Optimal Movement of Mobile Sensors for Barrier Coverage of a Planar Region (Extended Abstract)

Transcription

1 Optimal Movemet of Mobile Sesors for Barrier Coverage of a Plaar Regio (Exteded Abstract) B. Bhattacharya M. Burmester Y. Hu E. Kraakis Q. Shi A. Wiese March 16, 2008 Abstract Itrusio detectio, area coverage ad border surveillace are importat applicatios of wireless sesor etworks today. They ca be (ad are beig) used to moitor large uprotected areas so as to detect itruders as they cross a border or as they peetrate a protected area. We cosider the problem of how to optimally move mobile sesors to the fece (perimeter) of a regio delimited by a simple polygo i order to detect itruders from either eterig its iterior or exitig from it. We discuss several related issues ad problems, propose two models, provide algorithms ad aalyze their optimal mobility behavior. 1 Itroductio Moitorig ad surveillace are two of the mai applicatios of wireless sesor etworks today. Typically, oe is iterested i moitorig a give geographic regio either for measurig ad surveyig purposes or for reportig various types of activities ad evets. Aother importat applicatio cocers critical security ad safety moitorig systems. Oe is iterested i detectig itruders (or movemets thereof) aroud critical ifrastructure facilities ad geographic delimiters (chemical plats, forests, etc). As a matter of fact, sice the iformatio security level of the moitorig system might chage rapidly because of hostile attacks targeted at it, research efforts are curretly uderway to exted the scalability of wireless sesor etworks so that they ca be used to moitor iteratioal borders as well. For example, [11] reports the possibility of usig wireless sesor etworks for replacig traditioal barriers (more tha a kilometer log) at both the buildig ad estate level. Also, Project 28 cocers the costructio of a virtual fece as a way to complemet a physical fece that will iclude 370 miles of pedestria fecig ad 300 miles of vehicle barrier (see [8] which reports delays i its deploymet alog the U.S.-Mexico border). School of Computig Sciece, Simo Fraser Uiversity, Vacouver, BC, Caada. Departmet of Computer Sciece, Florida State Uiversity, Talahassee, Florida, USA. School of Computer Sciece, Carleto Uiversity, Ottawa, Otario, Caada. Istitut für Mathematik, Techische Uiversität Berli, Berli, Germay. Research supported i part by NSERC ad MITACS. 1

2 To begi, we say that a poit is covered by a sesor if it is withi its rage. I this paper we will use the cocept of barrier coverage as used i [11] ad which differs from the more traditioal cocept of full coverage. I the latter case oe is iterested i coverig the etire regio by the deploymet of sesors, while i the former all crossig paths through the regio are covered by sesors. Thus, oe is ot iterested i coverig the etire deploymet regio but rather to detect potetial itruders by guarateeig that there is o path through this regio that ca be traversed udetected by a itruder as it traverses the border. Clearly, barrier coverage is a appropriate model of movemet detectio that is more efficiet tha full coverage sice it requires less sesors for detectig itruders (this is the case, for example, whe the width of the deploymet regio is three times the rage of the sesors). I [3] the authors cosider the problem of how idividual sesors ca determie barrier coverage locally. I particular, they prove that it is possible for idividual sesors to locally determie the existece of barrier coverage, eve whe the regio of deploymet is arbitrarily curved. Although local barrier coverage does ot always guaratee global barrier coverage, they show that for thi belt regios, local barrier coverage almost always provides global barrier coverage. They also cosider the cocept of L-local barrier coverage whereby if the boudig box that cotais the etire trajectory of a crossig path has legth at most L the this crossig path is guarateed to be detected by at least oe sesor. Motivatio, model ad problem statemet. Motivated from the works of [3] ad [11], i this paper we go beyod by askig a differet questio ot examied by ay of these papers. More precisely, give that the mobile sesors have detected the existece of a crossig path (e.g., usig ay of the above algorithms) how do they repositio themselves most efficietly withi a specified regio so as to repair the existig security hole ad thereby prevet itruders. Further, we stipulate the existece of a geometric plaar regio (the critical regio to be protected) delimited by a simple polygo ad mobile sesors are lyig i the iterior of this polygo. We cosider a set of mobile sesors (or robots) lyig withi a regio that ca move autoomously i the plae. Each sesor has kowledge of the regio to be barrier-covered, of its geographic locatio ad ca move from its startig positio p to a ew positio p o the perimeter of this polygo. For each sesor, we look at the distace d(p, p ) betwee the startig ad fial positios of the sesors, respectively, ad ivestigate how to move the sesors withi this regio so as to optimize either the miimum sum or the miimum maximum of the distaces covered by the respective sesors. I the sequel we ivestigate the complexity of this problem for various types of regios ad types of movemet of the mobile sesors. Related work. A iterestig research article is by [1] which surveys the differet kids of holes that ca form i geographically correlated problem areas of wireless sesor etworks. The authors discuss relative stregths ad short-comigs of existig solutios for combatig differet kids of holes such as coverage holes, routig holes, jammig holes, sik/black holes, worm holes, etc. [2] looks at critical desity estimates for coverage ad coectivity of thi strips (or auli) of sesors. I additio, [5] ad [6] desig a distributed self deploymet algorithm for coverage calculatios i mobile sesor etworks ad cosider various performace metrics, like coverage, uiformity, time ad distace traveled till the algorithm coverges. Related is also the research o art gallery theorems (see [14]) which is cocered with fidig 2

3 the miimal umber of positios for guards or cameras so that every poit i a gallery is observed by at least oe guard or camera. I additio to the research o barrier coverage already metioed there is extesive literature o detectio ad trackig i sesor etworks. [12] cosiders the problem of evet trackig ad sesor resource maagemet i sesor etworks ad trasforms the detectio problem ito fidig ad trackig the cell that cotais the poit i a arragemet of lies. [9] addresses the problem of trackig multiple targets usig a etwork of commuicatig robots ad statioary sesors by itroducig a regio-based approach for cotrollig robot deploymet. [16] cosiders the problem of accurate mobile robot localizatio ad mappig with ucertaity usig visual ladmarks. Fially, related to the problem of detectig a path through a regio that ca be traversed udetected by a itruder is the paper [15] which gives ecessary ad sufficiet coditios for the existece of vertex disjoit simple curves homotopic to certai closed curves i a graph embedded o a compact surface. Outlie ad results of the paper. Sectio 2 gives the formal model o a circle ad defies the mi-max (miimizig the maximum) ad mi-sum (miimizig the sum) problems for a set of sesors withi a circle or a simple polygo. Sectio 3 looks at the simpler oe dimesioal case ad derives simple optimal algorithms for the case the sesors either all lie o a lie or o the perimeter of circle. Sectio 4 ad Sectio 5 are the core of the paper ad provide algorithms solvig the mi-sum ad mi-max problems, respectively. That is, i Sectio 4, a O( 3.5 log )-time algorithm for the mi-max problem o a circle ad a O(m 3.5 log )-time algorithm for the mi-max problem o a simple polygo are proposed (m is the umber of edges of the simple polygo). Our approximatio algorithms for mi-sum problems o a circle or a simple polygo are preseted i Sectio 5. Fially, Sectio 6 gives the coclusio. 2 Prelimiaries ad Formal Model First we describe the formal model o a circle ad provide the basic defiitios ad prelimiary cocepts. 2.1 Optimizatio o a circle The simpler sceario we evisio cocers mobile sesors which are located i the iterior of a uit-radius circular regio. A set of sesors are located iside the disk. Further, assume that the sesors are locatio aware (i.e., they kow their geometric coordiates) ad also kow the locatio of the ceter of the disk. We would like to move all the sesors from their iitial positios to the perimeter of the circle so as to 1) form a regular -go, ad 2) miimize the total/maximum distace covered. The motivatio for placig the sesors o the perimeter is because it provides the most efficiet way to protect the disk from itruders. Observe that whe all sesors lie equidistat o the vertices of a regular -go, they each eed to cover a circular arc of size 2π/ so as to be able to moitor the etire perimeter. Usig elemetary trigoometry, it follows easily that the trasmissio rage of each sesor must be equal to r = si(π/). More formally, for give sesors i positios A 1, A 2,..., A, respectively, which move to ew positios A 1, A 2,..., A at the corers of a regular -go the total distace covered 3

4 is d(a i, A i ). Every sesor moves from its curret positio A i to a ew positio A i. It is clear that the sum is miimized whe each sesor moves to its ew positio i a straight lie. The reaso for havig the sesors at the corers of a regular -go is because this is evidetly the optimal fial arragemet that will eable them to detect itruders (i.e., by beig equidistat o the perimeter). Thus, sice the fial positio A 1 A 2 A of the sesors forms a regular -go it is clear that all possible solutios ca be parametrized by usig a sigle agle 0 θ 2π. However, a difficulty arises i view of the fact that we must also specify a permutatio σ : {1, 2,..., } {1, 2,..., } of the sesors such that the i-th sesor moves from positio A σ(i) to the ew positio A i. Let the sesors have coordiates (a i, b i ), for i = 1, 2,...,. Let us parametrize the regular polygo with respect to the agle of rotatio say θ. The vertices of the regular -go that lie o the perimeter of the disk ca be described by ( ( ) ( (a i (θ), b i, (θ)) = cos θ +, si θ + (i 1)2π (i 1)2π )), (for i = 1, 2,..., ), (1) respectively, where (a i (θ), b i (θ)) are the vertices of the regular -go whe the agle of rotatio is θ. Miimizig the sum The optimizatio problem is mi θ S (θ), where the fuctio S (θ) (a i a i (θ)) 2 + (b i b i (θ)) 2, as a fuctio of the agle θ. This is defied by S (θ) := of course assumes that the i-th sesor is assiged to positio (cos(θ + (i 1)2π/), si(θ + (i 1)2π/)) o the perimeter. I geeral, we have to determie the miimum over all possible permutatios σ of the sesors. If for a give agle θ ad permutatio σ we defie S (σ, θ) := (aσ(i) a i (θ) ) 2 + ( bσ(i) b i (θ) ) 2 the the more geeral optimizatio problem is mi σ,θ S (σ, θ). Miimizig the maximum The previous problem was cocered with miimizig the sum of the distaces of the robots to their fial destiatios. I view of the fact that the robots are movig simultaeously it makes sese to ask for miimizig the maximum of the distaces of the robots to their fial destiatios max 1 i d(a i, A i ). The optimizatio problem is mi θ M (θ), where M (θ) := max 1 i (a i a i (θ)) 2 + (b i b i (θ)) 2, as a fuctio of the agle θ. This of course assumes that the i-th sesor is assiged to positio (cos(θ + (i 1)2π/), si(θ + (i 1)2π/)) o the perimeter. I geeral, we have to determie the miimum over all possible permutatios σ. If for a give permutatio σ we defie the followig maximum M (σ, θ) := max 1 i (aσ(i) a i (θ) ) 2 + ( bσ(i) b i (θ) ) 2 the the geeral optimizatio problem is mi σ,θ M (σ, θ). 2.2 Optimizatio o a simple polygo We similarly defie the problem of miimizig the sum ad miimizig the maximum o a simple polygo as follows. Let P be a simple polygo. (From ow o, a polygo is always assumed to be simple.) We deote the boudary of P by P. We assume that P is orieted Although the approach proposed later (parametric search) will also work for arbitrary simple curves, we refrai from such a geeralizatio so as to avoid uecessary complicatios. 4

5 i the clockwise (also called positive) directio. For ay two poits a, c P, we write ˆπ P (a, c) to deote the set of all poits b P such that whe startig after a i positive directio alog P, b is reached before c. Let p 0, p 1,..., p m 1 deote the vertices o P ordered i the positive directio. The edges of P are e 0, e 1,..., e m 1, where edge e i has edpoits p i ad p i+l, where 0 i < m (i.e., the idices are computed modulo m; e.g., p 0 = p m ). We deote by l(e i ) the legth of edge e i, 0 i < m, ad by ˆd P (a, b) the legth of ˆπ P (a, b) for ay two poits a ad b o P (called polygoal distace betwee a ad b). Let L(P ) = m 1 i=0 l(e i). We are give mobile sesors which are located i the iterior of P. Each sesor has the kowledge of its geometric coordiates ad the simple polygo (i.e., the geometric coordiates of all vertices p i, 0 i < ad the clockwise orderig of these vertices). The objective is to move all the sesors from their iitial positios to P such that 1) the polygoal distace betwee ay two cosecutive sesors o the polygo is L(P )/, ad 2) miimize the total/maximum distace covered. We postulate that if give sesors are located at positios A 1, A 2,..., A, ad the destiatio positios are A 1, A 2,..., A, respectively, the ˆd P (A i, A i+1 ) = L(P )/, 0 i <. 3 Mobile Sesors i Oe Dimesio I this sectio we look at the oe dimesioal problem ad provide efficiet algorithmic solutios. I particular, sice optimizatio for the miimum maximum is similar (ad simpler tha the two dimesioal aalogue) we provide algorithms oly for the miimum sum. 3.1 Sesors o a lie segmet I this model we suppose that the sesors ca move o a lie segmet. Further, istead of protectig a circular rage the sesor ca ow protect a iterval of a give size cetered at the sesor. Cosider the miimum sum optimizatio problem for the case of sesors o a lie. Without loss of geerality assume the segmet has legth 1 ad let the sesors be at the iitial locatios x 0 < x 1 < < x 1, respectively. The destiatio locatios are for i = 0, 1,..., 1. Theorem 1 The optimal arragemet is obtaied by movig poit x i to positio i = 0, 1,..., 1, respectively. 3.2 Sesors o the perimeter of a circle i 1, i 1, for I this model we suppose that the sesors ca move o the perimeter of a circle. Further, istead of protectig a circular rage the sesor ca ow protect a arc o the perimeter of a give size cetered at the sesor. The same idea as for a lie segmet should work for the case of a uit circle whe the sesors lie o the perimeter of the circle. The mai difficulty here is that we o loger have a uique destiatio. Istead, we ca parametrize all possible destiatios of the poits by φ + 2jπ, for j = 0, 1,..., 1, usig a fixed agle 0 φ < 2π. Theorem 2 There is a algorithm that computes a optimal cost arragemet of the sesors Whe the sesors movemet is i the iterior of the circle I this model we suppose that the sesors ad their destiatio positios are located o the perimeter of a circle ad the sesors ca move to their destiatio positios alog a straight 5

6 lie. Here we have the same result as the oe i Lemma 14 (see Appedix). Its proof is based o the fact there is a optimal solutio i which oe sesor does ot move at all. Theorem 3 There is a algorithm that computes a optimal cost arragemet of the sesors. 4 Mi-max problem i 2D I this sectio we study the problem of miimizig the maximum (mi-max problem) o a uit circle ad a simple polygo, ad provide efficiet algorithmic solutios. 4.1 O a circle Let λ m,c be the optimal value of the mi-max problem o a circle C, i.e., λ m,c = mi σ,θ M (σ, θ). It is easy to see that λ m,c is o more tha the diameter of the circle C, i.e., λ m,c 2. I this sectio we propose a parametric-searchig approach [13] to compute λ m,c. A o-egative value λ is feasible i the mi-max problem if all the sesors ca move from their iitial positios to the perimeter of the circle such that the ew positios form a regular -go ad the maximum movig distace is o more tha λ, otherwhise λ is ifeasible. Clearly, the mi-max problem is to compute the miimum feasible value, which is equal to λ m,c. The remaiig part of this sectio is orgaized as follows. We first show that a feasibility test of a give value λ(0 λ 2) ca be performed i time O( 3.5 ). The, a parametricsearchig approach for the mi-max problem is preseted, which rus i O( 3.5 log ) time Algorithm to check the feasibility test of λ For each i, 1 i, we costruct a circle of radius λ cetered at positio A i, deoted by C i. If a circle C i for some i is cotaied i C, the λ is ifeasible sice sesor A i caot move to the perimeter of C withi distace λ. We therefore assume that for each i, 1 i, either circle C i cotais C or C i itersects with C. Refer to Figure 3. For each i, 1 i, we deote by Q i the arc of C that lies i C i. Let q i(1), q i(2) be the agles of two edpoits of arc Q i i clockwise order, i = 1,...,. We let q i(1) = 0 ad q i(2) = 2π if C i cotais C. The followig property is importat to our algorithm for the feasibility test of λ. Its proof is omitted here. Lemma 4 A give o-egative value λ is feasible if ad oly if there exists a regular -go o the perimeter of C such that oe of its corer poits is a edpoit of arc Q i for some i(1 i ). The algorithm (Algorithm Check) to check the feasibility of λ is described i the appedix. It is easy to see that the sortig i the first step ca be doe i O( log ) ad the computatio of S j, j = 1,..., ca be doe i O( 2 ). I the third step, the process might try all O() regular -gos. For each regular -go, it takes O( 2.5 ) time (see [7]). Therefore, we have the followig lemma. Lemma 5 Whether a give positive value λ is feasible i the mi-max problem ca be determied i O( 3.5 ) time. 6

7 4.1.2 A parametric-searchig approach Our approach for the solutio to the mi-max problem is to ru Algorithm Check parametrically, which has a sigle parameter λ, without specifyig the value of λ m,c a priori. Note that for a fixed value of the parameter, the algorithm is executed i O( 3.5 ) steps. Imagie that we start the algorithm without specifyig a value of the parameter λ. The parameter is restricted to some iterval which is kow to cotai the optimal value λ m,c. (Iitially, we may start with the iterval [0, 2].) As we go alog, at each step of the algorithm we update ad shrik the size of the iterval, esurig that it icludes the optimal value λ m,c. The fial iterval cotais λ m,c ad ay value i it is feasible. Therefore, the miimum value of the fial iterval is the optimal value λ m,c. Theorem 6 The mi-max problem o a circle ca be solved i O( 3.5 log ) time. Note that our algorithm ca be easily exteded to the model i which all sesors are arbitrarily located o the plae (ot restricted to the iterior of the circle C). 4.2 O a simple polygo The parametric-searchig approach for a circle (described i sectio 4.1) should work for the case of a polygo where the destiatio positios of all sesors lie o the perimeter of the polygo. The mai difficulty here is that to check the feasibility of a positive value λ, there might be O(m) isolated polygoal chais of P withi the circle C i (of radius λ cetered at positio A i ) for each sesor A i. I other words, for a give positive value of λ each sesor will cotribute O(m) cadidate sets of destiatio positios o P istead of at most two cadidate sets o a circle. Hece, whether a give positive value λ is feasible i the mi-max problem o a simple polygo ca be determied by solvig O(m) matchig problems of size. Therefore, the feasibility test of the mi-max problem o a simple polygo ca be solved i O(m 3.5 ) time. Theorem 7 The mi-max problem o a simple polygo ca be solved i O(m 3.5 log ) time where m is the size of the simple polygo. 5 Approximatio algorithms for the mi-sum problem i 2D I this sectio we discuss the problem of miimizig the sum (mi-sum problem) o a circle ad a simple polygo, ad provide approximatio solutios for them. 5.1 O a circle Let λ s,c be the optimal value of the mi-sum problem o a circle, i.e., λ s,c = mi σ,θ S (σ, θ). We preset two approximatio algorithms for the mi-sum problem. Oe algorithm (labeled as the first approach) has a approximatio ratio π + 1 (sectio 5.2). The other oe (labeled as the secod approach) uses the first approach as a subroutie to obtai lower ad upper bouds of λ s,c ad has a approximatio ratio 1+ɛ, where ɛ is a arbitrary costat (Sectio 5.3). More otatios are itroduced as follows. Let ˆd C (x, y) deote the arc distace betwee two poits x ad y o the boudary if the cycle C ad let ˆπ C (x, y) deote the arc of legth 7

8 ˆd C (x, y) betwee x ad y. For a poit x o C, we deote by ˆQ x (r) the arc cosistig of all poits y o C such that ˆd C (x, y) r. For each i = 1,...,, let ω i be the smallest distace betwee A i ad a poit o the cycle C, ad we deote by B i the poit o C such that the distace d(a i, B i ) = ω i. We ote that for each i = 1,...,, B i is uique if A i is ot located at the ceter of C. I the case whe A i is located at the ceter of C, a arbitrary poit o C is selected to be B i. Let Ω = ω i. Obviously, we have the followig lemma. Lemma 8 Ω λ s,c. 5.2 The first approach The first approach (called Algorithm 1) cosists of three steps. Step 1 For each sesor A i, 1 i, compute B i. Step 2 Compute a destiatio regular -go for the set of poits B 1,..., B, ad fid the optimal arragemet of the poits to the vertices of the -go, by usig the algorithm for sesors o the perimeter of a circle described i Sectio 3.2. Let B i be the destiatio vertex of B i, 1 i. Step 3 Move A i to B i, 1 i, ad compute S1 = d(a i, B i ). I sectio 3.2 we showed that step 2 of Algorithm 1 ca be implemeted i O( 2 ) time. Thus the above algorithm ca be solved i O( 2 ) time Approximatio boud of Algorithm 1 I this sectio, we show that S 1 computed by the first approach is bouded by (π + 1) λ s,c. Suppose that A i is the destiatio of sesor A i, i = 1,...,, i a optimal solutio. Clearly, A 1,..., A lie o C ad form a regular -go. Obviously, ˆd C (B i, B i ) ˆd C (B i, A i ) sice {B 1,..., B } is a optimal solutio for the oe dimesioal mi-sum problem with the iput {B 1,..., B }. The followig lemma is easy to show. Lemma 9 For ay two poits x, y o C, ˆd C (x, y) π 2 d(x, y). Theorem 10 Algorithm 1 ca be implemeted i O( 2 ) time ad its approximatio ratio is o more tha π The secod approach The followig lemma is crucial for the secod approach (Algorithm 2) described i the appedix. Lemma 11 I a optimal solutio, there exists at least oe sesor A i (1 i ) such that its destiatio A i o C is o the arc Qˆ Bi ( π 2 S1 ). Proof. It is clear that S 1 λ s,c. Let A i be the destiatio of sesor A i i a optimal solutio, i = 1,...,. The there is at least oe sesor, say A k (1 k ), such that the distace d(a k, A k ) is o more tha S1. Accordig to Lemma 9, all poits o C with the distace to A k of o more tha S1 lie o the arc Qˆ Bk ( π 2 S1 ) (recall that B k is the poit o C closest to A k ), which completes the proof of Lemma 11. 8

9 5.3.1 Aalysis of the secod approach First, it is evidet that the ruig time of the secod approach is determied by the time eeded for solvig ( 1 ɛ + 1) O( ɛ ) bipartite matchig problems. Accordig to Lemma 11, there exists a optimal solutio i which oe of the corers of the correspodig regular -go is located at a poit o the arc Qˆ Bk ( π 2 S1 ) for some k, 1 k. I Step 2, the arc Qˆ Bk ( π 2 S1 ) is partitioed ito 1 ɛ pieces, ad therefore, the legth of each piece is o more tha πs1 ɛ (ote that the legth of Qˆ Bk ( π 2 S1 ) is πs1 ). Sice all possible values of k are cosidered, the differece betwee S 2 (computed by the secod approach) ad λ s,c (the optimal cost) is o more tha 1 2 πs1 ɛ = πs1 ɛ 2 = S1 ɛ π+1 ɛλ s,c, bytheorem 10). Therefore, we have the followig theorem. Theorem 12 The approximatio ratio of Algorithm 2 is o more tha 1 + ɛ for a give costat ɛ, ad the ruig time of the secod approach is O( 1 ɛ 4 ). 5.4 O a simple polygo Let λ s,p be the optimal value of the mi-sum problem o a polygo P. I this subsectio we preset a approximatio algorithm for the mi-sum problem o P, which has a approximatio ratio 1 + ɛ (ɛ is a arbitrary costat). Our algorithm for a simple polygo is very similar to the secod approach for a circle. I the secod approach for a circle, we use Algorithm 1 as a subroutie to obtai lower ad upper bouds of λ s,c. However, our approximatio algorithm for a simple polygo will use the solutio for the mi-max problem o the polygo to obtai lower ad upper bouds of λ s,p. Let λ m,p be the optimal value of the mi-max problem o P. It is easy to see that λ m,p λ s,p λ m,p. Our algorithm for a simple polygo P ad details of proofs are described i the appedix. Theorem 13 The approximatio ratio of the approach for a simple polygo is o more tha 1 + ɛ for a give costat ɛ, ad the ruig time of the secod approach is O( 1 ɛ m5 ). 6 Coclusio ad Ope Problems I this paper we gave a algorithm for solvig the mi-max problem ad a PTAS (Polyomial Time Approximatio Scheme) for the mi-sum problem i both oe ad two dimesios. Although it is ukow whether the mi-sum problem is NP -hard, we cojecture that it ca be solved i polyomial time. Evidece for this also comes from experimetal results o fidig the umber of differet couter-clockwise orderigs of sesors o the perimeter of a circle whe we sweep a regular -go alog the perimeter (see Appedix). I additio, several other variats of the problem o simple polygos ad regios are of iterest for further ivestigatio, icludig k-barrier coverage, regios with holes, ad various types of sesor placemets ad motios. Thus, i Subsectio 2.2, i order to miimize the umber of sesors used whe scaig the perimeter oe should take ito accout sectios already scaed. For example, this is the case if the polygo is a arrow rectagle of height less tha the rage of a sesor; this i itself is a iterestig optimizatio problem which is worth of further ivestigatio. Also of iterest is to refie the sesor motio model, the etwork model, ad the commuicatio model i order to eable effective itrusio detectio ad barrier coverage. 9

10 For example, the commuicatio model becomes crucial whe assumig the sesors either do ot have kowledge of the regio or do ot kow their coordiates. Refereces [1] N. Ahmed, S.S. Kahere, ad S. Jha. The holes problem i wireless sesor etworks: a survey. ACM SIGMOBILE Mobile Computig ad Commuicatios Review, 9(2):4 18, [2] P. Balister, B. Bollobas, A. Sarkar, ad S. Kumar. Reliable desity estimates for coverage ad coectivity i thi strips of fiite legth. Proceedigs of the 13th aual ACM iteratioal coferece o Mobile computig ad etworkig, pages 75 86, [3] A. Che, S. Kumar, ad T.H. Lai. Desigig localized algorithms for barrier coverage. Proceedigs of the 13th aual ACM iteratioal coferece o Mobile computig ad etworkig, pages 63 74, [4] R. Cole. Slowig dow sortig etworks to obtai faster sortig algorithms. Joural of the ACM (JACM), 34(1): , [5] N. Heo ad PK Varshey. A distributed self spreadig algorithm for mobile wireless sesor etworks. Wireless Commuicatios ad Networkig, WCNC IEEE, 3, [6] N. Heo ad PK Varshey. Eergy-efficiet deploymet of Itelliget Mobile sesor etworks. Systems, Ma ad Cyberetics, Part A, IEEE Trasactios o, 35(1):78 92, [7] J.E. Hopcroft ad R.M. Karp. A 2.5 algorithm for maximum matchigs i bipartite graphs. SIAM Joural o Computig, 2(4): , [8] S. S. Hu. Virtual Fece alog border to be delayed. Washigto Post, Thursday, February 28, [9] B. Jug ad G.S. Sukhatme. Trackig Targets Usig Multiple Robots: The Effect of Eviromet Occlusio. Autoomous Robots, 13(3): , [10] H. W. Kuh. The Hugaria Method for the assigmet problem. Naval Research Logistics Quarterly, 2:83 97, [11] S. Kumar, T.H. Lai, ad A. Arora. Barrier coverage with wireless sesors. Wireless Networks, 13(6): , [12] J. Liu, P. Cheug, F. Zhao, ad L. Guibas. A dual-space approach to trackig ad sesor maagemet i wireless sesor etworks. Proceedigs of the 1st ACM iteratioal workshop o Wireless sesor etworks ad applicatios, pages , [13] N. Megiddo. Applyig Parallel Computatio Algorithms i the Desig of Serial Algorithms. Joural of the ACM (JACM), 30(4): , [14] J. O Rourke. Art gallery theorems ad algorithms. Oxford Uiversity Press, Ic. New York, NY, USA, [15] A. Schrijver. Disjoit circuits of prescribed homotopies i a graph o a compact surface. Joural of Combiatorial Theory Series B, 51(1): , [16] S. Se, D. Lowe, ad J. Little. Mobile Robot Localizatio ad Mappig with Ucertaity usig Scale-Ivariat Visual Ladmarks. The Iteratioal Joural of Robotics Research, 21(8):735,

11 Appedix Proof. (Theorem 1) It is clear that the possible destiatios are of the form i 1, for i = 0, 1,..., 1. For ay poit x let d(x) be its destiatio. Recall that our goal is to determie the destiatio of each poit x so that the sum x d(x) (2) x is miimized. We call the route from x to d(x) the directed path o the lie from x to d(x) which is traced by the sesor located at x. The legth of this path is equal to x d(x). For ay two pits x < y we say that their paths iversely ovelap if they are i opposite directios ad they also overlap. As depicted i Figure 1, it is clear that for two poits x < y there x d(y) y d(x) x d(y) d(x) y d(y) x y d(x) Figure 1: Three possible cases for overlappig paths betwee the two poits x < y. are three possible cases of overlappig paths. It is easy to show that i each case we ca improve o the optimizatio sum (Equatio 2) by merely switchig the destiatios of x ad y, respectively. We call such a cofiguratio betwee two poits iversio. i Now cosider the cofiguratio resultig by movig poit x i to positio 1, for i = 0, 1,..., 1, respectively. Observe that for this arragemet there exist o two pairs x < y of sesors which iversely overlap. Moreover, we claim that ay two possible assigmets that do ot have pairs x < y that iversely overlap must have idetical sum (Equatio 2). Ideed, the sum of the top cofiguratio depicted i Figure 2 is (d(y) x) + (d(x) y) ad of the bottom it is (d(x) x) + (d(y) y). Sice (d(y) x) + (d(x) y) = (d(x) x) + (d(y) y) the result of the claim follows. This completes the proof of the theorem. Proof. (Theorem 2) First of all we prove the followig lemma. Lemma 14 The miimal cost assigmet of destiatios for the give poits, must be amog the assigmets ( (x 0, x 1,..., x j,..., x 1 ) x i, x i + 2π,..., x i + j 2π,..., x i + ( 1) 2π ), 11

12 x y d(y) d(x) x y d(y) d(x) Figure 2: A o-overlappig trasformatio leaves the sum uchaged. for i = 0, 1,..., 1. Proof. (Lemma 14) Cosider the miimal cost assigmet based o the agle x. Suppose that i the correspodig -go oe of the poits x 0, x 1,..., x j,..., x 1 is a vertex. Now rotate this -go ad observe that either its clockwise or its couter clockwise rotatio decrease the cost or remais the same. Now we prove the mai result. Suppose we are give poits x 0, x 1,..., x 1 i this cyclical order alog the perimeter of the circle. The mai steps of the algorithm are the followig. 1. For each poit x i {x 0, x 1,..., x 1 }, map all poits x j to destiatios x i + j 2π, for j = 0, 1,..., 1 (this implies that x i is mapped to itself). 2. Select the poit x i {x 0, x 1,..., x 1 } that optimizes the sum 1 x j x i j 2π (3) j=0 It is easy to show that i this mappig each poit x is mapped to a destiatio d(x) such that x d(x) π. Now just like the case of a lie (imagie Figure 1 o a circle) we ca show that the optimal assigmet caot have paths that iversely overlap. Moreover, accordig to Figure 2 (imagie it o a circle) the assigmets will be optimal each for the give startig positio x i. Proof. (Theorem 3) Assume to the cotrary that all sesors move i a optimal solutio. For each i, 1 i, let α i be the agle of rotatio of the sesor A i to its ew positio A i : positive agle deotes clockwise rotatio, while egative agle deotes couter-clockwise rotatio. For sesors which are moved to the opposite side of the circle we assume that α = π. From the law of cosie we derive that d(a i, A i ) = 2 2 cos α i. Let f(α) = 2 2 cos α. Trivially, f(α) is cocave over [0, π) ad [ π, 0]. We defie α := max αi 0 α i ad α := mi αi >0 π α i. Note that α < 0 sice all α i 0 ad α > 0 sice all α i < π. We defie the fuctio g : [a, b] R: g(β) := f (α i + β). 12

13 The value of g(β) represets the total cost of the solutio which we get by takig the origial solutio α i ad chagig the positios of the sesors o the perimeter by the agel β. Note that g is cocave for all β (a, b). We wat to show that there is a β 0 such that g (β) < g(0). If g (0) 0 the there is clearly a β with g (β) < g(0). This is a cotradictio. If g (0) = 0 the β = 0 is a local maximum sice g (0) < 0. So there is a β with g (β) < g(0). This is a cotradictio. Sice Lemma 14 is still valid i the model usig the straight-lie distace metric, the same idea as for the arc distace metric ca be applied here. C q i(1) Ai q i(2) Q i Figure 3: Notatios Q i, q i(1) ad q i(2). Proof. (Theorem 6) The whole approach for the mi-max problem is described as follows. The sortig step (the first step) of Algorithm Check with ukow λ m,c ca be performed by solvig O(log 2 ) feasibility tests if a parallel sortig etwork that rus O(log ) steps with processors is used [13]. Actually, it ca be reduced to O(log ) feasibility tests if Cole s result is applied [4]. The secod step does ot eed to solve feasibility tests sice the order of edpoits of arcs Q i, 1 i, provides eough iformatio to compute sets S j, 1 j We ow show how to ru the third step with ukow λ m,c. We ca see that o feasibility test is eeded for the steps 3(b) ad 3(c). However, to fid the itervals where the agles q j, (q j + 2π ) mod 2π,..., (q j + ( 1) 2π ) mod 2π, j = 1,..., 2 lie, we solve some feasibility tests sice these agles ad the edpoits of itervals are low-degree polyomials of ukow value λ m,c. It is iefficiet to execute step 3(c) oe by oe. To speed up the computatio, we ca compute them i a parallel way. For each agle, we ca locate the iterval cotaiig it i a biary-search fashio. Clearly, it ca be doe i O(log ) steps, ad each step is a compariso betwee two low-degree polyomials that ca be achieved by solvig a costat umber of feasibility tests. Therefore, the computatio of the step 3(a) for all possible j [1, 2] ca be doe i O(log ) parallel steps by usig O( 2 ) processors (there are 2 2 queries i total), where each 13

14 step is a compariso betwee two low-degree polyomials with ukow λ m,c. By applyig the idea of Cole [4], the computatio of the step 3(a) for all possible j [1, 2] ca be doe by solvig O(log ) feasibility tests. Proof. (Theorem 10) S 1 = d(a i, B i) [d(a i, B i ) + ˆd C (B i, B i)] d(a i, B i ) + ˆd C (B i, A i) d(a i, B i ) + π 2 d(b i, A i) (Lemma 9) d(a i, B i ) + π 2 [d(b i, A i ) + d(a i, A i)] (π + 1) λ s,c (Lemma 8). Algorithm Check 1. The first step is to sort the agles of edpoits of arcs Q i, 1 i. Let q 1,..., q 2 be the agles i icreasig order. These agles partitio the iterval [0, 2π] ito at most pairwise disjoit itervals, deoted by I 1,..., I For each iterval I j, 1 j 2 + 1, we determie the set of sesors, deoted by S j, that lie withi distace λ to its correspodig arc o C. 3. I the third step, we do the followig for a regular -go with rotatio q j, i = 1,..., 2. (a) It is easy to see that the agles of corer poits of such regular -go are q j, (q j + 2π ) mod 2π,..., (q j + ( 1) 2π ) mod 2π. We compute the itervals where these agles lie. Let B i, i = 1, 2,..., be the corer poits. (b) Costruct a bipartite graph betwee the set of corer poits of the regular -go ad the set of sesors. A edge is liked betwee corer poit B k ad sesor A i if d(a i, B k ) λ (1 i, k ). The bipartite graph ca be obtaied from the steps 2 ad 3(a). (c) Check if there exists a perfect matchig. If it is so, termiate the process ad retur Feasible. 4. Retur Ifeasible. Algorithm 2 Step 1 Usig Algorithm 1, compute S 1 defied above. Step 2 For each i = 1,...,, fid the arc Qˆ Bi ( π 2 S1 ) ad compute a set of poits that partitios the arc ito 1 ɛ pieces of equal legth where ɛ = 2ɛ 14 π(π+1).

15 Step 3 Clearly, there are ( 1 ɛ + 1) poits i total. For each poit x, costruct a regular -go P x such that oe of the corers of P x is located at x, ad fid the optimal arragemet of the sesors (A 1,..., A ) to the vertices of the -go by solvig a weighted bipartite matchig problem. (The Hugaria method to solve the weighted matchig problem i a complete bipartite graph of size takes O( 3 ) time (see [10])). Step 4 Amog all ( 1 ɛ + 1) regular -gos thus costructed, fid the oe with the miimum cost (deoted by S 2 ) ad output the optimal arragemet of the sesors to the vertices of the -go. Experimetal Results o the complexity of the mi-sum problem It is ot kow whether the mi-sum problem ca be solved optimally. Two related problems which could help clarify the issue are the followig. 1. Give a couter-clockwise orderig of sesors o the perimeter of C, solve the mi-sum problem. 2. Fid the umber of differet couter-clockwise orderigs of sesors o the perimeter of C whe we sweep a regular -go alog the perimeter of C. Table 1 shows our experimetal results to resolve the secod problem described above. The Table 1: Experimetal result for the umber of differet orderigs (over 20 test sets) # of sesors () # of regular -gos (t) average # of orderigs 10 1, , , , , , , , , , first colum i the table represets the umber of sesors cotaied i C (the sesors are radomly geerated), the secod colum represets the umber t of regular -gos beig tested (i.e., the arc distace betwee two cosecutive regular -gos is 2π t ), ad the third colum represets the average umber of differet couter-clockwise orderigs of the sesors to the vertices of the t -gos (for each value of, 20 differet sets of sesors are geerated). From this table, we ca see that there are o more tha differet couter-clockwise orderigs i the experimet. Mi-sum algorithm o a simple polygo Step 1 Usig the approach for the mi-max problem o P, compute λ m,p described above. 15

16 x 1 A k B k x 2 Figure 4: Lemma 11, ˆd C (x 1, B k ) = ˆd C (x 2, B k ) = π 2 S1 d(x 1, x 2 ) 2S1 (Lemma 9). Step 2 For each i, j where 1 i ad 0 j <, fid the sub-edge e i,j of edge e j that is withi the circle of radius λ m,p cetered at positio A i, ad compute a set of poits that partitios the the sub-edge ito ɛ pieces of equal legth. Step 3 Clearly, there are m ( m2 ɛ +1) O( ɛ ) poits i total. For each poit x, costruct a set of positios o P such that oe of them is located at x ad the polygoal distace betwee ay two cosecutive positios is L(P )/, ad fid the optimal arragemet of the sesors (A 1,..., A ) to the set of positios by usig the algorithm [10]. Step 4 Amog all O( m2 ɛ ) cadidate sets of positios thus costructed, fid the oe with the miimum cost. It is evidet that the ruig time of the above approach is determied by the time eeded for solvig O( m2 ɛ ) weighted bipartite matchig problems. Proof. (Theorem 13) The reaso why the approximatio ratio of the above approach is bouded by 1 + ɛ, is as follows. Sice λ s,p λ m,p, there is at least oe sesor whose movig distace to its destiatio is o more tha λ m,p i a optimal solutio. Let A i be oe such sesor ad its destiatio positio lies o edge e j i that optimal solutio. I Step 2, the sub-edge e i,j is partitioed ito ɛ pieces, ad therefore, the legth of each piece is o more tha 2ɛλ m,p. Sice all possible values of i ad j are cosidered, the differece betwee the value computed by the above approach ad λ s,p (the optimal cost) is o more tha 1 2 2λ m,p = ɛλ m,p ɛλ s,p. 16