Distributed Threshold Selection for Aggregate Threshold Monitoring in Sensor Networks

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1 Dstrbuted Threshold Selecton for Aggregate Threshold Montorng n Sensor Networks Mohammad S. Taleb School of Computer Scence, IPM, Iran mstaleb@pm.r Ahmad Khonsar School of Computer Scence, IPM, Iran ECE Department, Unversty of Tehran ak@pm.r Al Abbas ECE Department, Unversty of Tehran a.abbas@ece.ut.ac.r Abstract Motvated by applcatons lke sensor, peer to peer, ad hoc networks there has been growng nterest n montorng large scale dstrbuted systems. In these applcatons typcally we wsh to montor a global system condton whch s defned as a functon of local network elements parameters. In ths paper, we study Aggregate Threshold Queres n sensor networks, whch are used to detect when an aggregate value of all sensor measurements crosses a predetermned threshold. The major constrant n desgnng montorng applcatons s reducng the amount of communcaton burden whch s the domnant factor of energy dran n wreless sensor networks. In ths study, we address the aggregate threshold montorng problem by proposng a dstrbuted algorthm to set local thresholds on each sensor node so that only those sensors whose measurements crosses ther local thresholds commence communcaton. We adopt the FPTAS optmzaton formulaton of the problem [1] and propose a dstrbuted algorthm as the soluton to the problem. Smulaton results demonstrate the valdty of the proposed dstrbuted algorthm n attanng very close performance as the centralzed scheme. Index Terms Aggregate Threshold Montorng, Sensor Networks, Optmzaton, Iteratve Methods. I. INTRODUCTION In recent years, wth the abundance of emergng large scale dstrbuted systems such as sensor networks, and peer to peer networks, montorng scenaros s progressvely more consdered vtal to track these systems. In montorng ether we are nterested n supervsng the network tself (e.g. traffc engneerng, routng optmzaton, anomaly detecton n data networks, or envronment that network deployed n t (e.g. wldlfe behavor, movng objects n sensor networks. Network montorng ncludes measurng system parameters to react to dfferent network condtons. There are two ways of gettng knowledge from the network; the frst s to send requests nto the network to poll all relevant nformaton. Another opton s that all network elements push all possbly mportant readngs to the management staton. The nature of network montorng applcatons mposes specfc requrements on desgn of montorng algorthm: 1- The algorthm should provde real-tme detecton of noteworthy events. 2- The algorthm must be scalable to large number of nodes. 3- The detecton process should ncur mnmum communcaton n the network. Thereby, a challenge n montorng applcatons s to devse plans to reduce communcaton whle fulfllng applcaton requrements. A common method s to nstall local constrants or flters at remote sources to flter out unnecessary updates. Local constrants should have the property such that preservng all of them ensures that there was not any anomalous event. Each new measurement s compared to the flter and n case t volates (exts the correspondng range of the flter an update s sent to the base staton. Apparently effcent decomposton of global system constrant nto local constrant has a great mpact on reducng communcaton. Flter settng s dependent on a number of parameters ncludng the assocated cost wth sendng an update from each node to the management staton and the changng patterns of measurements. Often the montorng task s the need to detect when a functon of ndvdual network element readngs crosses a gven threshold. Typcally, montored functons are aggregate functons lke SUM, or AVG whch gves global nsghts concernng the state of the network. Consder the followng queres: Report when the number of enemy troops detected n a regon of the network crosses 20. Report when varance of temperature n a buldng crosses 3. Aggregate functons lke SUM, AVG are lnear functons of ndvdual sensor readngs, thus the global property of A T T can be decomposed to a set of x T constrants whch checked locally at each node. In ths work, we concentrate on the problem of determnng optmal values for T s. Prevous work [1] assumed that the montorng staton computes the optmal thresholds based on ndvdual nodes probablty dstrbuton functons and assgns thresholds for nodes. Thus, each node should update ts hstogram constructed over recent measurements ether perodcally or based on a change detecton algorthm to the base staton. But there s lmtaton on the applcablty of such an approach for sensor networks. The structure of such networks has necesstated the desgn of asynchronous, dstrbuted and faulttolerant computaton and nformaton exchange algorthms [4]. In ths work, we propose a dstrbuted algorthm for optmal threshold assgnment so as to mnmze the probablty of global system pollng. The remander of the paper s organzed as follows: Secton II revews related works. Secton III defnes the system model

2 and problem defnton. Secton IV provdes the optmal soluton to the threshold assgnment problem. Secton V presents a dstrbuted threshold assgnment algorthm based on the problem s optmal soluton. Secton VI valdates expermental evaluaton of the proposed method. Fnally, Secton VII concludes the study. II. RELATED WORKS In recent years, contnuous query processng for montorng dstrbuted data streams has attracted much research. Montorng aggregate threshold queres n networks ntally mentoned by the poneerng work of Raz et al [8]. They ntroduced nstallng local mathematcal constrants at remote stes and present a smple approach for threshold assgnment assumng unform data dstrbutons of system varables. Keralpera et al [12] presented several algorthms for statc and adaptve threshold settng for montorng thresholded counts queres and analyzed the communcaton complexty of each algorthm. Sharfman et al [16] ntroduced a geometrc approach for montorng arbtrary threshold functons. Recent work of Kashyap et al [10] consdered the problem of nonzero slack threshold assgnment whch adaptvely dedcates the fracton of total threshold to montorng node to absorb small local threshold crossng that elmnates the need for global system pollng. III. SYSTEM MODEL AND PROBLEM FORMULATION We model the topology of the sensor network by an undrected graph G =(V, E. The vertex set V = {1, 2,...,n} and edge set E V Vdenotes the set of nodes and lnks, respectvely. For the edge set, we have (, j E f and only f there s a connecton between node and j. Wealsorefer to the neghbors of node as N = {j V (, j E}. We assume that there s a base staton beng responsble for montorng the network. The base staton dssemnates queres to sensors and responds to user queres. Each sensor node contnuously measures (reads the requested local phenomenon x (for example, humdty, lght whch s n the range [0,M ] at a fxed samplng rate. The goal s to detect when the aggregate value of all the measurements n A x crosses a predetermned threshold T. To compute the aggregate we can perform value montorng va global pollng or algorthm lke [14]. Accordng to [5] value montorng drectly solves threshold montorng, and runnng O( 1 ɛ log T nstances of a threshold montorng algorthm for thresholds 1, 1+ɛ, (1 + ɛ 2,...,T solves value montorng wth relatve error ɛ. So, the two varants dffer by at most a factor of O( 1 ɛ log T. Thus determnng whether an aggregate has crossed a threshold does not requre knowng the exact value of the aggregate. A common method to reduce communcaton s to nstall local constrants or flters (n the form of an nterval [0,T ]at remote sources. Local constrants settng should mantan the property that preservng all of them, ensures that the aggregate value has not exceeded the threshold (coverng property. Each sensor after measurng the new value of x, checks condton L (x [0,T ], and the condton upon beng volated, an update s sent to the base staton to ntate global aggregate computaton. Usng ths method, vast amount of updates are fltered out at the source and are not transmtted to the base staton. The effcency of ths approach s largely dependent on flter settng method. But what values should we choose for T s? Selecton of values must satsfy the global property n A T T to conform coverng property. However, ths equaton leads to great flexblty n choosng T s. In ths respect, we face to another queston: what s the best selecton? Our goal s to mnmze communcaton cost, towards whch we should mnmze the probablty of global pollng. Every local flter volaton leads to global computaton, thus we should maxmze the probablty of preservng all local constrants max F (x 1 T 1,...,x n T n, where F s the jont cumulatve frequency dstrbuton over all the sensors. Computng the mult-dmensonal hstograms s somewhat cumbersome n terms of communcaton burden, and hence we consder more communcaton convenent assumpton of ndependence between sensor s data dstrbutons. Wth ths smplfyng assumpton, we have Pr(L = true = n F (T F (M Snce F (M s are constant, the optmzaton problem can be formulated as n max F (T (2 T subject to: (1 n A T T (3 where T =(T,..n s the vector representaton of local thresholds. Accordng to [1] local threshold selecton problem s NP-hard. They ntroduced a centralzed scheme, referred to as FPTAS, to solve the problem wthn ɛ relatve error for arbtrarly small ɛ. The soluton was based on the assumpton that all the values of F (T are ntegral powers of a constant α, whch s slghtly greater than 1. Thus, each F (T wll correspond to some α r and maxmzng n F (T wll be equvalent to maxmzng α r.lett (r be the local threshold of node such that α r F (T (r <α r+1 (4 Then, the optmzaton problem becomes n max r (5 r subject to: n A T (r T (6 where r =(r,..n s the vector representaton of powers. In order for (5 to admt a unque maxmzer, we argue that F must be an ncreasng and strctly concave functon. Indeed,

3 we assume that optmzaton varable, r,..n, belongs to a doman n whch F satsfes these assumptons. IV. OPTIMAL SOLUTION In ths secton, we solve problem (5. Problem (5 s a constraned problem, whose constrant (6 s coupled across the network. Such a constraned optmzaton problem can be effcently solved usng Interor Pont Method [4], whch necesstates the coordnaton among possbly all nodes of the networks, whch s undesrable or nfeasble. However, n the context of wreless ad-hoc and sensor networks, we are nterested n dstrbutve algorthms to solve (5. Towards ths end, we am at solvng problem (5 through ts dual. In ths respect, problem (5 s called prmal problem. In the sequel, we proceed to derve the dual problem of (5 and then present a dstrbutvely teratve algorthm as the soluton to the dual problem. A. Dervng Dual Problem We start by wrtng the Lagrangan of problem (5, as follows ( n n L(r,μ= r μ A T (r T (7 where μ > 0 s the Lagrange multpler assocated wth constrant (6. Usng Karush-Kuhn-Tucker (KKT condtons for convex optmzaton, to fnd optmal prmal varables,.e. optmal powers r, we should fnd the statonary ponts of the Lagrangan and satsfy complementary slackness condtons. The complementary slackness condtons for optmal prmal varable r and dual varable μ,are μ 0; (8 n A T (r T ; (9 ( n μ A T (r T =0 (10 In order to fnd the statonary ponts of the Lagrangan, we solve L r (r,μ =0 (11 where 0 s a vector wth all zero. For the th element of (11 we have L dt (r =1 μa (12 r dr From (4, recall that T (r s selected so that (4 s satsfed. Also, recall that α s a constant slghtly greater than 1. Therefore, the lower and upper bounds n (4 are suffcently close to each other, leadng us to approxmate F (T (r, usng the concept of geometrc mean as followng: F (T (r α r α r+1 αα r α r (13 Recallng the monotoncty assumpton of CDFs, each CDF admts a unque nverse, whch yelds the explct expresson for T as T (r =F 1 (α r (14 Substtutng (14 n (12, yelds L df 1 (α r = 1 μa (15 r dr dα r df 1 = 1 μa dr dr α r 1 df = 1 μa α r ln α dr (16 α r Settng (16 to zero and dong some algebrac manpulatons, gves an explct expresson for the optmal power r n terms of optmal Lagrange multpler μ. For the sake of presentaton, we defne 1 df G (z = dr (17 z Substtutng G (. n (16, we come up to the followng mplct equaton to obtan r 1 α r G (α r = μ (18 A ln α In order to solve problem (5 through ts dual, we need to obtan the Lagrange dual functon, or smply dual functon. The Lagrange dual functon D(μ s defned as the maxmum of the Lagrangan L(r,μ over the prmal varable r, fora gven μ. Thus, D(μ can be expressed as D(μ = max L(r,μ (19 r Based on the KKT condton and usng (18, for D(μ we get The dual problem s formulated as D(μ =L(r,μ (20 mn D(μ mn μ 0 μ 0 L(r,μ (21 Dual problem defned above can be solved usng teratve methods. In order to obtan a dstrbuted algorthm, we solve the dual problem (21 usng Gradent Projecton algorthm. We postpone solvng (21 to the next subsecton. B. Solvng The Dual In ths subsecton, we solve the dual problem usng Gradent Projecton Algorthm. To solve the dual problem, Gradent Projecton Algorthm adjusts μ n opposte drecton to the Gradent of dual functon,.e. D(μ. Precsely speakng, n the kth teraton step, μ (k s updated as follows [ ] + μ (k+1 = μ (k γ dd(μ(k (22 dμ

4 where [z] + = max{z,0} and γ s a suffcently small constant step sze. Usng the Danskn s Theorem [3], the dervatve of D(μ s gven by dd(μ dμ = T n A T (r (23 Substtutng (23 n (22, yelds [ ( n ]+ μ (k+1 = μ (k + γ A T (r (k T (24 where r (k s the soluton to (18 for a gven μ (k.inths equaton, γ s chosen suffcently small so as to guarantee the convergence. In the economcs lterature, Lagrange multpler or dual varable, μ s called shadow prce [11] and accordngly, (24 s called shadow prce update. Ths stems from the nterpretaton of ts role n solvng the prmal problem va ts dual. From (18 t s apparent that r s a decreasng functon of μ; therefore μ can be construed as the prce whch must be pad by node to acheve the threshold T (r. As the nature of such a prce s hdden to nodes from the prmal problem perspectve, t s called shadow prce. (18 and (24 form an teratve soluton to problem (21 and thereby problem (5. At each teraton step k, dual varable μ wll be updated based on the hstory of tself and the prmal varables r. Then, t would be utlzed by each node to update prmal varable r, accordngly. Therefore, after spendng enough teraton steps, prmal and dual varables tends to prmal-optmal r and dual-optmal μ, respectvely. Based on the above teratve soluton, we propose a dstrbuted algorthm as a dstrbuted soluton to the threshold selecton problem. We deffer the algorthm untl Secton V. V. ALGORITHM In ths secton, we propose a dstrbuted algorthm based on the teratve soluton obtaned n Secton IV. Consderng (24 and (18, t s clear that the teratve soluton to problem (5 can be exploted to desgn a dstrbuted algorthm. We elaborate to employ a heurstc mechansm, proposed by Jadbabae et al [9] to come up wth a fully dstrbuted algorthm whch only requres local nformaton of the network. Snce each sensor s supposed to update μ locally, the realzed update evoluton wll dffer between sensor nodes. To avod confuson between the updated value at each node, we ntroduce ˆμ (k notaton, by whch we mean the updated value of the shadow prce at node n teraton step k. The algorthm evolves as followng: At each teraton step k, each sensor exchanges ts data, ˆμ (k and r (k, wth ts neghborng nodes,.e. those belongng to N. Then, t proceeds to update ts estmaton of the shadow prce ˆμ (k, accordng to the nearest neghbor rule, proposed by Jadbabae et al [9], whch s modfed as followng: NDTA Neghbor-based Dstrbuted Threshold Assgnment Algorthm Intalzaton Intalze the followng tems: 1. A sandt. 2. S = n A. 3. =1..n, S = N A. Man Loop Do untl max r (k+1 r (k <ɛ At each sensor node, 1. Update the shadow prce as followng: ˆμ (k+1 [ = ˆμ (k + γ 2. Update r (k ( N A T (r (k α r(k+1 G (α r(k+1 = T S S ] + accordng to the followng equaton: 1 ˆμ (k A ln α NDTA. Neghbor-based Dstrbuted Threshold Assgnment Algorthm [ ( n ˆμ (k+1 = ˆμ (k + γ A T (r (k N A n A T ]+ (25 where N A / n A denotes the fracton of total threshold T assgned to the vrtual subnetwork establshed by nodes {} N. We wll refer to ths algorthm as Neghborbased Dstrbuted Threshold Assgnment Algorthm or for short NDTA Algorthm. The NDTA Algorthm s stated below. VI. EXPERIMENTAL EVALUATION We have conducted smulaton experments to evaluate the performance of our proposed algorthm. We verfy that NDTA Algorthm, locally executed on each node, may ndeed acheve the desred global optmal threshold assgnment. In our smulaton scenaro, we consder a sensor network consstng of 100 sensor nodes. We assume that each node ncessantly takes measurements of a physcal phenomenon, whose CDF obeys an exponental dstrbuton wth the exponent parameter λ. Although such a dstrbuton may sound to be of lmted nterest, t s worth mentonng that many sgnfcant real world applcatons mght fall wthn such a framework. Amongst such applcatons are montorng the dwell tme of a traffc flow whch pursue a Posson dstrbuton. The correspondng coeffcent A s assumed to be randomly drawn from a unform dstrbuton over [0, 5]. Step sze s chosen to be γ =1.2and the total threshold T s set to 10. The most sgnfcant ssues of nterest are the evolutons of prmal and dual (shadow prce varables. Evoluton of assgned thresholds T s and shadow prce μ for NDTA Algorthm are depcted n Fg. 1 and 2, respectvely. It s apparent from these fgures that by spendng less than 300 teraton steps, convergence was acheved and thereafter, μ and T s had ntangble varatons. NDTA s ndeed a suboptmal algorthm. In order to compare the performance of NDTA wth the optmal scheme, threshold assgnment for both NDTA and optmal scheme

5 Fg. 1. Evoluton of The Assgned Thresholds T (r for Some Nodes Usng NDTA Algorthm Fg. 3. Comparson Between NDTA and Centralzed Scheme whch acts usng only local nformaton of the network usng the data whch has been provded by the neghborng nodes. The results extracted from the expermental evaluaton was promsng and demonstrated the acheved performance of the NDTA algorthm s qute comparable to the results of the centralzed approach. Fg. 2. Evoluton of The Shadow Prce μ for Some Nodes Usng NDTA Algorthm whch s obtaned usng a centralzed algorthm s depcted n Fg. 3, whch shows that the result of NDTA s qute close to the optmal case. VII. CONCLUSION There have been many studes explorng varous applcatons of WSNs such as montorng. In ths paper, we addressed the problem of dstrbuted threshold selecton for aggregate threshold montorng n WSNs. The nature of such networks has nduced the desgn of dstrbuted algorthms for exchangng the nformaton among the sensor nodes. We formulated threshold selecton problem as an optmzaton problem that consders the data dstrbuton of dstnct montorng varables, and whose objectve s to mnmze the probablty of global pollng. The orgnal problem was non-convex, thus we adopted the so called FPTAS reformulaton whch was convex and has been solved usng a centralzed approach. We elaborated ths method and solved the problem va ts dual so as to acheve a dstrbuted soluton. Our dstrbuted soluton leads to the fully dstrbuted algorthm called NDTA, REFERENCES [1] S. Aggrawal, S. Deb, K. V. M. Nadu and R. Rastog, Effcent Detecton of Dstrbuted Constrant Volatons, In ICDE, pp , [2] B. Babcock and C. Olston, Dstrbuted Top-k Montorng, In SIGMOD, pp , [3] D. Bertsekas, Nonlnear Programmng, Athena Scentfc, [4] S. Boyd, A. Ghosh, B. Prabhakar and D. Shah, Gossp Algorthms: Desgn, Analyss, and Applcatons, In INFOCOM, pp , [5] G. Cormode, S. Muthurkshnan, K. Y. Algorthms for Dstrbuted Functonal Montorng, In SODA, [6] A. Deshpande, C. Guestrn, S. Madden, J. M. Hellersten and W. Hong, Model-Drven Data Acquston n Sensor Networks, In VLDB, pp , [7] A. Deshpande, C. Guestrn and S. Madden, Usng Probablstc Models for Data Management n Acqustonal Envronments, In CIDR, pp , [8] M. Dlman and D. Raz, Effcent Reactve Montorng, In INFOCOM, pp , [9] A. Jadbabae, J. Ln and A. Stephen Morse, Coordnaton of Groups of Moble Autonomous Agents Usng Nearest Neghbor Rules, In IEEE Transactons on Automatc Control, vol. 48, no. 6, pp , June [10] S. Kashyap, J. Ramamrtham, R. Rastog and P. Shukla, Effcent Constrant Montorng Usng Adaptve Thresholds, In ICDE, pp , [11] F. P. Kelly, A. Maulloo and D. K. H. Tan, Rate Control for Communcaton Networks: Shadow Prces, Proportonal Farness, and Stablty, Operatonal Research Socety, vol. 49, no. 3, pp , [12] R. Keralapura, G. Cormode and J. Ramamrtham, Communcaton- Effcent Dstrbuted Montorng of Thresholded Counts, In SIGMOD, pp , [13] D. Kfer, B. Sha and J. Gehrke, Detectng Change n Data Streams, In VLDB, pp , [14] C. Olston, J. Jang, and J. Wdom. Adaptve flters for contnuous queres over dstrbuted data streams. In SIGMOD, [15] V. Poosala and Y. Ioannds, Selectvty Estmaton Wthout The Attrbute Value Independence Assumpton, In VLDB, pp , [16] I. Sharfman, A. Schuster and D. Keren, A Geometrc Approach to Montorng Threshold Functons over Dstrbuted Data Streams, In SIGMOD, pp , 2006.