Coverage Problem for Sensors Embedded in

Temperature Sensitive EnvironmentsArunabha Sen*, Nibedita Das*, Ling Zhou*, Bao Hong Shen*, Sudheendra Murthy* and Prajesh

Bhattacharyat *Department of Computer Science and EngineeringArizona State University, Arizona 85281

Email: {asen, nibedita.maulik, ling.zhou, bao, sudhi}@asu.edutG.W. Woodruff School of Mechanical EngineeringGeorgia Institute of Technology, Georgia 30332Email: prajesh.bhattacharya@me.gatech.edu

Abstract- The coverage and connectivity problem insensor networks has received significant attention of theresearch community in the recent years. In this paper, westudy this problem for sensors deployed in temperaturesensitive environments. This paper is motivated by theissues encountered during deployment of bio-sensors ina human/animal body. Radio transmitters during oper-ation dissipate energy and raise the temperature of itssurroundings. A temperature sensitive environment likethe human body can tolerate such increase in temperatureonly up to a certain threshold value, beyond which seriousinjury may occur. To avoid such injuries, the sensorplacement must be carried out in a way that ensures thesurrounding temperature to remain within the threshold.Using a thermal model for heat distribution from multipleheat sources (radio transmitters), we observed that if thesensor nodes are placed sufficiently apart from each other,then the temperature of the surrounding area does notexceed the threshold. This minimum separation distanceconstraint gives rise to a new version of the sensor coverageproblem that has not been studied earlier. We prove thatboth the optimization version and the feasibility versionof the new problem are NP-complete. We further showthat an c-approximation algorithm for the problem cannotexist unless P = NP. We provide two heuristic solutions forthe problem and evaluate the efficacy of these solutionsby comparing their performances against the optimalsolution. The simulation results show that our heuristicsolutions almost always find near optimal solution in afraction of the time needed to find the optimal solution.Finally, an algorithm for forming a connected sensornetwork with minimum transmission power in such ascenario is provided.

I. INTRODUCTION

A biomedical sensor is a device, which is implantedin a human or animal body to monitor and transmitbiological information such as retinal pressure [1], oxy-gen level on the surface of exteriorized tissues [2]

etc. Recently, there has been an immense interest innew sensing, monitoring, wearable wireless devices andsensor networks for healthcare and clinical applications.This increased interest and importance of the emergingfield is demonstrated by the successful conclusion ofthe third IEEE International Workshop on Wearable andImplantable Body Sensor Networks [3]. Among manyother applications of biomedical sensor networks, theone that pertains to artificial retina [4] deserves specialattention. The authors in [5] note that "organs that areespecially sensitive to any temperature increase due to alack of blood flow to them are prone to thermal damage(e.g., lens cataracts)". The authors in [6] also emphasizeon the importance of considering possible health hazardsfor individuals exposed to EM field and identify the EMfield values that is safe for human body. The authorsin [1] note that heat build-up from the sensor electronicscan jeopardize the implantation of the sensor, as elevatedtemperature may cause infection, especially when theimplanted sensor becomes a haven for bacteria.An example of a bio-medical sensor network currently

used in clinical situations [7] is shown in figure 1.As seen in the figure, the geodesic Sensornet has alarge number of wires connecting the sensors to thecontroller. This situation is clearly unwieldy. Efforts arecurrently underway to replace the wired sensor networkby a wireless one in many universities and researchlaboratories. The formation of a network with implantedsensors on human or animal body poses a number ofchallenges. Firstly, the biomedical sensors cannot beimplanted in any arbitrary location of the body. Theplacement of the sensors has to be confined within aset of potential locations. Secondly, the placement mustbe done in such a way that the increase in temperaturedue to the operation of the sensors (radio transmitters) iswithin an acceptable limit. Although such implantationposes many challenges, the researchers in our bioengi-

1-4244-1268-4/07/$25.00 t2007 IEEE

Thiisfull textpaper waspeer reviewed at the direction ofIEEE Communications Society subject matter expertsforpublication in the IEEESECON 2007proceedings.520

neering department have already implanted such sensorsin monkey brain. It is anticipated that such implantationswill become fairly regular in the next few years.

Although the coverage and connectivity problems insensor networks have received considerable attentionfrom the research community in recent years [8]-[13],to the best of our knowledge, sensor placement andcoverage problems for a temperature sensitive environ-ment have not been studied earlier. Clearly, the areasurrounding the location of a sensor will observe anincrease in temperature due to the operation of thesensor and its radio transmitter. However, increase intemperature in such sensitive areas cannot be allowedto exceed a specified threshold. The introduction of thisthermal constraint makes the coverage problem aloneconsiderably more complex than similar problems stud-ied in [8], [10], [12].

In the sensor network literature, there exists two differ-ent versions of the coverage problem [11]. In the region-coverage version, the problem is to find the optimalplacement of the sensors so that the given region(s) ofinterest can be sensed. On the other hand in the point-coverage version, the problem is to find the optimalplacement of the sensors so that a set of pre-specifiedpoints can be sensed. The version of the coverageproblem discussed in this paper is different from eitherthe region-coverage or the point-coverage problem. Newconstraints are imposed due to the fact that (i) the sensorscannot be placed in any arbitrary location if the sensingregion happens to be the human/animal body and (ii) thesensors cannot be placed very close to each other as theincrease in temperature due to their joint operation mayexceed the acceptable threshold temperature and causethermal impairment.

Fowler et. al. in [14] showed that sensor coverageproblems are NP-complete. Hochbaum in [15] developedapproximation schemes for these NP-complete problems.Coverage and connectivity problem was studied in anintegrated fashion in [10], [12], [13]. The authors in[12] presented a Coverage Configuration Protocol thatprovides different degrees of coverage depending on theneeds of the applications and explored the relationshipbetween coverage and connectivity. Abrams et. al. in [8]developed a strategy for energy efficient monitoring ofwireless sensor networks.

In this paper, we present our findings on the placementand coverage problem for medical bio-sensors implantedin human/animal body. We thoroughly analyzed theheat distribution phenomenon in a temperature sensitiveenvironment like human body. Our analysis indicates thateven if the temperature increase due to the operation ofan isolated sensor remains below the allowable thresh-

Fig. 1. Geodesic Sensor Network for measurement of EEG

old, the surrounding temperature can still exceed thethreshold if multiple sensors operate at close proximityof one another. However, if the sensors are placedsufficiently apart from each other (beyond a criticalseparation distance), then the temperature increase willremain below the threshold. Due to the page limitationsand the focus of this conference, detailed discussionon the heat transfer process in the human body is notincluded in this paper. Interested readers are referred toour report [16]. The main conclusion of our heat transferanalysis is the following: Corresponding to the sensorpower dissipation Pdi,s, there exists a critical inter-sensor distance, dcr, such that if the distance betweenany two deployed sensors is less than dcr, then thetemperature in the vicinity of the sensors exceeds themaximum allowable temperature Tthreshold. Therefore,attention must be paid during sensor deployment toensure that the distance between any two sensors is atleast as large as dcr,

This conclusion from our thermal analysis leads tothe formulation of a new version of the sensor coverageproblem. In this version, one would like to find outthe fewest number of sensors that have to be placedin the region of interest, such that (i) the entire regionof interest is sensed and (ii) the distance between anytwo sensors is at least as large as the critical separationdistance. The introduction of the requirement (ii) adds anew dimension to the sensor coverage problem resultingin additional complexity.

The contributions of this paper are as follows:1) We introduce a new version of the placement and

coverage problem for sensors in a temperaturesensitive environment.

2) We prove that both the optimization version andthe feasibility version of the new problem are NP-complete.

3) We show that an E-approximation algorithm for theproblem cannot exist unless P = NP.

4) We provide an integer linear programming formu-lation for the optimal solution of the problem.

5) We provide two heuristic solutions for the problemand evaluate the efficacy of these solutions bycomparing their performances against the optimal

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solution. The simulation results show that ourheuristic solutions almost always find near optimalsolution in a fraction of the time needed to find theoptimal solution.

6) We also propose an algorithm to maintain a con-nected network with the required coverage in suchtemperature sensitive environments.

Before proceeding further into the details of our work,we would like to draw the reader's attention on threepoints regarding the results presented in this paper.Firstly, in our simulation experiments, we have assumeda circular sensing region associated with each sensor.However, the heuristic solutions proposed in this paperdo not assume the sensing region to be circular andcan be used with any irregular shaped sensing region.Secondly, although the problem studied in this paperwas motivated by an application of sensors in a tem-perature sensitive environment, our results are equallyapplicable in other domains that have a requirement fora minimum separation distance between any sensor nodepair. Finally, we point out that due to the minimumseparation distance constraint, performance bound of theform (O(log N)) given in [10] for the connected sensorcover problem cannot be obtained for this problem unlessP = NP.

II. PROBLEM FORMULATION FOR SENSORCOVERAGE

In the sensor placement and coverage problem dis-cussed in this paper, we have (i) a set of locations (orpoints pi) to be sensed, (ii) a set of potential locations (orpoints qi) for placements of sensors and (iii) a minimumseparation distance (dcr) between each pair of sensors.The goal is to deploy as few sensors as possible inpotential placement locations such that all points pi aresensed and no two sensors are closer than dcr Like mostof the previous studies in this area [9], [11], [13], weassume that each sensor is capable of sensing a circulararea (disk) of radius rsen} with the location of the sensorbeing the center of the circle. We assume that the sensingradius of all the sensors are identical. An example ofthe coverage problem is shown in figure 2, where thecircular nodes represent the points to be sensed (bluepoints), the square nodes represent the potential locationsof the sensors (green points) and circles represent thearea sensed by the sensor located at the center of thecircle. A line connecting two square nodes indicates thatthe distance between them is less than the minimumseparation distance dcr and a sensor can be deployedin at most one of these two locations. A solution to theproblem consists of selecting the locations b, c, d, f, g

for sensor deployment whose coverage is shown by theshaded region in the figure. It may be noted that no twonodes in this set have a line between them, indicatingthat the distance between them is at least as large as dcr

* h

I Point to be seaPotential senlso lcation \ /D Senlsor covei-age _=., Node pair withili imiiliLu-n separat:ionl distalice

Fig. 2. Potential sensor locations, sensing points and sensing disk

Formally, the decision version of our sensor coverageproblem can be stated as follows:Problem 1: Sensor Coverage Problem (SenCov)INSTANCE: Given (i) a set of points P = {Pl, . ,Pm}to be sensed (we will refer to these points as blue points)(ii) a set of points Q = {ql, . .., qn }, the potentialplacement locations of the sensors (we will refer to thesepoints as green points)(iii) sensing radius rsen of the sensors(iv) critical separation distance dcr(v) an integer KQUESTION: Is there a subset Q' C Q, such that(i) Q'I < K,(ii) Vpi c P, 3qj c Q' such that dist(pi, qj) < rsen,(iii) Vqi, qj C Q', dist(qi, qj) > dcrwhere dist(x, y) represents the Euclidean distance be-tween the points x and y.

In the optimization version of the problem, the inputdoes not have the parameter K and the objective is tofind the smallest K such that the conditions (ii) and (iii)are satisfied.

A. Sensor Coverage as Generalized Set Cover Problem

It may be noted that the sensor coverage problemdescribed above can be viewed as a generalized versionof the set cover problem as well as the independent setproblem [17]. These two problems have been studiedextensively. However, to the best of our knowledge, thegeneralized version of the problems as presented in thispaper have not been studied before.The sensor coverage problem can be transformed to

the generalized set coverage in the following way. Eachblue point pi in the instance of the sensor coverageproblem can be viewed as the element si to be coveredin the instance of the set cover problem. Corresponding

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to green point qj in the instance of the sensor coverageproblem, we can construct a subset Sj in the instance ofthe set cover problem. The set Sj will comprise of theelements si, if the dist(pi, qj) < rsen. The generalizedversion of the set cover introduced in this paper also hasa notion of a a collection of incompatible subsets. Twosubsets Si and Sj are said to be incompatible, if theEuclidean distance between the corresponding points qiand qj is less than the critical separation distance dcrOtherwise, the subsets Si and Sj are compatible. Thegeneralized set cover problem is specified as follows:Problem 2: Generalized Set Cover Problem (GSC)INSTANCE: (i) a set of elements S = {sI, , smm},(ii) a collection of subsets Si C S, S3 {SI .... , S(iii) a collection of tuples of incompatible subsetsINC = {(Stl jSl)7 (Si27 Sj2) **(Sit, Sjt)}(iv) an integer KQUESTION: Is there a subset S' C S, such that(i) 3S'l < K(ii) VsiCS,SSj 'SI, such that si C Sj,(iii) if Si, Sj c S' then (Si, Sj) , INC.

B. Sensor Coverage as Generalized Independent SetProblem

The Sensor Coverage problem can also be viewed asa generalization of the Independent Set problem [17].The sensor coverage problem can be transformed to

the generalized independent set problem in the followingway. From an instance of the sensor coverage problem,we can construct a graph G = (V, E) where each node inV represents a green point qi and two nodes vi, vj C Vhave an edge between them if the distance between thecorresponding points qi and qj is less than the criticalseparation distance dcr In addition, with each node vi CV, we associate a list of blue points pj. A blue point pjwill be in the list associated the node vi (representinga green point qi), if and only if the Euclidean distancebetween the points pj and qi is less than the sensingradius rsen. The generalized independent set problem isspecified as follows:Problem 3A: Generalized Independent Set Problem(GIS)INSTANCE: (i) a graph G = (V, E)(ii) a set of elements A = {a,... , a} and(iii) a subset Ai C A associated with each node vi C V.We will refer to the subset Ai as list associated with viand will denote it by L(vi). We assume Uv,vL(vi) = A.(iv) an integer K.QUESTION: Is there an independent set V' in the graphG = (V, E) such that(i) V' <K and(ii) Uvicv,L(vi) = A

C. Feasibility issue in GSC and GIS Problems

In section II-B, we formulated the SenCov problem asthe GSC problem. In the optimization version of the GSCproblem, the goal is to find the smallest subset S' C S,such that (i) Vsi C S, 3Sk c S', such that si C Sk and (ii)if Si, Sj c S' then (Si, Sj) , INC. Since the optimiza-tion version of the SC problem is NP-complete [17], andGSC reduces to SC when INC = 0, we can concludethat the GSC is also NP-complete. This conclusion leadsus to look for approximation algorithms for GSC withguaranteed performance bound, especially because it iswell known that such approximation algorithms withguaranteed performance bound exists for SC [15]. It maybe noted however that although the problems SC andGSC are very similar, they have a major difference. Toillustrate the point, we introduce the feasibility versionof the SC (SCF) and GSC (GSCF) problems. The SCFand GSCF problems are special cases of the SC andGSC problems respectively, when K= oc. In a similarfashion, we can consider the feasibility version of theGIS (GISF) and SenCov (SenCovF) problems as specialcases of the GIS and SenCov problems respectively,when K= oc.

It may be noted that the SCF problem can be solvedeasily by including in the set S' all the elements ofthe set S and checking if it covers all the elements ofthe set S. However, GSCF cannot be solved in such atrivial fashion because of the incompatibility among theelements of the set S. In fact, in the next section we showthat the GISF problem (and equivalently the GSCF andSenCovF problems) is NP-complete.The NP-completeness of GISF (together with equiva-

lent GSCF and SenCovF) problem puts a brake on ourattempt to develop an approximation algorithm for theGSC problem with a guaranteed performance bound. Inorder to avoid the problem, we introduce a modifiedversion of the GIS problem (GISM) whose goal is to findthe smallest independent set V', whose [Uv,cv' L(vi) isthe largest. Informally, the goal of the GISM problem isto find an independent set that covers the largest numberof blue points (points to be sensed) with smallest numberof green points (sensors). It may be noted that unlike theGISF problem, feasibility is not an issue for the GISMproblem. Before we formally define the GISM problem,we introduce the notion of cost of an independent set V'as follows:C(V') = (I UviLvL(vi) LJ-UvipEV L(vi)l + a) x VI,where av is a number much smaller than 1/n. (note: Vis the set of nodes in the graph with IVI = n, V' is anindependent set. av is introduced to handle the case whenUv,E]V L(vi) = Uv,E]V' L(vi) I)

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Problem 3B: Generalized Independent Set Problem(Modified) (GISM)INSTANCE: (i) a graph G = (V, E), ( V n)(ii) a set of elements A = {a, ..., a } and(iii) a subset Ai C A associated with each node vi C VWe will refer to the subset Ai as list associated with viand will denote it by L(vi). We assume Uv,vL(vi) = A.In the optimization version of the GISM problem wetry to find an independent set V' whose cost C(V) isthe smallest and in the decision version of the problem,we ask if there exists an independent set V' whose costC(V) is less than 1. Formally,QUESTION: Is there an independent set V' in the graphG = (V, E) whose cost is less than 1 (i.e., C(V') < 1)?

III. COMPUTATIONAL COMPLEXITY OF THE SENSORCOVERAGE PROBLEM

In this section, we first show that the GISF problemis NP-complete.Theorem 1: The GISF problem is NP-complete.

Proof: We give a transformation from the 3-Satisfiability (3-SAT) problem [17]. It may be notedthat an instance of the 3-SAT problem is made by aset of variables U = {U, .... ug} and a set of clausesC = {Cl,1.. ., ChI and want to find out if there is a truthassignment to the variables in U such that all the clausesin the set C are satisfied?From an instance of the 3-SAT problem, we generate

an instance of the feasibility version of the GISF problemand show that the GISF problem has a feasible solution,if and only if the instance of the 3-SAT problem is satis-fiable. The graph in the instance of GISF is constructedin three phases.Phase I: For each variable ui c U, we construct part ofthe graph G by introducing two nodes Vi = {ui, ui} andone edge Ei = {{ui, uv}i}, that is two nodes joined by asingle edge. Each node in the instance of GIS will havea list of elements associated with it. In the instance weare creating, the list of elements associated with boththe nodes ui and ui will contain a single element xi,i.e., L(ui) = L(ui) = {fx}. It may be noted that anyindependent set will contain at most one node from theset {ui, ui}-Phase II: For each clause cj c C, we construct a partof the graph with three nodes and three edges.V = {al[j], a2[j], a3[j}EJ, {{al [j] a2[j} {al [j] a3[j} {a2[j], a3[j}}

In addition, we assign L(al[j]) = L(a2[jl) = L(a3[ijl)= {Yi}.It may be noted that an independent set will contain atmost one node from the set {aI[j], a2[jl, a3[ij]}-

Phase III: For each clause cj c C, let the three literalsin cj be denoted by cj1, cj2, cj3. In this phase, we donot introduce any nodes, but introduce three additionaledges, corresponding to each clause cj.EJ' {{al[j] Cj}, a2[j], Cj2} {a3[j] Cj3}}

This concludes the construction of our instance of theGIS, with graph G = (V, E), whereV = (ug 1Vi) U (U$ VJ) andE = (Ug= E) U (Uh= E) U (UL iES)

where g and h corresponds to the number of variablesand clauses of the instance of the 3-SAT problem respec-tively. The instance of the graph G = (V, E) constructedusing the rules specified above will have 2g + 3h nodesand g + 6h edges. Figure 3 shows the instance of theGISF constructed from an instance of 3-SAT where U{U1, U2, U3, U4} and C = {{ul, u3, U4}, {U1, 1U2, U4}}- Itmay be noted that this construction can be completed inpolynomial time.

Fig. 3. Instance of the GISF constructed from an instance of 3-SATwhere U U{1i, 112, 113, 114} and C ={{ 1 13,11-4,11 , U2, X4}}

Claim 1: The instance of the 3-SAT problem has asatisfying truth assignment if and only if the instance ofthe GISF has a feasible solution.

Definition: The node cover of a graph G = (V, E) isa subset V' C V, such that all edges in E have at leastone end point in the set V'.As a step towards showing that the instance of the

3-SAT problem has a satisfying truth assignment if andonly if the instance of the GISF has a feasible solution,we first establish that the instance of the 3-SAT problemhas a satisfying truth assignment if and only if theinstance of the GISF has a node cover of size at mostg + 2h.

Claim 2: The instance of the 3-SAT problem has asatisfying truth assignment if and only if the instanceof the GISF has a node cover of size at most g + 2h.Moreover, this node cover must contain exactly one nodefrom each Vi, 1 < i < g and exactly two nodes from eachVJ/ I < i < h.Proof of Claim 2: The proof of this claim is given in

[17] (pages 54-56) and is not repeated here for brevity.Proof of Claim 1: In any graph G = (V, E), if V' is a

node cover, then V -V' must be an independent set. Ifthe instance of the GISF has a node cover of size g + 2h

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that contains exactly one node from each Vi, 1 < i < gand exactly two nodes from each V9', 1 < i < h, thenthe instance of the GIS must also have an independentset of size g + h that contains exactly one node fromeach Vi, 1 < i < n and exactly one node from eachV', 1 < i < m. It can be easily verified that the unionof the lists associated with the nodes belonging to thisindependent set will be equal to Uq 1X, U Uh lyi. Thisindependent set thus constitutes a feasible solution forthe GISF. Thus we can conclude that if the instance ofthe 3-SAT problem has a satisfying truth assignment, theinstance of the GISF problem has a node cover of sizeg+2h, which in turn implies that there is an independentset of size g+ h and the union of the lists associated withthe nodes in this independent set is equal to Ug 1Xi U

iYt.~~~ ~~~~~~~~=

Conversely, if the instance of the 3-SAT problemhas no satisfying truth assignment, then the size of thesmallest node cover in the instance of the GISF problemis greater than g + 2h. This implies that in the instanceof the GISF problem, size of the smallest independentset is smaller than g + h. It may be observed thatthe union of the list associated with the nodes of anindependent set of size smaller than g + h cannot beequal to Ug=1X U Uh lyi. Thus, if the instance of the3-SAT problem has no satisfying truth assignment, theinstance of the GISF problem has no feasible solution.Theorem 2: The GISM problem is NP-complete.Proof: The transformation is from the GISF problem.The instance of the GISM problem constructed is exactlythe same as the instance of the GISF problem. It is notdifficult to verify that the instance of the GISF problemwill have an independent set V' in the graph G = (V, E)such that U,,cv,L(vj) = A, if and only if the instanceof the GISM problem has an independent set V' whosecost is less than 1. This proves that the GISM problemis also NP-complete.

A. Hardness of approximation of the GISM problem

In this subsection we show that no polynomial timeE-approximation algorithm [17] can be developed for theGISM problem unless P= NP.Theorem 3: Unless P = NP, no E-approximation algo-rithm exists for the GISM problem with E < r (av is anumber much smaller than 1/n).Proof: Suppose that there exists a polynomial time E-approximation algorithm APP for the GISM problemwith E < 1 . This implies that for any instance I of the

-nra

GISM problem, ratio between the approximate solutionfor the instance I, APP(I), and the optimal solution,OPT(I), is bounded by E.

Claim: If there exists a polynomial time e-approximationalgorithm APP for the GISM problem with e < 1,then the GISF problem (proven to be NP-completeearlier) can be solved in polynomial time.The optimization version of the GISM problem finds

an independent set V' of minimum cost C(V'). Theapproximation algorithm APP returns an independentset V" with a cost value C(V"). If C(V") < 1, we knowthat the instance of the GISF problem has an independentset V" such that U,,Ev,,L(vi) = A. If C(VV") > 1,we know that the instance of the GISF problem has noindependent set U such that Uv,EuL(vi) = A.

The reason for the last statement is the follow-ing. Suppose (if possible) C(V") returned by the E-approximation algorithm APP (with e < n'a) is greaterthan 1, but the instance of the GISF problem has anindependent set V' such that Uv,,v,L(vi) = A. If theinstance of the GISF problem has an independent set V'such that Uv,Ev,L(vi) = A, then the optimal algorithmOPT would have returned an independent set with costat most na, where n is the number of nodes in thegraph G = (V, E) and av is a number much smaller than1/n. In this case, the ratio between the objective valuereturned by the approximate algorithm APP (greaterthan 1) and the objective value returned by the optimalalgorithm OPT (at most na), is not bounded by e ase < r1a, contradicting the existence an e-approximationalgorithm.

IV. OPTIMAL SOLUTION FOR THE SENSORCOVERAGE PROBLEM

From the discussion in section 2, it is clear that theSensor Coverage problem is equivalent to GeneralizedSet Cover problem. The optimal solution for the Gener-alized Set Cover problem can be obtained by solving anInteger Linear Program (ILP). The ILP formulation ofthe GSC problem is given below.From an instance of the GSC, first construct a n x m

matrix A whose entries are either 0 or 1. A(i, j) = 1 ifthe element si is a member of the subset Sj, otherwiseA(i, j) = 0. We will use an indicator vector x =

{x1i ...* Xn }I to indicate if the subset Si is in the finalsolution set S' or not. Accordingly, xi 1 iff Si C S'and xi = 0 otherwise. The ILP formulation of GSC isMinimize Zi=1 Xi, subject to the constraints(i) Ax > 1, and(ii) Vxi,xj,xi + xj < 1 if (Si, Sj) c INC.The optimal solution to the GISM problem can be

found by solving the following integer linear program-ming problem. Corresponding to every node vi, we willhave one binary variable xi. The variable xi takes value

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1, if vi is part of the independent set V' and 0 otherwise.Corresponding to every element ai c A, we will haveone binary variable yi. The variable yi takes value 1, ifai c L(vi) and vi is part of the independent set V'. Thevariable yi is U otherwise. The ILP for the GISM is thefollowing:Minimize a En xi-ZY I yi (a is a number muchsmaller than 1/n), subject to the following constraints(i) >n7I X, >1(ii) Vi, j, xi + xj < 1 if (vi, vj) c E.(iii) Vi, Yi < Zcvi xi where Vi {vi: ai c L(vi)}.(iv) Vi, Xi, Yi =/01.

V. HEURISTICS FOR THE SENSOR COVERAGEPROBLEM

We showed that the Sensor Coverage Problem canbe viewed as a Generalized Set Cover Problem. Thereexists a number of heuristic solutions for the Set Coverproblem. We will tailor one such algorithm [18] to suitour needs to solve the sensor coverage problem stated inthe form of Generalized Set Cover problem. It may be re-called that the Generalized Set Cover (GSC) Problem isspecified by (i) a set of elements S {s. sm} (ii)a collection of subsets Si C S, S = .SI, , Sn }, (iii)a collection of tuples of incompatible subsets INC =

{(Silt Sj1l), (Si2, Sj2), ., (Sit, Sjt)} and the goal is tofind the smallest subset S' C S, such that(i) Vsi C S, 3Sk C S', such that si C Sk and(ii) if Si, Sj C S', then (Si,Sj) V INC.From an instance of the GSC, we can construct an n x

m matrix A whose entries are either 0 or 1. A(i, j) = 1if the element si is a member of the subset Sj, otherwiseA(i, j) = 0. We present two different greedy heuristicsfor the solution of the GSC problem. Both the heuristicsselect one column after another of matrix A with a goalof covering the largest number of element of the set S.

It may be noted during execution of the algorithmeach column of matrix A (corresponding to each subsetSI, ... , Sr) can be in exactly one of the following threestates: selected, blocked, free. A column is classified asselected if it is selected by the algorithm to be partof the cover. A column is classified as blocked if itis incompatible with a selected column. A column isfree if it is neither selected nor blocked. The benefitassociated with a column is measured by the number ofuncovered elements (at that time) that will be coveredby the selection of that column. The penalty of columnCi is measured by the number of columns that will beblocked, which were not blocked earlier by the columnsselected prior to selection of Ci. Both the heuristics useselection-merit associated with a column for selecting the

Algorithm 1 heuristic generalized set cover1: initialize

Set-of-Free-Columns SI,..., S,k}Set-of-Blocked-Columns<= 0Set-of-Selected-Columns<= 0

2: repeat3: Find the column with highest selection merit in Set-of-

Free-Columns4: Identify the set of incompatible columns for this column5: Move the highest selection merit column to Set-of-

Selected-Columns6: Move the incompatible columns to Set-of-Blocked-

Columns7: if addition of this new column to Set-of-Selected-

Columns creates any redundant columns then8: Move the redundant columns to Set-of-Free-

Columns in decreasing order of their redundancymerit

9: Move the incompatible columns corresponding tothe set of redundant columns from Set-of-Blocked-Columns to Set-of-Free-Columns

10: end if11: if all elements of the set S are covered then12: Go to step 2113: end if14: if Set-of-Free-Columns is empty then15: Go to step 2116: end if17: if Set-of-Free-Columns contains only columns with

zero merit then18: Go to step 2119: end if20: until Set S is covered or

Set-of-Free-Columns 0 orSet-of-Free-Columns contains only columns withzero merit

21: print Set-of-Selected-Columns and the elements of the setS covered by these columns

next column to be included in the cover. The selection-merit of a column used by the first heuristic is the sameas its benefit value. The selection-merit of a column usedby the second heuristic the ratio of its benefit to penalty,if penalty is greater than zero. If penalty is equal to zerothe selection-merit is taken to be equal to its benefit.

During each iteration, the algorithm chooses the freecolumn with the highest selection-merit to be includedin the cover. This column is moved from the set of freecolumns to the set of selected columns. If in the set offree columns, there exists columns that are incompatiblewith this selected column, they are moved to the setof blocked columns. A column in the set of selectedcolumns may become redundant, if addition of a newcolumn during one iteration renders its presence in the

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set of selected columns unnecessary, i.e., all the elementscovered by this column is covered by some other selectedcolumns. A redundant column is removed from the setof selected columns and is returned to the set of freecolumns. In addition, some of the columns blocked dueto inclusion of this column in the set of selected columns,may also be returned to the set of free columns at thistime (if they are not blocked by some other columns, stillpart of the selected set). The redundancy-merit of a col-umn is determined by the number of uncovered elementsthat can potentially be covered by the movement of theset of columns from the set of blocked columns to the setof free columns. In case of multiple redundant columns,the one with the highest redundancy merit is removedfirst. Redundant columns are removed one after anotheruntil no redundant columns are in the set of selectedcolumns.

VI. SIMULATIONS RESULTS AND DISCUSSION

We conducted extensive simulations to evaluate theefficacy of the two heuristics proposed in the section VIby comparing their performances with the optimal solu-tion. For the simulation experiments, we generated twosets of uniformly distributed random points on a plane.In the absence of the information about the distributionof these points in practice, we have assumed a randomdistribution in our simulations. The first set representsthe points to be sensed (blue points) and the secondrepresents the potential location of the sensors (greenpoints). In addition, we generated the sensing radius ofthe sensors rsen and the minimum separation distancebetween the sensors dcr We are particularly interestedin exploring the impact of dcr in finding solution to theGISM problem and its relation with rsen. If dcr 2rsen,then we may have a situation where a part of the sensingregion cannot be covered by any sensors. This is depictedin figure 4(b) where the area enclosed by the three pointswhere the circles meet cannot be covered by any sensors.However, as shown in figure 4(a), this problem does notexist if dcr < 3rsen. For this reason, in our simulationexperiments we have dcr < 3rsen.

* Sensor node

'I's3r

( / \31 P-skn

/ 1.1 \

(a) d,r < v/3-rsen

.a/ \

"C,\ia,I/

R i

)end rsen

(b) d,r = 2r,,en

Fig. 4. Relationship of dcr with rsen (a) dcr = Orsen (b) dcr2rSen

We conducted the simulation experiments by varyingthe minimum separation distance parameter dcr from 0through 50 while keeping the sensing radius parameter(rsen) fixed at 30. The number of potential sensor loca-tions (green points) were considered to be significantlylower than that of the points to be sensed (blue points).In our experiments we kept the number of blue points tobe thrice the number of green points. Five data sets (I1through 15) were generated for each combination of (i)the number of blue points (B), (ii) the number of greenpoints (G), (iii) rsen, and (iv) dcr The goal of the optimalalgorithm as well as the two heuristics were to find thelargest number of blue points that can be covered withthe smallest number of green points. It may be noted thatour optimal algorithm is guaranteed to find a solutionthat covers all the blue points with the smallest numberof green points (subject to minimum separation distanceconstraint), if such a solution exists. The results of oursimulation are presented in table I (next page). It may benoted that some entries in the table have a number withinparentheses associated with it while many other entriesdo not. The implication of an entry (say X) having nonumber within parentheses associated with it is that Xgreen points (sensor locations) are sufficient to cover all(150) blue points. If an entry (say Y) has a number (sayZ) within parentheses associated with it, it implies thatthe largest number of blue points that can be covered isZ (not 150) and it can be done using Y green points.The evaluation of the heuristics show that both of

them produce near optimal solution most of the time.Moreover, they produce such high quality solution in afraction of the time needed to find the optimal solution.While some problem instances needed more than 24hours of computing time to find the optimal solution ona Pentium IV machine running CPLEX version 8.0, theheuristics produced near optimal solutions in only a fewseconds. From their performance, we can conclude thatboth heuristics are quite effective. Between heuristics 1and 2, the second heuristic is somewhat more "intelli-gent" in the sense that its benefit function not only takesinto account the number of uncovered blue points beingcovered by the selection of a green point, but also takesinto account the number of green points that are blockedfrom future consideration due to selection of this greenpoint. However in practice, the seemingly "unintelligent"heuristic 1 seem to be outperforming heuristic 2 in manyinstances. Since this is somewhat counter-intuitive, wetry to explain the phenomenon below.

In the example shown in figure 5(a), we have dcr = 15and rsen 30. If we use heuristic 1, whose selec-tion metric is the number of uncovered blue points,we have Selection-merit(A) = 5, Selection-merit(B) =3,

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TABLE IPERFORMANCE COMPARISON OF OPTIMAL AND HEURISTICS SOLUTIONS WITH B = 150, G = 50 AND r,en 30. (IN THE TABLE, OPT =

OPTIMAL, HI = HEURISTIC 1, H2 = HEURISTIc 2.)

Data Set 1 Data Set 2 Data Set 3 Data Set 4 Data Set 5de ropt Hi H2 IOpt] Hi H2 ] Opt Hi H2 Opt Hi H2 Opt Hi H20 f 6 8 8 7 7 7 6 7 7 6 7 7 7 10 105 6 8 8 7 7 7 6 7 7 6 7 7 7 10 1010 6 8 8 7 7 7 6 7 8 6 7 8 7 10 815 6 8 7 7 8 10 6 7 8 6 8 7 7 9 1020 6 8 8 7 8 9 6 7 7 6 8 9 7 9 825~ 6 8 8 7 7(149)~ 9 6 7 7 6 8 8 7 8(146)~ 930~ 6 8 8 7 6(145) 8(149)~ 6 6(141) 9(144)~ 6 7 1 0 7 6(144) 8(147)~35 6 5(143) 7(142) 7 5(142) 7(148) 6 6(130) 6(144) 6 6(145) 7(143) 7(148) 5(141) 6(140)40 6 5(143) 4(131) 6(149) 5(140) 6(145) 6 5(127) 6(143) 6(148) 5(129) 5(132) 6(147) 5(141) 5(128)45 6(145) 4(130) 4(99) 6(143) 5(134) 5(140) 6(145) 3(105) 4(114) 6(146) 4(115) 5(130) 5(139) 4(127) 4(120)50 L4(138) 4(128) 4(98) [5(139) 14(121) 4(113) L6(143) 4(111) 4(112) 4(136) 4(115) 4(131) 4(134) 3(112) 4(125)

...Node pair within minimum separation distance FEPotential sensor locations* Points to be sensed Sensor locations not selected

(a) ~~(b) (c)

Fig. 5. Comparison of Heuristics 1 and 2

Selection-merit(C) = 1. So, heuristic 1 chooses the greenpoint A first, which is enough to cover all the bluepoints as shown in 5(b). This is the optimal solution forthis problem instance. However if we use the heuristic2, whose selection merit is the ratio of the number ofuncovered blue points to the number of green pointsblocked, we have Selection-metric(A) = 5/2 = 2.5,Selection-merit(B) = 3/1 = 3, Selection-metric(C) = 1/1= 1. In this case, the algorithm will choose green pointB first, which only covers the blue points 2, 3 and 4,and blocks green point A. Then the only green pointthat the heuristic can choose is point C. This scenariois depicted in figure 5(c). So, the final solution usingthe heuristic 2 will comprise of green points B and Ccovering only blue points 1, 2, 3 and 4. The reason whyheuristic 2 fails to find as good a solution as heuristic 1in this case is the fact that the separation distance (dcr)is much smaller than the sensing radius 'rsen.

It may be recalled that both the heuristics use theselection-merit to identify the green point to be includedin the cover. In case of heuristic 1, the selection-metricis equal to the benefit value and in case of heuristic 2,it is equal to the benefit to penalty ratio. When rsen ismuch larger than dc, the benefit associated with a greenpoint is much larger than the penalty and in such casesthe heuristic 1 seems to be doing better than heuristic 2.On the other hand when d,r is much larger than 'rsen,the penalty associated with a green point is much largerthan the benefit and in such cases the heuristic 2 seems

to be doing better than heuristic 1. These observationsare consistent with the simulation results.

VII. CONNECTED SENSOR NETWORK

The previous section presented optimal and heuristicsolutions to find the minimum number of sensors re-

quired to cover a set of points in temperature sensitiveenvironments. In applications where these sensors are notdirectly connected to the controller, they should forma connected network so that the data sensed by anysensor can be delivered to the controller (possibly bymultiple hops through other sensor nodes). We provide a

two-phase solution for such applications, where the firstphase optimizes on the coverage (using the heuristics ofthe previous section) while the second phase determinesthe minimum transmission range Tmtn, necessary so thatthe selected sensors can form a connected communica-tion network. We assume all sensors have same trans-mission range and two sensors can communicate if theyare within the transmission range of one another. Recallthat the minimum separation distance d,r between thesensors depends on the power dissipation of the sensors

Pdiss. The power dissipation of a sensor comprises ofthe power dissipation due to sensing and communica-tion, of which the dissipation due to communication isdominant. As a result, the minimum separation distanced,r mainly depends on the the power dissipation dueto communication which determines the communicationrange rcom. If the minimum transmission range Tmtn, issuch that Tmtn, < rcom, then the sensors selected in phase1 already form a connected communication network. Onthe other hand if Tmtn, > rcom, we need to select moresensor locations to ensure connectivity while maintainingthe minimum separation distance. We have investigatedthe problem of selecting minimum number of additionalsensor locations. We do not provide those results heredue to the space limitations. We provide the algorithmto compute the minimum transmission range Tmtn, below.Transmission Range Problem: Given a complete graphG=(V, E) where V represents the set of sensors selected

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in the first phase and non-negative edge weights w(e) fore = (u, v), u, v c V representing the Euclidean distancebetween sensor u and v, find a spanning tree T of G suchthat maxeET w(e) is minimized. This minmax spanningtree problem is also known as the bottleneck spanningtree problem in the literature [19].

Algorithm 2 compute smallest transmission-range1: Sort the edges of E such that w(el) < ... < w(em)2: Set E' = ell}3: for i = 2 to m do4: if E' U {ei} is acyclic then5: E' =E'U{ei}6: end if7: end for8: return maxeEE/ w(e)

The algorithm computes the edges E' that form theminmax spanning tree and returns the largest edgeweight in the tree. The transmission range of the sensorsset to the largest edge weight ensures that the resultingcommunication network is connected. The algorithmruns in O( E log( E )) time.Theorem 2 Any spanning tree T*=(V, E') where

E'={e'1, e ... 1e' I} constructed using the above algo-rithm produces a minmax spanning tree (MMST).Proof: The proof is by contradiction. Suppose that forany spanning tree T of G other than T*, f (T) denotesthe smallest value of i such that e' is not in T. Assumethat T* is not a MMST and T is a MMST such thatf (T) is as large as possible. Suppose that f (T)=k. Thisimplies that e .,es e. are in both T and T* but e/is not in T. Clearly, adding the edge e/ to the edge setE' of the tree T creates a unique cycle C, i.e., T + ekcontains C. Suppose that e' is an edge in C that is inT but not in T*. Since e' is not a cut edge of T + e/,T + e- e' is a connected graph with m -1 edges andtherefore another spanning tree of G. Clearly w(T')max(w(T -e'/), w(el)The algorithm chose the edge e/ before it chose the

edge e'/, even though neither e/ nor e' would havecreated a cycle with the edges e/, es,.. e,e,-/. Thisimplies w(e') > w(el). This observation, together withthe fact that w(T')=max(w(T -e), w(e/)), concludesthat w(T') < w(T). Thus T' is also a MMST. However,f (T') > k = f (T), contradicting the choice of T as theMMST with the largest f (T). Therefore T=T* and T*is indeed a MMST.

VIII. CONCLUSION

In this paper, we have introduced a new version ofthe sensor placement and coverage problem. We have

shown that both the optimization and the feasibilityversions of the problem are NP-complete. Moreover, ane-approximation algorithm for the problem cannot bedeveloped unless P = NP. Our heuristics produce nearoptimal solution for most of the problem instances in afraction of the time needed to find an optimal solution.

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