Achieving Crossed Strong Barrier Coverage in Wireless Sensor Network

Transcription

1 sensors Artcle Achevng Crossed Strong Barrer Coverage n Wreless Sensor Network Rusong Han 1 ID, We Yang 1, * and L Zhang 2 1 School of Electronc and Informaton Engneerng, Bejng Jaotong Unversty, Bejng , Chna; hanrusong@bjtu.edu.cn 2 School of Electronc and Electrcal Engneerng, Unversty of Leeds, Leeds LS2 9DX, UK; l.x.zhang@leeds.ac.uk * Correspondence: wyang@bjtu.edu.cn; Tel.: Receved: 24 January 2018; Accepted: 5 February 2018; Publshed: 10 February 2018 Abstract: Barrer coverage has been wdely used to detect ntrusons n wreless sensor networks (WSNs). It can fulfll the montorng task whle extendng the lfetme of the network. Though barrer coverage n WSNs has been ntensvely studed n recent years, prevous research faled to consder the problem of ntruson n transversal drectons. If an ntruder knows the deployment confguraton of sensor nodes, then there s a hgh probablty that t may traverse the whole target regon from partcular drectons, wthout beng detected. In ths paper, we ntroduce the concept of crossed barrer coverage that can overcome ths defect. We prove that the problem of fndng the maxmum number of crossed barrers s NP-hard and nteger lnear programmng (ILP) s used to formulate the optmzaton problem. The branch-and-bound algorthm s adopted to determne the maxmum number of crossed barrers. In addton, we also propose a mult-round shortest path algorthm (MSPA) to solve the optmzaton problem, whch works heurstcally to guarantee effcency whle mantanng near-optmal solutons. Several conventonal algorthms for fndng the maxmum number of dsjont strong barrers are also modfed to solve the crossed barrer problem and for the purpose of comparson. Extensve smulaton studes demonstrate the effectveness of MSPA. Keywords: barrer coverage; crossed; wreless sensor networks; branch and bound; shortest path 1. Introducton Wreless sensor networks (WSNs), whch have the outstandng advantages of easy confguraton, flexblty n shrnkng or expandng, strong fault-tolerance, and moblty, have played an mportant role n montorng and analyzng dynamc, hostle, unfamlar, and unexplored envronments. In some montorng and early warnng applcatons, such as border protecton, battlefeld survellance, and anmal mgraton observaton, the prmary objectve s to detect the ntruders who penetrate the target regons wth sensor nodes. In WSNs, a seres of sensor nodes, whose sensng regons overlap, form a sensor barrer for ntruders and can guarantee the detecton of penetratng behavor n partcular drectons. Thus, desgnng strateges for formng sensor barrers has great mportance and s often referred to as the barrer coverage problem [1 4]. In contrast to the full coverage presented n [5,6], whch requres detectng ntruders at every pont n ther trajectory, barrer coverage only ensures that the ntruson behavor s detected. The man concerns of the two coverage problems n coverage ntensty and the movement of montorng targets are dfferent. Thus, wth a lower number of sensor nodes, barrer coverage stll acheves a satsfactory level of ntruder detecton. Bascally, barrer coverage can be classfed nto weak barrer coverage and strong barrer coverage [1]. In weak barrer coverage, the horzontal projectons of sensng regons overlap, only guaranteeng to detect movements along vertcal traversng paths, as llustrated by the dash lnes n Sensors 2018, 18, 534; do: /s

2 Sensors 2018, 18, of 17 Fgure 1a. Meanwhle, as shown n the same fgure, f an ntruder knows where the sensor nodes are, t may adopt a polygonal path, ndcated by the sold lne, wthout beng detected. In contrast, strong barrer coverage, whch provdes contnuous coverage, ensures that every ntruson s detected snce any crossng path needs to traverse a barrer. As shown n Fgure 1b, despte followng a polygonal path, the ntruder can be detected by the strong barrer on the top. (a) (b) (c) (d) Fgure 1. Varous types of barrer coverage n wreless sensor networks (WSNs): (a) Weak barrer coverage. (b) Strong barrer coverage. (c) Hybrd barrer coverage. (d) Barrer coverage n drectonal sensor networks (DSNs). Addtonally, based on the characterstcs of barrers, barrer coverage can be classfed n other ways: local barrer coverage and global barrer coverage, 1-barrers and k-barrers, coverage wth moble sensors and coverage wth statonary sensors, etc. As llustrated by Fgure 1c, f a sensor wth moblty (ndcated by the dark crcle) s used and assgned to move to the poston ndcated by the dash lnes, t would form a 2-barrer coverage n the hybrd WSN, whch employs both moble and statonary sensors. Moreover, wth the rapd development of drectonal sensor networks (DSNs), barrer coverage wth drectonal sensng models (shown n Fgure 1d) has attracted a great deal of nterest. Unfortunately, although strong barrer coverage can detect ntruders who traverse the target regon vertcally, a securty vulnerablty exsts snce a WSN cannot handle the traverse ntruson. For the applcaton of observng anmal mgraton or battlefelds, t s also of great mportance to detect traversng targets snce t helps us to grasp the moton law of ntruders and develop strateges to enhance the securty level of a system. When usng prevous research results to desgn and confgure a sensor barrer, people are ncapable of detectng ntruson behavor n both horzontal and vertcal drectons. Thus, we need to fnd a seres of sensor nodes, whose sensng regons overlap and can form two contnuous paths that lnk the boundares of the horzontal and vertcal drectons. In ths paper, that seres of sensor nodes s referred to as a crossed strong barrer (crossed barrer for short). In bref, crossed barrer coverage can ncrease the possblty of detectng ntruson and thus mprove system securty. Thus, we ntroduce the concept of crossed barrer coverage, whch conssts n the detecton of both horzontal and vertcal ntrusons, to tackle the challenges mentoned above. Crossed barrer coverage apples to more complex stuatons where the conventonal barrer cannot satsfy the securty requrements of a system. Moreover, as argued n [1,7], schedulng sensor nodes to work n barrers leads to both savngs n node numbers and extensons n network lfetme. The savngs are acheved snce redundant sensor nodes can go nto sleep mode or be removed. Thus, we attempt to determne the

3 Sensors 2018, 18, of 17 maxmum number of crossed barrers n order to extend the lfetme of a WSN. Note that maxmzng the number of crossed barrer s not merely the superposton of fndng and addng up the maxmum barrer n ndvdual drectons. We demonstrate t through analyss and smulatons n the followng sectons. The man contrbutons of our work are descrbed as follows: 1. To the best of our knowledge, we are the frst to ntroduce and study crossed barrer coverage n WSNs. We also show that the problem of fndng the maxmum number of crossed strong barrers s NP-hard. 2. We provde an nteger lnear programmng (ILP) formulaton to better descrbe the optmzaton problem of fndng the maxmum number of crossed barrers, and we use the branch-and-bound algorthm to obtan the optmal soluton, whch wll serve as a benchmark for other algorthms. 3. We propose a heurstc algorthm called a mult-round shortest path algorthm (MSPA) to fnd the maxmum number of crossed barrers. The MSPA can acheve near-optmal solutons n polynomal tme. 4. We modfed the algorthms used n the conventonal barrer coverage problem to make them sutable for the new problem and conducted extensve smulatons to evaluate ther performance. The remander of ths paper s organzed as follows. We provde a bref dscusson about related lterature and work n Secton 2. Secton 3 ntroduces the system model and the problem statement. In Secton 4, we present the ILP formulaton for the optmzaton problem and solve t usng the branch-and-bound algorthm. In Secton 5, we present our MSPA method and several modfed algorthms. Smulaton work and numercal results are presented n Secton 6, and we conclude the paper n Secton Related Work The concept of barrer coverage was orgnally proposed n the context of robotcs sensors [8] and frst ntroduced nto WSNs n [1]. Snce then, actve research has been carred out n ths area n the followng aspects. Note that we summarzed these research works not only to provde the prelmnary knowledge about barrer coverage, but also to ndcate further research drectons for the crossed barrer coverage problem. Sensng model: In conventonal studes of the barrer coverage problem [9 14], the Boolean dsc sensng model [15], whch s expressed by an omn-drectonal crcle, s wdely used. In ths model, a target nsde or outsde the sensng range of a sensor node s detected by the sensor wth probablty one or zero. Although t s smple for analyss, the model cannot fully characterze the sensor measurements, whch are usually affected by nose and vary wth the dstance between the sensor and the target. Thus, the work n [16 18] assumed the probablstc sensng model. Moreover, state-of-the-art research work consders more practcal ssues such as the utlzaton of the three-dmensonal (3D) sensng model n camera sensor networks [19] and the jont probablty model n the compound event barrer coverage problem [20]. However, we adopt the Boolean dsc sensng model n our research snce we are the frst to study the crossed barrer coverage problem. Deployment method: There are two man approaches to barrer coverage n WSN: random deployment and determnstc deployment [21]. The former nvolves schedulng randomly dstrbuted sensor nodes n a WSN to buld sensor barrers, whle the latter ams at globally optmzng the locatons of sensors to mnmze the total number of sensors, whle ensurng the performance of barrer coverage. For determnstc deployment, based on the geometrc shape, the curve-based deployment and lne-based deployment are respectvely presented n [21 24]. We analyzed the scenaro wth random deployment snce t s more common for large-scale WSNs. Coverage ntensty: Coverage ntensty s measured by the barrer number, the contnuty of barrers, and the probablty of detectng a target. Based on the coverage ntensty that a WSN provdes, barrer coverage s categorzed n the followng ways: strong barrer coverage and weak barrer coverage, 1-barrers and k-barrers, and worst- and best-case coverage and exposure path coverage [15].

4 Sensors 2018, 18, of 17 Here, strong barrer coverage s studed, and barrer number s adopted as the measurement of coverage ntensty. Sensor moblty: Deployng moble sensors n a WSN can be extremely valuable n hostle envronments and has recently attracted great nterest. Barrer coverage wth the consderaton of sensor moblty [25 29] ams at effcently mprovng barrer coverage under the constrants of the avalable moble sensors and ther movng range. For example, n [28,29], the authors tred to effcently form barrer coverage by leveragng moble sensors to fll n the gaps between pre-deployed statonary sensors, whle ensurng that the number or the total movng cost of moble sensors s at a mnmum. Nonetheless, sensor moblty s not consdered n ths paper snce we am at solvng the barrer coverage problem by makng the most of statonary sensors. Network lfetme: The ntal am of ntroducng barrer coverage was to mantan a certan ntruder-detectng capacty and extend the lfetme of the whole WSN. In [7], a wakeup schedule for ndvdual nodes was proposed to maxmze network lfetme, utlzng the redundancy of sensors. In [30], the authors focus on the effect of the lfetme of sensor nodes and prove that the heterogenety of sensor lfetme affects the barrer number. Thus, to acheve more effcent barrer coverage n practce, lfetme ssues should be consdered n future research. As mentoned n the prevous secton, the traversal ntruson s a securty vulnerablty for strong barrer coverage. Based on the above-mentoned aspects, t can be seen as a coverage ntensty problem. Meanwhle, maxmzng the number of crossed barrers helps to extend the lfetme of the WSN. The followng lterature provdes a rudmentary knowledge of the work related to ours. In [1], Kumar et al. demonstrated how to use a centralzed method to fnd dsjont barrers. The core dea s to use the maxmum flow algorthm n graph theory, and ths algorthm s wdely used n research on the strong barrer coverage problem. In [31], the authors present heurstc ways to elmnate strong barrers that have conflcts, but the barrer coverage problem they solved s lmted to the horzontal drecton. In [13], whch nspres our research most, Lu et al. derved crtcal condtons for strong barrer coverage and devsed an effcent dstrbuted algorthm to construct dsjont barrers. In ther algorthm, they constructed vertcal barrers n vertcal strps to connect the horzontal barrers n the adjacent segments. Vertcal barrers help to prevent ntruders from movng between segments, resultng n a more robust network. However, the authors dd not provde further detals on how to choose a sngle barrer by combnng several local horzontal barrers. In [32,33], the authors proposed the concepts of an event-drven partal barrer and a renforced barrer. These two knds of barrers can detect movements n varous drectons, whch enhances the montorng capablty of the conventonal sensor barrer. The deas of these barrers resemble ours n that we both ntend to strengthen barrer coverage by buldng complex barrers wth more than one par of start and end ponts. Nevertheless, n ths paper, we utlze and evaluate more methods of selectng approprate sensor nodes n order to buld the maxmum number of crossed barrers. Thus, nhertng the dea of usng the barrers n both horzontal and vertcal drectons to renforce the robustness of a network, we consder ntroducng crossed barrers as a substtute for conventonal barrers to strengthen the montorng capablty of a sngle barrer. We wll further explan the detaled models and problem formulatons n the followng sectons. 3. Models and Problem Statement We consder a WSN wth n omn-drectonal sensors S = {s 1, s 2,..., s n } randomly deployed to montor a rectangular regon B wth length L x and wdth L y. As shown n Fgure 2, we represent a senor node wth a sector denoted by a quadruple s = P, R, a, V, where P s the coordnate of the sensor n a two-dmensonal montorng plane; R s the maxmum sensng radus and a s the sensng offset angle; V s a unt vector, representng the sensng orentaton of the node. The shaded sector shown n Fgure 2 represents the feld of vew (FoV) of a sensor node. In addton, the Boolean sensng model [23] s used. In ths model, all the space

5 Sensors 2018, 18, of 17 ponts wthn the FoV can be detected wth possblty one and can be sad to be covered by the sensor. The ponts outsde the FoV have no possblty of beng detected and cannot be sad to be covered by the sensor. Y V O a a Px (, y) R X Fgure 2. The quadruple sensng model n a WSN. The conventonal omn-drectonal sensng model s a specal case of the quadruple sensng model when a = and V s an arbtrary unt vector. Note that we adopt ths model snce t can also be used n applcatons of DSNs where the drectonal sensor s modeled; however, n the followng analyss and smulatons, we contnue to use the omn-drectonal sensng model for smplcty. Thus, we ntroduce some defntons here to better descrbe the problem. Defnton 1. Strong barrer. A strong barrer s a contnuous horzontal or vertcal path that conssts of a seres of sensor nodes whose sensng regons overlap wth adjacent ones and guarantee detecton of ntrudng paths n both vertcal and horzontal drectons. Defnton 2. Crossed barrer. A crossed barrer conssts of both a vertcal strong barrer and a horzontal one that do not share a common sensor, ensurng detecton of any ntrudng path n both vertcal and horzontal drectons. Defnton 3. Drectonal coverage graph G(V,E). A drectonal coverage graph of a wreless sensor network S s constructed as follows: G represents the graph made up by the sets of vertexes V and drected edges E. The set V conssts of vertexes correspondng to the sensors n the network. In addton, V has four vrtual nodes, s1, t1, s2 and t2, that correspond to the left, rght, upper, and lower boundares, respectvely. A drected edge exsts between two nodes f ther sensng regons overlap n the deployment regon B. The drecton of the edge s from the node that s closer to s1 (or s2) to the node closer to t1 (or t2). An edge exsts between a vertex and a partcular vrtual node f the sensng regon of the correspondng sensor overlaps wth a boundary of the regon. An llustraton of crossed barrer coverage and a drectonal coverage graph s presented n Fgure 3. As shown n Fgure 3a, the horzontal strong barrer, whch connects the leftmost border s1 and the rghtmost border t1 of the survellance regon, and the vertcal one, whch connects s2 and t2, together consttute a crossed barrer. In Fgure 3b, a drectonal coverage graph s gven to model the network shown n Fgure 3a. The dots represent sensor nodes or boundares, whle the drected edges wth arrows show the overlappng relatonshp between sensor nodes. Snce we always buld barrers from partcular sdes of a network for convenence, we choose s1 and s2 as the start ponts of buldng horzontal and vertcal barrers. The drecton of an edge ndcates the relatve order of two connected nodes. The graph n the context of graph theory provdes a clear descrpton of network topology.

6 Sensors 2018, 18, of 17 t2 t2 s1 t1 s1 t1 s2 (a) s2 (b) Fgure 3. An llustratons of a crossed barrer and a drectonal coverage graph. (a) A crossed barrer. (b) The drectonal coverage graph showng the overlappng relaton between nodes. In the coverage graph of Fgure 3b, f we can fnd two paths that lnk (s1, t1) and (s2, t2), respectvely, and are dsjont wth respect to each other, we can easly obtan a crossed barrer n Fgure 3a. In fact, [7,14] provde a method for fndng node-dsjont paths. However, snce there are two pars of vrtual nodes n our problem, the prevous method cannot be drectly appled. The man target of our research s to fnd the maxmum number of crossed strong barrers, snce schedulng the sensor barrers to work alternately can effcently prolong network lfetme. Thus, we need to fnd more pars of dsjont paths n the correspondng drectonal coverage graph. The followng theorem s presented to analyze the computatonal complexty. Theorem 1. The problem of fndng the maxmum number of crossed barrers s NP-hard. Proof of Theorem 1. By defnton, a crossed barrer s made up of two barrers that do not have common sensors and le n perpendcular drectons. From the drectonal coverage graph, we need to fgure out the maxmum pars of dsjont horzontal and vertcal paths. Thus, we turn the problem nto a flow problem n graph theory to solve t. We construct a new auxlary graph to assst the proof. Every vertex v (except for the four vrtual nodes) n the drectonal graph s dvded nto two sub-vertexes v n and v out. The nward edges go nto v n and the outward ones come from v out. There are nner edges lnkng the sub-vertexes wth capacty set to one. The capacty of other edges s also set to one. The transformed drectonal coverage graph s equvalent to the prevous one snce every vertex should only be used once, as we have mentoned. Assumng that the maxmum number of crossed barrers s R c Z +, then the decson verson of the problem can turn nto the drected two-commodty ntegral flow problem n [34] wth R 1 = R 2 = R c, where R 1 and R 2 are the flow from s 1 to t 1 and from s 2 to t 2. Snce the equvalent drected two-commodty ntegral flow problem s NP-complete, then the optmzaton verson of our problem, whch s fndng the maxmum number of crossed barrers, s NP-hard. 4. ILP Formulaton & the Branch-and-Bound Method In ths secton, we present an nteger lnear programmng formulaton whose soluton wll serve as a benchmark, and we use the branch-and-bound algorthm to solve t ILP Formulaton Based on the drectonal coverage graph, we develop an ILP formulaton for our problem. In the followng formulaton, fe hor and fe ver represent the flow on edge e, where subscrpts hor and ver ndcate that a flow s a part of the horzontal or vertcal paths; Ev n j and Ev out j stand for the sets of nwards and outward edges of vertex v j n V.

7 Sensors 2018, 18, of 17 s.t. max mn ( e E n t1 fe hor, fe ver ) (1) e Et2 n ( fe hor + fe ver ) = ( fe hor e Ev n e j Ev out j + f ver e ), v j V\{s1, t1, s2, t2} (2) ( fe hor + fe ver ) 1, v j V\{s1, t1, s2, t2} (3) e Ev n j ( f hor e + f ver e ) 1, e E (4) fe hor, fe ver = {0, 1}. (5) To be specfc, Equaton (1) concerns fndng the maxmum number of crossed barrers. Snce a crossed barrer consttutes barrers n two drectons, the optmzaton target s maxmzng the mnmum value of the barrer numbers n horzontal and vertcal drectons, rather than maxmzng the barrer number n a sngle drecton. Equaton (2) s a flow conservaton equaton, whch arses from the fact that the flows on the nward edges of a vertex (except for s1, s2, t1 and t2) ncreases equally wth those on the outward edges. Furthermore, wth the requrement of gettng dsjont paths n the drectonal coverage graph, Equaton (3) guarantees that the maxmum tme a sensor node can be used are always bounded by one. Fnally, Equatons (4) and (5) ensures that the maxmum tme an edge can be used are less than one. However, t seems that Equaton (1) s not n the standard form of ILP, whch means we cannot use general methods to analyze and solve t. Thus, some further work s done here to transform object functon nto a standard one. We ntroduce an nteger f c Z +, whch satsfes the followng nequatons: f c f c Then, Equaton (1) can be replaced by the followng functon: fe hor (6) e Et1 n fe ver. (7) e Et2 n max f c. (8) Here, f c can be seen as the mnmum value of the flows n two drectons. Therefore, Equaton (1) s replaced by Equaton (8) wth two extra condtons, Equatons (6) and (7), added to guarantee that f c s not bgger than the flow value n any drecton The Branch-and-Bound Method Snce fndng the maxmum number of crossed barrers s NP-hard, there s no algorthm that can obtan the optmal solutons n polynomal tme. We have the ILP formulaton for the problem, so a smple dea s to use brute-force enumeraton, whch nvolves systematcally enumeratng all possble solutons and checks whether each canddate satsfes the problem s statement. However, the exponental growth of effort requred to examne all possble canddates s unacceptable for a WSN wth many nodes. Thus, we consder solvng t usng the branch-and-bound algorthm [35,36], whch uses selectve enumeraton rather than complete enumeraton.

8 Sensors 2018, 18, of 17 For the branch-and-bound algorthm, there are two essental components: branchng and boundng. Branchng s the operaton that dvdes a problem nto two or more subproblems such that the soluton of the orgnal problem can be acheved from the solutons of the subproblems. For the above ILP, a smple branchng operaton s to pck a varable ( fe ver or fe hor ) and to replace the current problem wth two subproblems. Thus, both subproblems wll be copes of the current problem, where the varable ( fe ver or fe hor ) s set to 0 n one copy and set to 1 n the other. Snce the varable ( fe ver or ) has to take ether 0 or 1 n an optmal soluton, the branchng scheme guarantees that an optmal f hor e soluton of the orgnal problem wll be an optmal soluton of one of the two subproblems. Boundng s a functon that returns a bound on the optmal soluton of the current subproblem. For the maxmzaton problem, the returned value wll be a number larger or equal to an optmal soluton. For our problem, a smple choce s to remove the ntegralty constrants on the varables by replacng them wth lower and upper bounds on the varables, and to transform the subproblem to a lnear programmng (LP) problem. Ths operaton s called the LP relaxaton of the subproblem, whch s qute easy to solve. Snce the feasble solutons of the ILP subproblem are all feasble for the LP problem, the optmal soluton of the former wll always be worse or equal to the optmal soluton of the latter. Moreover, an addtonal use of the bound ganed s that t s possble to dscard subproblems that have a bound value worse than the value of the best currently known soluton of the orgnal problem. Although the worst-case complexty of the branch-and-bound algorthm s stll an exponental tme algorthm, t can speed up the calculaton for average cases. 5. Proposed Algorthms In ths secton, we present a mult-round shortest path algorthm (MSPA), whch works heurstcally to guarantee effcency whle mantanng good solutons. Heurstc methods are wdely used to solve dfferent coverage problems, e.g., barrer coverage n [31] and target coverage n [37], to mprove calculaton effcency. In addton, for the purpose of comparson, some conventonal algorthms of fndng maxmum strong barrers are also modfed here to adjust to the new problem The Mult-Round Shortest Path Algorthm (MSPA) As mentoned before, the horzontal strong barrer should be vertex-dsjont or sensor-dsjont wth a vertcal one to consttute a crossed barrer. Thus, we propose choosng the sub-barrer alternately and removng nodes on a sub-barrer from the drectonal coverage graph G(V, E) once the sub-barrer s chosen. The pseudo code of MSPA s presented n Algorthm 1. In the pseudo code, Djkstra(G, vertex1, vertex2) represents the functon to calculate the shortest path from vertex1 to vertex2 n G usng Djkstra s algorthm [38], and Remove(G, vertex_set) s a functon whch removes the vertexes and edges related to the vertexes n vertex_set. Snce the functon Remove() s executed after a sngle-drecton path s found, t ensures that all sub-barrers are node-dsjont wth each other. Djkstra s algorthm can fnd the shortest paths between nodes n a graph. In our algorthm, the length of a path s counted by the hop counts of sensor nodes on t. Snce a path wth the least hop counts may avod usng extra sensor nodes, we choose the horzontal and vertcal barrers usng the metrc of hop counts. The computatonal complexty of Djkstra s algorthm s O( V log( V ) + E ), where V and E are the number of vertexes and edges, respectvely. Snce V = n and E = n 2 at most, the worst-case tme complexty of the MSPA s O(n 2 ), whch s sgnfcantly lower than that of the branch-and-bound algorthm.

9 Sensors 2018, 18, of 17 Algorthm 1: The Mult-Round Shortest Path Algorthm Input: The drectonal coverage graph G of a WSN; Output: The maxmum number of crossed barrers N max ; The set of nodes on the th crossed barrer Q ; 1 0; N max 0; Q hor ; Q ver ; 2 whle True do 3 f there exst a path from s1 to t1 on G(V, E) then 4 Q hor Djkstra(G,s1,t1); 5 G Remove(G, Q hor ); 6 else 7 break; 8 end 9 f there exst a path from s2 to t2 on G(V, E) then 10 Q ver Djkstra(G,s2,t2); 11 G Remove(G, Q ver ); 12 else 13 break; 14 end 15 N max N max + 1; ; Q Q hor Q ver ; 17 end 5.2. Max-Flow-Least-Conflcts Algorthm and Max-Flow-Least-Counts Algorthm The maxmum flow algorthm such as the Edmonds Karp algorthm [39] can calculate the maxmum flow of a network n polynomal tme, and t s chosen n many lteratures to fgure out the maxmum node dsjont barrers. Takng advantage of the maxmum flow algorthm, we present the max-flow-least-conflcts algorthm and the max-flow-least-counts algorthm n ths secton. As an auxlary graph s constructed when proofng Theorem 1, we can obtan the maxmum numbers of dsjont strong barrers on horzontal and vertcal drectons ndvdually by usng the maxmum flow algorthms to the auxlary graph. Moreover, the horzontal (or vertcal) strong barrers are node-dsjont wth each other snce the capactes for all the nner edges n the auxlary graph are set to 1, whch restrcts the maxmum tme that sensor nodes can be used. For the max-flow-least-conflcts algorthm, we present a step-by-step heurstc method to combne the horzontal and vertcal barrers to make crossed barrers. The pseudo code s presented n Algorthm 2. In each step, a horzontal (or vertcal) path, whch conflcts wth the fewest number of the other paths n the vertcal (or horzontal) drecton, s selected, and the conflctng paths and the path selected are removed from the auxlary graph. In Lnes 6 and 17 of the pseudo-code, M s a matrx showng the conflctng relatons between horzontal flows and vertcal flows. The columns represent the horzontal flows, whle the rows represent the vertcal flows. An element n the matrx M s set to 1 when the correspondng flows n the two drectons have common nodes. By searchng the matrx, we can quckly determne whether a path conflcts wth others. The path selecton procedure s executed alternately between horzontal and vertcal drectons, and a par of these operatons leads to a crossed barrer.

10 Sensors 2018, 18, of 17 Algorthm 2: The Max-Flow-Least-Conflcts Algorthm Input: The drectonal coverage graph G of a WSN; Output: The maxmum number of crossed strong barrers N max ; The set of nodes on the th crossed strong barrer Q ; 1 Construct an auxlary graph G of G; 2 Calculate the sets of horzontal flows F hor through Maxflow(G, s1, t1), and the sets of vertcal flows F ver through Maxflow(G, s2, t2); 3 0; N max 0; Q hor ; Q ver ; 4 whle True do 5 f F hor = then 6 Calculate the conflctng matrx M, and pck a flow f hor n F hor wth least conflcts n M; 7 Q hor f hor ; F con the flows that conflct wth f hor 8 f F con then 9 Remove f hor from F hor ; 10 else 11 Remove f hor from F hor and F con from F ver ; 12 end 13 else 14 break; 15 end 16 f F ver = then 17 Update the conflctng matrx M, and pck a flow f ver n F ver wth least conflcts; 18 N max N max + 1; + 1; Q ver f ver ; Q Q hor Q ver ; 19 F con the flows that conflct wth f ver ; 20 f F con then 21 Remove f ver from F ver ; 22 else 23 Remove f ver from F ver and F con from F hor ; 24 end 25 else 26 break; 27 end 28 end The basc dea of the max-flow-least-counts algorthm s smlar to that of the max-flow-least-conflcts algorthm. However, the path selecton procedure s dfferent. In Lnes 6 and 17, the max-flow-least-counts algorthm wll choose the flows wth the least hop counts. The pseudo-code for the ths algorthm s omtted to save space. For a network wth n sensor nodes, there are 2n + 4 nodes n ther auxlary graph. The computatonal complexty of usng the Edmonds-Karp algorthm s O( V E 2 ), where V and E are the number of vertexes and edges, respectvely. Thus, the worst-case computatonal complexty of gettng the maxmum flows by usng the Edmonds-Karp algorthm s O(n 3 ). For the operaton n calculatng the conflctng matrx, the worst-case complexty s O((2n + 4) ((2n + 4) 1)/2) = O(n 2 ). Thus, the worst-case computatonal complextes for both the max-flow-least-conflcts algorthm and the max-flow-least-counts algorthm are O(n 3 )-slghtly hgher than that of the MSPA. Overall, the dea of choosng the flows wth the least conflcts or counts s smlar to the work n [31]. However, we have modfed ther rules of choosng vertex-dsjont paths wth a sngle ;

11 Sensors 2018, 18, of 17 source snk par so that the new ones can ft n wth the crossed barrer problem, where paths from two path groups that have dfferent sources and snks need to be chosen. Algorthm 3: The Max-Flow-Maxmum-Independent-Set Algorthm Input: The drectonal coverage graph G of a WSN; Output: The maxmum number of crossed strong barrers N max ; The set of crossed strong barrer Q; 1 Construct an auxlary graph G of G; 2 Calculate the sets of horzontal flows F hor through Maxflow(G, s1, t1), and the sets of vertcal flows F ver through Maxflow(G, s2, t2); 3 Calculate the conflctng matrx M for F hor and F ver ; 4 Combne the non-conflct barrers n F hor and F ver to obtan the set of potental crossed barrers F pon ; 5 Construct an auxlary graph G for F pon ; 6 Q MaxISalgorthm(G ); 7 N max Q ; 5.3. The Max-Flow-Maxmum-Independent-Set Algorthm In the two algorthms above (Algorthms 2 and 3), barrers from the sets of horzontal and vertcal barrers are chosen to buld a crossed barrer on the bass of the number of conflcts or hop counts. In ths secton, we consder selectng proper sub-barrers by ntroducng the concept of a maxmum ndependent set. In graph theory, an ndependent set (IS) s a set of vertexes n a graph where no two vertexes are adjacent. To be specfc, t s a set S v of vertexes such that, for every two vertexes n S v, there s no edge connectng them. A maxmal ndependent set (MIS) s an ndependent set that s not a subset of any other ndependent set. In other words, there s no vertex outsde the ndependent set that may be added, snce t s maxmal wth respect to the ndependent set property. A maxmum ndependent set (MaxIS) s a MIS wth maxmum cardnalty n a graph. Snce we can obtan the maxmum number of node-dsjont barrers n horzontal and vertcal drectons, we consder combnng the non-conflct barrers to make potental crossed barrers and then utlze the algorthms of calculatng the MaxIS to obtan a set of crossed barrers that has maxmum cardnalty and no conflct. We present a smple example n Fgure 4 to llustrate how the max-flow-maxmum-ndependent-set algorthm works. Horzontal barrers f f1 2 f 14 f Vertcal barrers f3 4 f 13 Crossed barrers f 24 (a) (b) Fgure 4. The llustratons of the max-flow-maxmum-ndependent-set algorthm. (a) The barrers n ndvdual drectons. (b) The maxmum ndependent set (MaxIS) of the graph s calculated to obtan the maxmum crossed barrers.

12 Sensors 2018, 18, of 17 In Fgure 4, f 1 and f 2 are the node-dsjont barrers n the horzontal drecton, whch s calculated by the maxmum flow algorthm, whle f 3 and f 4 are those n the vertcal drecton. If f 2 conflcts wth f 3, whch means that f 2 has one or more common nodes wth f 3, we can obtan the relatons shown n Fgure 4a, where the edges n the graph ndcate that the barrers they have lnked can be combned together to consttute a crossed barrer. Thus, we can obtan three potental crossed barrers: f 13, f 14 and f 24. An auxlary graph as shown n Fgure 4b s then constructed n whch the vertexes represent potental crossed barrers and the edges ndcate that there are conflcts between vertexes. For our example, snce f 14 shares the barrers wth f 13 and f 24, by utlzng a heurstc algorthm [40], we can obtan the maxmum ndependent set of the graph, whch ncludes f 13 and f 24. Consequently, we can obtan the crossed barrers f 13 and f 24, and the maxmum number of crossed barrers s two. The core purpose of ntroducng the MaxIS s to model the conflctng relatons n real world applcatons n a vrtual graph and then use heurstc algorthms to handle t. In an auxlary graph, two conflctng barrers can be mapped to two vertexes wth an edge lnkng them. Thus, when usng the algorthms for MaxIS, the two vertexes cannot be n an ndependent set at the same tme wth respect to the ndependent set property. We present the pseudo-code for the max-flow-maxmum-ndependent-set algorthm n Algorthm 3. In Step 6, MaxISalgorthm(G ) s a functon to obtan the maxmum ndependent set by usng the heurstc algorthm called the vertex support algorthm (VSA) [40]. The tme complexty for Steps 1 to 2 s O(n 3 ). For Steps 3, 4, and 5, the worst-case tme complexty s O(n 2 ). Therefore, the tme complexty of the max-flow-maxmum-ndependent-set algorthm s O(n 3 ). The worst-case computatonal complextes for t are not as good as that of MSPA but stll outperform those of the branch-and-bound algorthm. Furthermore, the connectvty ssue of sensor nodes s already taken nto account n all these three algorthms, snce, when barrers are bult, adjacent nodes need to able to communcate and have overlapped sensng areas wth each other. 6. Evaluaton In ths secton, we evaluate the performance of the proposed algorthms va extensve smulatons n terms of the number of crossed barrers. The purpose of the smulatons s twofold: to prove the effectveness of the MSPA n comparson wth other algorthms when consderng both tme complexty and soluton accuracy, and to evaluate the effect of mportant parameters (such as the sze of target regon, the number of sensor nodes, the sensng radus, and the sensng offset angle) on the crossed barrer number. The followng smulaton experments were performed on a laptop computer equpped wth an Intel(R) Core(TM) MQ processor, an 8-GB memory, and the 64-bt Wndows 10 operatng system. The smulaton codes are based on the Matlab language. Assume that, for defnteness and wthout loss of generalty, omn-drectonal sensors are unformly deployed n a square regon wth dmensons of m. A ddactc example s ncluded n Scenaro 1 to llustrate the algorthms. In Scenaro 1, R = 40 m and n = 120. As shown n Fgure 5, the deployment map shows the locatons of sensor nodes n the square regon. The areas n dark blue ndcate that they are well montored or covered by sensor nodes and that some sensors may be redundant. In order to acheve crossed barrers n the network, branch-and-bound, MSPA, and other proposed algorthms are executed to compare the maxmum number of crossed barrers they can obtan. Frst, one of the crossed barrers calculated by the MSPA s llustrated n Fgure 5. The sensor nodes on the barrer, represented by the green crcles, form two contnuous strong barrers lnkng two pars of boundares that can further consttute a crossed barrer. Thus, our MSPA can dentfy crossed barrers correctly from the network.

13 Sensors 2018, 18, of 17 Fgure 5. Scenaro 1: The deployment map of a WSN and a crossed barrer calculated by the MSPA. Then, we present the crossed barrer numbers calculated by dfferent algorthms. The branch-and-bound algorthm (B&B for short) can obtan 10 crossed barrers, whch s optmal among these algorthms. For the MSPA, the max-flow-least-conflcts algorthm (least-conflcts, for short), the max-flow-least-counts algorthm (least-counts for short), and the max-flow-maxmum-ndependent-set algorthm (MaxIS for short), the maxmum numbers of crossed barrers are 6, 4, 4, and 5, respectvely. Although B&B outperforms MSPA n the number of crossed barrers, the MSPA stll acheves a good soluton, wth the worst-case tme complexty beng kept at a reasonable level. For a network wth a substantal number of sensor nodes, tme complexty needs to be taken nto consderaton. Usng heurstc algorthms are mportant for balancng the precson of solutons and the computaton complexty. In the followng scenaros, we manly conduct smulatons to evaluate the effects of node number (n), sensng radus (R), and sensng offset angle (a) on the maxmum number of crossed barrers by settng them to dfferent values. All smulatons are done n two confguratons, where the szes of the target regon are set to m and m, respectvely, and the results are the statstcal average of 100 smulatons. In Scenaro 2, R = 20 m and n ranges from 50 to 350, wth 50 as the ncrement. As can be observed n Fgure 6, the crossed barrer numbers of all the algorthms ncrease monotoncally and lnearly wth the ncrease n node number. Ths s because deployng more sensor nodes n a fxed-area regon leads to a hgher densty for sensor nodes, whch n turn helps to enhance the connectvty of sensor nodes and provdes more paths for constructng crossed barrers. Addtonally, the number of crossed barrers n Fgure 6b s hgher than that n Fgure 6a wth the same node number, whch s also related to the densty of the sensor nodes, rather than the sze of the regon. B&B provdes optmal solutons that wll serve as benchmarks for other algorthms. The MSPA outperforms other proposed algorthms but acheves a sub-optmal soluton. Consderng, however, that B&B s an enumeraton method whle MSPA s a heurstc one, the performance of the MSPA s acceptable. Furthermore, the MSPA has better performance than the other three algorthms n whch the maxmum flow algorthm s used to calculate the sub-barrers. The MSPA dentfes more crossed barrers wth lower computatonal complexty. Thus, the operaton of decdng the sub-barrers frst may lmt the maxmum number of crossed barrers. In addton, the ndependent-set performs better than the least-conflcts and the least-counts, snce, after determnng the maxmum flows, the ndependent-set provdes more combnatons of sub-barrers. Least-counts yelds the worst results snce the hop counts may be too smple to be a good ndcator of whch path should be chosen.

14 Sensors 2018, 18, of C r o s s e d b a r r e r n u m b e r B & B M S P A L e a s t-c o n flc t L e a s t-c o u n t In d e p e n d e n t-s e t C r o s s e d b a r r e r n u m b e r B & B M S P A L e a s t-c o n flc t L e a s t-c o u n t In d e p e n d e n t-s e t N o d e n u m b e r (n ) (a ) N o d e n u m b e r (n ) (b ) Fgure 6. Scenaro 2: The effect of node number on crossed barrer number: (a) regon sze: m; (b) regon sze: m. In Scenaro 3, n = 200 and the sensng radus R s changed from 10 to 35 m wth 5 m as the ncrement. Observng Fgure 7, we can obtan the smlar monotoncally ncreasng trend as that n Scenaro 2. However, n Fgure 7b, the solutons of the MSPA, the least-conflcts, and the least-counts become closer as sensng radus ncreases. The reason for ths phenomenon s that there are lmtatons to the solutons of all fve algorthms when the sensng radus reaches a relatvely hgh value. Consderng an extreme case n whch R reaches a value such that all the sensors can cover the whole regon, then there s no use of ncreasng the maxmum number of crossed barrers when ncreasng R. The solutons of all algorthms wll approach ther lmt when R s hgh enough. Addtonally, the MSPA has a better performance than other algorthms. 5 0 C r o s s e d b a r r e r n u m b e r B & B M S P A L e a s t-c o n flc t L e a s t-c o u n t In d e p e n d e n t-s e t C r o s s e d b a r r e r n u m b e r B & B M S P A L e a s t-c o n flc t L e a s t-c o u n t In d e p e n d e n t-s e t S e n s n g r a d u s ( R ) (a ) S e n s n g r a d u s ( R ) (b ) Fgure 7. Scenaro 3: The effect of sensng radus on crossed barrer number: (a) regon sze: m; (b) regon sze: m. In Scenaro 4, we consder changng the sensng offset angle a to evaluate ts effect. Snce there are many applcatons related to utlzng a drectonal sensng model nstead of an omn-drectonal one, the smulatons may attest to the effectveness of algorthms usng dfferent sensng models. In ths scenaro, n = 200, R = 20 m, and a s changed from to 0 wth a decrement of 1/6. A random unt vector s assgned to each sensor node as the sensng orentaton. As shown n Fgure 8, all algorthms have a downward trend when decreasng the sensng offset angle. The reason for ths s that, wth the lmtaton n the sensng offset angle, the area that a sensor node can cover becomes

15 Sensors 2018, 18, of 17 smaller, whch consequently leads to a lower chance for sensor nodes to have an overlapped sensng area and form barrers. We have noted that, n Fgure 8a, when a s close to or 0, the slopes of the curves are relatvely gentle. The probablty of two sensor nodes to have overlapped sensng regons becomes ether too hgh or too low when a s close to and 0. Thus, the curves n the fgure have dfferent slopes. In Fgure 8b, the slope between 1/6 and 0 s stll substantal. The reason les n the fact that a greater densty of sensor nodes may slow down convergency. When deployng drectonal sensors to acheve crossed barrers, we may need to ncrease the number of sensor nodes to acheve an adequate number of crossed barrers C r o s s e d b a r r e r n u m b e r 1 5 A t-c o n flc t t-c o u n t p e n d e n t-s e t B & B M S P L e a s L e a s In d e 2 5 C r o s s e d b a r r e r n u m b e r B & B M S P L e a s L e a s In d e A t-c o n flc t t-c o u n t p e n d e n t-s e t / 6 2 / 3 1 / 2 1 / 3 S e n s n g o ffs e t a n g le ( a ) (a ) 1 / / 6 2 / 3 1 / 2 1 / 3 S e n s n g o ffs e t a n g le ( a ) (b ) 1 / 6 0 Fgure 8. Scenaro 4: The effect of sensng offset angle on crossed barrer number: (a) regon sze: m; (b) regon sze: m. Through the smulatons above, we evaluated the effect of some mportant parameters on crossed barrer coverage. We found that all parameters have a drect effect on the coverage ntensty of the entre regon, whch wll further affect the maxmum number of crossed barrers that can be found. 7. Conclusons We heren studed the problem of achevng the maxmum number of crossed barrers n a wreless sensor network. We frst defned a crossed barrer, whch can overcome the defect of traversal ntruson, and presented the problem of maxmzng the number of crossed barrer. Then, we proved that the problem s NP-hard n computaton complexty. We further developed the ILP formulaton for the optmzaton problem and used the branch-and-bound algorthm to obtan an optmal soluton. Moreover, we proposed an effcent MSPA algorthm along wth some other heurstc algorthms. Through theoretcal analyss and smulatons, we found that the MSPA outperforms other algorthms n both the worst-case computatonal complexty and the maxmum number of crossed barrers acheved. Acknowledgments: Ths work was supported by the Key Projects of Natonal Key Research and Development Program (2016YFC ), the Natural Scence Foundaton of Chna under Grant ( ), and the Natonal Scence&Technology Pllar Program of Chna (2013BAK06B03). Author Contrbutons: Rusong Han proposed the dea of ths paper and conducted the theoretcal analyss and smulaton work. We Yang and L Zhang proposed many useful suggestons n the process of theoretcal analyss. Rusong Han wrote the paper, and L Zhang revsed t. Conflcts of Interest: The authors declare no conflct of nterest.

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