(a) (b) (c)

Fig. 3: Coverage near a part of the bound-ary: (a) the contour-based placement witha constant distance: 5 sensors, (b) the bestfor the contour-based methods: 2 sensors,(c) the optimal placement: 1 sensor.

2 Problem Definition

The region of interest, denoted as A, is a 2D finite area that has an arbitraryboundary and contains arbitrary obstacles. It can not have any isolated subre-gions. The obstacles can not partition the region or have holes inside themselves.Otherwise, we don’t treat the region as a whole but as separated ones. The areathat needs to be covered is the region excluding obstacles. Both the region andobstacles are modeled as simple polygons of finite sizes on a 2D plane.

The sensors are static and homogenous with a fixed sensing radius rs. Herewe use the binary sensor model, which means the sensing range of s is a diskcentered at s with a radius of rs. A point is covered by a sensor if it’s withina distance of rs and a line of sight exists from the sensor. The coverage of asensor with obstacles, Cov(s), may be part of the disk (Figure 2). If there aren obstacles inside A, and the obstacle i occupies an area Obsi, the total areaoccupied can be denoted as O = ∪i∈[1,n]Obsi. And Cov(s) is defined as:Cov(s) = {u ∈ A−O| ‖ u− s ‖≤ rs, ku + (1− k)s ∈ A−O,∀k ∈ [0, 1]}Full coverage for a sensing region A means every point in the area A−O mustbe within Cov(s) of at least one sensor s. Our sensor placement problem canbe defined as placing the minimum number of sensors in a finite region witharbitrary boundary and obstacles to achieve full coverage.

3 Sensor Placement Algorithms

To achieve full coverage for a region A, our method works in the following4 procedures: 1) deploying a regular pattern over the region; 2) finding theuncovered holes; 3) partitioning the holes into triangulations; and 4) placingsensors to cover the holes. In the following, we explain the procedures in details.

3.1 Optimal Regular Pattern Deployment (ORPD)

We firstly deploy the regular pattern in Figure 1, which is optimal for coveringa plane, to the region A regardless of obstacles or the region boundary. The

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starting position can be randomly chosen at (x, y) ∈ A. The sensors are placedat (x, y) as well as its six neighbors with distance

√3rs in the pattern. Note

that we assume a sensor could be ‘placed’ inside an obstacle or outside A inthis procedure. For each sensor, as long as it is in A, sensors are added at itsneighbors. We continue doing this until each neighbor of the sensors inside Ahas been occupied by one sensor. In addition, we use three lists Lobs, Lin andLout to store the sensors placed inside obstacles O, inside A−O, and outside Arespectively. Figure 4 gives an example.

Fig. 4: the region after theORPD: the white area isA−O

Fig. 5: Finding a hole(gray areas are hole(s))

Fig. 6: Holes are foundand merged (gray areasare holes)

3.2 Finding Holes

After the ORPD, there may be some subareas, called holes, uncovered nearobstacles and the region boundary due to the following two reasons: 1) the lackof sensors: some sensors are ‘placed’ into obstacles or outside the region, such ass1 and s2 in Figure 4; 2) blocks: some sensors’ coverage is blocked by obstaclesor the region boundary, such as s3 and s4. The arbitrary shapes of obstacles andthe region make it difficult to cover these holes efficiently. Our method overcomesthis by firstly accurately finding the holes.

The ORPD can be regarded as a tessellation of regular hexagons with edgers inscribed in the circles (Figure 1). Therefore, we model the sensing range of asensor s to be its effective coverage, the regular hexagon, and denote it as H(s).Then, it’s easy to find the holes caused by the lack of sensors. For each sensordeployed inside obstacles or outside A, the hole of s, hole(s) is the overlap,Hin(s), between H(s) and A−O. It’s a bit difficult to calculate the holes causedby blocks (Figure 5). For a sensor deployed inside A−O, we also first calculateHin(s). If Hin(s) is nonempty, holes may exist inside it. We denote the perimeterof an area as P (area). P (Hin(s)) consists of PH(s), the blue lines in Figure 5,and PA(s), the red lines excluding the intersection points with PH(s). Note thatPH(s) is a part of P (H(s)), and PA(s) is a part of P (A − O). Next, we drawdirected lines from s to each vertex v of Hin(s) and extend them until reachingP (H(s)). The line −→sv crosses P (Hin(s)) at points p1, p2, ..., pn (n > 0) listedin the increasing distances from s. The segment of −→sv inside H(s) is denoted as(s, p1, p2, ..., pn). We set p0 = s and get Theorem 1 to find the vertices of hole(s).

Theorem 1. Given (s, p1, p2, ..., pn), for ∀k ∈ [1, n), pk is a vertex of hole(s) ifand only if one of the following conditions is satisfied:1)pk+1 ∈ PH(s) 2)pk+1 ∈ PA(s) AND pk+pk+1

2 ∈ Hin(s)

Arbitrary Obstacles Constrained Full Coverage in WSNs 5

3)pk−1 ∈ PA(s) AND pk+pk−12 ∈ Hin(s)

4)pk−1 ∈ PA(s) AND pk+1 ∈ PA(s)For k = n, pn is a vertex of hole(s) if and only if pn ∈ PH(s) and pn−1 6= s.

Proof. As each −→sv stops at P (H(s)), we can get k = n ⇐⇒ pk ∈ P (H(s)), andk ∈ [1, n) ⇐⇒ pk ∈ PA(s). P (H(s)) − PH(s) is inside obstacles or outside A.Therefore, pn is a vertex of hole(s) if and only if pn ∈ PH(s) and pn−1 is nots. As for k ∈ [1, n), the theorem is proved by enumerating all the cases of thepositions of pk, pk−1 and pk+1 as follows:

pk pk−1 pk+1 pk is a vertex of hole(s)PA(s) s PH(s) TruePA(s) s P (H(s))−PH(s) FalsePA(s) s PA(s) True iff pk+pk+1

2 ∈ Hin(s)PA(s) PA(s) PH(s) TruePA(s) PA(s) P (H(s))−PH(s) True iff pk+pk−1

2 ∈ Hin(s)PA(s) PA(s) PA(s) True

utAs the perimeter of hole(s) can only consist of parts of P (Hin(s)) and segmentson each directed line −→sv, we have Theorem 2:

Theorem 2. Connecting the vertices of hole(s) along P (Hin(s)) and −→svi, i =1, 2, ..., m, where m is the number of vertices in Hin(s), we get hole(s).

For a single sensor, hole(s) may be composed of a set of subholes. We mergethe subholes as long as they share a point. For all the sensors placed, we continuemerging their holes, and denote the final set of holes as HOLE . According toour finding process, we can see the holes have good properties:1) narrow in width: the width of the hole for a single s is no larger than rs.After merging, each part of a hole in HOLE has a width no larger than

√3rs,

otherwise there must be some sensors inside the hole, which is a contradiction;2) accurate in size: If the sensing range of s is simulated as H(s) of area 3

√3

2 r2s

as we do now, HOLE is exactly the uncovered areas. Actually the range is a diskof area πr2

s . Then hole(s) is at most (π− 3√

32 )r2

s larger than the real uncoveredarea in the sensing range of s.Algorithm 1 describes the whole procedure, and Figure 6 gives an example.

3.3 DT-based Partition of Holes

To cover a hole in HOLE , we partition it into triangles whose edges are nolonger than rs, so that it can cover a triangle to place a sensor at any of its threevertices. We achieve this by the following three steps:Step 1: Partitioning the long edges: If an edge of the hole is longer than rs,d l

rse−1 points are added to partition it evenly, such as AB in Figure 7. We also

treat the newly added points as vertices of the hole.Step 2: Applying delaunay triangulation: We apply DT over the vertices of thehole, and get a triangulation of the hole by keeping triangles inside it. Here DT

6 H. Tan et al.

Algorithm 1 Finding HolesInput:the region A, obstacles O, Lin,Lout and Lobs Output: the holes HOLE1: For each item s ∈ Lout ∪ Lobs do2: hole(s) = H(s) ∩ (A−O)3: if hole(s) 6= ∅, HOLE = HOLE ∪ hole(s)4: End For5: For each item s ∈ Lin do6: Hin(s) = H(s) ∩ (A−O)7: If Hin(s) 6= ∅, Then8: Finding the vertices of hole(s) /* Theorem 1 */

Compute and merge subholes to get hole(s); update HOLE /* Theorem 2 */9: End If

10: End For11: For each pair of items hole1 and hole2 in HOLE12: if hole1 ∩ hole2 6= ∅, hole = hole1 ∪ hole2 and update HOLE13: End For

is chosen because 1) it can perform quickly within O(nlogn) time, where n isthe number of vertices; 2) the minimum angle of all the triangles is maximizedso that it tends to avoid the skinny triangles [14]. Based on the advantage of DTand the property that our holes are narrow, the triangulation will add no or fewlong edges that need to be handled in Step 3.Step 3: Partitioning triangles with an edge longer than rs, such as CD in Fig-ure 7: Also, to avoid skinny triangles, we always partition the longest edge amongall the triangles. This step stops until all the edges don’t exceed rs and we finallyget a partition, Thole, over the hole.The partition for a hole from Figure 6 is illustrated in Figure 7. After han-dling all the holes in HOLE , we get a set of partitions, Tholes, as described inAlgorithm 2.

3.4 Placing Sensors to Cover Holes

Based on the partition, full coverage is equivalent to the requirement that at leastone vertex of each triangle in Tholes is placed a sensor. For each Thole in Tholes,we first compute its dual graph, D(Thole), which has a node v for every trianglet(v) in Thole. Two nodes u and v form an edge if t(v) and t(u) share at least apoint. If D(Thole) is a tree, it is 3-colorable. When performing DFS (Depth-FirstSearch) on the tree, for each v except the root visited, there must be one or two(when t(v) shares only a vertex with the triangle visited in last step) free nodesuncolored in t(v). However, in our case, D(Thole) may be a graph with cyclesand not 3-colorable. In order to ensure that every triangle has 3 different coloredvertices, we allow a vertex to be ‘doubly colored’. When performing DFS on thegraph, for the last node v in a cycle, the free vertex in t(v) has been colored.We check whether the three vertices of t(v) have had 3 different colors. If yes,continue to visit the next node directly; If no, append the absent color to acolored free vertex and make it doubly colored, such as the vertex D in Figure 7.

Arbitrary Obstacles Constrained Full Coverage in WSNs 7

Algorithm 2 DT-based partition of HolesInput: the set of holes HOLE Output: a partition Tholes over HOLE1: For each hole in HOLE do2: Initialize an array Lp, a partition Thole and a sorted array Le empty3: Lp= Lp∪ vertices of hole4: For each edge e of hole do /* Step 1 */

add d length(e)rs

e − 1 points evenly on e and append the points to Lp

End For5: Do DT over Lp and store the triangulation of hole to Thole /* Step 2 */6: Insert edges in Thole longer than rs to Le in decreasing order of lengths7: While(Le is nonempty) do /* Step 3 */

8: add d length(e)rs

e − 1 points evenly on the first item e and remove it9: For each triangle containing e do

connect the points added to the vertex that is not on e; update Thole

10: if there are edges added longer than rs, insert them to Le

11: End For12: End While13: Tholes = Thole ∪ Tholes

14: End For

After coloring, we choose the minimal group of the vertices with the same colorto place our sensors. Because for each cycle in D(Thole), there is at most onevertex in Thole doubly colored, we get Theorem 3:

Theorem 3. For a hole h, if it has n vertices in Th and c cycles in D(Th),bn+c

3 c sensors can always fully cover it.

After placing sensors to cover all the holes, together with the sensors in Lin, wecan get full coverage over the region A (Figure 8).

Fig. 7: Partition a holeinto triangles and colorthe vertices.

Fig. 8: Full coverage: redand green points are sen-sors

Fig. 9: Cover a rectangle oflong edges: red and greenpoints are sensors

4 Analysis and Experiments

Because of the arbitrary obstacles and the region, to analyze the bounds of thenumber of sensors for full coverage, we need use parameters from both the input

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region and our placement algorithms. For the region A, nv is the number ofvertices on P (A − O), and the length of each edge i on P (A − O) is leni. Forthe holes in HOLE , the number of vertices locate on {PH(s)|hole(s) 6= ∅} isnvph, and the number of vertices on {PA(s)|hole(s) 6= ∅} is nvpa. The numberof sensors to cover HOLE is nch. k is the number of vertices to divide long edgesafter applying DT over HOLE . Recall that there are no or few edges longer thanrs after DT, so k is small. m is the total number of cycles in the dual graphsof Tholes. m is also small and closely related to the shapes of the region andobstacles, and importantly, also closely related to the positions of the obstacles.The size of Lin is nin, and the number of vertices in Tholes is nvt. Then thenumber of sensors to achieve full coverage of A is:

N = nin + nch ≤ nin + b(nvt + m)/3c / ∗ Theorem 3 ∗ / (1)

= nin + bnvph + nvpa +∑

i(d leni

rse − 1) + k + m

3c (2)

≤ nin + bnvph + nv +∑

i(d leni

rse − 1) + k + m

3c / ∗ nvpa ≤ nv ∗ / (3)

N ≥ area(A−O)area(H(s))

=area(A−O)

3√

3r2s/2

(4)

nin is the asymptotically optimal number of sensors to cover the region excludingHOLE by the ORPD. nvph approximates to nvpa +

∑i(d leni

rse−1) in most cases,

and for irregular obstacles and regions, nvph can be much smaller. So, the upperbound of the sensors for holes is nearly 2

3 (nv +∑

i(d leni

rse−1)). For contour-based

methods, the sensors placed along boundaries can reach nv +∑

i(d leni

r e − 1),where r is the constant placement distance set not larger than

√2rs.

The experiments also validate the above analysis and show our efficiencyfor both regular and irregular cases. The example in Figure 8 illustrates our effi-ciency to handle comb-shaped boundaries. Figure 9 is to cover a rectangle, whichillustrates our efficiency to handle long straight lines. Moreover, we conduct sim-ulations in 4 types of regions (Figure 10) using 3 types of sensors of different rs.Figure 11 compares our results with the contour-based ones’. Note that ‘Contour-based-1’ is to first place sensors along the boundaries, and then add extra sensorsfor full coverage by supposing a near-optimal placement. ‘Contour-based-2’ ap-plies the ORPD and then places sensors along the boundaries to cover holes. Inthe OPRD, we try a set of different locations for the first sensor, and choose theone leading to the fewest sensors for full coverage. Greedily, we place the firstsensor rs

2 away from the longest boundary and rotate the regular pattern to fullycover the longest boundary. We can see the more irregular of the obstacles andregion, the more sensors our method saves.

5 Discussion

When talking about obstacles above, we actually mean opaque obstacles, whichneither allow sensors to be deployed inside nor allow the sensing signal to pass

Arbitrary Obstacles Constrained Full Coverage in WSNs 9

(a) (b) (c) (d)

Fig. 10: Four types of regions: (a) a regular region and obstacles (b) an indoor-likeregion (c) an outdoor-like region (d) an extremely irregular region and obstacles

Fig. 11: Comparison of the contour-based methods with ours in above 4 regions

through. However, the transparent obstacles, such as ponds, don’t allow sensorsplaced inside but allow the signal to pass through. When the obstacles andregion boundary are all transparent, the holes appear after the ORPD only dueto the lack of sensors. We can compute Hin(s) for each s ∈ Lout ∪ Lob. Foreach nonempty Hin(s), we need to add at most 5 sensors to cover it [5]. Settingn = |{s ∈ Lout ∪ Lob|Hin(s) 6= ∅}|, we get the upper bound of the sensorsneeded as nin + 5n. Note that the upper bound in [5] is wrong because it sumsup each Hin(s) without considering their positions are as crucial as their sizes.One region to violate their upper bound is a w × l rectangle while w → 0.

In practice, due to the inherent uncertainty of sensor’s sensing ability, theprobabilistic sensor model is more precise than the binary model [7]. As fullcoverage is achieved in our placement, for any point in A−O, there must be atleast one sensor s within a distance rs. Therefore, our coverage probability forany point is not smaller than e−λrβ

e , where λ, re and β are parameters in themodel. If a higher probability is required, we need to use a virtual sensing radiusr, where rs − re ≤ r < rs, so that the probability e−λ(r+re−rs)β

is guaranteed.For arbitrary obstacles, the effective coverage of a sensor can be infinitesimal.

The number, shapes, sizes as well as relative positions of obstacles make it hardto design and analyze algorithms for full coverage. In practice, the obstacles arecommonly not extremely irregular, such as buildings in an urban sensing region.Therefore, we may choose to study coverage constrained with limited number ofspecial sizes and shapes of obstacles. Moreover, instead of full coverage, we mayalso choose to ignore holes smaller than a predefined threshold. This could bean important extension of our current work.

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6 Conclusion and Future work

In this paper, we propose an efficient full coverage method for a sensing regionwith arbitrary boundary and obstacles. We achieve this by carefully studyinghow the obstacles and the region boundary block coverage. The holes after theoptimal regular pattern deployment are efficiently and accurately calculated. Al-gorithms based on delaunay triangulation and 3-coloring are designed to coverthese holes. Compared with other methods, the proposed one can achieve fullcoverage while dramatically saving sensors especially when the region and obsta-cles are irregular. Besides the extension mentioned in the previous section, thefuture work can be carried out in many directions, such as multiple connectiv-ity and coverage with obstacles, 3D coverage and the surface coverage [15] withobstacles, coverage by mobile and heterogeneous sensors.

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