Autonomous Deployment for Load Balancing k-surface Coverage in Sensor Networks

Transcription

1 1 Autonomous Deployment for Load Balancng k-surface Coverage n Sensor Networks Feng L, Jun Luo, Wenpng Wang, and Yng He Abstract Although the problem of k-area coverage has been ntensvely nvestgated for dense wreless sensor networks (WSNs), how to arrve at a k-coverage sensor deployment that optmzes certan objectves n relatvely sparse WSNs stll faces both theoretcal and practcal dffcultes. Moreover, only a handful of centralzed algorthms have been proposed to elevate 2D area coverage to 3D surface coverage. In ths paper, we present a practcal algorthm APOLLO (Autonomous deployment for Load balancng k-surface coverage) to move sensor nodes toward k-surface coverage, amng at mnmzng the maxmum sensng range requred by the nodes. APOLLO enables purely autonomous node deployment as t only entals localzed computatons. We prove the termnaton of the algorthm, as well as the (local) optmalty of the output. We also show that our optmzaton objectve s closely related to other frequently consdered objectves for 2D area coverage. Therefore, our practcal algorthm desgn also contrbutes to the theoretcal understandng of the 2D k-area coverage problem. Fnally, we use extensve smulaton results to both confrm our theoretcal clams and demonstrate the effcacy of APOLLO. Index Terms Wreless sensor networks, autonomous deployment, k-area/surface coverage, load balancng. I. INTRODUCTION One of the major functons of wreless sensor networks (WSNs) s to montor a certan area n terms of whatever physcal quanttes demanded by applcatons [2]. In achevng ths goal, a basc requrement mposed onto WSNs s ther area/surface coverage: 1 t ndcates the montorng qualty of WSNs. Whereas many research proposals focus on ether analyzng the performance of statc sensor deployments (e.g., [5], [25]) or schedulng sensor actvty to retan the coverage of gven deployments (e.g., [33], [45]), there exsts an unfalng trend n seekng autonomous deployments asssted by moble sensor nodes to arrve at certan predefned objectves (e.g., [6], [34]). Our proposal n ths paper falls nto ths later trend. Due to the vulnerablty of sensor nodes, multple-coverage (k-coverage) s often appled to enhance the fault tolerance The prelmnary conference verson of ths paper has been publshed by IEEE ICDCS 2012 [28], where our focus was only on 2D k-area coverage. In ths paper, we generalze [28] to 3D k-surface coverage. Ths work s supported n part by AcRF Ter 2 Grant ARC15/11. Feng L, Jun Luo, and Yng He are wth School of Computer Engneerng, Nanyang Technologcal Unversty, Sngapore. E-mal: {fl3, junluo, yhe}@ntu.edu.sg Wenpng Wang s wth Department of Computer Scence, The Unversty of Hong Kong, Chna. E-mal: [email protected] 1 For WSNs deployed on 2D planes, we only focus on approaches concernng area coverage (e.g., [5], [25]), as opposed to the pont coverage (e.g., [8], [14], [44]). Surface coverage s an elevaton of area coverage from 2D planes to 3D surfaces (terrans), makng the deployment more useful n practce (e.g., for montorng volcanos). n face of node falures (e.g., [5], [45]). In addton, k- coverage may yeld hgher sensng accuracy through data fuson [13]. Exstng approaches n achevng k-coverage rely on ether randomzed (e.g., [15], [25]) or regular (e.g., [3], [5]) deployments. Whereas randomzed deployments requre a substantally denser network (e.g., [15], [25]), regular deployments serve only as theoretcal gudelnes [3], [5] as they often requre centralzed coordnatons and may not accommodate rregular network regons. Also, f the physcal phenomena under survellance change, the cost for re-deployment can be huge. Therefore, autonomous deployments, when movable nodes [10] are avalable, are good complements to the randomzed or regular deployments: they may acheve a densty comparable to that of regular deployments, whle beng more adaptve to rregularty and varatons of the network regons. However, exstng technques for autonomous deployments may only handle 1-coverage, and extendng them to k- coverage s hghly nontrval. Frst of all, autonomous deployments through (node) moton control requre each node to compute ts coverage n a localzed manner (.e., relyng as much as possble on close-by nodes). Although qute a few localzed algorthms have been proposed to perform such computatons for 1-coverage (e.g., [7], [34]), no algorthm, to the best of our knowledge, exsts for localzed k-coverage computatons. Secondly, even f the k-coverage computatons can be performed locally, there s no guarantee whether a moton control strategy may converge, due to the sgnfcant dfference between 1-coverage and k-coverage. Thrdly, exstng approaches are almost heurstcs that offer no provable guarantee on the qualty of the eventual deployment. Fnally, as ndcated n [43], extendng 2D area coverage to 3D surface coverage for WSNs deployed on 3D terrans may ncur new challenges even for 1-coverage. By far, only a handful of proposals employ (partally) centralzed algorthms to handle the deployment for 1-surface coverage [18], [43]; no pure autonomous deployment mechansm has ever been proposed. These are exactly the problems we want to tackle n our paper. In ths paper, we consder the problem of movng sensor nodes towards k-coverage. In partcular, we assume that n- odes have tunable sensng ranges and are randomly deployed ntally. Our goal s to cover a certan montored area or surface to the extent that every pont n ths area/surface s at least montored by k sensor nodes and that the maxmum sensng range used by the nodes s mnmzed. As a larger sensng range mples a larger energy consumpton of a node, our APOLLO (Autonomous deployment for Load balancng k-surface coverage) approach ams at balancng the sensng load (thus prolongng network lfetme) whle guaranteeng

2 2 k-coverage, wth the help of moble nodes. The man contrbutons we are makng n ths paper are: We desgn the APOLLO algorthm such that t executes n a localzed manner,.e., relyng only on nformaton from close-by nodes. We prove the termnaton of APOLLO as well as the (local) optmalty of ts output. We dscuss the relaton between APOLLO and other commonly used optmzaton objectves for 2D area coverage, whch provdes a better understandng of optmal k- coverage deployments whose theoretcal characterzatons are hard to obtan under general settngs. We show that, by usng geodesc dstance [9] to replace the conventonal Eucldean metrc, APOLLO can be naturally extended to handle autonomous deployments on 3D surfaces. Snce moton and communcaton cost cannot be neglected, our algorthm allows for a tradeoff between the two factors. Through extensve smulatons drven by realstc power consumpton data, we show that the exstng hardware platform can afford the energy consumpton of our algorthm n terms of moton and communcaton. To the best of our knowledge, we are the frst to tackle the problem of k-coverage autonomous deployment, for both 2D areas and 3D surfaces. The remanng of our paper s organzed as follows. We brefly survey the closely related lterature n Sec. II. We formally defne our model and problem n Sec. III, n whch we also revew the basc mathematcal tools we need n our later algorthm desgn. In Sec. IV, we present our APOLLO algorthm for 2D k-area coverage and analyze ts performance, we also nterpret our soluton wth respect to other optmzaton frameworks. We then extend APOLLO to handle 3D k- surface coverage n Sec. V. The effcacy of APOLLO s further confrmed by extensve smulaton results reported n Sec. VI. We fnally conclude our paper n Sec. VII. II. RELATED WORK Before surveyng the proposals related to area/surface coverage and moble asssted autonomous deployment, we frst revew another nterestng topcs, pont (or target) coverage and area coverage wth random deployments, from whch we may gan some hnts for our proposal. Pont coverage problem has been extensvely studed n the past decade. Besdes provdng coverage servce, ther another concern s the lmted energy supply of sensor nodes. Gven a random deployment wth statc sensor nodes, they ether dvde the nodes nto multple sets and schedule the dutes of these set (e.g. [8], [11], [44]), or mnmze the number of the actve sensor nodes (e.g., [14]), to guarantee the network lfetme. The energy lmtaton s also taken nto account n area coverage wth random deployments, e.g. [21], [25], [38], [41], [45]. Inspred by them, maxmzng the network lfetme s also one of the man objectves of our proposal but n a qute dfferent way. The statc and determnstc area coverage problem s essentally a geometry problem; the results for 1-coverage wth a mnmum number of nodes can be drectly taken from pure mathematcal research [22]. In later research proposals for WSN coverage, the focus s more on mnmum node 1-coverage wth certan connectvty requrement (e.g., [4], [5], [19]). Whle t s known that a 1-covered WSN s also connected f the transmsson range R t and the sensng range R s satsfy R t 3R s, a strp-based deployment strategy s proposed n [19] for other values of R t, whch s proven to be asymptotcally optmal n [4]. In fact, more strps allows hgher degree of connectvty (or k-connectvty). Compared wth k-connectvty, the progress on k-coverage appears to be relatvely slower. A few 3-coverage heurstcs that am at boundng the mnmum separaton among sensor nodes s proposed n [23]; the paper also shows that boundng the mn-separaton may lead to lower coverage redundancy and s hence a good approxmaton to mnmum node 3-coverage. To the best of our knowledge, the only optmalty result n terms of mnmum node k-coverage s presented n [5], where k = 2. It appears that mnmum node k-coverage (for k > 2) s better to be tackled ndrectly due to ts hardness. As we wll show n Sec. IV-C, our objectve of a k-coverage wth mnmax sensng range may also mply mnmum node k-coverage. Deployng WSNs for k-coverage usng moble nodes s also reported n [3], [36], but ther approaches are not autonomous as they all rely on a blueprnt to gude the node moblty. Consderng that a WSN can be deployed on a 3D terran for some applcaton scenaro, the surface coverage problem has also been nvestgated. Zhao et al. [24], [43] show that extendng just 1-area coverage to 1-surface coverage already poses great challenges, and they provde approxmaton algorthms to the resultng NP-complete problem. Recently, Jn et al. [18] apples Centrod Vorono Tessellaton (CVT) [12] and dscrete Rcc flow [17] to obtan some form of optmal surface 1-coverage by defnng sensng range upon geodesc dstance. However, the dscrete Rcc flow s a global parametrzaton process, so the moton control cannot be mplemented n a dstrbuted manner usng only local nformaton. Our work s also related to the sensng heterogenety ssue [6], [35]. Unlke the prevous proposals that am to cope wth sensng heterogenety or evaluate ts mpact, we actvely explot the sensng heterogenety to construct our algorthm that gudes the autonomous deployment. III. PROBLEM DEFINITION AND MATHEMATICAL BACKGROUND In ths secton, we frst present the system model and defne our optmzaton problem, and then ntroduce the relevant mathematcal bascs. To smplfy the exposton, the above dscussons are all for 2D plane wth Eucldean metrc. We then show that a 2D plane can be straghtforwardly generalzed to a 3D surface by replacng Eucldean metrc wth geodesc dstance, and we also ntroduce the mathematcal tools to perform ths map locally. A. System Model We assume a WSN consstng of a set N = {n 1,, n N } of sensor nodes, and N = N. Let U = {u 1,, u N } denote the locatons of sensor nodes. The nodes are ntally deployed arbtrarly on a 2D targeted area A. Each node n s equpped

3 3 wth certan mechansms (e.g., motors plus wheels) that allow t to gradually change ts locaton u [10]. We also suppose that nodes are equpped wth bumper sensors to detect and avod obstacles n the targeted area [30]. All nodes have an dentcal transmsson range γ, and we denote by N (n ) = {n j u u j 2 γ, j} the one-hop neghbors of n. We defne the omndrectonal sensng model as a dsk centered at u wth sensng range r. We assume the sensng ranges are adjustable accordng to dfferent applcaton requrements [42], [45]. A pont v A s sad to be covered by node n ff the Eucldean dstance between v and node locaton u s no longer than r,.e., v u 2 r. We use f(v, u, r ) to ndcate f v s covered by node n : f(v, u, r ) = 1 f v s covered by n ; otherwse, f(v, u, r ) = 0. We apply the MDS-based technque [30] that reles on the rangng ablty of each node to construct a local coordnate system for moton control, but we do not requre global locaton nformaton as t s mmateral to our algorthm. In terms of energy cost, we consder only the cost nduced by the sensng actvtes of a node. Because our network deployment strategy ams to acheve a constant (and long-term) coverage by movng sensor nodes n the ntal phase, the communcaton cost becomes neglgble as the data transmsson actvtes only take place sporadcally, whle the energy spent n movng s only a one-tme nvestment. We assume that the energy consumed by a sensor node n s an ncreasng functon E(r ) of ts sensng range r, and ths functon s dentcal for all nodes. In other words, wth a certan amount of energy E(r ), n can only cover the ponts {v A v u 2 r }. B. Problem Formulaton The node locatons and sensng ranges defne a network deployment wth a certan coverage. Defnton 1. A network deployment {u, r } s sad to acheve k-coverage ff for any pont v A, there exst at least k sensor nodes coverng t, or f(v, u, r ) k. To allow the sensor nodes k-cover the targeted area, we dvde A nto several dsjont areas {A k j } j=1,2,..., and at least k sensor nodes are allocated to take care of each area. In other words, each sensor node n takes care of multple subareas, and we ndcate ths coverng relaton by n (A k j ): t equals 1 f n covers A k j ; otherwse 0. We also denote by Ak n the area covered by n : we have A k n = n (A k j )=1 Ak j. The sensng range r of n s determned by the farthest pont n A k n from u,.e., r = max v A k n v u 2, so that A k n can be totally covered by n. Our k-coverage sensor deployment problem (k-csdp) can be formulated as follows. mnmze R (1) {u },{A k )} j },{n(ak j subject to N =1 f(v, u, r ) k, v A (2) v u 2 R,, v A k n (3) A k j 1 A k j2 =, j Ak j = A (4) Lterally, k-csdp ams at determnng the node locatons {u }, the area partton {A k j }, and the coverng relatons { n (A k j )}, such that the targeted area A s k-covered, whle the maxmum sensng range among all nodes s mnmzed. As energy consumpton s an ncreasng functon of sensng range, k-csdp s equvalently balancng the energy consumpton over a whole WSN and hence maxmzng the lfetme of the WSN. As the problem s generally not convex due to ts nonconvex feasble regon, we have to be contented wth local mnmum (.e., a locally mnmzed maxmum sensng range). C. Hgh Order Vorono Dagram We hereby brefly ntroduce the deas and theores on hgh order Vorono dagram [31]. They are key to our autonomous deployment strategy. In a k-order Vorono dagram, the targeted area A s segmented nto ˆN k dsjont areas {Vj k} j=1,..., ˆN, 2 each of k whch s assocated wth k closest generators (sensor nodes n our case),.e., a subset Nj k N wth Nj k = k. The k-order Vorono cell Vj k s defned as { Vj k = v A v u } 2 v u 2, n Nj k, n N /N j k (5) The set Nj k s called the generator set of Vj k. It s straghtforward to see that each sensor node n s assocated wth multple Vorono cells. Let Vn k denote the unon of the Vorono cells for whch n serves as a generator; we term Vn k the domnatng regon of n (hence n the domnator of Vn k ). We also have the followng proposton. Proposton 1. A pont v A s sad to belong to V k n ff there exst at most k 1 other generators such that ther dstance to v s less than v u 2. Proof: Assume v Vn k but there were another k nodes {n j } j J, J, J =k such that u j v 2 < u v 2. Then the pont v would strctly belong to the set of k-order Vorono cells generated by {n j }, whch does not nclude n as a generator; a contradcton. Conversely, f there are at most k 1 nodes, {n j } j J, J, J k 1, closer to v than n, we can fnd the set of v s k-nearest nodes (whch obvously contans {n j } {n }) to generate a set of k-order Vorono cells contanng v. As n s a generator, v Vn k. Based on the above proposton, assumng Sn k (v) = {n j N u j v 2 < u v 2, j }, we can re-defne the domnatng regon of n as V k n = {v A S k n (v) k 1} (6) We llustrate k-order Vorono partton (k = 1, 2, 3, 4) generated by 30 nodes n Fg. 1. The cells shown n each fgure are {Vj k }. Takng 2-order Vorono partton for example, as shown n Fg. 1(b), the area enclosed by red (resp. green) polygon s actually the domnatng regon of the red node (resp. green node). The hatched area s the Vorono cell generated by the two nodes,.e., the ponts n ths area are closer to the two nodes than any other nodes. 2 In 1-order Vorono dagram, the number of Vorono cells equals the number of generators (.e., ˆN 1 = N), whle n generalzed k-order Vorono dagram (k 1), ˆN k s O(k(N k)) [31].

4 4 (a) 1-order (c) 3-order (b) 2-order (d) 4-order Fg. 1. k-order Vorono partton for k = 1, 2, 3, 4. The dsks at the backdrop represent the (overlappng) sensng ranges of ndvdual sensor nodes. D. Generalzaton to 3D Surfaces For WSN deployed on a 3D surface, the conventonal Eucldean metrc s no longer approprate. Servng as the generalzaton of straght lne n curved space (e.g., 3D surfaces), geodesc s the shortest path between two gven ponts on the surface [9]. Therefore, geodesc dstance metrc s a natural choce for measurng dstance on a 3D surface. By replacng Eucldean metrc wth geodetc dstance, we may mgrate the aforementoned model and problem defntons drectly from 2D planes to 3D surfaces. 1) Problem Descrpton on 3D Surfaces: Consder the case where a WSN N s deployed on the 3D surface M, we use geodesc dstance as the dstance metrc. In partcular, we replace all the dstant metrc u v 2 used for 2D planes by g(u, v), the geodesc dstance between u and v. As a result, all the defntons n Sec. III-A to III-C can be drectly mgrated to 3D surface. For example, k-csdp and hgh order Vorono dagram can be redefned usng geodesc metrc. The only dfference here s that, whle A does not need an explct characterzaton, M s often represented by a trangular mesh that s gven to all nodes durng the ntalzaton phase. In order to compute geodesc dstance on M, we adopt the Improved Chen & Han s (ICH) algorthm [39]. The ICH algorthm handles the sngle source, all destnatons geodesc problem, amng at computng the geodesc from the source u M to any destnaton pont v M wthn a certan geodesc radus r from near to far wth a tme complexty of O(n 2 log n) where n s the number of vertces of the trangular mesh M wthn r. It mantans a prorty queue of geodesc wndows on the edges of the trangular mesh, and outputs a poly-lne geodesc path for each par of source and destnaton. The ICH algorthm also can be extended to other types of geodesc problem [40], e.g., sngle source, sngle destnaton and multple sources, any destnaton. Such a package of geodesc computaton tools s suffcent for our 3D extenson. 2) Logarthm and Exponental Maps: As we need a local coordnate system to compute k-order Vorono dagrams n a localzed manner for every node, we cannot rely on a parametrzaton method such as Rcc flow due to ts global nature and hgh computatonal cost. We nstead apply the ICH algorthm to compute an logarthm/exponental map [9] around a certan node, whch constructs a (local) geodesc polar coordnate system on the curved surface as follows. Gven a pont u M, we desgnate a local coordnate system on ts tangent plane T u centered at u. A logarthm map : M T u maps the ponts on M to T u. In partcular, for any other pont v M n a suffcently small neghborhood of u, there s a unque geodesc G u (v) extendng from u to v. As G u(v), the tangent of G u (v), s obvously tangent to M at u, G u(v) T u. Therefore, v can be mapped to T u wth geodesc polar coordnates {g(u, v), θ u,v } where g(u, v) s the geodesc dstance between u and v and θ u,v s the polar angle of G u(v) on T u. Consequently, the geodesc coordnates on M around u can be transformed to normal coordnates under any orthogonal bass {e 1, e 2 } wth orgn u. The nverse of logarthm map s called exponental map. We llustrate the log/exp map n Fg. 2. Here the ICH algorthm [39] s used exp 1 u (a) Coordnate system n T u (b) Coordnate system n M Fg. 2. The log/exp maps at pont u ntroduces a mappng between a vector orgnatng at u on T u and a geodesc begnnng at u on M. The crcles on T u are mapped to the contours of the geodesc functon on M. to trace the geodesc G u (v) such that any pont v n the neghborhood of u on M can get ts geodesc polar coordnates on T u. As mentoned above, the ICH algorthm mantans a wavefront and propagates outward from u, fnally outputtng a local geodesc coordnate system wthn a radus r. IV. APOLLO: LOCALIZED k-coverage NODE DEPLOYMENT ALGORITHM In ths secton, we frst develop two optmalty condtons for k-csdp. Then we present the APOLLO algorthm detals. The correctness of APOLLO s then proven, and we fnally dscuss the relaton between k-csdp and other optmzaton problems related to k-coverage deployment, along wth the correspondng propertes of APOLLO. A. Optmal Condtons To motvate our algorthm, we develop two optmalty condtons for k-csdp. Frstly, we show f we fx {u }, then k-order Vorono dagram s the optmal soluton to k-csdp. Proposton 2. If we fx the sensor locatons {u }, the k- order Vorono dagram {V k j } generated by {u } s an optmal

5 5 partton of A. Also, n (Vj k) = 1 f Vk j n (Vj k) = 0. V k n ; otherwse Proof: The proof s by contradcton. Suppose for fxed {u } =1,,N, there exsts an optmal soluton to k-csdp denoted by R, {Āk j }, { n (Āk j )}. Let r = max v u v Āk n 2 and r V = max v V k n v u 2. Also assume that the optmal value s obtaned for nî,.e., R = r. If î rv = r, then t î î s straghtforward to see that r V = max î {r V }, otherwse a contradcton to the defnton of Vn k : some regons are not covered by the k-closest nodes. Therefore, n ths case the k-order Vorono dagram s equally optmal. If r V > r, t î î means that nî could cede a certan regon to have t covered by other nodes whle reducng max {r V }. However, ths agan contradcts the defnton of Vn k : as nî s already one of the k-closest nodes that can cover the ceded regon, cedng ths regon to some other nodes would ncrease max {r V }. Before statng the second optmalty condton, we need the followng defnton. Defnton 2. Gven an arbtrary set S n Eucldean s- pace, the Chebyshev center u c s defned as u c = arg mnû (max u S u û 2 ) Gven an area partton {A k j } and ts domnator allocaton (or coverng relatons) { n (A k j )}, the optmal locatons of {n } can be obtaned accordng to the followng proposton. Proposton 3. If we fx the partton {A k j } and ts domnator allocaton { n (A k j )}, the optmal sensor locaton u for k- CSDP s gven by the Chebyshev center of A k n. Proof: As n needs to cover A k n and the objectve of k- CSDP s to mnmze the maxmum sensng range among all sensors, the optmal soluton under a fxed partton s acheved f each sensor ndvdually mnmzes ts own sensng range. Ths exactly concdes wth the property of Chebyshev center, hence the proposton follows. B. The Algorthm Gven the two optmalty condtons stated n Sec. IV-A, we mmedately have an teratve algorthm to solve k-csdp. The pseudo-codes of our APOLLO algorthm are presented n Algorthm 1. The algorthm proceeds n rounds. At the Algorthm 1: APOLLO Input: For each n N, ntal locaton u (0), stoppng tolerance ε Output: {u } and {r } 1 For every node n N perodcally (every τ ms): 2 Compute ts domnatng regon V k n 3 Compute the Chebyshev center c of V k n 4 f u c 2 > ϵ then 5 u + u + α(c u ) /*α s the step sze*/ 6 end 7 u c, r max u V k n u u 2 begnnng of each round, the k-order Vorono dagram s computed for the whole WSN, resultng n {V k 1,, V kˆn k } along wth {V k n 1,, V k n N } (Lne 2). Then each node computes the Chebyshev center of ts domnatng regon (Lne 3), and moves to that locaton to end ths round (Lne 4-6). The algorthm termnates f each node s ndeed located at the Chebyshev center of ts domnatng regon. As a perfect matchng s mpossble n face of numercal errors, we use a small value ε as the stoppng tolerance: the algorthm termnates f the dstance from the node s current locaton to the Chebyshev center of ts domnatng regon s smaller than ε. Also, n order to avod oscllaton, a step sze α < 1 s chosen to confne the moton of the nodes. At the termnaton, each node tunes ts sensng range to be the mnmum value (the crcumradus of ts domnatng regon) that covers ts domnatng regon. As a domnatng regon s a polygon, we apply Welzl s algorthm [37] to compute the Chebysehev center by takng the vertces of the regon as the nput. Smlar algorthms have been appled n [6], [34], but they were used to ndrectly optmze a dfferent objectve (see our dscussons n Sec. IV-C). Consequently, ther approaches do not abde by the optmalty condtons and employ a very dfferent termnaton condton. Therefore, ther termnaton proofs do not apply to our case even for k = 1. Most mportantly, as our algorthm deals wth a more general k- coverage, we are facng the followng new challenges n understandng our algorthm: 1) How to compute Vn k n a localzed manner wthout nvolvng all nodes? 2) Does the algorthm termnate for k 1? 3) What s the complexty of computatons? We tackle these challenges n the followng. 1) Localzed Algorthm for Computng Vn k : Unlke 1-order Vorono dagram that can be computed (mostly) by only nteractng wth one-hop neghbors N (n ) of a gven node n, N (n ) may not be suffcent to obtan k-order Vorono cells, especally when k s large. The reason s smple: at least k +1 nodes should be nvolved to compute a domnatng regon of n [26]. Therefore, we propose Algorthm 2 to locally calculate Vn k n an expandng rng manner. Algorthm 2: Localzed Vn k Computaton Input: For each n N, ntal rng radus ρ = 0 Output: Vn k 1 repeat 2 ρ ρ + γ; out true 3 N (n, ρ) {n j u j u 2 < ρ} 4 Construct a local coordnate system usng N (n, ρ) 5 foreach v A s.t. v n 2 = ρ/2 do 6 Ŝn k (v) {n j N (n, ρ) u j v 2 < u v 2, j } 7 f Ŝk n (v) < k then out false; break 8 end 9 untl out = true; 10 Compute Vn k based on N (n, ρ) Bascally, we expand the search rng ρ wth a granularty of the transmsson range γ (lne 2). As expendng ρ beyond γ wll need mult-hop communcaton and the hop number s always an nteger, t makes no sense to apply a smaller granularty. We use the embeddng algorthm proposed n [32] to construct a local coordnate system (lne 4). If the locaton

6 6 nformaton s avalable, ths step s not necessary. Under the constructed coordnate system, we check whether the crcle centered at u wth a radus ρ/2 s not domnated by n anymore (lnes 5 to 8, based on Proposton 1). Because the Vorono edges computed gven N (n, ρ) dvde the crcle wth radus ρ/2 nto a fnte number of arcs, each of whch ether fully domnated by n or not at all, we only need to check an arbtrary pont on each arc n the actual mplementaton. The rng expendng termnates f the answer becomes true. Fnally, we compute Vn k usng only nodes fallng nto the current search rng (lne 10). We need another lemma to show that Vn k computed by Algorthm 2 s ndeed the one that would be computed n a centralzed manner usng global nformaton. Lemma 1. If the domnatng regon of n s enclosed by a crcle centered at u wth a radus of ρ/2, then t s fully determned by all the nodes located wthn another crcle centered at u wth a radus of ρ. Proof: For a dsk (u, ρ/2) centered at u wth radus ρ/2, f Vn k (u, ρ/2), the boundary of Vn k also belongs to (u, ρ/2). Accordng to the defnton of Vorono cells, the cell boundary conssts of bsectors, each of whch s determned by two generators. For Vn k, one generator s n, and all other generators can be obtaned by gong through each segment (or bsectors) on the boundary of Vn k and dentfyng another generator that determnes ths bsector along wth n. Snce the boundary of Vn k belongs to (u, ρ/2), all generators of Vn k belong to (u, ρ). thus s qute sutable for our autonomous deployment. Based on the boundary awareness, the boundary node executes the crcle checkng procedure (lnes 5 to 8) only for the arc that les wthn the network coverage area, as shown n Fg. 3. Fnally, the boundary node calculates ts domnatng regon, usng N (n, ρ) as well as the searchng rng to determne the boundary of the domnatng regon. For an ntal random deployment n whch nodes only occupy a small fracton of A, ths procedure has the effect of pushng boundary nodes outwards, hence expandng the network coverage to the whole A. In fact, such a constraned checkng procedure should always be used by nodes on the boundary of A, the dfference s that A s boundary, known n advance to the nodes, serves as a natural boundary for a domnatng regon. 2) Termnaton Analyss: Showng the termnaton of Algorthm 1 appears to be hghly non-trval, as many k- order Vorono cells are concernng a certan node, and the domnatng regon of a node s mostly probably a non-convex regon. Fortunately, the termnaton can be shown by focusng on the boundary of a domnatng regon. Proposton 4. Algorthm 1 termnates for α (0, 1]. Proof: As shown n Fg. 5, u l and Vk,l n are the locaton and domnatng regon of node n at the begnnng of the l-th round, respectvely. We also denote by c l and Rl the Chebyshev center and the crcumradus of Vn k,l computed Fg. 5. Notatons n the proof for Proposton 4. Fg. 3. Recognzng the network boundary (the green dashed curve), a boundary node (dark) determnes a searchng rng (the outer crcle) and check the half-radus arc wthn the network coverage area (the nner red arc). It then calculates ts domnatng regon (the blue area) wth N (n, ρ), whle the searchng rng helps to determne part of the boundary. The correctness of our algorthm s mmedate from ths lemma: as the algorthm termnates when n s not domnatng the crcle centered at u wth a radus ρ/2 anymore, the nodes fallng nto (u, ρ) are suffcent to compute V k n. In Fg. 4, we demonstrate ths suffcency usng k-order domnatng regon (k = 1 to 12) n a regularly deployed WSN. The regular deployment s chosen to facltate exposton, our algorthm works for any arbtrary deployments. For a node n on the boundary, the search rng wll never stop expandng, as the arc that s out of the network coverage wll always need the domnaton of n. To cope wth ths ssue, we frst refer to our prevous work [29] for an on-lne boundary detecton servce. Sne each sensor node can dentfy f t s on the network boundary only relyng on local postons of t onehop neghbors, ths boundary detecton s hghly effcent and by n durng the l-round. Let ˆRl = max u V k,l u u l n 2 be the farthest dstance from u l n Vk,l n. Fnally, we defne R l = max {R l} and ˆR l = max { ˆR l}. We frst prove the termnaton for α = 1 by contradcton. For each n, we put a dsk (c l, Rl ) centered at c l wth radus R l N. Obvously, =1 (cl, Rl ) form a k-coverage over the targeted area A. The termnaton s naturally justfed f we can prove that Vn k,l+1 s nsde (c l, Rl ) after u l s updated to c l (.e., ul+1 = c l ). For each pont q on the boundary of Vn k,l+1, t s straghtforward to see that c l s the locaton of the k-th nearest nodes. Assume that q s outsde (c l, Rl ), there would be only k 1 dsks coverng q, whch contradcts the fact that N =1 (cl, Rl ) consttute a k-coverage over A. We then prove the termnaton for 0 < α < 1. Accordng to lne 5 of Algorthm 1, durng the l-th round, u l+1 = u l +α(cl ul ). We put dsks (ul, ˆR l ) and (c l, ˆR l ) centered at u l and cl, respectvely. Obvously, Vk,l n s nsde (u l, ˆR l ) (c l, ˆR l ), whch mples Vn k,l (u l+1, ˆR l ). N Hence, =1 (ul+1, ˆR l ) consttute a k-coverage over A.

7 7 (a) 1-order (b) 2-order (c) 3-order (d) 4-order (e) 5-order (f) 6-order (g) 7-order (h) 8-order () 9-order (j) 10-order (k) 11-order (l) 12-order Fg. 4. The domnatng regon of the central node n k-order Vorono dagram k = 1,, 12. The central node needs to collect locaton (or range) nformaton from ts neghborng nodes (the dark nodes) va mult-hop communcaton accordng to Algorthm 2. Addtonally, we llustrate mult-hop transmsson range usng red crcles n (a). Whle the cases for k = 1 can be handled by nvolvng only the 6 closest nodes (1-hop neghbors) to the central node, computng the 2-, 3- and 4-order domnatng regons requres 2-hop neghbors. When k > 4, all sensor nodes wthn 3 hops are nvolved. Followng a smlar argument as for α = 1, we have Vn k,l+1 s nvolved n (u l+1, ˆR l ), whch completes the proof. In summary, our APOLLO algorthm termnates for any α (0, 1]. Usually, smaller α leads to slower convergence but smoother movng locus. As a byproduct of the proof, we also conclude that ˆR s non-ncreasng teratvely and fnally equvalent to R. Especally, when α = 1, R s also nonncreasng n the teratve process. Therefore, Corollary 1. Algorthm 1 termnates at a local mnmum of k-csdp. It s mportant to note that R l and ˆR l are ntroduced only for our proof. Durng the algorthm executon, each node n can only compute ts own R l and ˆR l. Accordng to our earler dscusson n Sec. IV-B1 (see Fg. 3 also), the evoluton of the node postons often takes two phases: an expandng phase and a convergng phase. The expandng phase exsts f the ntal node dstrbuton s non-unform, our APOLLO algorthm wll force the node to spread out durng ths phase, as dscussed at the end of Sec. IV-B1. Durng ths phase, both R l and ˆR l are most probably acheved by boundary nodes. The expandng phase ends when all the boundary nodes are at the boundary of A, ths s when the convergng phase starts. 3) Computatonal Complexty Analyss: Each teraton of APOLLO conssts of two major steps: computng domnatng regons and Chebyshev centers. The domnatng regons are calculated n an expandng rng manner (see Algorthm 2). We suppose N (ρ) = N (n, ρ) s the number of the neghborng nodes of n wthn the searchng rng ρ. Accordng to [26], each sensor node computes local k-order Vorono edges (as well as vertces) wth a complexty of O(k 2 N (ρ) log N (ρ)). Recall that N (ρ) leads to O(k(N (ρ) k)) Vorono edges [31], the followng checkng operaton has a complexty of O(kN (ρ)(n (ρ) k)) n the worst case. The underlyng boundary detecton servce requres only one-hop neghbors, thus merely results n a neglgble cost of O(N (γ) log N (γ)) [29]. Assumng up to H-hop neghbors are requred (.e., ρ expands from γ to Hγ wth a granularty of γ), Algorthm 2 has a complexty of O(k 2 HN (Hγ) log N (Hγ) + khn (Hγ)(N (Hγ) k)). For certan coverage order k, the overall complexty of Algorthm 2 s hghly lmted by the number of requred communcaton hops H. Accordng to our experments shown n Fg. 4, even the transmsson range s restrcted (only avalable for computng 1-order Vorono domnatng regon), nodes wthn two or three hops are suffcent n most cases. Fnally, gven O(k(N (Hγ) k)) Vorono vertces outputted by Algorthm 2, we spend O(k(N (Hγ) k)) on computng ther Chebyshev center [37]. The total number of teratons of APOLLO depends on ndvdual cases, and thus cannot be derved by extng analyss technques. Smlar wth [6], [18], we wll employ extensve smulatons to reveal APOLLO s convergence rate as well as the nduced tme cost n Sec. VI. C. Dscussons In ths secton, we show the relatons between the output of our APOLLO algorthm and other optmzaton objectves often consdered for area coverage problems n WSNs. Mn-Node k-coverage: One type of problem that s commonly tackled n the research communty s to acheve k-coverage usng a mnmum number of nodes (e.g., [3], [5], [43]). As ths problem often assumes that all nodes have a fxed and dentcal sensng range r s, t appears that APOLLO may not suggest a drect soluton to t. However, we can transform our algorthm to delver a good approxmaton to ths mn-node k-coverage problem as follows. APOLLO s called teratvely 3 and R s compared wth r s at the end of each teraton. Nodes are added (resp. reduced) f R > r s (resp. R < r s ), untl R r s but addng one more node would make R > r s. Although the soluton may not be optmal, t yelds very good approxmaton to the optmal soluton, as we wll demonstrate n Sec. VI. If fact, as the 3 If an applcaton only requres a one-tme (rather than autonomous) deployment, we may use APOLLO n a centralzed fashon.

8 8 up-to-date algorthms are all approxmatons for k > 2 and they are not autonomous (e.g., [3]), our algorthm s also the frst autonomous deployment for approxmatng mn-node k- coverage wth an arbtrary k. Maxmum k-coverage: Another type of problem ams at maxmzng coverage under fxed sensng ranges, but exstng proposals only focus on 1-coverage [6], [18], [34]. A natural defnton of the general maxmum k coverage problem s to maxmze the area that s k-covered under a fxed sensng range. The major dfference between k = 1 and k > 1 s that the former acheves maxmum coverage f nodes are far apart from each other whereas the same prncple does not apply to the latter. An obvous example s the followng: assume only 3 nodes are used to 3-cover an area, the maxmum coverage s acheved only f all three nodes are put at the same locaton. Consequently, the heurstc of boundng the mnmum separaton among nodes [23] fals. Intutvely, APOLLO may delver a good approxmaton to the maxmum k-coverage problem, e.g., APOLLO termnates at the optmal soluton for the aforementoned 3-node case. Connectvty: Although mantanng network connectvty s not our concern n desgnng APOLLO, t appears to be a natural outcome of achevng k-coverage for k 2. Under k- coverage, t s easy to see that there should be at least k nodes n the sensng range r of a node n (ncludng n ), otherwse u s not k-covered. In realty, as shown n Fg. 4, there are at least 7 nodes n a certan sensng range for k 2. Gven the common assumpton n the lterature that γ r (e.g., [6], [34], [44]), each node n a WSN has at least a degree of 6, whch s suffcent to guarantee connectvty n the WSN. Mn-Max Far: Whle our k-csdp only requres that the maxmum sensng range s mnmzed and hence does not concern nodes wth non-maxmum sensng ranges, the mnmax far utlty (a Pareto optmal pont) requres that a node n cannot further reduce ts sensng ranges r wthout ncreasng the sensng range r j (r j r ) of another node n j. Accordng to the property of k-order Vorono dagram, the output of APOLLO s at least locally optmal wth respect to the mnmax far utlty,.e., f we reduce r, another node n j that shares domnatng regon boundary wth n should ncrease r j to mantan k-coverage, but we know r j r before r gets reduced. In fact, our smulaton results n Sec. VI show that, after APOLLO termnates, the maxmum and mnmum sensng ranges are almost the same for k > 2. V. APOLLO ON 3D SURFACES Though replacng Eucldean metrc by geodesc dstance yelds a straghtforward extenson of our problem from 2D planes to 3D surfaces (as we dscussed n Sec. III-D), our APOLLO algorthm has to be slghtly tuned to adapt to the local coordnate maps (.e., the log/exp maps). As APOLLO (Algorthm 1) nvolves two man computatons: domnatng regon and Chebyshev center, we present the APOLLO 3D extenson wth respect to these two separately. A. Computng Domnatng Regons By redefnng the k-order Vorono dagram based on geodesc dstance. Algorthm 2 could be extended to handle computatons on 3D surfaces, whle stll guaranteeng the localty of the computatons. After constructng a local coordnate system on the 3D surface (see Sec. III-D2), each node n expands ts searchng rng ρ wth a granularty of transmsson range γ, untl the geodesc dsk = {v M g(u, v) ρ/2} s not domnated by n anymore. n needs only to communcate wth N (n, ρ) = {n j g(u, u j ) ρ} f ts domnatng regon les n, accordng to Lemma 1. Ths extenson s pretty straghtforward; the only dfference s that, whle we compute u u j 2 on 2D planes, we use the ICH algorthm to obtan g(u, u j ) on 3D surfaces. A seemng dscrepancy here s that, whle the sensng range s mostly determned by Eucldean metrc, APOLLO operates on geodesc dstance. Fortunately, based on [16], t can be g(u,v) u v 2 derved that 1 β, where β s a constant determned by the geometrc propertes of a 3D surface (.e., ts maxmum Gaussan curvature). Consequently, the sensor nodes smply assgn the geodesc dstance g(u, v) (the upper bound of the Eucldean dstance) to ther Eucldean sensng ranges (.e., r = g(u, v)). Ths leads to a feasble soluton that does not compromse much of the optmalty. For brevty, we omt ths step n the later presentatons. B. Computng Chebyshev Centers After determnng the domnatng regon Vn k, the next step s for n to compute the Chebyshev center of ts domnatng regon. The problem s reduced to that, gven a set of ponts (.e., the vertces of Vn k n our case) on a surface, how to compute ther Chebyshev center. Unfortunately, compared wth ts 2D counterpart, computng Chebyshev centers on 3D surface (.e., under geodesc destance) appears to be hghly non-trval; t has not been addressed n the lterature to the best of our knowledge. We thus propose an teratve algorthm to calculate an approxmate Chebyshev center c M of Vk n usng log/exp map (see the pseudo-codes n Algorthm 3): the basc dea s to frst compute the mass center of the domnatng regon by teratvely applyng log/exp map (lnes 1 6), and then determne the Chebyshev center (lnes 7 8). Algorthm 3: Approxmatng a Chebyshev Center on a 3D Surface Input: For each n N, a local geodesc coord. system on the surface, the domnatng regon V k n, stoppng tolerance ε Output: c 1 Intalze the mass center of Vn k as ω u 2 repeat 3 Compute the logarthm map, exp 1 ω (Vn k ) T ω, of Vn k at ω 4 Compute the mass center ω T ω of exp 1 ω (Vn k ) 5 f ω 2 > ϵ then ω δ [ ] exp ω ( ω ) ω 6 untl ω 2 ϵ; 7 Ṽk n exp 1 ω (Vn k ); compute the Chebyshev center, c, of Ṽk n n the 2D tangent plane T ω 8 c exp ω ( c ) As the dffculty n computng the Chebyshev center s the local shape dstorton resultng from any 3D-to-2D map,

9 9 we want to fnd a log/exp map that yelds the smallest dstorton wthn V k n. Intutvely, the mass center of V k n, ω = arg mn u V k n g 2 (u, v)dv, may yeld a log/exp map that has the smallest shape dstorton. Therefore, we take Vn k to the tangent plane T ω by applyng the logarthm map to Vn k at ω, and we compute the Chebyshev center of exp 1 ω (Vn k ) (lne 7), and we fnally determne the Chebyshev center of Vn k usng the exponental map (lne 8). As the shape dstorton has been suppressed as far as possble, we beleve that c s a good approxmaton to the real Chebyshev center of Vn k. Here δ [ ] exp ω ( ω ) ω n lne 5 s computed wth respect to a geodesc from exp ω ( ω ) to ω on the 3D surface. The queston s whether replacng {c } by {c } n Algorthm 1 stll allows termnate. Fortunately, we have Proposton 5. Algorthm 1 termnates wth {c } f the step sze α s suffcently small. Proof: Let V on M be contaned n a geodesc ball π B(y, r) centered at y M wth radus r < 2 max{0,κ}, where κ s the maxmal Gaussan curvature of ponts nsde B(y, r). It s proven n [20] that the functon Φ(ω) = V g2 (ω, v)dv for ω V s convex and acheves a unque mnmum ω B(y, r). A smple computaton shows that 0 = Φ(ω ) = 2 V exp 1 ω (v)dv. In other words, the mass center of V s unquely defned and ndependent of the ntal value, hence the teratve computaton (lnes 1-6) converges to the mass center. Wth the exponental map at the mass center ω, a tangent plane T ω s constructed and the Chebyshev center c s computed on that plane. Ths ends one round for Algorthm 1. Suppose we take a suffcently small step sze α, the next round for Algorthm 1 wll be done on almost the same tangent plane T ω. Therefore, from a node n s pont of vew, the computatons nvolved n two consecutve rounds l and l+1 are done n 2D. So we can apply the provng method for Proposton 4 to show Ṽk,l+1 n (c l, Rl ). As the log/exp map s a bjecton wthn B(y, r), we have Vn k,l+1 (c l, Rl ), whch completes the proof. The computatons of Algorthm 3 are done by ndvdual nodes wth nether communcatons nor motons. Snce Φ(ω) has C 2 smoothness, the algorthm has a quadratc convergence rate, causng a neglgble computatonal cost. α < 1 almost always guarantees the overall convergence. VI. SIMULATIONS In ths secton, we report our smulaton results. We frst present the convergence of APOLLO. Studyng the energy consumptons durng and after the autonomous deployments, we also evaluate the performance of APOLLO n Mn- Node k-coverage and Maxmum k-coverage, followed by the adaptablty to network rregulartes. Fnally, we valdate the effectveness of APOLLO 3D extenson. A. Convergence As convergence results we obtan from our extensve experments are all smlar, we present only two case to demonstrate the convergence of our algorthm. We consder a targeted area of 1 km 2, and ntally deploy 100 sensor nodes ether at the bottom-left corner (see Fg. 6(a)), or separated nto two dsjont groups located at the bottom-left and upper-rght corners (see Fg. 6(f)). Accordng to the followng four subfgures for both cases, our algorthm apparently leads to an even node dstrbuton n the sense of multple coverage. Specfcally, n the multple coverage cases wth k = 2, 3, 4, nodes tend to cluster n groups of sze k, n contrast to the pure even dstrbuton for k = 1. Ths s not a surprse as such an even clusterng dstrbuton yelds more overlaps of the domnatng regons among every cluster, whch n turn reduces the requred sensng range. Interestngly, ths appears to also meet the needs of maxmum k-coverage. As we dscussed n Sec. IV-C, APOLLO leads to a co-locaton deployment for the extreme example of usng three nodes to acheve 3-coverage. The second case also shows that two dsconnected clusters wll eventually merge nto a concocted network. Our extensve smulatons show that the ntal deployment does not have sgnfcant mpact on the algorthm output. We show the convergence process of APOLLO n Fg. 7. Snce a sensor node fnally reaches the Chebyshev center of Fg. 7. Crcumradus (km) Coverage Max Crcumradus 1 Coverage Mn Crcumradus 2 Coverage Max Crcumradus 2 Coverage Mn Crcumradus 3 Coverage Max Crcumradus 3 Coverage Mn Crcumradus 4 Coverage Max Crcumradus 4 Coverage Mn Crcumradus # of rounds The convergence of APOLLO. ts domnatng regon and the sensng range s equvalent to the crcumradus of the domnatng regon, we llustrate the relatonshp between executon rounds (of length τ ms each) and the maxmum/mnmum crcumrad. As the nodes are deployed at the corner of the targeted area ntally, the maxmum crcumccle usually appears on the network boundary, whch s manly determned by the searchng rng (Fg. 3). Consequently, the maxmum crcumrad for k = 1, 2, 3, 4 are almost the same at the begnnng. Correspondng to our proof of termnaton, the maxmum crcumradus s monotoncally decreasng wth the executon rounds of APOLLO, whle the mnmum radus s ncreasng n general. In the end, the maxmum and mnmum rad are very close to each other, especally for larger k. Whle the monotonc decreasng n maxmum crcumradus shows the termnaton of APOLLO, the fact that the mnmum crcumradus concdes wth the maxmum one further confrms the balanced sensng load. B. Energy Consumpton durng Deployments In ths secton, we use TOSSIM [27] smulatons drven by realstc power consumpton data to evaluate the energy consumpton of the whole deployment process. We assume that a moble sensor node s equpped wth a Mcromo coreless D- C motor ( aspx). It moves a McaZ Mote n a speed of 0.2 m/s wth an energy consumpton of 120 mw. We get the communcaton cost data from the specfcaton of the popular CC2420 rado [1]: transmt power 52.5 mw and recevng (or dle-lstenng)

10 10 (a) Intal deployment I (b) 1-coverage I (c) 2-coverage I (d) 3-coverage I (e) 4-coverage I (f) Intal deployment II (g) 1-coverage II (h) 2-coverage II () 3-coverage II (j) 4-coverage II Fg. 6. Intal deployments and k-coverage (k = 1, 2, 3, 4) deployments as the output of APOLLO. Comm. consumpton (J) Combned consumpton (J) Net 1 Net 2 Net 3 Net 4 Net 5 Net Step sze α (a) Total communcaton consumpton x 10 4 Net 1 Net 2 Net 3 Net 4 Net 5 Net Step sze α Fg. 8. (c) Total combned consumpton Moton consumpton (J) Combned node consumpton (J) Net 1 Net Net 3 Net Net 5 Net Step sze α (b) Total moton consumpton Net 1 Net 2 Net 3 Net 4 Net 5 Net Step sze α (d) Node combned consumpton Energy consumpton durng the autonomous deployments power 56.4 mw. We assume that, durng the -th round, the rados are dsabled when nodes move, and the communcaton (and computaton) sesson starts only when nodes have moved to ther new locatons {u + } (see Algorthm 1). Based on the same scenaro studed n Sec. VI-A (.e., 100 nodes on 1 km 2 area), Fg. 8 demonstrates the actual energy consumpton of sx autonomous deployments. It s evdent that a smaller step sze α results n more rounds but shorter total movng dstances; ths s shown by a decreasng communcaton consumpton n Fg. 8(a) and an ncreasng moton consumpton n Fg. 8(b) as functons of α. Therefore, gven certan power consumpton specfcatons for moton and communcaton, we may tune α to obtan an energy effcent deployment. For our current settngs, the best step sze shown by Fg. 8(c) s around 0.3 to 0.7. We also pck up nodes that consume the hghest energy n each WSN to llustrate the energy consumed by ndvdual nodes n Fg. 8(d). To put these consumptons nto perspectve, a 2450mAh Energzer ( AA battery contans 13kJ, so the (maxmum) ndvdual node consumpton of 200J only accounts from a small part of the node s energy storage. We also report the tme cost of APOLLO n Fg. 9. As the tme cost stems from both communcaton and moton, the general trend s smlar to the total combned consumpton shown by Fg 8(c): the tme cost s mnmzed around α [0.3, 0.7] as well, for whch APOLLO termnates wthn around 25 mnutes and s reasonable for practcal applcatons. Fg. 9. Tme cost (s) Net 1 Net 2 Net 3 Net 4 Net 5 Net Step sze α Total tme cost. C. Energy Consumpton after Deployments In ths secton, we show the sensng energy consumpton after APOLLO completes the deployments. We agan consder Maxmum sensng load Coverage 2 Coverage 3 Coverage 4 Coverage # of nodes (a) Maxmum sensng load Total sensng load Coverage 2 Coverage 3 Coverage 4 Coverage # of nodes (b) Total sensng load Fg. 10. Energy consumpton n the fnal deployments of 100 nodes a targeted area of 1 km 2, whle scalng the network sze from 20 to 180. As data processng and memory accessng consume

11 11 most of power n sensng and ther frequency depends on the covered area, we model the energy consumpton n sensng to be proportonal to the area of the sensng dsk centered at the sensor node wth a radus r. In partcular, we defne the energy consumpton functon as E(r ) = πr 2 : an ncreasng functon of r We llustrate the sensng energy consumpton n terms of maxmum load max{e(r )} =1,...,N and total load N =1 E(r ) n Fg. 10. As we deploy more sensor nodes to a gven targeted area, each node takes care of less area when achevng a certan coverage. The maxmum sensng load s thus decreasng wth the ncreasng number of nodes. Gven a certan number of nodes, to acheve hgher coverage degree, each sensor node s supposed to cover larger area thereby enhancng the maxmum sensng load. We also observe that for k 1 -coverage and k 2 -coverage, the rato of maxmum loads between them s roughly k 1 /k 2, whch can be explaned as follows. Snce APOLLO makes the mnmum sensng range very close to the maxmum one, each sensor node roughly covers the same area k A /N,.e., E(r ) = k A /N where A s the area of the targeted regon. Nevertheless, ncreasng the number of nodes does decrease the total sensng load of the WSN, shown by Fg. 10 (b). Snce usng a less number of nodes mples a larger sensng dsk for each node, ths n turn yelds more overlap between sensng ranges (.e., a larger sensng redundance), thus a hgher total load. D. Comparsons wth Mn-Node k-coverage As mentoned n Sec. IV-C, our APOLLO algorthm results n a good approxmaton to mn-node k-coverage problem (where all nodes have the same sensng range and the objectve s to mnmze the number of nodes used to acheve k- coverage). In ths secton, we compare our algorthm wth the deployment strateges proposed n [5] and [3], n terms of the requred number of nodes guaranteeng k-coverage (k 2). As we can ncrease the mnmum sensng range to the maxmum one n the output of APOLLO wthout compromsng coverage, we assgn an dentcal sensng range to every node as the maxmum sensng range R n our case. Ba et al. have proven n [5] that, wthout consderng boundary effect and wth an dentcal sensng range r, the optmal congruent deployment densty 4 for 2-coverage s 4π/3 3. Gven a targeted area A, we thus compute the mnmum number of sensor nodes Nk=2 for 2-coverage as: Nk=2 4π A 3 = 3 πr = 4 A 2 3, here we use A to replace the 3R 2 area of Vorono polygons generated by sensor nodes, whch leads to an under-estmaton of Nk=2 due to the boundary effect. We smulate large-scale WSNs wth sze rangng from 1000 to 1600 n a 1 km 2 targeted area. The result s shown n Table I. In general, the number of nodes deployed by APOLLO s about 15% hgher than the mnmum value, and t s obvous that the boundary effect s the man reason for ths dfference. As the boundary effect s not consdered n [5], extra nodes are needed to cover the vacances on the boundary due to the msmatch between the congruent deployment and an arbtrarly 4 Deployment densty s defned as a rato of the area of sensng dsks to the area of Vorono polygons generated by sensor nodes [5]. shaped targeted area. Therefore, we conclude that APOLLO actually leads to a very good approxmaton of the mn-node 2-coverage problem. TABLE I THE MINIMUM NUMBER OF SENSOR NODES TO ACHIEVE 2-COVERAGE N R (m) Nk= In [3], Ammar et al. propose to decompose a targeted area nto adjacent Reuleaux trangles, and nodes are deployed n the ntersecton areas between these trangles (so-called lens 6k A n [3]). Accordng to ther dervaton, (4π 3 nodes are 3)r 2 requred to k-cover A where k 3 and r s the sensng range. Here we compare ths feasble deployment wth APOLLO. We deploy 180 nodes n a 1 km 2 area and let all nodes have the same sensng range Rk. We also compte the number of nodes that deployed accordng to the strategy proposed n [3] 6k (4π 3 3)Rk 2 as Nk =. From the results shown n Table II, t s very clear that APOLLO can use much less nodes to acheve the same level of coverage compared wth [3]. TABLE II THE NUMBER OF SENSOR NODES TO ACHIEVE k-coverage WITH THE STRATEGY PROPOSED IN [3] FOR k = 3, 4,..., 8 R k k (m) Nk E. Performance n Maxmum k-coverage In addton to the analyss n Sec. IV-C, we evaluate the performance of APOLLO n solvng maxmum k-coverage problem. Due to space lmtaton, we only llustrate the results of 4-coverage. We assume sensor nodes have fxed sensng ranges of 135m and are deployed n a square targeted regon of 1 km 2. We also vary the network scale from 80 to 110 wth a step of 10. The 4-coverage rato (.e., the rato between the 4-covered area and the whole targeted regon) n each round s demonstrated n Fg. 11. It s shown the area 4-covered by the sensor nodes s ncreased wth the executon of APOLLO, and reaches maxmum when APOLLO converges. Snce the sensng ranges of the sensor nodes are fxed, a network wth s- mall scale may not be able to fully 4-cover the targeted regon, e.g., 80, 90 or 100 nodes n our case. By gradually deployng more sensor nodes (e.g., 110 nodes), we can have the whole targeted area 4-covered. Recall that N = 110 sensor nodes wth a fxed sensng range of r = 135m can 4-cover (hence k = 4) an area of up to N(4π (3))r 2 6k = 0.9km 2. Therefore, we beleve that, APOLLO provdes a good approxmaton to the maxmum k-coverage problem. F. Adaptablty to Obstacles We demonstrate the autonomous adaptablty of APOLLO to arbtrarly shaped targeted area (wth obstacles nsde) n

12 12 (a) Intal deployment I (b) 2-coverage I (c) 4-coverage I (d) 6-coverage I (e) 8-coverage I (f) Intal deployment II (g) 2-coverage II (h) 4-coverage II () 6-coverage II (j) 8-coverage II Fg. 12. Adaptablty of APOLLO to arbtrarly shaped areas and obstacles as well. Fg coverage rato nodes 90 nodes 100 nodes 110 nodes # of rounds 4-coverage rato. Fg. 12. The holes wthn the network regon represent obstacles that moble sensor nodes cannot move upon. Obvously, APOLLO adapts well to these rregulartes and agan acheves the even clusterng dstrbuton as f the area were regular. G. Extenson to 3D Surfaces We apply the 3D APOLLO extenson dscussed n Sec. V for WSN deployments on varous 3D terran surface models. The three terran models are approxmated by 5k, 20k and 130k trangles, respectvely. In Fg. 13, each row shows one terran where we deploy the sensor nodes. Snce a large-scale sensor network are usually ar-dropped n the applcatons of terran montorng, we ntally deploy 400, 400 and 800 moble sensor nodes for these three terrans respectvely n a random manner. Fg. 13 shows the outputs also reflect clusterng dstrbutons n the multple coverage cases wth k = 2, 3, 4. The smlarty between Fg. 13 and Fg. 6 clearly demonstrates that our deployment algorthm desgned for 2D deployments has been successfully extended to handle 3D surface deployments. Consderng space lmt, we hereby omt the llustraton of the convergence process of APOLLO n 3D deployment, as t s very smlar to ts 2D counterpart. In Table III, we use the maxmum and mnmum (Eucldean) sensng ranges resulted from the autonomous deployments to show the qualty (n terms of load balancng) of the coverage. In order to make the numbers comparable to each other, we normalze the three surfaces such that ther 2D projectons are all on a 1 km 2 area. A drecton nterpretaton, TABLE III THE MAXIMUM AND MINIMUM SENSING RANGES FOR THREE SURFACE DEPLOYMENTS Surface 1 Surface 2 Surface 3 k Max range Mn range by comparng Table III wth Fg. 7, the dfferences between the maxmum sensng ranges and the mnmum sensng ranges n 3D deployments are a lttle larger than the ones delvered by 2D deployments. In another word, qualty of APOLLOs output n terms of load balancng s worse n 3D than n 2D. Ths s expectable as the problem becomes sgnfcantly harder to handle on 3D surfaces. However, the results n Table III do not ndcate a worse performance of APOLLO n terms of solvng the k-coverage optmzaton problem, because t has never been proven that an optmal k-coverage soluton (for k > 2) for ether 2D planes or 3D surfaces can be or have to be acheved by dsks wth an dentcal radus. It s hghly possble that, on 3D surfaces, an optmal k-coverage soluton ndeed accommodates varable rad. Another reason s that, we employ an approxmated Chebyshev centers n 3D APOLLO whch may lead to compromse n terms of load balancng. VII. CONCLUSION In ths paper, we have focused on mnmzng the maxmum sensng range to acheve load balancng k-coverage through autonomous deployments (.e., relyng on moble sensors n- odes and the wreless communcatons among them). We have

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Senor Networks, 5(1):8:1 8:36, Feng L receved hs BS degree n Computer Scence from Shandong Normal Unversty, Chna, n 2007, and the MS degree n Computer Scence from Shandong Unversty, Chna, n He s currently a PhD student at School of Computer Engneerng, Nanyang Technologcal Unversty, Sngapore. Hs research nterests are computatonal geometry and ts applcaton n wreless sensor networks. Jun Luo receved hs BS and MS degrees n Electrcal Engneerng from Tsnghua Unversty, Chna, and the PhD degree n Computer Scence from EPFL (Swss Federal Insttute of Technology n Lausanne), Lausanne, Swtzerland. From 2006 to 2008, he has worked as a post-doctoral research fellow n the Department of Electrcal and Computer Engneerng, Unversty of Waterloo, Waterloo, Canada. In 2008, he joned the faculty of the School of Computer Engneerng, Nanyang Technologcal Unversty n Sngapore, where he s currently an assstant professor. Hs research nterests nclude wreless networkng, moble and pervasve computng, appled operatons research, as well as network securty. More nformaton can be found at Wenpng Wang s a professor of computer scence at the Unversty of Hong Kong. Hs research covers computer graphcs, vsualzaton, and geometrc computng. He has recently focused on mesh generaton and surface modelng for archtectural desgn. He s journal assocate edtor of Computer Aded Geometrc Desgn (CAGD), Computers and Graphcs (CAG), IEEE Transactons on Vsualzaton and Computer Graphcs (TVCG, ), Computer Graphcs Forum (CGF), and IEEE Computer Graphcs and Applcatons (CG&A). He has been the program char of several nternatonal conferences, ncludng Pacfc Graphcs 2003, ACM Symposum on Physcal and Sold Modelng (SPM 2006), Internatonal Conference on Shape Modelng (SMI 2009), and the conference char of Pacfc Graphcs 2012, SIAM Conference on Geometrc and Physcal Modelng 2013 (GD/SPM13), and SIGGRAPH Asa 2013 Yng He receved the BS and MS degrees n Electrcal Engneerng from Tsnghua Unversty, Chna, and the PhD degree n Computer Scence from the State Unversty of New York (SUNY), Stony Brook, USA. He s currently an assocate professor at the School of Computer Engneerng, Nanyang Technologcal Unversty, Sngapore. Hs research nterests are n the broad areas of vsual computng, wth a focus on the problems that requre geometrc computaton and analyss. More nformaton can be found at