An Energy Efficient Hierarchical Clustering Algorithm for Wireless Sensor Networks

Transcription

1 An Energy Effcent Herarchcal Clusterng Algorthm for Wreless Sensor Networks Seema Bandyopadhyay and Edward J. Coyle School of Electrcal and Computer Engneerng Purdue Unversty West Lafayette, IN, USA {seema, Abstract A wreless network consstng of a large number of small sensors wth low-power transcevers can be an effectve tool for gatherng data n a varety of envronments. The data collected by each sensor s communcated through the network to a sngle processng center that uses all reported data to determne characterstcs of the envronment or detect an event. The communcaton or message passng process must be desgned to conserve the lmted energy resources of the sensors. Clusterng sensors nto groups, so that sensors communcate nformaton only to clusterheads and then the clusterheads communcate the aggregated nformaton to the processng center, may save energy. In ths paper, we propose a dstrbuted, randomzed clusterng algorthm to organze the sensors n a wreless sensor network nto clusters. We then extend ths algorthm to generate a herarchy of clusterheads and observe that the energy savngs ncrease wth the number of levels n the herarchy. Results n stochastc geometry are used to derve solutons for the values of parameters of our algorthm that mnmze the total energy spent n the network when all sensors report data through the clusterheads to the processng center. Keywords- Sensor Networks; Clusterng Methods; Vorono Tessellatons; Algorthms. I. INTRODUCTION Recent advances n wreless communcatons and mcroelectro-mechancal systems have motvated the development of extremely small, low-cost sensors that possess sensng, sgnal processng and wreless communcaton capabltes. These sensors can be deployed at a cost much lower than tradtonal wred sensor systems. The Smart Dust Proect at Unversty of Calforna, Berkeley [4, 5, 6] and WINS Proect at UCLA [, 7], are two of the research proects attemptng to buld such low-cost and extremely small (approxmately cubc mllmeter) sensors. An ad-hoc wreless network of large numbers of such nexpensve but less relable and accurate sensors can be used n a wde varety of commercal and mltary applcatons. These nclude target trackng, securty, envronmental montorng, system control, etc. To keep the cost and sze of these sensors small, they are equpped wth small batteres that can store at most Joule []. Ths puts sgnfcant constrants on the power avalable for communcatons, thus lmtng both the transmsson range and the data rate. A sensor n such a network can therefore communcate drectly only wth other sensors that are wthn a small dstance. To enable communcaton between sensors not wthn each other s communcaton range, the sensors form a mult-hop communcaton network. Sensors n these mult-hop networks detect events and then communcate the collected nformaton to a central locaton where parameters characterzng these events are estmated. The cost of transmttng a bt s hgher than a computaton [] and hence t may be advantageous to organze the sensors nto clusters. In the clustered envronment, the data gathered by the sensors s communcated to the data processng center through a herarchy of clusterheads. The processng center determnes the fnal estmates of the parameters n queston usng the nformaton communcated by the clusterheads. The data processng center can be a specalzed devce or ust one of these sensors tself. Snce the sensors are now communcatng data over smaller dstances n the clustered envronment, the energy spent n the network wll be much lower than the energy spent when every sensor communcates drectly to the nformaton processng center. Many clusterng algorthms n varous contexts have been proposed [-7, -8]. These algorthms are mostly heurstc n nature and am at generatng the mnmum number of clusters such that any node n any cluster s at most d hops away from the clusterhead. Most of these algorthms have a tme complexty of O (n), where n s the total number of nodes. Many of them also demand tme synchronzaton among the nodes, whch makes them sutable only for networks wth a small number of sensors. The Max-Mn d-cluster Algorthm [5] generates d-hop clusters wth a run-tme of O(d) rounds. But ths algorthm does not ensure that the energy used n communcatng nformaton to the nformaton center s mnmzed. The clusterng algorthm proposed n [7] ams at maxmzng the network lfetme, but t assumes that each node s aware of the whole network topology, whch s usually mpossble for wreless sensor networks whch have a large number of nodes. Many of these clusterng algorthms [, 6, 7, 8] are specfcally desgned wth an obectve of generatng stable clusters n envronments wth moble nodes. But n a typcal wreless sensor network, the sensors locatons are fxed and /0/$7.00 (C) 00 IEEE 7

2 the nstablty of clusters due to moblty of sensors s not an ssue. For wreless sensor networks wth a large number of energy-constraned sensors, t s very mportant to desgn a fast algorthm to organze sensors n clusters to mnmze the energy used to communcate nformaton from all nodes to the processng center. In ths paper, we propose a fast, randomzed, dstrbuted algorthm for organzng the sensors n a wreless sensor network n a herarchy of clusters wth an obectve of mnmzng the energy spent n communcatng the nformaton to the nformaton processng center. We have used results n stochastc geometry to derve values of parameters for the algorthm that mnmze the energy spent n the network of sensors. II. RELATED WORK Varous ssues n the desgn of wreless sensor networks desgn of low-power sgnal processng archtectures, lowpower sensng nterfaces, energy effcent wreless meda access control and routng protocols [, 6, 0], low-power securty protocols and key management archtectures [9-0], localzaton systems [, ], etc. have been areas of extensve research n recent years. Gupta and Kumar have analyzed the capacty of wreless ad hoc networks [8] and derved the crtcal power at whch a node n a wreless ad hoc network should communcate to form a connected network wth probablty one [9]. Many clusterng algorthms n varous contexts have also been proposed n the past [-7, -8], but to our knowledge, none of these algorthms am at mnmzng the energy spent n the system. Most of these algorthms are heurstc n nature and ther am s to generate the mnmum number of clusters such that a node n any cluster s at the most d hops away from the clusterhead. In our context, generatng the mnmum number of clusters mght not ensure mnmum energy usage. In the Lnked Cluster Algorthm [], a node becomes the clusterhead f t has the hghest dentty among all nodes wthn one hop of tself or among all nodes wthn one hop of one of ts neghbors. Ths algorthm was mproved by the LCA algorthm [8], whch generates a smaller number of clusters. The LCA algorthm elects as a clusterhead the node wth the lowest d among all nodes that are nether a clusterhead nor are wthn -hop of the already chosen clusterheads. The algorthm proposed n [9], chooses the node wth hghest degree among ts hop neghbors as a clusterhead. In [4], the authors propose a dstrbuted algorthm that s smlar to the LCA algorthm. In [8], the authors propose two load balancng heurstcs for moble ad hoc networks. The frst heurstc, when appled to a node-d based clusterng algorthm lke LCA or LCA, leads to longer, low-varance clusterhead duraton. The other heurstc s for degree-based clusterng algorthms. Degree-based algorthms, n conuncton wth the proposed load balancng heurstc, produce longer clusterhead duraton. The Weghted Clusterng Algorthm (WCA) elects a node as a clusterhead based on the number of neghbors, transmsson power, battery-lfe and moblty rate of the node [7]. The algorthm also restrcts the number of nodes n a cluster so that the performance of the MAC protocol s not degraded. The Dstrbuted Clusterng Algorthm (DCA) uses weghts assocated wth nodes to elect clusterheads [5]. These weghts are generc and can be defned based on the applcaton. It elects the node that has the hghest weght among ts -hop neghbors as the clusterhead. The DCA algorthm s sutable for networks n whch nodes are statc or movng at a very low speed. The Dstrbuted and Moblty-Adaptve Clusterng Algorthm (DMAC) modfes the DCA algorthm to allow node moblty durng or after the cluster set-up phase [6]. All of the above algorthms generate -hop clusters, requre synchronzed clocks and have a complexty of O (n). Ths makes them sutable only for networks wth a small number of nodes. The Max-Mn d-cluster Algorthm proposed n [5] generates d-hop clusters wth a run-tme of O(d) rounds. Ths algorthm acheves better load balancng among the clusterheads, generates fewer clusters [5] than the LCA and LCA algorthms and does not need clock synchronzaton. In [7], the authors have proposed a clusterng algorthm that ams at maxmzng the lfetme of the network by determnng optmal cluster sze and optmal assgnment of nodes to clusterheads. They assume that the number of clusterheads and the locaton of the clusterheads are known a pror, whch s not possble n all scenaros. Moreover the algorthm requres each node to know the complete topology of the network, whch s generally not possble n the context of large sensor networks. McDonald et al. have proposed a dstrbuted clusterng algorthm for moble ad hoc networks that ensures that the probablty of mutual reachablty between any two nodes n a cluster s bounded over tme []. Henzelman et al. have proposed a dstrbuted algorthm for mcrosensor networks n whch the sensors elect themselves as clusterheads wth some probablty and broadcast ther decsons [6]. The remanng sensors on the cluster of the clusterhead that requres mnmum communcaton energy. Ths algorthm allows only -hop clusters to be formed, whch mght lead to a large number of clusters. They have provded smulaton results showng how the energy spent n the system changes wth the number of clusters formed and have observed that, for a gven densty of nodes, there s a number of clusters that mnmzes the energy spent. But they have not dscussed how to compute ths optmal number of clusterheads. The algorthm s run perodcally, and the probablty of becomng a clusterhead for each perod s chosen to ensure that every node becomes a clusterhead at least once wthn / P rounds, where P s the desred percentage of clusterheads. Ths ensures that none of the sensors are overloaded because of the added responsblty of beng a clusterhead. In [], the authors have consdered a -level herarchcal telecommuncaton network n whch the nodes at each level are dstrbuted accordng to two ndependent homogeneous Posson pont processes and the nodes of one level are connected to the closest node of the next hgher level. They 74

3 have then studed the moments and tal of the dstrbutons of characterstcs lke the number of lower level nodes connected to a partcular hgher level node and the total length of segments connectng the lower level nodes to the hgher level node n the herarchy. We use the results of ths paper to obtan the optmal parameters for our algorthm. Baccell and Zuyev have extended the above study to herarchcal telecommuncaton networks wth more than two levels n []. They have consdered a network of subscrbers at the lowest level connected to concentraton ponts at the hghest level, drectly or ndrectly through dstrbuton ponts. The subscrbers, dstrbuton ponts and the concentrators form the three levels n the herarchy and are dstrbuted accordng to ndependent homogeneous Posson processes. Assumng that a node s connected to the closest node of the next hgher level, they have used pont processes and stochastc geometry to determne the average cost of connectng nodes n the network as a functon of the ntensty of the Posson processes governng the dstrbuton of nodes at varous levels n the network. They have then derved the ntensty of the Posson process of dstrbuton ponts (as a functon of the ntenstes of the Posson processes of subscrbers and concentraton ponts) that mnmzes ths cost functon. They have also extended the above results for non-purely herarchcal models and have derved the optmal ntensty of Posson process of dstrbuton ponts numercally, gven the ntenstes of other two processes. They have then generalzed the cost functon for networks wth more than three levels. The algorthm proposed n ths paper s smlar to the clusterng algorthm n [6]. In [6], the authors have assumed that the sensors are equpped wth the capablty of tunng the power at whch they transmt and they communcate wth power enough to acheve acceptable sgnal-to-nose rato at the recever. We, on the other hand, assume a network n whch the sensors are very smple and all the sensors transmt at a fxed power level; data between two communcatng sensors not wthn each other s rado range s forwarded by other sensors n the network. The authors, n [6], have observed n ther smulaton experments that n a network wth one level of clusterng, there s an optmal number of clusterheads that mnmzes the energy used n the network. In ths paper, we have used the results provded n [] to obtan the optmal number of clusterheads at each level of clusterng analytcally, for a network clustered usng our algorthm to generate one or more levels of clusterng. III. A NEW, ENERGY-EFFICIENT, SINGLE-LEVEL CLUSTERING ALGORITHM A. Algorthm Each sensor n the network becomes a clusterhead (CH) wth probablty p and advertses tself as a clusterhead to the sensors wthn ts rado range. We call these clusterheads the volunteer clusterheads. Ths advertsement s forwarded to all the sensors that are no more than k hops away from the clusterhead. Any sensor that receves such advertsements and s not tself a clusterhead ons the cluster of the closest clusterhead. Any sensor that s nether a clusterhead nor has oned any cluster tself becomes a clusterhead; we call these clusterheads the forced clusterheads. Because we have lmted the advertsement forwardng to k hops, f a sensor does not receve a CH advertsement wthn tme duraton t (where t unts s the tme requred for data to reach the clusterhead from any sensor k hops away) t can nfer that t s not wthn k hops of any volunteer clusterhead and hence become a forced clusterhead. Moreover, snce all the sensors wthn a cluster are at most k hops away from the cluster-head, the clusterhead can transmt the aggregated nformaton to the processng center after every t unts of tme. Ths lmt on the number of hops thus allows the cluster-heads to schedule ther transmssons. Note that ths s a dstrbuted algorthm and does not demand clock synchronzaton between the sensors. The energy used n the network for the nformaton gathered by the sensors to reach the processng center wll depend on the parameters p and k of our algorthm. Snce the obectve of our work s to organze the sensors n clusters to mnmze ths energy consumpton, we need to fnd the values of the parameters p and k of our algorthm that would ensure mnmzaton of energy consumpton. We derve expressons for optmal values of p and k n the next subsecton. B. Optmal parameters for the algorthm To determne the optmal parameters for the algorthm descrbed above, we make the followng assumptons: a) The sensors n the wreless sensor network are dstrbuted as per a homogeneous spatal Posson process of ntensty λ n -dmensonal space. b) All sensors transmt at the same power level and hence have the same rado range r. c) Data exchanged between two communcatng sensors not wthn each others rado range s forwarded by other sensors. d) A dstance of d between any sensor and ts clusterhead s equvalent to d / r hops. e) Each sensor uses unt of energy to transmt or receve unt of data. f) A routng nfrastructure s n place; hence, when a sensor communcates data to another sensor, only the sensors on the routng path forward the data. g) The communcaton envronment s contenton- and error-free; hence, sensors do not have to retransmt any data. The basc dea of the dervaton of the optmal parameter values s to defne a functon for the energy used n the network to communcate nformaton to the nformaton-processng center and then fnd the values of parameters that would mnmze t. 75

4 ) Computaton of the optmal probablty of becomng a clusterhead: As per our assumptons, the sensors are dstrbuted accordng a homogeneous spatal Posson process and hence, the number of sensors n a square area of sde a s a Posson random varable, N wth mean λ A, where A 4a. Let us assume that for a partcular realzaton of the process there are n sensors n ths area. Also assume that the processng center s at the center of the square. The probablty of becomng a clusterhead s p ; hence, on average, np sensors wll become clusterheads. Let D be a random varable that denotes the length of the segment from a sensor located at ( x, y ),,,..., n to the processng center. Wthout loss of generalty, we assume that the processng center s located at the center of the square area. Then, D N n] x + y da a. () 4a A Snce there are on an average np CHs and the locaton of any CH s ndependent of the locatons of other CHs, the total length of the segments from all these CHs to the processng center s 0.765npa. Now, snce a sensor becomes a clusterhead wth probablty p, the clusterheads and the non-clusterheads are dstrbuted as per ndependent homogeneous spatal Posson processes PP and PP0 of ntensty λ pλ and λ ( )λ respectvely. 0 p For now, let us assume that we are not lmtng the maxmum number of hops n the clusters. Each non-clusterhead ons the cluster of the closest clusterhead to form a Vorono tessellaton [0]. The plane s thus dvded nto zones called the Vorono cells, each cell correspondng to a PP process pont, called ts nucleus. If N v s the random varable denotng the number of PP0 process ponts n each Vorono cell and L v s the total length of all segments connectng the PP0 process ponts to the nucleus n a Vorono cell, then accordng to results n [], λ0 E [ N v N v ] () λ λ E [ Lv Lv ]. () λ 0 / Defne C to be the total energy used by the sensors n a Vorono cell to communcate one unt of data to the clusterhead. Then, Lv C. (4) r Defne C to be the total energy spent by all the sensors communcatng unt of data to ther respectve clusterheads. Because, there are np cells, the expected value of C condtoned on N, s gven by E C np C N ]. (5) [ n If the total energy spent by the clusterheads to communcate the aggregated nformaton to the processng center s denoted by C, then, 0.765npa C. (6) r Defne C to be the total energy spent n the system. Then, C C + C np ( p) 0.765npa +. / r p λ r Removng the condtonng on N yelds: C] C ] p pa N ] + r pλ r p pa λa +. r pλ r E [C] s mnmzed by a value of p that s a soluton of / cp p 0. (9) The above equaton has three roots, two of whch are magnary. The second dervatve of the above functon s postve for the only real root of (9) and hence t mnmzes the energy spent. The only real root of (9) s gven by (7) (8) 76

5 + c c( + 7c p ( + 7c + where c. 06a λ. + + c c c 7c 7c + 4) + 4). (0) ) Computaton of the maxmum number of hops allowed from a sensor to ts clusterhead: Tll now we have not put any lmt on the number of hops ( k ) allowed between a sensor and ts clusterhead. Our man reason for lmtng k was to be able to fx a perodcty for the clusterheads at whch they should communcate to the processng center. So, f we can fnd the maxmum possble dstance (call t R max ) at whch a PP0 process pont can be from ts nucleus n a Vorono cell, we can fnd the value of k by assumng that a dstance R max from the nucleus s equvalent to R max / r hops. Settng k Rmax / r wll also ensure that there wll be very few forced clusterheads n the network. Snce t s not possble to get a value of R max such that we can say wth certanty that any pont of PP0 process wll be at the most Rmax dstance away from ts nucleus n the Vorono Tessellaton, we take a probablstc approach; we set Rmax to a value such that the probablty of any pont of PP0 process beng more than R max dstance away from all ponts of PP process s very small. Usng ths value of R max, we can get the value of parameter k that would make the probablty of any sensor beng more than k hops away from all volunteer clusterheads very small. Let ρ M be the radus of the mnmal ball centered at the nucleus of a Vorono cell, whch contans the Vorono cell. We defne p R to be the probablty that ρ M s greater than a certan value R,.e. p R P( ρ M > R). Then, t can be proved that p R 7 exp(.09λ R ) []. If R α s the value of R such that p R s less than α, then, R α 0.97 ln( α / 7). () p λ Ths means that the expected number of sensors that wll not on any cluster s n α f we set 0.97 ln( α / 7) k. () r p λ To ensure mnmum energy consumpton, we wll use a very small value for α, whch mples that the probablty of all sensors beng wthn k hops from at least one volunteer clusterhead s very hgh. For α and values of p and k computed accordng to (0) and (), for a network of 000 sensors, on an average sensor wll not on any volunteer clusterheads and wll become a forced clusterhead. The optmal value of p for a network wth 000 nodes n an area of 00 sq. unts s 0.08, whch means 80 nodes wll become volunteer clusterheads on an average. Hence, for a network of 000 nodes n an area of 00 sq. unts, only. % of all clusterheads are forced clusterheads. C. Smulaton Experments and Results We smulated the algorthm descrbed n Secton III for networks wth varyng sensor densty ( d ) and dfferent values of the parameters p and k. In all these experments, the communcaton range of each sensor was assumed to be unt. Fg. shows the output of one of these smulatons of our algorthm wth parameters p and k set to 0. and on a network of 500 sensors dstrbuted unformly n a square area of 00 square unts. To verfy that the optmal values of the parameters p and k of our algorthms computed accordng to (0) and () do mnmze the energy spent n the system, we smulated our clusterng algorthm on sensor networks wth 500, 000 and 000 sensors dstrbuted unformly n a square area of 00 sq. unts. Wthout loss of generalty, t s assumed that the cost of transmttng unt of data s unt of energy. The processng center s assumed to be located at the center of the square area. For the frst set of smulaton experments, we consdered a range of values for the probablty ( p ) of becomng a clusterhead n the algorthm proposed n Secton III. For each of these probablty values, we computed the maxmum number of hops ( k ) allowed n a cluster usng () and used these values for the maxmum number of hops allowed n a cluster n the smulatons. The results of these smulatons are provded n Fg.. Each data pont n Fg. corresponds to the average energy consumpton over 000 experments. It s evdent from Fg. that the energy spent n the network s ndeed mnmum at the theoretcally optmal values of the parameter p computed usng (0) (let us call ths optmal value p opt ), whch are gven n Table I for 500, 000 and 000 sensors n the network. 77

6 4500 T o t a l E n e r g y S p e n t n000 n000 n Probablty of becomng a clusterhead Fgure. Output of smulaton of the sngle level clusterng algorthm Most of the clusterng algorthms n the lterature (LCA [], LCA [8] and the Hghest Degree [9, 4] algorthms) have tme complexty of O (n), whch makes them less sutable for sensor networks that have large number of sensors. The Max- Mn d-cluster Algorthm [5] has a tme-complexty of O (d), whch may be acceptable for large networks. Hence, we have compared the performance of our proposed algorthm (wth optmal parameter values) and the Max-Mn d-cluster algorthm (for d,,, 4 ) n terms of the energy spent n the system usng smulaton. The experments were conducted for networks of dfferent denstes. For each network densty we used our algorthm (descrbed n Secton III) to cluster the sensors, wth the probablty of becomng a clusterhead set to the optmal value ( p opt ) calculated usng (0) and maxmum number of hops ( k ) allowed between any sensor and ts clusterhead equal to the value calculated usng p opt n (). Fgure. Total Energy Spent vs. probablty of becomng a clusterhead n algorthm n Secton III. T o t a l E n e r g y S p e n t d Densty of Sensors d d Our Algorthm d4 TABLE I. Number of Sensors ( n ) ENERGY MINIMIZING PARAMETERS FOR THE ALGORITHM Densty ( d ) Probablty ( p opt ) Maxmum Number of Hops ( k ) The computed values of p opt and the correspondng values of maxmum number of hops ( k ) n a cluster for networks of varous denstes are provded n Table I. The results of the smulaton experments are provded n Fg.. We observe that the proposed algorthm leads to sgnfcant energy savngs. The savngs n energy ncreases as the densty of sensors n the network ncreases. Fgure. Comparson of Our Algorthm and the Max-Mn D-Cluster Algorthms. IV. A NEW, ENERGY-EFFICIENT, HIERARCHICAL CLUSTERING ALGORTHM In Secton III, we have allowed only one level of clusterng; we now extend the algorthm to allow more than one level of clusterng. Assume that there are h levels n the clusterng herarchy wth level beng the lowest level and level h beng the hghest. In ths clustered envronment, the sensors communcate the gathered data to level- clusterheads (CHs). The level- CHs aggregate ths data and communcate the aggregated data or estmates based on the aggregated data to level- CHs and so on. Fnally, the level-h CHs communcate the aggregated data or estmates based on ths aggregated data to the processng center. The cost of communcatng the nformaton from the sensors to the processng center s the energy spent by the sensors to communcate the nformaton to level- clusterheads (CHs), plus the energy spent by the level- 78

7 CHs to communcate the aggregated nformaton to level- CHs,, plus the energy spent by the level-h CHs to communcate the aggregated nformaton to the nformaton processng center. A. Algorthm The algorthm works n a bottom-up fashon. The algorthm frst elects the level- clusterheads, then level- clusterheads, and so on. The level- clusterheads are chosen as follows. Each sensor decdes to become a level- CH wth certan probablty p and advertses tself as a clusterhead to the sensors wthn ts rado range. Ths advertsement s forwarded to all the sensors wthn k hops of the advertsng CH. Each sensor that receves an advertsement ons the cluster of the closest level- CH; the remanng sensors become forced level- CHs. Level- CHs then elect themselves as level- CHs wth a certan probablty p and broadcast ther decson of becomng a level- CH. Ths decson s forwarded to all the sensors wthn k hops. The level- CHs that receve the advertsements from level- CHs ons the cluster of the closest level- CH. All other level- CHs become forced level- CHs. Clusterheads at level, 4,..., h are chosen n smlar fashon, wth probabltes p, p4,..., ph respectvely, to generate a herarchy of CHs, n whch any level- CH s also a CH of level (-), (-),,. B. Optmal parameters for the algorthm The energy requred to communcate the data gathered by the sensors to the nformaton processng center through the herarchy of clusterheads wll depend on the probabltes of becomng a clusterhead at each level n the herarchy and the maxmum number of hops allowed between a member of a cluster and ts clusterhead. In ths secton, we obtan optmal values for the parameters of the algorthm descrbed n Secton IV-A that would mnmze ths energy consumpton. To do so, we make the same assumptons as n Secton III- B. Snce we have assumed that the sensors are ponts of a homogeneous Posson process of ntensty λ, the number of sensors n a square area of sde a s a Posson random varable (let s call ths N ) wth mean λ A, where A 4a s the area of the square. Let us assume that for a partcular realzaton of the process, there are n sensors n ths area. Let us also defne: N : the number of members n a level- cluster, L : the sum of dstances between the members of a level- cluster and ther level- CH, H : the number of hops from a member to ts CH n a typcal level- cluster, TCH : the total number of level- CHs, C : the total cost of communcatng nformaton from all level- CHs to the level-(+) CHs, and C : the total cost of communcatng nformaton from the sensors to the data processng center through the herarchy of clusterheads generated by the clusterng algorthms. In the proposed algorthm, the sensors elect themselves as level- CH wth probabltes p and the level- CHs elect themselves as level-(+) CHs wth probablty,,,..., ( ). Hence, by propertes of the p + h Posson process, level- CHs, homogeneous Posson processes of ntenstes,,,..., h are governed by λ p λ. By arguments smlar to those n Secton III-B., the sum of dstance of level-(-) CHs from a level- CH,,,..., h n a typcal level- cluster or the sum of dstance of sensors from a level- CH s gven by L ( p ) λ p λ p /. () The expected number of level-(-) CHs n a typcal level- cluster s gven by p N. (4) p Therefore, the expected number of hops between a level-(- ) CH and ts level- CH n a typcal level- cluster s gven by H L r N r λ p. (5) The expected number of level- CHs s gven by TCH n. (6) p 79

8 Hence, the expected total cost of communcatng nformaton from all the level-(-) CHs to ther respectve level- CHs,,..., ( h ), h s gven by C TCH N H N n]. (7) The expected value of the total cost of communcatng nformaton from all the sensors to ther level- CHs s gven by E [ C0 E TCH N H N ]. (8) [ n Hence, the expected total cost of communcatng nformaton from sensors to the processng center n the clustered envronment s gven by: C h 0.765a h n p + C ] N n r 0 h 0.765a n p r h + n ( p ) r ( p ). By un-condtonng on N, we fnd: C] C ] h λ A p h 0.765a r + λa ( p ) r λ p ( p ). λ p (9) (0) As apparent from Fg. 6 and Fg. 7, the functon n (0) has a very complex form wth many local mnma. Even f the celng of an expresson s approxmated by ust the expresson n (0), closed-form solutons for probabltes p,,,..., h that mnmze the resultng cost of communcaton E [C] have not been obtaned, but can be found numercally. Once the optmal probabltes are obtaned, followng the same arguments as n secton III-B., k,,,..., h can be calculated accordng to the equaton, k r 0.97 ln( α / 7). () λ p In the above equaton, α denotes the probablty that the number of hops between a member and the clusterhead n a level- cluster s more than k,,,..., h. C. Numercal Results and Smulatons We smulated the algorthm descrbed n Secton IV-A on networks of sensors dstrbuted unformly wth varous spatal denstes. In all cases, we assumed that unt of energy spent n communcatng unt of data. We use the algorthm to generate a clusterng herarchy wth dfferent number of levels n t to see how the energy spent n the network reduces wth the ncrease n number of levels of clusters. In these smulatons, we have used the numercally computed set of optmal probabltes (that mnmzes E [C] gven by (0)) of becomng clusterheads at each level n the clusterng herarchy. Fg. 4. and Fg. 5 show how the energy consumpton decreases as the number of levels n the herarchy ncreases. e(total Energy Spent) Log r r4 r n 5,000 Area 5,000 sq. unts Number of levels n the clusterng herarchy Fgure 4. Total Energy Spent vs. number of levels n the clusterng herarchy n a network of 5000 sensors wth communcaton rad r dstrbuted n a square area of 5000 sq. unts. 70

9 e(total Energy Spent) Log λ.5 λ 5 λ 0 n 5,000 r unts Number of levels n the clusterng herarchy Fgure 5. Total Energy Spent vs. number of levels n the clusterng herarchy n a network of 5000 sensors of communcaton radus dstrbuted wth spatal densty λ. In Fg. 4, we observe that the energy savngs are hgher for networks of sensors wth lower communcaton radus. These results can be explaned as follows. In networks of sensors wth hgher communcaton radus, the dstance between a sensor and the processng center n terms of number of hops s smaller than the dstance n networks of sensors wth lower communcaton radus and hence there s lesser scope of energy savngs. The energy savngs wth ncrease n the number of levels n the herarchy are also observed to be more sgnfcant for lower densty networks. Ths can be attrbuted to the fact that among networks of same number of sensors, the networks wth lower densty has the sensors dstrbuted over a larger area. Hence, n a lower densty network, the average dstance between a sensor and the processng center s larger as compared to the dstance n a hgher densty network. Ths means that there s more scope of reducng the dstance traveled by the data from any sensor n a non-clustered network, thereby reducng the overall energy consumpton. Snce data from each sensor has to travel at least one hop, the mnmum possble energy consumpton n a network wth n sensors s n, assumng each sensor transmts unt of data and the cost of dong so s unt of energy. From Fg. 4 and Fg. 5, t s apparent that the energy consumpton s very close to ths value when the number of levels n the herarchy s 5, rrespectve of the densty of sensors and ther communcaton radus. Hence, f one chooses to store the numercally computed values of optmal probablty n the sensor memory, only a small amount of memory would be needed. Hence, they may run out of ther energy faster than other sensors. As proposed n [6], the clusterng algorthm can be run perodcally for load balancng. Instead of runnng the algorthm perodcally, another possblty s that clusterheads trgger the clusterng algorthm when ther energy levels fall below a certan threshold. Among many other ssues, the behavor of the proposed clusterng algorthm and the herarchy generated by t n event of sensor falures s worth nvestgatng. VI. CONCLUSIONS AND FUTURE WORK We have proposed a dstrbuted algorthm for organzng sensors nto a herarchy of clusters wth an obectve of mnmzng the total energy spent n the system to communcate the nformaton gathered by these sensors to the nformaton-processng center. We have found the optmal parameter values for these algorthms that mnmze the energy spent n the network. In a contenton-free envronment, the algorthm has a tme complexty of O ( k + k k h ), a sgnfcant mprovement over the many O (n) clusterng algorthms n the lterature [,,4,8,9]. Ths makes the new algorthm sutable for networks of large number of nodes. In ths paper, we have assumed that the communcaton envronment s contenton and error free; n future we ntend to consder an underlyng medum access protocol and nvestgate how that would affect the optmal probabltes of becomng a clusterhead and the run-tme of the algorthm. V. ADDITIONAL CONSIDERATIONS The sensors whch become the clusterhead n the proposed archtecture spend relatvely more energy than other sensors because they have to receve nformaton from all the sensors wthn ther cluster, aggregate ths nformaton and then communcate to the hgher level clusterheads or the nformaton processng center. 7

10 Fgure 6. Plot of the energy functon n (0) when there are two levels of clusterheads n a network of 0000 sensors of communcaton range of 4 unts dstrbuted n an area of 500 sq. unts. Fgure 7. Contour plot of the energy functon n (0) when there are two levels of clusterheads n a network of 0000 sensors of communcaton range of 4 unts dstrbuted n an area of 500 sq. unts. 7

11 REFERENCES [] G. J. Potte and W. J. Kaser, Wreless Integrated Network Sensors, Communcatons of the ACM, Vol. 4, No. 5, pp 5-58, May 000. [] D. J. Baker and A. Ephremdes, The Archtectural Organzaton of a Moble Rado Network va a Dstrbuted Algorthm, IEEE Transactons on Communcatons, Vol. 9, No., pp , November 98. [] B. Das and V. Bharghavan, Routng n Ad-Hoc Networks Usng Mnmum Connected Domnatng Sets, n Proceedngs of ICC, 997. [4] C. R. Ln and M. Gerla, Adaptve Clusterng for Moble Wreless Networks, Journal on Selected Areas n Communcaton, Vol. 5 pp , September 997. [5] A. D. Ams, R. Prakash, T. H. P. Vuong and D. T. Huynh, Max-Mn D-Cluster Formaton n Wreless Ad Hoc Networks, n Proceedngs of IEEE INFOCOM, March 000. [6] W. R. Henzelman, A. Chandrakasan and H. Balakrshnan, Energy- Effcent Communcaton Protocol for Wreless Mcrosensor Networks, n Proceedngs of IEEE HICSS, January 000. [7] C.F. Chassern, I. Chlamtac, P. Mont and A. Nucc, Energy Effcent desgn of Wreless Ad Hoc Networks, n Proceedngs of European Wreless, February 00. [8] A. Ephremdes, J.E. Weselther and D. J. Baker, A Desgn concept for Relable Moble Rado Networks wth Frequency Hoppng Sgnalng, Proceedng of IEEE, Vol. 75, No., pp. 56-7, 987. [9] A. K. Parekh, Selectng Routers n Ad-Hoc Wreless Networks, n Proceedngs of ITS, 994. [0] A. Okabe, B. Boots, K. Sughara and S. N. Chu, Spatal Tessellatons: Concepts and Applcatons of Vorono Dagrams, nd edton, John Wley and Sons Ltd. [] S.G.Foss and S.A. Zuyev, On a Vorono Aggregatve Process Related to a Bvarate Posson Process, Advances n Appled Probablty, Vol. 8, no. 4, pp ,996. [] J. M. Kahn, R. H. Katz and K. S. J. Pster, Next Century Challenges: Moble Networkng for Smart Dust, n the Proceedngs of 5th Annual ACM/IEEE Internatonal Conference on Moble Computng and Networkng (MobCom 99), Aug. 999, pp [] F. Baccell and S. Zuyev, Posson Vorono Spannng Trees wth Applcatons to the Optmzaton of Communcaton Networks, Operatons Research, vol. 47, no. 4, pp. 69-6, 999. [4] B. Warneke, M. Last, B. Lebowtz, Krstofer and S. J. Pster, Smart Dust: Communcatng wth a Cubc-Mllmeter Computer, Computer Magazne, Vol. 4, No., pp 44-5, Jan. 00. [5] J. M. Kahn, R. H. Katz and K. S. J. Pster, Next Century Challenges: Moble Networkng for Smart Dust, n the 5th Annual ACM/IEEE Internatonal Conference on Moble Computng and Networkng (MobCom 99), Aug. 999, pp [6] V. Hsu, J. M. Kahn, and K. S. J. Pster, "Wreless Communcatons for Smart Dust", Electroncs Research Laboratory Techncal Memorandum M98/, Feb [7] [8] P. Gupta and P. R. Kumar, The Capacty of Wreless Networks,, IEEE Transactons on Informaton Theory, vol. IT-46, no., pp , March 000. [9] P. Gupta and P. R. Kumar, Crtcal Power for Asymptotc Connectvty n Wreless Networks, pp , n Stochastc Analyss, Control, Optmzaton and Applcatons: A Volume n Honor of W.H. Flemng. Edted by W.M. McEneany, G. Yn, and Q. Zhang, Brkhauser, Boston, 998. ISBN [0] W. Ye, J. Hedemann, and D. Estrn, An Energy-Effcent MAC Protocol for Wreless Sensor Networks, In Proceedngs of the st Internatonal Annual Jont Conference of the IEEE Computer and Communcatons Socetes (INFOCOM 00), New York, NY, USA, June, 00. [] N. Bulusu, D. Estrn, L. Grod, and J. Hedemann, Scalable Coordnaton for Wreless Sensor Networks: Self-Confgurng Localzaton Systems, In Proceedngs of the Sxth Internatonal Symposum on Communcaton Theory and Applcatons (ISCTA 00), Amblesde, Lake Dstrct, UK, July 00. [] N. Bulusu, J. Hedemann, and D. Estrn, Adaptve beacon Placement, Proceedngs of the Twenty Frst Internatonal Conference on Dstrbuted Computng Systems (ICDCS-), Phoenx, Arzona, Aprl 00. [] A. B. McDonald, and T. Znat, A Moblty Based Framework for Adaptve Clusterng n Wreless Ad-Hoc Networks, IEEE Journal on Selected Areas n Communcatons, Vol. 7, No. 8, pp , Aug [4] M. Gerla, and J. T. C. Tsa, Multcluster, Moble, Multmeda Rado Networks, Wreless Networks, Vol., No., pp , 995. [5] S. Basagn, Dstrbuted Clusterng for Ad Hoc Networks, n Proceedngs of Internatonal Symposum on Parallel Archtectures, Algorthms and Networks, pp. 0-5, June 999. [6] S. Basagn, Dstrbuted and Moblty-Adaptve Clusterng for Multneda Support n Mult-Hop Wreless Networks, n Proceedngs of Vehcular Technology Conference, Vol., pp , 999. [7] M. Chatteree, S. K. Das, and D. Turgut, WCA: A Weghted Clusterng Algorthm for Moble Ad hoc Networks, Journal of Cluster Computng, Specal ssue on Moble Ad hoc Networkng, No. 5, 00, pp [8] A.D. Ams, and R. Prakash, Load-Balancng Clusters n Wreless Ad Hoc Networks, n Proceedngs of ASSET 000, Rchardson, Texas, March 000. [9] A. Perrg, R. Szewczyk, V. Wen and J. D. Tygar, SPINS: Securty protocols for Sensor Networks, n 7th Annual Internatonal Conference on Moble computng and Networkng, 00, pp [0] D. W. Carman, P. S. Kruus, and B. J. Matt, Constrants and approaches for dstrbuted sensor network securty, NAI Labs Techncal Report 00-00, September