Particle Swarm Optimization for the Clustering of Wireless Sensors

Transcription

1 Rochester Insttute of Technology RIT Scholar Works Presentatons and other scholarshp Partcle Swarm Optmzaton for the Clusterng of Wreless Sensors Jason C. Tllett Rochester Insttute of Technology Raghuveer Rao Rochester Insttute of Technology Ferat Sahn Rochester Insttute of Technology T. Rao SUNY Brockport Follow ths and addtonal works at: Recommended Ctaton Jason C. Tllett, Raghuveer M. Rao, Ferat Sahn, T. M. Rao, "Partcle swarm optmzaton for the clusterng of wreless sensors", Proc. SPIE 5100, Dgtal Wreless Communcatons V, (23 July 2003); do: / ; Ths Conference Proceedng s brought to you for free and open access by RIT Scholar Works. It has been accepted for ncluson n Presentatons and other scholarshp by an authorzed admnstrator of RIT Scholar Works. For more nformaton, please contact rtscholarworks@rt.edu.

2 Copyrght 2003 Socety of Photo-Optcal Instrumentaton Engneers. Ths paper was publshed by SPIE and s made avalable as an electronc reprnt (preprnt) wth permsson of SPIE. One prnt or electronc copy may be made for personal use only. Systematc or multple reproducton, dstrbuton to multple locatons va electronc or other means, duplcaton of any materal n ths paper for a fee or for commercal purposes, or modfcaton of the content of the paper are prohbted.

3 Partcle Swarm Optmzaton for the Clusterng of Wreless Sensors Jason Tllett a, Raghuveer Rao a, Ferat Sahn a, TM Rao b a Rochester Insttute of Technology, Rochester NY b SUNY at Brockport, Brockport, NY ABSTRACT Clusterng s necessary for data aggregaton, herarchcal routng, optmzng sleep patterns, electon of extremal sensors, optmzng coverage and resource allocaton, reuse of frequency bands and codes, and conservng energy. Optmal clusterng s typcally an NP-hard problem. Solutons to NP-hard problems nvolve searches through vast spaces of possble solutons. Evolutonary algorthms have been appled successfully to a varety of NP-hard problems. We explore one such approach, Partcle Swarm Optmzaton (PSO), an evolutonary programmng technque where a swarm of test solutons, analogous to a natural swarm of bees, ants or termtes, s allowed to nteract and cooperate to fnd the best soluton to the gven problem. We use the PSO approach to cluster sensors n a sensor network. The energy effcency of our clusterng n a data-aggregaton type sensor network deployment s tested usng a modfed LEACH-C code. The PSO technque wth a recursve bsecton algorthm s tested aganst random search and smulated annealng; the PSO technque s shown to be robust. We further nvestgate developng a dstrbuted verson of the PSO algorthm for clusterng optmally a wreless sensor network. Keywords: Partcle Swarm Optmzaton, sensor networks, clusterng 1. INTRODUCTION Consder a sensor network of several sensors deployed to gather data and transmt t to a central staton. For example t could consst of chemcal sensors deployed to generate a map of the level of a substance n an envronment and to relay the map at perodc ntervals to a base staton. In such a scenaro, one possble mplementaton would be for the base staton to communcate drectly wth all of the sensors nodes. Havng all nodes communcatng over long ranges wll quckly deplete the energy stores of the nodes. Nodes far from the base staton wll de frst and the area beng montored wll no longer be concdent wth the orgnal target area. They wastefully communcate over long dstances when t would be benefcal to have many smaller communcaton schedules dstrbuted throughout the network wth nodes communcatng over dstances that are much less than the full spatal extent of the sensor network. These communcaton cells are clusters. In a cell, a central clusterhead node can receve data from the other cluster nodes and relay t to the base staton so that total energy consumpton s reduced. Thus, clusterng can promote energy conservaton. Clusterng can promote coverage maxmzaton. It s desrable for the sensor network to be able to montor consstently and unformly over the entre target area for the maxmum tme. To evenly dstrbute energy dsspaton by the nodes so that they all de at about the same tme, energy ntensve roles, lke the role of nodes that make up a routng backbone, must be shared among nodes n the sensor network. When a node takes on ths specal role t s called a clusterhead. Clusterng can promote resource conservaton. Wth all nodes ether synchronzng to the same communcaton schedule or sharng the same frequency band or codes, the effcency of the communcaton protocol wll degrade for larger sensor networks. Reusng frequency bands or codes wthn spatally dstnct clusters or synchronzng schedules of logcally related nodes may conserve resources. Clusterng makes effcent, ntellgent data aggregaton possble. Sensor networks are unque n that they may gather nformaton that s correlated for nearby nodes. Data compresson or aggregaton at clusterheads reduces the amount of data to be njected nto the routng backbone. For effcent or maxmum compresson, t s necessary to form logcal clusters of nodes whose data are most closely correlated. Dgtal Wreless Communcatons V, Raghuveer M. Rao, Sohel A. Danat, Mchael D. Zoltowsk, Edtors, Proceedngs of SPIE Vol (2003) 2003 SPIE X/03/$

4 Effcent routng s not possble wthout clusterng. All nodes would then be a sngle hop from the base staton. It s possble to enable routng wthout the use of clusters, but each node would need a complete routng table, whch would not scale well to large numbers of sensor nodes. To mantan scalablty, clusters of nodes are formed to reduce the sze of the node s routng tables. Thus clusterng s necessary for scalable routng of data. Much theoretcal work has already been publshed on clusterng n the graph theoretcal framework 1-8. Most of the work has been n the context of developng effcent routng protocols for wreless, ad hoc, sensor networks 9. However, clusterng need not be for the beneft of routng alone. If sensed data s correlated over an area covered by multple sensors, data aggregaton and compresson can reduce the number of packets that need to be forwarded to the end user. Clusterng s now a wdely accepted method for optmzng sensor network performance 10. To perform effcently, sensor networks must often optmze global parameters wthout global knowledge and control. Examples of optmzaton objectves relevant to sensor networks nclude Maxmzng data packets extracted from the sensor network over ts lfetme. Maxmzng the sensor network s effectve lfetme. Mnmzng delay between a user query and a sensor network response. Maxmzng sensor coverage. The clusterng problem can be vewed as a search through a typcally NP-Hard soluton space. Mnmzng the dstance between a node and ts clusterhead 11 s one possble way of clusterng. The parameter to optmze vares wth applcaton and some common parameters were lsted n the bullets above. The problem s that although a global parameter optmzaton s requred, nodes may not have global knowledge, and therefore must partcpate n solvng the problem usng only local nteractons. Clusterng of nodes has been proposed as an approach for maxmzng the lfespan of a group of wreless sensors 12. It must be emphaszed that ths s an applcaton specfc result where we are assumng that energy can be conserved by combnng redundant data from nearby sensors at data aggregaton nodes, or clusterheads, before forwardng the data to the fnal destnaton. In the Low-Energy Adaptve Clusterng Herarchy (LEACH) protocol 13, the clusterheads are determned randomly each round. The authors also consder havng a central computaton determne the clusterhead dentfcaton for each round (LEACH-C). All of the nodes send ther poston and energy level data to a central computer. The central computer calculates the average energy level of the whole sensor group. All nodes wth energy reserves n excess of the average energy of the nodes are consdered as nodes that are avalable to take on the clusterhead role. It then solves the NP hard problem 11 of fndng k optmal clusters and sends back to the cluster the clusterhead ID of each node n the sensor network. They report usng a smulated annealng approach 14. The Partcle Swarm Optmzaton (PSO) 15 technque has been appled to a varety of combnatoral problems 16. We approach the problem of clusterng a wreless sensor network n an ad hoc data-aggregaton type deployment usng PSO. The rest of ths paper s organzed as follows. Secton 2 detals a centralzed approach to applyng PSO for clusterng a wreless sensor network and presents some results of prevous work. In secton 3, the results of ns-2 17 smulatons usng PSO to handoff clusterheads n the LEACH-C 12 protocol are presented. Secton 4 contans an analyss of the recursve bsecton algorthm usng dfferent approaches for fndng the optmal dvdng lnes. Secton 5 outlnes a dstrbuted verson of PSO for the clusterng of wreless sensors. The paper concludes wth a summary and future work secton. 2. ALGORITHM FOR CLUSTERING RECURSIVE BISECTION WITH CONTRAINTS 18 The problem s to fnd n a group of N nodes, where A nodes are avalable for the clusterhead role, the k nodes that mnmze the average dstance between a node and ts assgned clusterhead, whle at the same tme, constranng the number of nodes n each cluster. Ths s an NP hard problem 11 of a dscrete and combnatoral nature. There s a wealth of lterature on the subject of clusterng algorthms, from books 4 to revews 19, to a paper on L-capactated clusterng 11 whch closely resembles our clusterng crtera. 74 Proc. of SPIE Vol. 5100

5 We ntroduce some defntons to assst n descrbng the clusterng algorthm presented here. We defne the Dstrbuted Sensor Network (DSN) as the entre node collecton. A Sensor Group () s a subset of the DSN. Each ncludng the DSN has N total nodes and A nodes avalable to take on the cluster head role. Here refers to a unque. Each also has M to dentfy the number of clusters that must be constructed n the. Sensor Groups are always subdvded, provded M > 1 nto M smaller s. When M = 1, then the no longer needs to be further dvded, and a clusterhead can be assgned. Each mantans a flag, call t Has_Been_Splt, to hold nformaton about whether or not the has been subdvded. When a has M = 1, t can be called a cluster. We use a smple clusterng strategy. We recursvely dvde each nto 2 s startng wth the DSN. We contnue to splt the resultng s untl all Has_Been_Splt flags are true. Then for each that has M = 1, we pck from the avalable cluster heads n the cluster, the one that mnmzes the dstance between t and the other nodes n the cluster. Recursve dvson of non-cluster s s only one pece of the algorthm. The other pece s coupled wth the objectve functon of our optmzaton. When we splt an, we need to ensure that each resultng has enough avalable clusterheads to allow for recursve subdvdng of the resultng s. Equally mportant s the condton that the number of nodes n the resultng s should be balanced such that at the end, when all s have been splt, the number of nodes n each cluster s about the same. Our am s to maxmze the lfetme and data throughput of the DSN. Gvng each clusterhead the same communcaton and processng burden each round should help acheve the goal. 2.1 Partcle Swarm Optmzaton for Clusterng We use PSO 15 to determne the parameters of a lne that dvdes optmally a. By optmally, we mean that the number of nodes, as well as the number of avalable clusterheads, n each resultng wll be balanced accordng to the number of clusters the must form. If the wll form an even number of clusters, then the approprate balance s that the number of nodes and avalable cluster heads n the 2 s resultng from the splt wll be equal. In general, the ftness of a soluton s measured by Ftness = ( a f A) + ( a f A) + ( c f N ) + ( c f N), (1) where a1 and a 2 are the number of avalable cluster heads n regon 1 and 2 respectvely. Regons 1 and 2 refer to the s that result from the optmal lne dvson. The symbols c 1 and c 2 are the number of nodes n regon 1 and 2 respectvely. The letters A, N, and M are as defned above and refer to quanttes n the that s beng dvded. Fnally f = M / M and f = M / M where M 1 and M 2 refer to the number of clusters that wll need to be formed n regons 1 and 2 respectvely. 2.2 Prevous Results Even though our ftness s not dentcal to the mnmum dstance between all nodes and ther clusterheads, our algorthm and optmzaton combne to approxmate the dstance ftness metrc. We compared our clusterng to k-means clusterng and found that the average dstance that a PSO clustered node could expect to be from ts cluster head to be only 2% hgher, n terms of the percentage of the lnear extent of the DSN, than the value acheved usng k-means. We found PSO clusterng to be able to cluster robustly for a wde range of DSN szes, cluster requrements and clusterhead avalablty. The PSO clusterng algorthm was able to dentfy natural clusters. Natural clusters were artfcally created just to explore whether or not the algorthm would dentfy them. The recursve bsecton algorthm usng PSO to determne the optmal splt at each dvson s fast. Wth 40 partcles n a swarm, the algorthm rarely requred more than a couple of steps to fnd a good lne for the splt. What remaned unknown was how good our algorthm was compared to the smulated annealng approach to clusterng 12. Proc. of SPIE Vol

6 3. LEACH-C WITH RECURSIVE BISECTION CENTRALIZED CLUSTERING Clusterng of nodes has been proposed as an approach for 12 maxmzng the lfespan of a group of wreless sensors. It must be emphaszed that ths s an applcaton specfc result where we are assumng that energy can be conserved by combnng redundant data from nearby sensors at data aggregaton nodes, or clusterheads before forwardng the data to the fnal destnaton. In the LEACH 12 protocol, the clusterheads are determned randomly each round. The dstrbuted selecton of clusterheads can result n an neffcent network confguraton. The authors 12 also consder havng a central computaton determne the clusterhead dentfcaton for each round. All of the nodes send ther poston and energy level data to a central computer. The central computer calculates the average energy level of the whole sensor group. The A nodes that are avalable to be clusterheads are those whose energy exceeds the average energy of the DSN. It the solves the NP hard problem 11 of fndng M optmal clusters and sends back to the cluster the clusterhead ID of each node n the sensor network. They report usng a smulated annealng approach 14 for determnng optmal clusterng based on a purely dstance-based metrc. We use the PSO approach to dvde a sensor node feld usng the recursve bsecton algorthm. To determne whether or not our sub-optmal clusterng s energy effcent for a data aggregaton type sensor network applcaton, we replaced the functon BSsetup n the object orented class BSAgent n the orgnal LEACH-C code, that s avalable from 20, wth our own verson that would perform the clusterhead dentfcaton usng clusters formed by recursve bsecton usng PSO to fnd splt lnes. We then ran the modfed code to generate plots of the energy use of the DSN as a functon of the data delvered by the DSN. Ths s a drect measure of the energy effcency of the DSN. Fgure 1 PSO recursve bsecton clusterng s nearly ndstngushable from optmal clusterng on a dstance-based metrc Fgure 1 s a plot of our results for a sngle experment. In ths experment, there are 100 nodes n a 100x100 unt area. The locaton of the base staton s at (50,175) as ndcated n the plot and each node begns wth 2J of energy. Plotted n the fgure s the amount of data receved at the base staton versus the energy expended by the sensor network. An energy effcent sensor network s one that can delver the most data at the expense of the least amount of energy. We 76 Proc. of SPIE Vol. 5100

7 fnd that the clusters formed usng PSO enabled recursve bsecton result n a energy effcency that s ndstngushable from the energy effcency of the clusters formed optmally usng a dstance metrc up to the pont at whch nodes start to de. At that pont, the PSO enabled recursve bsecton clusterng outperforms the dstance-based optmal clusterng up untl near the end where the dstance-based optmal clusterng overtakes and ends wth only a slghtly hgher data delvery value. The PSO enabled recursve bsecton outperforms the dstance-based optmal clusterng n terms of the number of objectve functon calculatons requred to perform the clusterng. The dstance-based optmal clusterng requres from 200 to 500 steps 13 to converge. The dstance-based optmal clusterng uses smulated annealng as the optmzaton technque. In each step of the smulated annealng algorthm, there s a sngle evaluaton of the objectve functon and the objectve functon evaluaton s O(n) where n s the number of nodes. For PSO enabled recursve bsecton, at each step, each partcle n the swarm performs an objectve functon evaluaton, therefore the number of objectve functon evaluatons s n p where n s the number of sensor nodes and p s the number of partcles n the swarm. The complexty of the objectve functon evaluaton for the PSO enabled recursve bsecton algorthm s also O(n). Snce the objectve functon evaluaton complextes of both methods are the same, we can compare the performance of the two algorthms smply by comparng the average number of objectve functon evaluatons requred for convergence. For the experment represented by Fgure 1, the total number of objectve functon evaluatons requred by the PSO enabled recursve bsecton, s on average about 50. That s 4 to 10 tmes fewer objectve functon evaluatons than requred by dstance-based optmal clusterng usng smulated annealng. 4. RECURSIVE BISECTION ANALYSIS Is the performance of the recursve bsecton clusterng usng PSO fast because of PSO or s t fast because of the recursve bsecton? To determne f the performance of PSO recursve bsecton s due to the algorthm or the method for fndng a good lne, PSO, we mplemented the same recursve bsecton algorthm wth two other methods of determnng the splt lne. The ftness was mantaned as n Equaton 1. In order for a lne to be accepted, Equaton 1 had to be satsfed along wth one other constrant, the recurson constrant. There needed to exst n each resultng enough avalable clusterheads so that t was possble to splt t agan f necessary, to form the desred number of total clusters. If the algorthm s unable to fnd a dvdng lne that satsfes both Equaton 1 and the recurson constrant n an allowed maxmum number of objectve functon evaluatons, the algorthm fals. These events are talled and become an mportant part of the analyss of the recursve bsecton algorthm. The recursve bsecton algorthm s mplemented usng smulated annealng, random search and PSO. 4.1 Random Search In the random search approach to fndng the optmal dvdng lne subject to ftness and constrants, we smply randomly pck the ( xyθ,, ) values of the dvdng lne on ther defned ranges at each step. If the new ftness s better than the best ftness found so far, the lne s retaned as the current soluton. At the next step, the parameters of a completely new random lne are generated and ts ftness tested. The process s repeated untl the full DSN clusterng results or untl the maxmum number of search steps s used durng a sngle bsecton, ndcatng falure. 4.2 Characterzng Smulated Annealng for Recursve Bsecton of a Sensor Network To mplement smulated annealng, we frst need to characterze the optmzaton for our specfc problem. Indeed, ths s one of the drawbacks of usng smulated annealng. Smulated annealng can be descrbed as a gradent followng strategy wth hll hoppng abltes. The algorthm s ntalzed wth a guess of the soluton. The ftness of the ntal soluton s evaluated and a new soluton s generated by perturbng the current soluton by a typcally small, random value. The perturbaton s typcally taken to be small compared to the range of the varable beng perturbed. The maxmum value of the perturbaton s a parameter of the smulated annealng that needs to be fxed. The maxmum value need not be constant over the steps of the algorthm but could be, for example, reduced as the algorthm progresses to suppress global exploraton. However, the temperature of the smulated annealng serves the same purpose so we wll fx the maxmum value of the permutaton to a constant value to be determned n our characterzaton. Proc. of SPIE Vol

8 The temperature T s used to promote hll hoppng. At each step of the algorthm, a new soluton s derved from the current one. If the ftness on the new soluton s better than the ftness of the current soluton, the new soluton becomes the current soluton. If the new soluton has poorer ftness than the current soluton, t can stll become the current soluton wth a probablty gven by e f Tn, (2) where n s the step of the algorthm, T n ndcates that the temperature vares as the algorthm progresses, also called the coolng schedule, and f s the dfference between the ftness of the new soluton and the ftness of the current soluton. Gven our selecton that lower ftnesses are better ftnesses, a postve f occurs when the new soluton s of lower qualty than the current soluton. When ths occurs, the transton s allowed wth a probablty gven by Equaton 2. The dependence of T on n s therefore an algorthmc parameter that must be chosen such that the smulated annealng algorthm has relatvely large probabltes of transtonng to hgher energy (poorer ftness) states for low n, but the probablty of such transtons approaches zero as n becomes large. Ths s analogous to coolng the smulaton. To characterze T n, we must, for a wde range of possble perturbaton parameters, look at the n dependence of f, and derve a sutable analytcal expresson for T n. The expresson for the coolng schedule s arrved at by frst plottng f as a functon of n for dfferent combnatons of ε x, εy, ε θ for smulated annealng search algorthm runs n whch the temperature s held at zero. We found εx = εy = 5 and ε θ = 1 to lead to values of f that converged to zero as the algorthm progressed. A plot of these values of the perturbaton parameters n Fgure 2. f vs. n s shown for Fgure 2 - f vs. n for zero temperature smulated annealng N=100, M=5, A=50 78 Proc. of SPIE Vol. 5100

9 We used the profle of Fgure 2 to defne a naïve coolng schedule for the smulated annealng. So for our mplementaton of smulated annealng of the optmal dvdng lne, we chose the perturbaton parameters as they are n Fgure 2, and a new lne s kept wth the followng probablty, f ì n ï αn P = í e f > , where α n = 2945e. (3) ï î 1 otherwse 4.3 Comparng PSO, Smulated Annealng and Random Search for Recursve Bsecton The recursve bsecton algorthm was mplemented usng PSO, smulated annealng and random search as the technques for generatng the optmal splt lne. To generate comparson data, a set of 100 nodes s randomly generated, and then the recursve bsecton algorthm s executed for each optmzaton technque on the same set of nodes. The experment s repeated 20 tmes, each tme generatng a new set of 100 nodes. The 20 trals were repeated for dfferent clusterng demands. Specfcally, we repeated the 20 tral runs for values of M = { 5,10,15,20, 25,30,35,40, 45}. In all experments, the value of N and A were fxed at 100 and 50 respectvely. Note that for M>25, the nodes per cluster s 3 or less and that the smallest cluster sze s 2. We present n Fgure 3 the average number of objectve functon evaluatons requred to successfully cluster a DSN usng recursve bsecton. Fgure 3 Comparson of PSO, Smulated Annealng and Random Search for Successful Clusterngs The results presented n Fgure 3 are remarkable but must be nterpreted properly. When successful, random search s always faster than the other two approaches. PSO appears to converge faster than smulated annealng over much of the range of nodes-per-cluster graphed. The feature at abscssa 6.67 for the smulated annealng data s probably related to the choce of the coolng schedule. A more detaled preparaton of the coolng schedule may help to mprove the performance of the smulated annealng algorthm over a wder range of nodes-per-cluster. Proc. of SPIE Vol

10 We have plotted n Fgure 4 the number of faled clusterngs, out of the 20 trals, for each of the optmzaton technques. The fgure exposes that the PSO technque s the most robust technque for determnng the optmal splt lne for recursve bsecton. In fact the PSO approach was the only approach that never faled n our trals to perform a clusterng on a range of node-per-cluster from 5 to 20. Fgure 4 must be taken nto consderaton when nterpretng Fgure 3. For node-per-cluster values below 5, the clusterng becomes more dffcult and wll requred more search. Therefore, the number of objectve functon evaluatons requred performng a successful clusterng rses for smaller cluster szes. Snce PSO was more lkely to be successful for clusterng the DSN for small cluster szes, t was more lkely to record a large value for the number of objectve functon evaluatons requred to perform the clusterng. Hence Fgure 3 has elevated values of the number of objectve functon evaluatons for the PSO approach because t s robust. Fgure 4 Evaluatng the robustness of the optmzaton approaches 5. A DISTRIBUTED VERSION OF PSO FOR SENSOR NETWORKS Can PSO clusterng be mplemented as a dstrbuted clusterng approach? So far we have consdered clusterng only n the sense that t can be performed by a centralzed entty, the base staton. For sensor networks, nodes wll typcally not have drect contact wth a base staton; therefore f clusterng s to be mplemented, the clusterng topology must be computed n a dstrbuted fashon. Most current clusterng approaches use localzed graph parttonng algorthms to form node clusters. These algorthms do not nternalze operatonal parameters of the sensor network that need to be optmzed. PSO can nternalze parameters to be optmzed n the ftness functons of nodes. We ntroduce here a smple extenson of PSO for dstrbuted computng. Example: Buldng an Optmal Communcatons Range, Fully Connected Graph, for Clusterng Assume that a sensor node s sensng range s fxed. Assume each node knows ts poston so that communcatng nodes can calculate dstances. Also assume that the sensor node can adjust ts power so t can vary ts reach or communcaton range. One framng of a problem statement could be how should nodes vary ther communcaton range optmally such that the number of lnks s mnmzed whle ensurng that the sensor network s not parttoned. By not parttoned, t s mpled that there s a d -hop route from any node n the network to any other node n the network, where d s an nteger. Ths s a clusterng problem based on a sngle optmzaton, specfcally, mnmzng the number of lnks. The 80 Proc. of SPIE Vol. 5100

11 soluton wll result n sets of connected nodes or a graph. In the lowest-id 21-23, 30 approach, for example, lnks are determned by communcaton ranges, but here n the PSO soluton, the communcaton range s optmally adjusted for a specfc purpose. Agan, our purpose here n ths llustratve example s to mnmze the number of lnks whle remanng fully connected. Wth ths problem statement t s possble to propose a soluton n the framework PSO. To apply PSO to ths problem, t s necessary to ntroduce an nnovatve extenson of the exstng algorthm. It s ths extenson of the PSO algorthm that allows one to dentfy as partcles n the swarm, nodes n the network. The extenson nvolves how the ftness of a partcle, or node s expressed. Tradtonally, a sngle functon takes the partcle test soluton and returns ftness. But now there s no sngle functon. Each node has ts own ftness functon, whch s dstnct from the ftness functons of the other nodes. For the concrete example presented here, the ftness of node could be wrtten as f å j 2 ( d r r ), (4) j j where the sum s over neghbor nodes of that are wthn communcaton range, d j s the dstance separatng the nodes, and r j s node j s currently mplemented representatve range. Representatve range means to represent the dstance you would need to travel from a node before passng nto another node s representatve range. The ftness functon nvolves another term but the mportant revelaton s that each node s tryng to optmze ts own value of r, but s takng cues about what ts value of r should be based on the ftnesses and r j s reported by ts neghbors. The exchanges of ftnesses and values could be executed durng normal operaton and nodes could use converged ranges durng re-clusterng. Ftness could represent not only communcaton range, but also energy level, connectvty, target sensng data densty (to ndcate how relevant or actve the sensor s data s) and other operatonal parameters of the sensor network. Exstng clusterng technques are not capable of nternalzng system parameters n ths way. Addtonally, sensors could have many levels of ftness, evaluated usng dfferent combnatons of system parameters, thereby allowng for more than one type of clusterng to exst smultaneously. 6. SUMMARY AND CONCLUSIONS We have begun an exploraton nto optmal clusterng of wreless sensor networks usng a relatvely new and novel evolutonary approach to optmzaton, PSO. A recursve bsecton algorthm for approxmatng a dstance based clusterng s mplemented wth an objectve functon that ams to equalze the number of nodes n each cluster. Ths ftness s well algned wth an applcaton specfc data-aggregaton type deployment of an ad hoc wreless sensor network. The algorthm s shown to conserve energy as well as to generate an optmal dstance-based metrc clusterng. Addtonally, the PSO approach to determnng optmal dvsons s shown to be more robust than other methods tested. 7. FUTURE WORK The coolng schedule of a smulated annealng approach to recursve bsecton of a DSN s mportant n determnng the convergence of the algorthm. The ntal temperature must be hot enough to allow lberal exploraton of bad ntal states and cool slow enough to allow the tral solutons to anneal at a gven temperature. Determnng ths schedule s complcated by the nature of the recursve bsecton algorthm. For example, f we are to splt 100 nodes nto 10 clusters, n the frst splt, the nodes cover the entre DSN. But as bsecton progresses, the spatal extent of a to be splt wll decrease. Ths wll lower the meltng pont of the smulated annealng algorthm. Ths s due to the expresson for the ftness of the optmal lne. Therefore, a complete characterzaton must nvolve dervng a coolng schedule, whch adjusts tself dependng on the nature or parameters of the partcular bsecton beng performed. Proc. of SPIE Vol

12 We wll mplement a dstrbuted verson of PSO to determne s the dstrbuted verson can ndeed perform ftness based clusterng of a DSN. It s crtcal that ths generalzaton be explored, as many envsoned deployments of ad hoc wreless sensor networks wll preclude havng a central authorty to manage clusterng. If the dstrbuted PSO can cluster, t wll be nterestng to explore the benefts of mult-levels of clusterng. Can a DSN beneft from havng one logcal clusterng that maxmzes data aggregaton effcency whle smultaneously mantanng another clusterng that maxmzes routng effcency? We beleve the PSO cooperatve paradgm can be used to explore ths queston. REFERENCES 1. T. F. Gonzalez, "On the Computatonal Complexty of Clusterng and Related Problems," presented at Proc. 10th IFIP Conference on System Modelng and Optmzaton, New York, New York USA, B. W. Kernghan and S. Ln, "An Effcent Heurstc Procedure for Parttonng Graphs," Bell Techncal Journal, pp , D. S. Johnson, C. R. Aragon, L. A. McGeoch, and C. Schevon, "Optmzaton by Smulated Annealng: An Expermental Evaluaton; Part I, Graph Parttonng," Operatons Research, vol. 37, pp , J. A. Hartgan, Clusterng Algorthms. New York, NY, USA: Wley, V. K. G. Karyps "Analyss of Mult-Level Graph Parttonng," presented at Proc. Supercomputng '95, San Dego, CA, USA, H. D. Smon and S. H. Teng, "How good s recursve bsecton?," SIAM Journal of Scentfc Computng, vol. 18, pp , P. Arabe, L. J. Hubert, and G. DeSoete, "Clusterng and Classfcaton," World Scentfc, L. Tao, Y. C. Zhao, K. Thulasraman, and M. N. S. Swamy, "Smulated Annealng and Tabu Search Algorthms for Multway Graph Partton," Journal of Crcuts, Systems and Computers, vol. 2, pp , S. Ramanathan and M. Steenstrup, "A survey of routng technques for moble communcatons networks," Moble Networks and Applcatons 1, pp , K. Sohrab, J. Gao, V. Alawadh, and G. J. Potte, "Protocols for Self-Organzaton of a Wreless Sensor Network," presented at Proc. 37th Allerton Conference on Communcaton, Computng and Control, P. Agarwal and C. Procopuc, "Exact and Approxmaton Algorthms for Clusterng," presented at Proceedngs of the Nnth Annual ACM-SIAM Symposum on Dscrete Algorthms, Baltmore, MD, W. Henzelman, A. Chandrakasan, and H. Balakrshnan, "An Applcaton-Specfc Archtecture for Wreless Mcrosensor Networks," IEEE Transactons on Wreless Communcatons, vol. 1, pp , W. B. Hezelman, "Applcaton-Specfc Protocol Archtectures for Wreless Networks," n Electrcal Engneerng and Computer Scence: MIT, 2000, pp T. Murata and H. Ishbuch, "Performance Evaluaton of Genetc Algorthms for Flowshop Schedulng Problems," presented at 1st IEEE Conference on Evolutonary Computaton, R. C. Eberhart and J. Kennedy, "A new optmzer usng partcle swarm theory," presented at Proceedngs of the Sxth Internatonal Symposum on Mcro Machne and Human Scence, Nagoya, Japan, K. E. Parsopoulos and M. N. Vrahats, "Recent approaches to global optmzaton problems through Partcle Swarm Optmzaton," Natural Computng, vol. 1, pp , UCB/LBNL/VINT, "The Network Smulator (ns-2)," J. Tllett, R. Rao, F. Sahn, and T. Rao, "Cluster Head Indentfcaton n an Ad Hoc Sensor Network Usng Partcle Swarm Optmzaton," presented at 2002 IEEE Internatonal Conference on Personal Wreless Communcatons, Inda, P. K. Agarwal and M. Sharr, "Effcent algorthms for geometrc optmzaton," ACM Comput. Surveys, MIT, "uamps Code," D. R. Ln and M. Gerla, "Adaptve Clusterng for Moble Wreless Networks," IEEE Journal on Selected Areas n Communcatons, vol. 15, pp , D. J. Baker, A. Ephremdes, and J. A. Flynn, "The desgn and smulaton of a moble rado network wth dstrbuted control," IEE JSAC, SAC-2(1), pp , A. Ephremdes, J. E. Weselther, and D. J. Baker, "A desgn concept for relable moble rado networks wth frequency hoppng sgnalng," presented at IEEE 75(1), Proc. of SPIE Vol. 5100

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