Relay Placement Based on Divide-and-Conquer

Transcription

1 Relay Placemet Based o Divide-ad-Coquer Ravabakhsh Akhlaghiia, Azadeh Kaviafar, ad Mohamad Javad Rostami, Member, IACSIT Abstract I this paper, we defie a relay placemet problem to cover a large umber of sesors accordig to multiple purposes usig a miimal umber of relays. Fidig the best solutio requires expoetial ru time that takes years i large etworks. Therefore, we divide the mai problem ito sub-problems ad desig a polyomial-time algorithm for fidig a approximate solutio. We developed a software tool for ruig the algorithm ad graphical represetatio of placemet. Usig this tool, our evaluatio experimets show the performace of the polyomial-time algorithm i compariso with the best solutio. Idex Terms Wireless sesor etwork, placemet, coverage, clusterhead, relay. I. INTRODUCTION Wireless sesor etworks are suitable for may applicatio icludig atioal security, military operatio ad eviromet moitorig. Node placemet is a importat area i these etworks that studies where to place sesor or relay odes to improve etwork performace ad reduce eergy cosumptio. Node placemet may be either radom or determiistic []. I this paper, we cosider determiistic placemet i which odes are placed at exact desired locatios. We cosider the etwork model (Fig. ) i which sesor moitors the eviromet ad seds the sesed iformatio to the base statio through relay odes. I a wireless sesor etwork (WSN), if a small set of special ode whose mai fuctio is packet forwardig, are deployed, the maagemet ad etwork operatios i the etwork is simplified. These odes are called relay odes or clusterheads [2]. As show i Fig., each clusterhead covers a umber of sesors determied by a virtual circle aroud it. A fudametal problem [] which arises whe establishig such a etwork is where to deploy those relay odes to achieve the required grade of service, while meetig the system costraits. Fig. 2 demostrates a example of clusterig affect o commuicatio eergy cosumptio. I this figure, two sample clusterig optio amely A-B ad C-D have bee show. The total square of distaces betwee clusterheads ad sesors i clusterig A-B is 65 uit while it is 7 uits for clusterig C-D. Cosiderig the relatio of eergy dissipatio ad this simplified metric (total square of distaces), Fig. 2 highlights the effect of clusterig quality o etwork's total eergy dissipatio. The placemet algorithm computes umber of required relays cosiderig the fact that a relay may simultaeously cover two or more eighbor sesors ad less umber of relays is used i this way. Optimal placemet is achieved whe all the sesors are covered with the miimum umber of relays. The Divide-ad-Coquer approach is a practical way to divide the problem space ad search i it for a good fast solutio. Based o this approach, we desig a algorithm that divides the physical area ito small sub area places clusterheads i each sub area, ad the combies the solutios of the sub areas. The algorithm tries to use the lowest umber of clusterheads i each sub area. Whe two sub areas are combied, the result should also cotai the miimum umber of clusterheads. The rest of this paper is orgaized as follows. Sectio II reviews the existig work o clusterhead placemet. I Sectio III, we defie a ovel placemet problem. I Sectio IV, we desig a approximate algorithm to solve the problem by breakig the problem ito sub-problems. Sectio V itroduces a algorithm to solve a sub-problem. Sectio VI cotais the umerical experimets ad results. Sectio VII fially cocludes the paper. Fig.. A clustered wireless sesor etwork Mauscript received October, 20; revised October 6, 20. Ravabakhsh Akhlaghiia is with the Departmet of Egieerig, Azad Uiversity of Gachsara, Gachsara, Ira ( akhlaghiia.r@gmail.com). Azadeh Kaviafar is with Guila Uiversity, Rasht, Ira ( azadeh_kaviyafar@yahoo.com). Mohamad Javad Rostami is with the Departmet of Computer Egieerig, Bahoar Uiversity, Kerma, Ira ( mjrostamy@yahoo.com). 7 Fig. 2. Two examples of clusterig II. RELATED WORK We review the prior work related to relay placemet. The trasmissio rages for relays ad ordiary sesors are deoted R ad r, respectively.

2 The settig that both relay odes ad sesors ca perform the packet forwardig is kow as the sigle-tiered ifrastructure. Cheg et al. [3] developed algorithms to place the miimum umber of relay odes ad maitai the coectivity of a sigle-tiered WSN, uder the assumptio that R = r. The problem was modeled by the Steier Miimum Tree with Miimum Number of Steier Poits ad Bouded Edge Legth (SMT-MSP) problem, which arose i the study of amplifier deploymet i optical etwork ad was proved to be NP-hard [4]. Based o a miimum spaig tree, Li ad Xue [4] developed a algorithm to solve the SMT-MST problem. They proved it to have a approximatio ratio of 5, which Che et al. [5] tighteed to 4. I the same paper, a 3-approximatio algorithm was also proposed. Based o Li ad Xue s algorithm, Cheg et al. [3] proposed a differet 3-approximatio algorithm ad a radomized 2:5-approximatio algorithm. I order to provide fault-tolerace, Kashyap et al. [6] studied how to place miimum umber of relays such that the resulted WSN is 2-coected, whe R = r. Zhag et al. [7] improved the results of Kashyap et al. by developig algorithms to compute the optimal ode placemet for etworks to achieve 2-coectivity, uder the more geeral coditio that R These algorithms aimed to miimize the umber of relay odes while providig fault-tolerace. The settig that oly relay odes ca perform the packet forwardig is kow as the two-tiered ifrastructure. Pa et al. [8] first ivestigated the two-tiered ifrastructure for optimal ode placemet. Further studies cosiderig a i.i.d. uiformly distributed sesor locatio with R 4r were give i [9] ad [0]. Lloyd ad Xue [] developed algorithms to fid optimal placemet of relay odes for the more geeral relatioship R r, uder sigle-tiered ad two-tiered ifrastructures. III. THE MPCHP PLACEMENT PROBLEM First, we defie a optimizatio problem called MiCHP (Miimum ClusterHead Placemet) that determies the miimum umber of clusterheads which are required to satisfy a umber of purposes. The, we defie a relay placemet problem called MPCHP (Multi-Purpose ClusterHead Placemet) i wireless sesor etworks that places the miimum umber of clusterheads determied by MiCHP i the etwork while cosiderig more purposes. We desig a polyomial-time algorithm called solvempchp to fid a approximate solutio to MPCHP. solvempchp is flexible i a way that it ca cosider more purposes. A. Problem Defiitio I this sectio, we defie ad formulate the placemet problem. I the ext sectio we refer to the coditios defied i this problem. Problem MiCHP: Give a umber of sesors radomly located i two dimesioal area A. Fid the miimum umber of cluster heads which are required to be mi placed i A such that the followig coditios are satisfied: Coditio : For each sesor there is at least oe 72 Coditio 2: No covex sub-area A ' exists i A such that there are more tha M sesors per clusterhead i A '. Coditio 3: Each clusterhead c has two disjoit paths to the base. Two cluster heads are able to directly commuicate if their distace is ot more tha R. I other word two clusterheads are able to directly commuicate if they are located o the border or iside of the commuicatio circle of each other. A clusterhead ad a sesor are able to directly commuicate if their distace is ot more tha r. Clusterhead c has two disjoit paths to the base, if at least two clusterheads are placed o the border or iside of the commuicatio circle of c. If the umber of sesors aroud c is high, the more tha two clusterheads may be required withi the commuicatio rage of c to cover the sesors ad c may get more tha two disjoit paths to the base. Now, we wat to exted MiCHP to a placemet problem that also cosiders the followig two purposes. Purpose : Clusterheads are placed as close as possible to sesors. Purpose 2: There exist similar umbers of sesors aroud clusterheads. We defie two parameters called t ad u for a deployed wireless sesor etwork cotaiig clusterheads c, c2,..., c. Let us assume that the etwork is clustered i a way that i sesors s i,, si,2,..., si, are members of clusterhead c i, for i,2,...,. d i equals the summatio of distaces betwee c i ad its sesors for i,2,...,. We defie e i () i The, we defie 2 u d i ad t ( i e) (2) i i Parameter u quatifies Purpose ad parameter t quatifies Purpose 2 i a case of ode placemet. Now, we are able to defie the MPCHP problem. Problem MPCHP: Give a umber of sesors radomly located i two dimesioal area A. Place the miimum umber mi of clusterheads determied by MiCHP i A i a way to maximize mi( t ) mi( u) (3) t u such that the followig coditios are satisfied: Coditio : For each sesor there is at least oe Coditio 2: No covex sub-area A ' exists i A such that there are more tha M sesors per clusterhead i A '. Coditio 3: Each clusterhead c has two disjoit paths to the base. where t ad u are the parameters of the placemet defied i (2), ad mi(t) ad mi(u) are the miimum values for t ad u amog possible cases of placemet. IV. AN APPROXIMATED SOLUTION TO MPCHP I this sectio, we break MPCHP ito smaller i

3 sub-problems ad desig a algorithm to fid a approximated solutio to it. A. Dividig the Problem Fidig the best solutio requires expoetial executio time ad is impossible if umber of sesors is large ad the etwork area is huge. To solve the problem, we use a divide-ad-coquer approach to desig a algorithm with polyomial time complexity. To do thi the mai problem is divided ito smaller sub-problems ad each sub-problem is solved separately. The combiatio of the solutios of the sub-problems is a approximate solutio to the mai problem. We defie the sub-problem of MPCHP called SubMPCHP to be somethig like MPCHP but try to optimize sesor coverage i the area limited by the commuicatio circle of a sigle clusterhead. The, SubMPCHP will be much simpler tha MPCHP to solve. First we defie a problem called SubMiCHP to determie the miimum umber of clusterheads i a sub-problem as follows. Problem SubMiCHP: Give a umber of sesors radomly located i two dimesioal area A. A umber of clusterheads are placed i A. Cosider the placed clusterhead C. Fid the miimum umber sub mi of clusterheads which are required to be placed o the border or iside the commuicatio circle of C such that the followig coditios are satisfied: Coditio 4: For each sesor s located o the border or iside the commuicatio circle of C, there is at least oe Coditio 5: No covex sub-area A ' exists i the commuicatio circle of C such that there are more tha M sesors per clusterhead i A '. Coditio 6: Clusterhead C fids two disjoit paths to the base. Let us cosider the sub-problem of clusterhead C i the sesor etwork depicted i Fig. 3. We ca achieve the followigs by placig two clusterheads at the two locatios show i Fig. 3. C fids two paths. All the sesors o the border or iside the commuicatio circle of C are covered. Some eighbor sesors outside the commuicatio circle of C are covered. Problem SubMPCHP: Give a umber of sesors radomly located i two dimesioal area A. A umber of clusterheads are placed i A. Cosider the placed clusterhead C. Place the miimum umber sub mi of clusterheads determied by SubMiCHP o the border or iside the commuicatio circle of C i a way to maximize mi( t ) mi( u) (4) t u Such that the followig coditios are satisfied: Coditio 4: For each sesor s located o the border or iside the commuicatio circle of C, there is at least oe Coditio 5: No covex sub-area A ' exists i the commuicatio circle of C such that there are more tha M 73 sesors per cluster head i A '. Coditio 6: Clusterhead C fids two disjoit paths to the base. where t ad u are the parameters of the placemet defied i (2), ad mi(t) ad mi(u) are the miimum values for t ad u amog possible cases of placemet. B. Solutio Based o the defiitio of SubMPCHP, we propose algorithm solvempchp to solve the MPCHP problem. Algorithm. solvempchp While there are still ucovered sesors i the etwork do ) Place a clusterhead at a poit i the etwork where it covers the highest umber of ucovered sesors. 2) While there is uvisited clusterhead C i the etwork do Make a SubMPCHP problem for C. Solve the SubMPCHP problem. Mark C as visited. Ed. V. SOLVING THE SUBMPCHP PROBLEM I this sectio, we propose a algorithm to solve the SubMPCHP problem. Accordig to the solvempchp algorithm, we place clusterheads oe by oe to reduce algorithmic complexity. Util all the clusterheads are ot placed, we ca ot guaratee that every clusterhead fids two disjoit paths to the base. We should place the miimum umber of clusterheads to achieve this property. Coditio 7: Each clusterhead cotais at least two clusterheads withi its commuicatio rage which are ot withi the commuicatio rages of each other. Fig. 3. Placig two clusterheads iside the commuicatio circle of C (a) C fids two differet paths (a) A rig of clusterheads (b) C may fid oly oe path Fig. 4. Makig two paths for clusterhead C (b) A umber of clusterheads without rig Fig. 5. A umber of placed clusterheads

4 (a) The virtual grid (b) Four clusterheads are placed i a sub-area Fig. 6. The virtual grid o the placemet area of a sub-problem If we follow coditio 7 while solvig SubMPCHP, the it guaratees that C fids two differet paths (Fig. 4(a)). Otherwise, the case of Fig. 4(b) may happe. I this case, C has two eighbor clusterhead but they both lead to oe sigle clusterhead at the right. I other word C fids oly oe path through two differet eighbors. If we follow coditio 7 while solvig the sub-problem the the resulted etwork cotais rigs of clusterheads (such as the oe depicted i Fig. 5(a)). Every clusterhead will be part of a rig such that we ca ot have a broke lie of clusterheads (such as the oe depicted i Fig. 5(b)). It is ot efficiet to check all possible poits withi the commuicatio rage of C for clusterhead placemet. To reduce umber of cadidate poit we cosider a virtual grid (Fig. 6(a)) o the commuicatio circle of C. Each grid poit is a cadidate for clusterhead placemet. The grid makes it possible to distribute the clusterheads as uiform as possible. We set the distace betwee grid poits at miimum to keep umber of poits small. Cosiderig too few grid poits leads to missig efficiet cadidate locatios. Sice the distace betwee a clusterhead ad a sesor must be o more tha r to commuicate, we set the distace o the grid to r. We may sometimes eed to place more clusterheads i a sub-area tha what the grid offers. This case happes oly because of high desity of sesors to satisfy Coditio 5. I this case, we place additioal clusterheads i sub-areas of the grid (such as Fig. 6(b)). Algorithm solvesubmpchp() Place a virtual grid GR o the etwork with a size ad locatio that satisfies the followig two coditios. ) Grid poits are oly located withi the distace of R from clusterhead C. 2) The distace betwee every two eighbor poits of GR is r. For every rxr square sub-area SA o GR, do ) While there are more tha M sesors per clusterhead i SA, do Place a ew clusterhead at a radom locatio i SA. Place ew clusterheads at all the poits of GR. Mark all the poits of GR as o-visited. While there is o-visited poits o GR do 2) Select o-visited poit gp which its clusterhead covers the least umber of sesors. I the case of a tie, select the poit which is closer to C. 3) Remove the clusterhead at gp if both Coditio 4 ad Coditio 7 remai satisfied after removig this clusterhead. 4) Mark gp as visited. Set c equal to the umber of clusterheads o GR. Remove all the clusterheads o GR. Amog all the cases of placig c clusterheads o GR, select the case that 5) satisfies both Coditio 4 ad Coditio 7 6) ad maximizes (4). Retur the locatios of the curretly-placed clusterheads i the etwork as the solutio. VI. NUMERICAL EVALUATION I this sectio, we evaluate the solvempchp algorithm comparig with two other relay placemet algorithm ad we review the results. We defie the sceario defied i Table I. I this sceario, umber of sesors is chaged while etwork dimesio is fixed. We evaluate the followig three relay placemet algorithms i our experimets. ) solvempchp: preseted i this paper 2) fidbestmpchp: the best solutio of MPCHP foud by testig all the cases of placemet leadig to expoetial ru time 3) Algorithm2.2: proposed i [0] TABLE I: EVALUATION PARAMETERS Parameter Value Network Dimesio 500m x 500m Clusterhead s Commuicatio Rage 50 m Sesor s Commuicatio Rage 30 m Number of Sesors variable from 00 to 800 Sesor Distributio i Network No-uiform Executio Time (s) Number of ClusterHeads solvempchp Algorithm2.2[6] fidbestmpchp Number of Sesors Fig. 7. Executio time versus umber of sesors solvempchp Algorithm2.2[6] fidbestmpchp Number of Sesors Fig. 8. Number of clusterheads versus umber of sesors A. Numerical Results Executio of solvempchp takes a umber of secods whereas evolutioary placemet algorithms such as [2] typically eed tes of miutes to complete. Fig. 7 compares solvempchp i executio time to fidbestmpchp ad Algorithm2.2. Accordig to this chart, solvempchp is averagely 5 times faster tha Algorithm2.2 ad 3000 times 74

5 faster tha fidbestmpchp. Fig. 8 shows umber of required clusterheads to solve MPCHP versus umber of sesors. We expect that by icreasig umber of sesor umber of clusterheads goes up, because a additioal sesor may require a ew clusterhead for coverage. But ot every sesor requires a dedicated clusterhead sice a clusterhead ca simultaeously cover multiple sesors. So, doublig umber i the horizotal axi umber i the vertical axis is averagely magified by. i the chart. That i umber of sesors has a slight effect o umber of required clusterheads. The solutio of solvempchp is close to the best solutio. The performace ratio of solvempchp to fidbestmpchp is averagely.06. B. Implemetatio We developed a graphical eviromet writte i Java to ru ad display clusterhead placemet ad implemeted the proposed algorithm i it. Fig. 9 shows a view of the software. I the etwork area, sesors are displayed as blue poits. Clusterhead is displayed as a red poit with the commuicatio circle aroud it. The software is able to iitially geerate sesors at either determiistic or radom locatios i the etwork. After geeratig sesor the placemet algorithm is executed. Part of the graphical eviromet of the software is developed i [2]. Now, we look at the iteral structure of the software which is developed as a Java project i Eclipse [3]. Fig. 0 shows the compoets of the software ad their relatio. Each compoet is defied as a Java class. The software is composed of the followig classes: ) Parameter: This class keeps the global parameters of the evaluatio which are cofigured at the begiig. 2) Sesor: This class defies a sesor ode ad stores its locatio. 3) ClusterHead: This class defies a clusterhead ad stores its locatio. 4) NetIfo: This class stores etwork iformatio icludig sesors ad clusterheads. 5) AreaPael: This is a class that is displayed as a white graphical area withi the mai frame i which sesors ad clusterheads are draw. Size of AreaPael is cofigured at the begiig as a parameter. 6) ParameterPael: This is a class that is displayed as a frame i the left corer of the mai frame i which the Ru butto ad a text box are placed. 7) Placemet: The placemet algorithm is implemeted i this class that reads etwork iformatio from the NetIfo class. It determies locatios of clusterheads ad the uses the DrawUtility class for drawig them. 8) DrawUtility: This class cotais routies for drawig sesors ad clusterheads o AreaPael. 9) GearateSesors: A routie i this class is called at the begiig of the evaluatio to geerate a umber of sesors i radom/determiistic locatios. 0) CoverageFrame: This class is the mai frame that cotais a object of the AreaPael class ad a object of the ParameterPael class. This class calls routie ruplacemet() from the Placemet class to ru the placemet algorithm. Fig. 9. A view of the clusterhead placemet software Fig. 0. Compoets of the clusterhead placemet software VII. CONCLUSION I this paper, we defie a clusterhead placemet problem called MPCHP to cover a large umber of sesors accordig to multiple purposes usig a miimal umber of clusterheads. Sice fidig the best solutio requires expoetial ru time, we divide MPCHP ito sub-problems ad desig a polyomial-time algorithm called solvempchp for fidig a approximate solutio to MPCHP. We developed a software tool for ruig the algorithm ad graphical represetatio of placemet. Our evaluatio experimets show the performace of the solvempchp algorithm i compariso with the best solutio. REFERENCES [] J. Li, L. L. H. Adrew, C. H. Foh, M. Zukerma, ad H.-H. Che, "Coectivity, Coverage ad Placemet i Wireless Sesor Networks", Sesor vol. 9, o. 0, pp , [2] H. Karl, ad A. Willig, Protocols ad Architectures for Wireless Sesor Network Chichester, U.K: Wiley, 2005, ch. 0, pp [3] X. Cheg, D. Du, L. Wag, ad B. Xu, "Relay Sesor Placemet i Wireless Sesor Networks", Wirel. Netw., vol. 4, pp , [4] G. Li, ad G. Xue, "Steier Tree Problem with Miimum Number of Steier Poits ad Bouded Edge-Legth", If. Proc. Lett., vol. 69, pp , 999. [5] D. Che, D. Du, X. Hu, G. Li, L. Wag, ad G. Xue, Approximatios for Steier Trees with Miimum Number of Steier Poits, J. Glob. Opt., vol.8, pp.7 33, [6] A. Kashyap, S. Khuller, ad M. Shayma, "Relay Placemet for Higher Order Coectivity i Wireless Sesor Networks", i Proc. 25th IEEE Iteratioal Coferece o Computer Commuicatio Barceloa, 2006, pp. -2. [7] W. Zhag, G. Xue, ad S. Misra, "Fault-Tolerat Relay Node Placemet i Wireless Sesor Networks: Problems ad Algorithms", i Proc. 26th IEEE Iteratioal Coferece o Computer Commuicatio Achorage, AK, USA, 2007, pp [8] J. Pa, Y. Hou, L. Cai, Y. Shi, ad S. She, "Topology cotrol for Wireless Sesor Networks", i Proc. ACM Mobicom 03, Sa Diego, CA, USA, 2003, pp [9] B. Hao, J. Tag, ad G. Xue, "Fault-Tolerat Relay Node Placemet i Wireless Sesor Networks: Formulatio ad Approximatio", i Proc. 75

6 IEEE Workshop High performace switchig ad routig, U.S.A., 2004, pp [0] J. Tag, B. Hao, A. Se, "Relay Node Placemet i Large Scale Wireless Sesor Networks", Comput. Comm. vol. 29, pp , [] E. Lloyd, ad G. Xue, "Relay Node Placemet i Wireless Sesor Networks", IEEE Tras. Comoput., vol. 56, pp , [2] K. Yıldırım, T. Kalaycı, A. Uğur, Optimizig Coverage i a K-Covered ad Coected Sesor Network Usig Geetic Algorithms, i Proc. 9th WSEAS Iteratioal Coferece o Evolutioary Computig, Sofia, Bulgaria, [3] Eclipse Java Software, Available: Ravabakhsh Akhlaghiia was bor i 978 i Ira. Mr. Akhlaghiia received his B.Egr. degree from Isfaha Uiversity Of Techology (IUT) i 2000, ad his M.S. degree from Azad Uiversity, Dezfol brach (Dezfol, Ira) i computer egieerig i 200. From 2004 to 20, he is workig as a etwork professioal at Natioal Iraia South Oil Compay (NISOC). He is iterested i the area of wireless ad hoc etworks. Azadeh Kaviafar was bor i 98 i Ira. Ms. Kaviafar received her B.Egr. degree from Azad Uiversity, Lahija brach (Lahija, Ira) i 2004 ad her M.S. degree from Sharif Uiversity, Kish brach (Kish, Ira) i IT egieerig i 2009 Sice 2006, she is workig as a lecturer at Azad Uiversity, ad as a admiistrator of educatioal system at Guila Uiversity.She is iterested i the area of wireless etworks ad etwork egieerig. Mohammad Javad Rostami was bor i 978 i Ira. He received his B.Sc i computer egieerig from Bahoar Uiversity (Kerma, Ira) i 200 ad M.Sc i computer egieerig from Amirkabir Uiversity of Techology (Tehra, Ira) i He has bee a faculty member of Bahoar Uiversity sice His mai research iterests iclude diverse routig ad QoS routig algorithm wireless sesor etwork ad heuristic etwork algorithms Mr. Rostami is a member of Iteratioal Associatio of Computer Sciece ad Iformatio Techology. 76