Constructing a Wireless Sensor Network to Fully Cover Critical Grids by Deploying Minimum Sensors on Grid Points Is NP-Complete

Constructing a Wireless Sensor Network to

Fully Cover Critical Grids by Deploying

Minimum Sensors on Grid Points

Is NP-Complete

Wei-Chieh Ke, Bing-Hong Liu, and

Ming-Jer Tsai, Member, IEEE

Abstract—This paper proves that deploying sensors on grid points to construct a wireless sensor network that fully covers critical grids using minimum sensors (Critical-Grid Coverage Problem) and that fully covers a maximum total weight of grids using a given number of sensors (Weighted-Grid Coverage Problem) are each NP-Complete.

Index Terms—NP-Complete, wireless sensor networks, coverage problem.

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Awireless sensor network is composed of several wireless sensors, each of which monitors specific environmental attributes, records sensing data, derives environmental conditions by aggregating the sensing data, and returns the aggregated data to the base station. The rapid development of wireless communications and em-bedded microsensing technologies facilitated the use of wireless sensor networks in our daily lives; a wide range of applications exist for wireless sensor networks, including fire control systems, environmental monitoring, battlefield surveillance, health care, nuclear, biological, chemical (NBC) attack detection, intruder detection, etc. Recently, studies of wireless sensor networks received considerable attention [8], [9], [11].

The deployment of sensors on a given field is an important issue that affects wireless sensor networks. Deploying sensors on a field can be separated into two steps: 1) A rudimentary deploy-ment is executed and 2) some recovery methods are adapted to face actual problems, such as signal scattering, fading, interference, etc. A deployment algorithm must provide a predetermined coverage level for a given application [1]. Chakrabarty et al. [2] addressed the full coverage problem. The Coverage-Centric Active Node Selection (CCANS) problem of determining the minimum connected dominating set in a dense sensor network such that the coverage probabilities of the given points each are larger than a given parameter pthwas shown to be NP-Complete [11]. CCANS is NP-Complete because it can be restricted to the problem of determining the minimum connected dominating set in a sensor network, known to be NP-Complete, by allowing only the instances of CCANS with pth¼ 0.

Sometimes, however, sensor fields are large or the available budget cannot provide enough sensors for full coverage on the sensor field, as in the instance of the number of sensors needed to deploy an intrusion alert network at a military installation. Another example is a wilderness ecological observation network; a wireless sensor network that fully covered the entire wilderness would be impossible to create. In some situations, sensors are constrained by limited sensor ranges and it is difficult to produce

full coverage on the sensor field. Bomb detection sensors [13] have very limited sensor ranges and it would be impossible to deploy as many sensors as would be necessary to protect an entire train station, government office, or other location against a bomb attack. In [5] and [12], the authors attempted to maximize the coverage area using a specified number of sensors.

In the literature, the proposed deployment algorithms provided coverage levels in which all areas of a field were considered to be equivalent. In many applications, however, critical areas and common areas must be distinguished adequately. For example, in a wilderness ecological observation network, the “hot spots” such as nests of animals may be assigned to critical areas. To date, the difficulty in constructing a wireless sensor network that fully covers critical areas using the minimum number of sensors was not determined precisely.

In this research, the sensor field is divided into grids consisting of squares or equilateral triangles. Using this format, the minimum number of sensors required to be deployed on grid points in order to construct a wireless sensor network that fully covers the critical grids was discussed. This problem was termed the Critical-Grid Coverage Problem. Consideration of the dual problem, that is, the Weighted-Grid Coverage Problem, addressed the maximum total weight of coverage grids that can be achieved by deploying a given number of sensors on grid points, assuming that grids are weighted based on priority of deployment, in which case, the more critical the grid, the greater the weight.

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We apply the unit disk graph model [3] as the communication model, where a sensor can send messages to another senor if the transmission range, Rt, reaches that sensor, and employ the binary sensor model [10], where a sensor detects an event with probability equal to 1 if the event is within the sensor range Rs and with probability equal to 0 otherwise. This assumption simplifies the linkage between the transmission range and the sensor range and has negligible impact on the performance of a wireless sensor network. Two problems we consider are the Critical-Grid Cover-age Problem and the Weighted-Grid CoverCover-age Problem:

Critical-Grid Coverage Problem.Given Rs, Rt, k, a sensor field divided into grids having length ‘, and a set of critical grids, does a wireless sensor network exist that fully covers all critical grids by deploying no more than k sensors on grid points?

Weighted-Grid Coverage Problem.Given Rs, Rt, k, w, and a sensor field divided into weighted grids having length ‘, does a wireless sensor network exist that fully covers grids having a total weight no less than w by deploying k sensors on grid points?

In this paper, the Planar 3SAT Problem [7], a known NP-Complete problem, is used to show the hardness of the Critical-Grid Coverage Problem and is illustrated below. If viis a variable in V , vi and vi are positive and negative literals over V , respectively. A 3-literal clause over V is the disjunction of three literals over V . A 3SAT Boolean formula is a collection of 3-literal clauses. GðBÞ denotes the graph of a 3SAT Boolean formula B¼ fc1; c2; . . . ; cmg, whose vertex set is fcij1  i  mg [ fvjj1  j ng and edge set is fðci; vjÞjvj2 cior vj2 cig [ fðvj; vjþ1Þj1  j < ng [ fðvn; v1Þg. A 3SAT Boolean formula B is a P-3SAT Boolean formula if GðBÞ is planar. Fig. 1 illustrates the graph of the P-3SAT Boolean formula B ¼ fc1; c2; c3g. The Planar 3SAT Problem is shown below.

Planar 3SAT Problem. Given a set V ¼ fv1;   ; vng and a P-3SAT Boolean formula B ¼ fc1; c2; . . . ; cmg, does a truth assign-ment exist for V that simultaneously satisfies all the clauses in B? A field serviced by sensors is commonly divided into square grids, as illustrated in Fig. 2a. Grids, composed of regular hexagons [4], illustrated in Fig. 2b, are also used because the

710 IEEE TRANSACTIONS ON COMPUTERS, VOL. 56, NO. 5, MAY 2007

. The authors are with the Department of Computer Science, EECS Rm 832, National Tsing Hua University, 101, Kuang Fu Rd., Sec. 2, HsingChu, Taiwan 30013 ROC.

E-mail: u882522@alumni.nthu.edu.tw, {lbh, mjtsai}@cs.nthu.edu.tw. Manuscript received 18 Apr. 2006; revised 29 Aug. 2006; accepted 2 Nov. 2006; published online 9 Feb. 2007.

For information on obtaining reprints of this article, please send e-mail to: tc@computer.org, and reference IEEECS Log Number TC-0150-0406. Digital Object Identifier no. 10.1109/TC.2007.1019.

regular hexagon is most like the circle among polygons capable of tiling the plane without overlap. Each square or regular hexagonal grid has a grid point located at its center. Each regular hexagon can be further divided into six equilateral triangles as illustrated in Fig. 2c. The deployment of sensors on grids composed of equilateral triangles is superior to the deployment of sensors on grids composed of regular hexagons if the sensors are deployed on the vertices of the equilateral triangles. For example, the deploy-ment of four sensors on the grid points B1, B2, B3, and B4 of regular hexagons is equivalent to the deployment of two sensors on the grid points C1 and C2 of equilateral triangles required to fully cover all critical areas if Rt¼ 3‘ and Rs¼ ‘, where ‘ denotes the grid length. In this paper, we consider grids composed of equilateral triangles instead of grids composed of regular hexagons. The grid point indicates the center of the square grid or the vertex of the equilateral triangle grid.

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The Planar 3SAT Problem can be reduced to the Critical-Grid Coverage Problem in polynomial time if the sensor field is divided into grids of equilateral triangles having length ‘ with ‘  Rs< 2‘=pffiffiffi3and 2‘  Rt. Let V ¼ fv1;   ; vng be a set of variables and B¼ fc1; c2; . . . ; cmg be a P-3SAT Boolean formula making up an arbitrary instance of the Planar 3SAT Problem. We must construct the output instance, consisting of the sensor field divided into grids of equilateral triangles having length ‘, the critical grids, and the values of Rs, Rt, and k, such that a wireless sensor network exists that fully covers all critical grids by deploying no more than ksensors on grid points if, and only if, a truth assignment exists for V that satisfies all clauses in B.

Once a suitably encoded planar graph of B, GðBÞ, is given in a plane, the output instance is constructed as follows: Let the plane be the sensor field. The sensor field is divided into grids of equilateral triangles having length ‘. Fig. 3 shows six types of 3-plug grids used in the constructed components of the output instance. A 3-plug grid has three grid vertices each associated with plug T (True), F (False), or R (Repulse). The plugs T , F , and R are used to determine the placement of the constructed components, as discussed later. The critical grids and the values of Rs, Rt, and k are assigned in a six phase process, described as follows:

1. Construction of Clause Components. A clause compo-nent ci consisting of seven 3-plug grids is constructed for each clause ci in GðBÞ. Figs. 4a, 4b, 4c, and 4d illustrate the ðT ; T ; T Þ-type, ðF ; T ; T Þ-type, ðT ; F ; F Þ-type, and ðF ; F ; F Þ-type clause components for the clauses that contain all positive literals, one negative literal, one positive literal, and all negative literals, respectively. Each type of clause component can be rotated. For example, ðT ; T ; T Þ-type, ðF ; T ; T Þ-type, and ðF ; T ; T Þ-type clause components are constructed for c1¼ ðv1þ v2þ v3Þ, c2¼ ðv1þ v2þ v4Þ, and c3¼ ðv2þ v3þ v4Þ, respectively. 2. Construction of Variable Components. The

vari-able component vj consisting of ðT ; F ; RÞ-type and ðF ; R; RÞ-type 3-plug grids is constructed for each variable vj in GðBÞ. Fig. 5 illustrates ðp; qÞ-type variable compo-nents. A ðp; qÞ-type variable component has p indentations on the left side and q indentations on the right side, where an indentation is composed of two ðF ; R; RÞ-type grids. In GðBÞ, a loop, termed the loop of variables, exists that is composed of all variables in B. In Fig. 1, ðv1; v2; v3; v4; v1Þ is the loop of variables. Clause c1 exists inside the loop of variables and clauses c2 and c3 are outside the loop of variables. A ðp; qÞ-type variable component is constructed for the variable that connects with p and q clauses outside and inside the loop of variables in GðBÞ, respectively. For example, (1, 1)-type, (2, 1)-type, (1, 1)-type, and (2, 0)-type variable components are constructed for variables v1, v2, v3, and v4, respectively.

3. Placement of Clause and Variable Components. The constructed variable components are placed on a vertical line in the sensor field such that the distance between variable components vi and viþ1 is

ffiffiffiffiffi 3‘ p

. The constructed clause components are properly located such that their

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Fig. 1. The graph of the P-3SAT Boolean formula B¼ fc1; c2; c3g, where c1¼ ðv1þ v2þ v3Þ, c2¼ ðv1þ v2þ v4Þ, and c3¼ ðv2þ v3þ v4Þ.

Fig. 2. The sensor fields are divided into grids of (a) squares, (b) regular hexagons, and (c) equilateral triangles in which critical areas are shown in gray and the circles denote the grid points.

Fig. 3. Six types of 3-plug grids. (a) ðT ; T ; T Þ-type, (b) ðF ; T ; T Þ-type, (c)ðT ; F ; F Þ-type, (d) ðF ; F ; F Þ-type, (e) ðF ; R; RÞ-type, and (f) ðT ; F ; RÞ-type.

Fig. 4. Four types of clause components, each consisting of seven 3-plug grids, are shown in horizontal stripe configurations. (a)ðT ; T ; T Þ-type, (b) ðF ; T ; T Þ-type, (c)ðT ; F ; F Þ-type, and (d) ðF ; F ; F Þ-type.

Theorem 4.The Weighted-Grid Coverage Problem is NP-Complete if the sensor field is divided into grids of squares having length ‘ with ‘=pffiffiffi2 Rs< ‘and j‘  Rt<

ffiffiffiffiffiffiffiffiffiffiffiffiffi j2_{þ 1} p

‘for some positive integer j.

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Previously, the hardness of the Critical-Grid Coverage Problem and the Weighted-Grid Coverage Problem were not precisely determined. In this paper, both problems are shown to remain NP-Complete if the sensor field is divided into grids of squares having length ‘ with ‘=pffiffiffi2 Rs< ‘ and j‘  Rt<

ffiffiffiffiffiffiffiffiffiffiffiffiffi j2_{þ 1} p

‘ for some positive integer j and if the sensor field is divided into grids of equilateral triangles having length ‘ with ‘  Rs< 2‘=

ffiffiffi 3 p

and 2‘ Rt. In the future, our work will focus on identifying the conditions under which the problems are solvable in polynomial time. Another possible extension is to design efficient heuristics for these problems.

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This work is sponsored by the Ministry of Economic Affairs of the Republic of China under grant 94-EC-17-A-04-S1-044.

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