WIRELESS sensor networks are used in a wide range of

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1 IEEE/ACM TRANSACTIONS ON NETWORKING, VOL X, NO X, MONTH 2X Optmzng Lfetme for Contnuous Data Aggregaton wth Precson Guarantees n Wreless Sensor Networks Xueyan Tang, Member, IEEE, and Janlang Xu, Member, IEEE Abstract Ths paper explots the tradeoff between data qualty and energy consumpton to extend the lfetme of wreless sensor networks To obtan an aggregate form of sensor data wth precson guarantees, the precson constrant s parttoned and allocated to ndvdual sensor nodes n a coordnated fashon Our key dea s to dfferentate the precsons of data collected from dfferent sensor nodes to balance ther energy consumpton Three factors affectng the lfetme of sensor nodes are dentfed: the changng pattern of sensor readngs; 2 the resdual energy of sensor nodes; and 3 the communcaton cost between the sensor nodes and the base staton We analyze the optmal precson allocaton n terms of network lfetme and propose an adaptve scheme that dynamcally adjusts the precson constrants at the sensor nodes The adaptve scheme also takes nto consderaton the topologcal relatons among sensor nodes and the effect of n-network aggregaton Expermental results usng real data traces show that the proposed scheme sgnfcantly mproves network lfetme compared to exstng methods Index Terms data aggregaton, data accuracy, energy effcency, network lfetme, sensor network I INTRODUCTION WIRELESS sensor networks are used n a wde range of applcatons to capture, gather and analyze lve envronmental data [], [2] A wreless sensor network typcally conssts of a base staton and a group of sensor nodes see Fgure The sensor nodes are responsble for contnuously samplng physcal phenomena such as temperature and humdty They are also capable of communcatng wth each other and the base staton through rados The base staton, on the other hand, serves as a gateway for the sensor network to exchange data wth applcatons to accomplsh ther mssons Whle the base staton can have contnuous power supply, the sensor nodes are usually battery-powered The batteres are nconvenent and sometmes even mpossble to replace When a sensor node runs out of energy, ts coverage s lost The msson of a sensor applcaton would not be able to contnue f the coverage loss s remarkable Therefore, the practcal Manuscrpt receved October 24, 26; revsed February 23, 27; approved by IEEE/ACM TRANSACTIONS ON NETWORKING Edtor N Shroff Ths work was supported n part by a grant from Nanyang Technologcal Unversty Project No RG47/6 Janlang Xu s work was supported n part by grants from the Research Grants Councl of the Hong Kong SAR, Chna Project Nos HKBU 25/5E and FRG/5-6/II-65 A prelmnary report of ths work was presented at IEEE INFOCOM 26 X Tang s wth the School of Computer Engneerng, Nanyang Technologcal Unversty, Sngapore e-mal: asxytang@ntuedusg J Xu s wth the Department of Computer Scence, Hong Kong Baptst Unversty, Kowloon Tong, Hong Kong e-mal: xujl@comphkbueduhk Dgtal Object Identfer Applcaton Base Staton Wreless Sensor Network Fg /$ c 27 IEEE System Archtecture value of a sensor network s determned by the tme duraton before t fals to carry out the msson due to nsuffcent number of alve sensor nodes Ths duraton s referred to as the network lfetme [] It s both msson-crtcal and economcally desrable to manage sensor data n an energyeffcent way to extend the lfetme of sensor networks The data captured by the sensor nodes are often converted nto an aggregate form requested by the applcatons eg, average temperature readng Prmarly desgned for montorng purposes, many sensor applcatons requre contnuous aggregaton of sensor data [3] Exact data aggregaton requres substantal energy consumpton because each sensor node has to report every readng to the base staton In wreless sensor networks, communcaton s a domnant source of energy consumpton [4], [5] To save energy, data semantcs can be relaxed to allow approxmate data aggregaton wth precson guarantees [6], [7], [8], [9] The precson can, for example, be specfed n the form of quanttatve error bounds: average temperature readng of all sensor nodes wthn an error bound of C In ths way, the sensor nodes do not have to report all readngs to the base staton Only the updates necessary to guarantee the desred level of precson need to be sent It s, however, a challengng task to optmze network lfetme under approxmate data aggregaton because the sensor nodes are nherently heterogeneous n energy consumpton Frst, when the data captured by dfferent sensor nodes change at dfferent magntudes and frequences, the sensor nodes may report data at dfferent rates Second, the wreless communcaton cost depends on the transmsson dstance [], [] Due to the geographcally dstrbuted nature of sensor networks, the sensor nodes are lkely to dffer sgnfcantly n the energy cost of sendng a message to the base staton Even f all sensor nodes report data at the same rate, ther energy consumpton can be hghly unbalanced, thereby reducng network lfetme In addton to reportng local sensor readngs, the ntermedate nodes n a mult-hop network are also responsble for relayng

2 2 IEEE/ACM TRANSACTIONS ON NETWORKING, VOL X, NO X, MONTH 2X the data orgnated from other nodes to the base staton The nodes closer to the base staton normally relay larger amounts of data than the nodes farther away from the base staton In ths paper, we nvestgate the optmzaton of network lfetme for approxmate data aggregaton We leverage the semantcs of approxmate data aggregaton n balancng the energy consumpton of the sensor nodes Our key dea s to dfferentate the qualty of data collected from dfferent sensor nodes by parttonng the precson constrant of data aggregaton among the sensor nodes n a coordnated fashon Our contrbutons are summarzed as follows: We dentfy three factors affectng the lfetme of sensor nodes n the context of approxmate data aggregaton: the changng pattern of sensor readngs; 2 the resdual energy of the sensor nodes; and 3 the communcaton cost between the sensor nodes and the base staton We then analyze the optmal precson allocaton n terms of network lfetme We develop a canddate-based method for precson allocaton and prove ts optmalty for sngle-hop networks Based on ths method, an adaptve scheme s proposed to dynamcally adjust the error bounds allocated to the sensor nodes The adjustment perod s also dynamcally set to control the communcaton overhead We derve the hardness results of canddate-based precson allocaton n mult-hop networks We extend the adaptve scheme to work n mult-hop networks by takng nto consderaton the effect of n-network aggregaton and the topologcal relatons among the sensor nodes We present an expermental evaluaton usng real data traces over a wde range of system confguratons The results show that the proposed scheme sgnfcantly mproves network lfetme compared to exstng methods The rest of ths paper s organzed as follows Secton II summarzes the related work Secton III descrbes the system model and gves some basc defntons Secton IV analyzes the optmal precson allocaton n sngle-hop networks and then proposes an adaptve precson allocaton scheme Secton V extends the adaptve scheme to mult-hop networks The expermental setup and results are dscussed n Secton VI Fnally, Secton VII concludes the paper II RELATED WORK Wreless sensor networks have attracted much research effort n recent years From the networkng perspectve, researchers have prmarly focused on optmzng network related operatons such as routng and meda access [2], [3], [4], [5] From the database perspectve, researchers have manly focused on query processng over sensor data [6], [7], [8], [9] However, not much work has looked nto tradng data qualty for energy effcency Recently, several approaches have been proposed to relax data semantcs and allow a specfed degree of naccuracy to be tolerated n sensor data collecton To acqure approxmate readngs of ndvdual sensor nodes, the precson constrants can be set ndependently for dfferent sensor nodes [8], [2] In contrast, to collect an aggregate form of sensor data over the network, the precson settngs of dfferent sensor nodes should be nter-related Olston et al [6] nvestgated burden-based precson adjustment for contnuous queres over dstrbuted data streams However, they dd not model nnetwork aggregaton, whch s a commonly used technque to reduce the traffc of data collecton n wreless sensor networks [2] Sharaf et al [7] mplemented a smple unform precson allocaton for n-network sensor data aggregaton Delgannaks et al [9] further optmzed the allocaton to reduce the number of messages transmtted n the network However, none of these studes has taken energy and lfetme models nto consderaton Thus, ther proposed technques are not effectve n handlng the energy constrants n wreless sensor networks As shall be shown by our expermental results, mnmzng the total network traffc does not necessarly optmze network lfetme Dfferent from exstng work, n ths paper, we am at extendng network lfetme for data aggregaton wth precson guarantees n sensor networks Consdne et al [22] and Nath et al [23] mplemented approxmate data aggregaton n the presence of mult-path routng by means of sketches and synopses However, they dd not make use of temporal localty to suppress data updates Deshpande and Chu et al [24], [25] appled statstcal technques to model the dstrbutons of sensor data for approxmate data collecton The performance of ths approach depends on the qualty of the models bult Dfferent from ths approach, we do not requre the constructon of statstcal models n advance Our proposed technques dynamcally adapt to the changng pattern of sensor readngs on the fly Related work on approxmate data collecton also ncludes representng sensor readngs wth sophstcated data structures [26], [27] and explotng the spatal correlaton between sensor readngs [28], [29] These studes are complementary to our work III PRELIMINARIES We consder data aggregaton wth precson guarantees n a network of n sensor nodes The sensor nodes are geographcally dstrbuted n an operatonal area They perodcally sample the local phenomena such as temperature and humdty Wthout loss of generalty, the samplng perod s assumed to be tme unt The base staton collects data from the sensor nodes and feeds them to an applcaton The applcaton specfes the precson constrant of data aggregaton by an upperbound E called the error bound on the quanttatve dfference between an approxmate result and the exact result [7], [9] That s, on recevng an aggregate result A from the sensor network, the applcaton would lke to be assured that the exact aggregate result A les n the nterval [A E,A + E] In approxmate data aggregaton, not all sensor readngs have to be sent to the base staton To reduce communcaton cost, the desgnated error bound on aggregate data can be parttoned and allocated to ndvdual sensor nodes we shall call t precson allocaton Each sensor node updates a new readng wth the base staton only when the new readng sgnfcantly devates from the last update to the base staton and volates the allocated error bound To guarantee the desgnated precson of aggregate data, the error bounds allocated

3 TANG et al: OPTIMIZING LIFETIME FOR CONTINUOUS DATA AGGREGATION WITH PRECISION GUARANTEES IN WIRELESS SENSOR NETWORKS 3 to ndvdual sensor nodes have to satsfy certan feasblty constrants Dfferent aggregaton functons mpose dfferent constrants In ths paper, we consder three commonly used types of aggregatons: SUM, COUNT and AVERAGE For SUM and COUNT aggregatons, to guarantee an error bound E on aggregate data, the total error bound allocated to the sensor nodes cannot exceed E, e, n e E, = where e s the error bound allocated to node For AVERAGE aggregaton, the total error bound allocated to the sensor nodes cannot exceed n E, e, n e n E, 2 = where n s the number of sensor nodes Elgble precson allocaton under the feasblty constrant s not unque For example, n a network of temperature sensor nodes, f the gven error bound on AVERAGE aggregaton s C, we can allocate an error bound of C to each sensor node Alternatvely, we can also allocate an error bound of 55 C to a selected node and an error bound 5 C to each of the remanng nodes Ths offers the flexblty to adjust the energy consumpton of ndvdual sensor nodes by careful precson allocaton In general, to collect the readngs of a sensor node at hgher precson e, smaller error bound, the sensor node needs to send data updates to the base staton more frequently, whch ntroduces hgher energy consumpton We denote the energy consumed by sensor node to send and receve a data update by s and v respectvely They can take dfferent forms to cater for a wde range of factors In the smplest case, f all sensor nodes use a default rado communcaton range, s s are the same for all nodes More sophstcatedly, f the sensor nodes know the locatons of the recevers [3], [3], [], they can adapt the power level to the transmsson dstance The sensor nodes wth longer transmsson dstances would be assocated wth hgher s s In addton, relablty can also be modeled n the energy cost The sensor nodes ncdent to less relable lnks are enttled to hgher s s and v s due to possble retransmssons The exact forms of s and v are orthogonal to our analyss and beyond the scope of ths paper We smply assume that each sensor node knows s and v Smlar to other studes [32], [33], [34], [35], we defne the network lfetme as the tme duraton before the frst sensor node runs out of energy Our analyss s also applcable to redundant sensor deployment where each locaton of nterest s covered by several sensor nodes From the vewpont of network lfetme, the set of sensor nodes montorng the same locaton can be converted to an equvalent sngle node by addng up the energy budgets of these sensor nodes More generally, f the network lfetme s defned as the tme duraton before a gven porton of sensor nodes run out of energy, our proposed scheme can be appled repeatedly after the exhauston of a sensor node s energy IV PRECISION ALLOCATION IN SINGLE-HOP NETWORKS We start by nvestgatng the precson allocaton n a snglehop network where each sensor node sends ts local readngs to the base staton drectly Sngle-hop networks are preferred n some stuatons due to a number of reasons [] Moreover, the analyss of precson allocaton n a sngle-hop network also provdes nsghts on the allocaton n a mult-hop network The adaptve precson allocaton scheme developed for sngle-hop networks wll serve as a buldng block of the scheme we shall propose for mult-hop networks n Secton V Note that constrants and 2 share the characterstc that the total error bound of the sensor nodes s capped by a gven value We shall focus on constrant n our dscusson The analyss and algorthms developed n ths paper can be adapted to handle constrant 2 n a straghtforward manner They are also drectly applcable to SUM and AVERAGE aggregatons over any fxed subset of the sensor nodes A Analyss of Optmal Precson Allocaton Consder a snapshot of the network Let e, e 2,, e n be the error bounds currently allocated to sensor nodes, 2,, n respectvely The quanttatve relatonshp between the rate of data updates sent by a sensor node and ts allocated error bound depends on the changng pattern of sensor readngs Wthout loss of generalty, we shall denote the update rate of each sensor node as a functon u e of the allocated error bound e u e s essentally the rate at whch node s readng changes by more than e Intutvely, u e s a non-ncreasng functon wth respect to e, and u = Snce the sensor nodes n a sngle-hop network are not nvolved n relayng data from other sensor nodes to the base staton, the energy consumpton rate of node s smply u e s, where s refers to the energy cost for node to send a data update to the base staton Suppose the resdual energy of node s p Then, the expected lfetme of node s p u e s Therefore, the network lfetme s gven by mn n p u e s The objectve of precson allocaton s to fnd a set of error bounds e, e 2,, e n that maxmze the network lfetme under the constrant n e E = We now analyze the optmal precson allocaton For smplcty, we shall assume functons u s are contnuous and denote the nverse functon of u by u Snce u s non-ncreasng, the mnmum lfetme of sensor node s gven by p l = u s

4 4 IEEE/ACM TRANSACTIONS ON NETWORKING, VOL X, NO X, MONTH 2X Wthout loss of generalty, suppose l l 2 l n For each par of nodes and j where j, consder the error bound u p l j s that makes the lfetme of node equvalent to the mnmum lfetme of node j Snce u s non-ncreasng, t follows from l j l j+ that u p u p l j s l j+ s Thus, gven any j < n, j = u p l j s j = u p l j+ s = u j+ uj+ + = u p j+ + l j+ s j+ = = u j = p l j+ s u p l j+ s j = u p l j+ s Ths mples j = u p l j s s non-decreasng wth ncreasng j Note that when j =, j = u p l j s = u u = Therefore, gven an error bound E > on data aggregaton, f n = u p l n s > E, there must exst a j, where j < n, such that Snce we have j = u p j + E < l j s j + = j = u p l j + s = u p E < l j s = j = j = u p l j + s u p, l j + s u p l j + s Hence, there also exsts an l, where l j l < l j +, such that j u p l = E 3 s = p On the other hand, f n = u l n s E, snce u s are non-ncreasng and u =, there exsts an l, where l l n, such that n u p l = E 4 s = For convenence, we shall denote j = n n ths case so that 4 s consstent wth 3 Theorem : An optmal precson allocaton s gven by { e u p = l j, s j < n Ths allocaton has a lfetme l Proof: It follows from 3 that e,e 2,,e n satsfes the feasblty constrant of precson allocaton Assume on the contrary that there exsts another precson allocaton e,e 2,,e n whch has a lfetme l > l The defnton of network lfetme mples that for any j, Thus, p p u e c l > l = u e c u e < u e Snce u s non-ncreasng, we have Therefore, n e = e > e j = e > j = e = E, whch contradcts the feasblty of e,e 2,,e n Hence, the theorem s proven Theorem mples that the sensor nodes wth hgh resdual energy p, slow change n readngs e, low u, and low communcaton cost s may be assgned zero error bounds The sensor nodes allocated non-zero error bounds n an optmal precson allocaton must be equal n the energy consumpton rate normalzed by the resdual energy: r = u e s p We shall call r the normalzed energy consumpton rate To extend network lfetme, t s mportant to balance the normalzed energy consumpton rates of the sensor nodes B Canddate-Based Precson Allocaton In practce, the exact forms of u s e, the changng patterns of sensor readngs may not be known a pror and they may even change dynamcally Thus, we propose a canddate-based method for precson allocaton The key dea s to let each sensor node estmate and report to the base staton the normalzed energy consumpton rates for a number of canddate error bounds based on hstorcal sensor readngs The base staton optmzes precson allocaton based on these canddates to extend network lfetme Snce the general relatonshps between error bounds and update rates are not known, we restrct the error bound of each sensor node to one of ts canddates Such allocatons are called canddate precson allocatons and the one that maxmzes network lfetme s called the optmal canddate precson allocaton Assume that each sensor node chooses m canddates For each node, let e, < e,2 < < e,m be the lst of canddate error bounds, and r,,r,2,,r,m be the correspondng normalzed energy consumpton rates It follows that

5 TANG et al: OPTIMIZING LIFETIME FOR CONTINUOUS DATA AGGREGATION WITH PRECISION GUARANTEES IN WIRELESS SENSOR NETWORKS 5 r, r,2 r,m Suppose the smallest canddate error bounds for the sensor nodes do not add up to the desgnated bound on data aggregaton, e, e, +e 2, + +e n, E Algorthm presents the pseudocode to compute the optmal canddate precson allocaton Algorthm Optmal Canddate Precson Allocaton n a Sngle-Hop Network Input: E: error bound of data aggregaton e,,r, : canddate error bounds and normalzed energy consumpton rates Output: e,x : error bound of each node n optmal allocaton : for = to n do 2: x = ; 3: end for 4: whle mn n x m do 5: j = arg max n,x m r,x ; 6: f e j,xj+ + j e,x > E then 7: break; 8: end f 9: x j = x j + ; : end whle Intally, the error bound of each sensor node s set to ts smallest canddate steps to 3 In each teraton of steps 4 to, the error bound of the node havng the hghest energy consumpton rate s replaced wth ts next smallest canddate The teraton stops f a new replacement would make the total error bound of the sensor nodes exceed the desgnated bound on data aggregaton steps 6 to 7 The worst-case tme complexty of Algorthm s Omn 2 We show that Algorthm produces an optmal canddate precson allocaton Theorem 2: The canddate precson allocaton computed by Algorthm maxmzes network lfetme Proof: Let e,x,e 2,x2,,e n,xn be the precson allocaton computed by Algorthm It s obvous that e,x,e 2,x2,,e n,xn s feasble Suppose under such allocaton, sensor node k has the hghest normalzed energy consumpton rate, e, r k,xk = max n r,x The network lfetme s then gven by = r k,xk max r,x n It s easy to nfer from Algorthm that: f x k < m, e k,xk + + k e,x > E; Our proposed canddate selecton method to be dscussed later n ths secton satsfes ths constrant 2 As shall be shown by our expermental results Secton VI, a small m lke 5 s suffcent to acheve near optmal network lfetme and for each x > where k, r,x r k,xk Assumng on the contrary that there exsts another canddate precson allocaton e,x,e 2,x 2,,e n,x n wth a longer network lfetme, e, It follows that Snce we have Therefore, and hence, max n r,x Based on property, > max r,x n max n r,x < max n r,x r k,xk = max n r,x, r k,x k max n r,x < r k,x k e k,x k + =k n > E x k < x k m, e k,xk + e k,x k e,x = e,x e k,xk + + =k = e k,x k + =k Thus, there must exst a j k such that whch mples e j,xj > e j,x j, x j x j It follows from property that Therefore, r j,x j r j,xj r k,xk e,x e,x max n r,x r j,x j r k,x k = max n r,x, whch contradcts the assumpton that e,x,e 2,x 2,,e n,x n has a longer network lfetme Hence, the theorem s proven

6 6 IEEE/ACM TRANSACTIONS ON NETWORKING, VOL X, NO X, MONTH 2X C Adaptve Precson Allocaton We now present an adaptve precson allocaton scheme that works by adjustng the error bounds of the sensor nodes perodcally The nterval between two successve adjustments s called an adjustment perod At the begnnng of an adjustment perod, each sensor node selects a lst of canddate error bounds e,,e,2,,e,m The node keeps track of the update counts under these error bounds as t captures new readngs 3 At the end of the adjustment perod, node normalzes the counts by the length of perod to obtan the data update rate u,j for each e,j Node then computes the normalzed energy consumpton rate r,j for each e,j by r,j = u,j s p, where p s the present resdual energy of node Node sends a canddate report message ncludng the e,j s and r,j s to the base staton On recevng the messages from all sensor nodes, the base staton computes the optmal precson allocaton e,x,e 2,x2,,e n,xn usng Algorthm In case n = e,x < E, the leftover error bound E n = e,x s smply allocated to the node wth the hghest normalzed energy consumpton rate snce dong so would only extend network lfetme Fnally, the base staton sends a precson allocaton message to the sensor nodes ncludng the new error bounds for ther adjustments Algorthm and Theorem 2 are generc n that they are applcable to any lst of canddates In ths paper, we propose to choose a set of canddate error bounds that are exponentally spaced The closer the canddates to the current error bound, the smaller the dfference between neghborng canddates The motvaton s to adjust the error bounds at coarse granularty when they are far away from the optmum, and adjust them at fne granularty when they are close to the optmum Let e be the current error bound of sensor node Then, the canddate error bounds of node range from 2 e to 3 2 e Gven the number of canddates m = 2k +, the canddate error bounds are selected as 2 e, 3 4 e,, 2k 2 k e,e, 2k + 2 k e,, 5 4 e, 3 2 e Note that the network lfetme s determned by the lfetme of the most energy-consumng node Thus, to control the energy overhead of adjustments, we propose to cap the energy overhead at the most energy-consumng node by a gven porton α of ts energy budget Ths s done by dynamcally adaptng the adjustment perod at each adjustment Specfcally, each sensor node counts the number of data updates sent to the base staton n the adjustment perods At an adjustment, node estmates ts energy consumpton rate by Ns /L, where N s the update count n the past adjustment perod, s s the energy cost for sendng, and L s the duraton of the past adjustment perod Note that at an adjustment, each sensor node needs to send a canddate report message to and receve a precson allocaton message from the base 3 Note that the sensor node does not actually send data updates based on these canddate error bounds It updates the readngs wth the base staton accordng to the currently allocated error bound only staton Thus, the energy cost at node due to an adjustment s s +v, where s and v are the sendng and recevng costs respectvely To lmt t at a porton α of the energy consumed by node, the duraton of the next adjustment perod L should be set such that e, s + v L = α Ns L, L = L s + v αns Each sensor node computes L and ncludes t n the canddate report message sent to the base staton at the end of an adjustment perod Among all L s receved, the base staton selects the lowest one L as the next adjustment perod so as to cap the adjustment overhead at a porton α of the energy consumed at the most consumng node L s then ncluded n the precson allocaton message sent by the base staton to all sensor nodes We shall nvestgate the mpact of α wth smulaton experments n Secton VI V PRECISION ALLOCATION IN MULTI-HOP NETWORKS A Modelng In-Network Aggregaton If the base staton s beyond the rado coverage of some sensor nodes, a mult-hop routng nfrastructure has to be set up to transport data from the sensor nodes to the base staton A common practce s to organze the sensor nodes nto a tree structure rooted at the base staton [2] In-network aggregaton s often used to reduce the network traffc of data collecton n mult-hop networks [2], [7], [9], [27] In ths approach, each ntermedate node aggregates the data receved from ts chldren before forwardng them upstream n order to cut down the volume of data sent over the upper-level lnks n the tree As a result, the data sent by an ntermedate node to ts parent s a partal aggregate result of the sensor readngs n the subtree rooted at the ntermedate node Lke that n a sngle-hop network, each sensor node s allocated an error bound e to control ts reportng of data updates to the parent We shall call t node s local error bound The operaton of a leaf node n a mult-hop network s the same as that n a sngle-hop network: t updates the parent node wth a new readng whenever the new readng dffers from the last reported readng by more than the local error bound For each ntermedate node, the local error bound s appled to the partal aggregate results at the node rather than ts local readngs [9] To do so, each ntermedate node mantans the latest data value reported by each chld At each samplng perod, the ntermedate node re-aggregates these data values together wth ts new local readng It sends the new partal aggregate result to the parent only when the result has changed beyond ts local error bound snce the last update to the parent For SUM aggregaton, the partal aggregate result s the sum of the sensor readngs n the subtree rooted at the ntermedate node In ths way, the aggregate result collected by the base staton s guaranteed to be wthn an error bound n = e from the exact aggregate result over the network [9] In fact, t can be shown by nducton that for each node, the data value mantaned by s parent node for dffers from

7 TANG et al: OPTIMIZING LIFETIME FOR CONTINUOUS DATA AGGREGATION WITH PRECISION GUARANTEES IN WIRELESS SENSOR NETWORKS 7 the exact aggregate result over the subtree T rooted at node by at most j T e j The correctness of ths clam s trval for any leaf node Suppose t s also true for any chld of an ntermedate node Let C be the set of s chldren Then, for any node c C, we have D,c d j e j, j T c j T c where d j s the sensor readng at node j, D,c s the data value mantaned by node for node c, and T c s the subtree rooted at node c Denote s parent node by p Accordng to the operaton of an ntermedate node presented above, we also have Dp, d e, c C D,c where D p, s the data value mantaned by node p for node Therefore, D p, d j j T = D p, c C D,c + c C D,c d c C j T c d j D p, d D,c + D,c d j c C j T c e + j T c e j = j T e j c C It follows from the above clam that the data value mantaned by the base staton for each of ts chld dffers from the exact aggregate result over subtree T by at most j T e j Therefore, the error of the aggregate result computed by the base staton s bounded by the sum of the local error bounds at all sensor nodes n = e We denote the rate of data updates sent by a sensor node to ts parent as a functon u e of s local error bound e 4 Takng nto consderaton the energy consumed n sendng and recevng data updates, the energy consumpton rate of node s then gven by u e s + c C u c e c v, where s refers to the energy cost for node to send a data update to s parent, and v refers to the energy cost for node to receve a data update from a chld Therefore, the expected network lfetme s gven by mn n p u e s + c C u c e c v, where p s the resdual energy of node The objectve of precson allocaton s agan to fnd a set of error bounds e,e 2,,e n that maxmze the network lfetme subject to the constrant n e E, = where E s the gven error bound on data aggregaton Smlar to sngle-hop networks, to extend network lfetme, t s mportant to balance the normalzed energy consumpton rates of the sensor nodes, e, to mnmze max n B Adaptve Precson Allocaton u e s + c C u c e c v p Adaptve precson allocaton n a mult-hop network also works by adjustng the error bounds of the sensor nodes perodcally Agan, we adopt the canddate-based method for precson allocaton Each sensor node selects a lst of m canddate local error bounds e, < e,2 < < e,m The sensor node keeps track of the data update rates to ts parent node u,, u,2,, u,m for these error bounds as t captures new readngs and produces new partal aggregate results At the adjustment, the local error bound of each node s to be set to one of ts canddates e,x where x m Followng the analyss n Secton V-A, the objectve of precson allocaton s to mnmze max n u,x s + c C u c,xc v p, subject to the constrant n e,x E = Ths s an NP-hard problem Theorem 3: The canddate precson allocaton problem defned above s NP-hard Proof: We show the allocaton problem s NP-hard by a polynomal reducton from the knapsack problem whch s known to be NP-complete [36] The knapsack problem s defned as follows: Gven a knapsack of capacty C, and n objects of szes c,c 2,,c n and profts w,w 2,,w n, the objectve of the knapsack problem s to fnd the largest total proft among all subsets of the objects that ft n the knapsack Let P be an nstance of the knapsack problem We frst construct a tree topology ncludng a base staton and n sensor nodes, where node s a chld of the base staton and the remanng nodes are chldren of node see Fgure 2 An nstance Q of the canddate precson allocaton problem s then constructed on ths tree by settng m = 2 e, each node has two canddate local error bounds; n, e, = x, e,2 = x + c, u, = y, u,2 = y w, s = v =, p = z, where x >, y > max n w, and z > are ntegral constants; and E = nx+c It s obvous that the constructon base staton 4 Recall that s local error bound e s appled to the partal aggregate results at node Therefore, strctly speakng, the update rate from node to ts parent also depends on how error bounds are allocated to s descendants To smplfy the analyss, we assume that the update rate reles on e only Fg n Instance Q of Canddate Precson Allocaton Problem

8 8 IEEE/ACM TRANSACTIONS ON NETWORKING, VOL X, NO X, MONTH 2X tme of nstance Q s polynomal to the sze of nstance P Next, we show that, for any ntegral bound K, there exsts a subset of the objects fttng n the knapsack wth a total proft at least K for nstance P f and only f there exsts a feasble canddate precson allocaton wth hghest normalzed energy consumpton rate at most ny K/z for nstance Q Note that all sensor nodes n nstance Q have the same energy cost to send or receve a data update Ths mples the energy consumpton rates of the leaf nodes n Fgure 2 e, nodes 2,3,,n cannot exceed that of node Also note that all sensor nodes have the same amount of resdual energy Therefore, regardless of precson allocaton, the hghest normalzed energy consumpton rate over all nodes s always that of node It s then easy to establsh a one-to-one correspondence between the object subsets fttng n the knapsack n nstance P and the feasble canddate precson allocatons n nstance Q In fact, for any object subset S, f S fts n the knapsack e, S c C, the correspondng canddate precson allocaton {e,2 S} {e, / S} s feasble and n ths case, node has a normalzed energy consumpton rate ny S w /z, where S w s the total proft of the object subset S Vce versa, for any feasble canddate precson allocaton e,x,e 2,x2,,e n,xn, node has a normalzed energy consumpton rate ny,x =2 w /z, and the correspondng object subset { x = 2} fts n the knapsack Thus, there exsts an object subset wth a total proft at least K for nstance P f and only f there exsts a canddate precson allocaton wth hghest normalzed energy consumpton rate at most ny K/z for nstance Q Hence, the theorem s proven In the followng, we present a dstrbuted algorthm to compute a suboptmal canddate precson allocaton We advocate dstrbuted algorthms because havng all nodes reportng the estmated update rates u,j s and resdual energy levels p s to the base staton places further burdens of energy consumpton on the nodes closer to the base staton whch are usually the energy bottlenecks n mult-hop networks To facltate presentaton, we shall refer to the sum of the local error bounds at the sensor nodes n the subtree T rooted at node as ts gross error bound In addton to the canddate local error bounds, each sensor node also selects a lst of m thresholds T, < T,2 < < T,m for ts gross error bound to assst the computaton For each threshold T,j, node computes a locally best precson allocaton we shall call t A,j among and ts chldren under the constrant that the gross error bound at node does not exceed T,j The computaton n our algorthm s carred out n a bottomup manner from the leaf sensor nodes to the base staton On computng the allocatons A,j s, each node sends a canddate report message ncludng a lst E,,U,,R,, E,2,U,2,R,2,, E,m,U,m,R,m to ts parent, where E,j s the gross error bound at node under A,j t s straghtforward that E, E,2 E,m, U,j s the rate of data updates sent by node to ts parent under A,j, and R,j s the hghest normalzed energy consumpton rate of the sensor nodes n subtree T under A,j A parent node performs the local computaton after recevng the canddate report messages from all chldren If s a leaf sensor node, the thresholds are set the same as ts canddate local error bounds, e, T,j = e,j Thus, E,j s are smply e,j s, and U,j s are smply the estmated update rates u,j s Snce does not receve data updates from any other node, R,j s are smply R,j = u,j s p If s an ntermedate sensor node, t collects the canddate report messages from all of ts chldren Together wth the locally estmated update rates u,, u,2,, u,m, node computes a locally best precson allocaton for each threshold T,j usng Algorthm 2 Gven a threshold T,j, only the canddate local error bounds e,h satsfyng e,h + c C E c, T,j are lkely to appear n a feasble precson allocaton step 3, otherwse the gross error bound at node would exceed T,j For each of these e,h s, the best allocaton of T,j e,h among s chldren s computed usng Algorthm step 4 Algorthm 2 Locally Best Precson Allocaton at Node n a Mult-Hop Network Input: T,j : a threshold for the gross error bound of node e,,u, : canddate local error bounds of node and estmated data update rates to s parent node E c,,u c,,r c, : gross error bounds, data update rates and hghest normalzed energy consumpton rates receved from each chld c of node Output: A,j : the computed best allocaton, whch ncludes the local error bound e,y allocated to node and the gross error bound E c,yc allocated to each chld c of node E,j : gross error bound at node under A,j U,j : data update rate from node to ts parent under A,j R,j : hghest normalzed energy consumpton rate of the nodes n subtree T under A,j : R,j = + ; 2: for h = to m do 3: f e,h + c C E c, T,j then 4: compute the optmal canddate precson allocaton for error bound T,j e,h among s chldren usng Algorthm based on E c, and R c, ; 5: for each chld c of, let E c,xc be the error bound of c n the optmal allocaton, then U c,xc s the correspondng data update rate from c to, and R c,xc s the correspondng hghest normalzed energy consumpton rate of the nodes n subtree T c ; u,h s + c C 6: R = max U c,xc v 7: f R < R,j then 8: R,j = R ; 9: U,j = u,h ; : E,j = e,h + c C E c,xc ; : y = h; 2: for each chld c of, y c = x c ; 3: end f 4: end f 5: end for p, max R c,xc ; c C

9 TANG et al: OPTIMIZING LIFETIME FOR CONTINUOUS DATA AGGREGATION WITH PRECISION GUARANTEES IN WIRELESS SENSOR NETWORKS 9 Suppose E c,xc s the gross error bound of each chld c n the best allocaton Then, U c,xc s the correspondng data update rate from c to, and R c,xc s the hghest normalzed energy consumpton rate of the nodes n subtree T c On computng s energy consumpton rate, the hghest normalzed energy consumpton rate of the nodes n subtree T can then be computed step 6 The canddate local error bound e,y that leads to the mnmum hghest energy consumpton rate s ncluded n the locally best precson allocaton A,j steps 7 to 3 The correspondng allocaton E c,yc s among s chldren are also recorded n A,j The worst-case tme complexty of Algorthm 2 s Om 2 C, 5 where C s the number of s chldren Node records the computed best allocaton A,j for each threshold T,j, and sends a canddate report message ncludng the lst E,,U,,R,, E,2,U,2,R,2,, E,m,U,m,R,m to ts parent node The base staton, on recevng the canddate report messages from all of ts chldren, computes a locally best precson allocaton among the chldren usng Algorthm The computed error bounds are then sent to the sensor nodes for ther adjustments n a top-down manner The base staton sends a precson allocaton message to ts chldren ncludng the gross error bounds allocated to them An ntermedate sensor node, on recevng ts allocated gross error bound, retreves the stored correspondng best allocaton whch contans a local error bound and a set of gross error bounds for ts chldren The ntermedate node apples the local error bound to ts partal aggregate results and sends the gross error bounds to ts chldren n a precson allocaton message A leaf sensor node, on recevng ts allocated gross error bound, smply takes t as the local error bound In case the total error bound n the precson allocaton computed by the base staton does not add up to E exactly, the leftover error bound s allocated to the node wth the hghest normalzed energy consumpton rate 6 Smlar to adaptve precson allocaton n a sngle-hop network, the canddate local error bounds of each sensor node and the thresholds for ts gross error bound are exponentally spaced around ts current local and gross error bounds respectvely Let e and E be the current local and gross error bounds of sensor node respectvely Gven the number of canddates m = 2k +, the canddate local error bounds are selected as 2 e, 3 4 e,, 2k 2 k e,e, 2k + 2 k e,, 5 4 e, 3 2 e, and the thresholds for s gross error bound are selected as 2 E, 3 4 E,, 2k 2 k E,E, 2k + 2 k E,, 5 4 E, 3 2 E Lke that n a sngle-hop network, we dynamcally adapt the adjustment perod to lmt the energy overhead of adjustments at the most energy-consumng node by a porton α of ts energy budget Note that at an adjustment n a mult-hop network, a sensor node receves a canddate report message 5 Agan, as wll be shown n Secton VI, a small m s suffcent to acheve near optmal network lfetme 6 To do so, each recorded allocaton A,j ncludes the chld node that roots the subtree contanng the node wth the hghest normalzed energy consumpton rate R,j The allocaton of the leftover error bound can then be routed to the ntended node along wth the precson allocaton message from each chld and sends one to ts parent It also receves a precson allocaton message from ts parent and sends one to ts chldren Thus, the energy cost at node due to an adjustment s s + s + C + v, where s and s are the sendng costs to the parent and chldren respectvely, v s the recevng cost, 7 and C s the set of s chldren At an adjustment, the energy consumpton rate of node s estmated by N s s + N v v /L, where N s and N v are the numbers of data updates sent to s parent and receved from s chldren respectvely n the past adjustment perod, and L s the duraton of the past adjustment perod Node suggests the duraton of next adjustment perod L as L = L s + s + C + v αn s s + N v v Each leaf node ncludes the suggested perod n the canddate report message sent to ts parent Each ntermedate node, on recevng the canddate report messages from ts chldren, chooses the shortest perod among that suggested locally and those receved from ts chldren Ths shortest perod s then ncluded n the canddate report message sent by the ntermedate node to ts parent Among all suggested perods receved, the base staton selects the shortest one L as the next adjustment perod so as to cap the adjustment overhead at a porton α of the energy consumed at the most consumng node L s then ncluded n the precson allocaton messages sent to all sensor nodes A Expermental Setup VI PERFORMANCE EVALUATION We developed a smulator based on ns-2 [37] and NRL s sensor network extenson [38] to evaluate the proposed adaptve precson allocaton scheme We used the followng energy models [] The energy consumed by a sensor node to send a message s s α+β d q, where s s the message sze, α = 5 nj/b s a dstance-ndependent term, β = pj/b/m 2 s the coeffcent for a dstance-dependent term, q = 2 s the exponent for the dstance-dependent term, and d s the transmsson dstance The energy consumed by a sensor node to receve a data update s s γ, where γ = 5 nj/b s a coeffcent ndependent of transmsson dstance In our experments, the default message sze was set at 48 bytes [6] The ntal energy budget at each sensor node was set at 5 Joule We smulated a sngle-hop network of sensor nodes and mult-hop networks of sensor nodes The layout of the sngle-hop network s shown n Fgures 3 The multhop network topologes were generated by randomly placng the base staton and sensor nodes n a 2m 2m area To smulate the spatal rregularty n sensor network deployment [39], we dvded the area nto a 4 4 grd The probabltes of deployng sensor nodes n the grd cells were assumed to follow a Zpf-lke dstrbuton That s, the 6 grd cells were randomly ordered nto a lst and the probablty to deploy sensor nodes n the th cell on the lst was set to θ /c, where θ s the Zpf parameter and c = 6 = θ s a normalzaton factor [4] The default value of θ was set at 7 The recevng cost s normally ndependent of the sender [], [], [33]

10 IEEE/ACM TRANSACTIONS ON NETWORKING, VOL X, NO X, MONTH 2X The sensor nodes were assumed to have a maxmum rado transmsson range of 4m If two sensor nodes were wthn the rado range of each other, they were consdered neghbors n the network connectvty graph The breadth frst search tree rooted at the base staton was then computed from the connectvty graph and used as the routng nfrastructure for data collecton [2], [27] We have expermented wth many randomly generated network topologes and observed smlar performance trends Due to space lmtatons, we shall only report the results of a sample network topology n ths paper The layout of the topology s shown n Fgure 4, where the sold crcle represents the base staton, the remanng crcles represent the sensor nodes and the lnes represent the lnks n the routng tree Fg 3 Fg 4 2 m 2 m 2 m 2 m m 8 m base staton m 2 m 2 m 2 m The Sngle-Hop Network Layout y m x m Sample Mult-Hop Network Layout 2 sensor node drecton of data update 6 sensor node We made use of the data provded by the LEM project [4] at the Unversty of Washngton to smulate the physcal phenomena n the mmedate surroundngs of sensor nodes Weather data were collected n the LEM project from several statons n the Washngton and Oregon states We used the temperature TEMP and solar radaton SOLAR traces logged by the staton at the Unversty of Washngton from August 24 to August 25 n our experments Each trace conssted of more than 5, readngs captured at a samplng perod of mnute Fgure 5 shows some representatve segments of these traces The TEMP and SOLAR data both fluctuate over tme ther readngs are hgher n the daytme and lower at nght In partcular, the SOLAR readngs reman unchanged regularly because the solar radaton s at nght For each of the TEMP and SOLAR traces, we extracted 2 dfferent subtraces startng at randomly selected tmeponts and assocated them wth the sensor nodes n our smulated network The samplng perod between two successve readngs n the trace was assumed to be tme unt Fg 5 Readng Sample Data Traces Temperature F Solar Radaton W/m 2 2 Tme The base staton computes the AVERAGE aggregaton of the readngs collected from all sensor nodes wth a desgnated error bound E As dscussed n Secton III, n ths case, the total error bound allocated to the sensor nodes should be capped by n E, where n s the number of sensor nodes The experments started wth the error bound unformly allocated to the sensor nodes, e, each node was allocated an error bound of E The followng precson allocaton schemes were smulated for performance comparson We measured the energy consumpton of each sensor node and the network lfetme n the experments Our Adaptve Precson Allocaton Adaptve-PA: Ths s the adaptve precson allocaton scheme proposed n Sectons IV-C and V-B By default, each sensor node selected m = 7 canddate error bounds and the energy cost due to adjustments was capped at α = 2% of the energy consumed at the most consumng node The performance mpacts of m and α are nvestgated n Secton VI-B We assumed that each data value n the message eg, sensor readng and canddate error bound took up 2 bytes In addton, a tmestamp of 2 bytes was ncluded n all messages for orderng and synchronzaton purposes The largest messages encountered n our experments were the canddate report messages n mult-hop networks Recall that the canddate report message ncludes a lst of E,,U,,R, s and a suggested next adjustment perod It requres a total of 6m + 4 = 46 bytes when m = 7, whch fts nto the default message sze Unform Precson Allocaton Unform-PA: The error bound s evenly parttoned among all sensor nodes [7], e, the precson allocaton remans the ntal one Ths s a smple and statc scheme whch does not dfferentate the sensor nodes by the changng pattern of sensor readngs, the resdual energy, and the communcaton cost wth the base staton Burden-based Precson Allocaton Burden-PA: Olston et al [6] presented a burden-based precson allocaton scheme for aggregate queres over dstrbuted data streams Ther objectve was to mnmze the total

11 TANG et al: OPTIMIZING LIFETIME FOR CONTINUOUS DATA AGGREGATION WITH PRECISION GUARANTEES IN WIRELESS SENSOR NETWORKS communcaton cost between data sources and the data snk In our experments, the energy consumed by each sensor node to send a data update to ts parent was taken as a measure of ts communcaton cost 8 Burden-PA works by perodcally reducng the error bound of each sensor node by a shrnk percentage and redstrbutng the leftover porton among the sensor nodes As suggested by [6], the shrnk percentage was set at 5% We smulated Burden-PA over a wde range of dfferent adjustment perods from 44 to 288 tme unts, whch correspond to to 2 days of data traces and found that no sngle perod provded the best performance for all expermental settngs Thus, to favor Burden-PA, for each expermental settng, we selected the best result obtaned over all adjustment perods tested and present t n ths paper Potental-Gan-based Precson Allocaton PGan- PA: To reduce the total number of messages n the network, Delgannaks et al [9] presented a precson allocaton scheme for sensor data aggregaton based on onlne estmaton of potental gans Smlar to Burden- PA, PGan-PA perodcally reduces the error bound of each sensor node by a shrnk percentage and redstrbutes the leftover porton among the sensor nodes As suggested by [9], the shrnk percentage was set at 4% Agan, we smulated PGan-PA over a wde range of adjustment perods from 44 to 288 tme unts and selected the best result obtaned to present n ths paper B Effect of m and α n Adaptve-PA Network Lfetme 3 Tme Unts Sngle-Hop Network Mult-Hop Network 5 Number of Canddate Error Bounds m Fg 6 Network Lfetme vs Number of Canddate Error Bounds n Adaptve- PA TEMP Trace, E = 6 F Frst, we nvestgate the performance mpact of the number of canddate error bounds m n the proposed Adaptve-PA scheme Fgure 6 shows the network lfetme for dfferent m values when the error bound E was set at 6 for the TEMP trace 9 Note that when m =, the current error bound s the only canddate Thus, the optmal canddate precson 8 We have also smulated Burden-PA wth the communcaton cost of each node set to the total energy consumed to send a data update to the base staton, whch ncludes the sendng and recevng costs at ntermedate nodes for relayng purposes f any Ths strategy was observed to perform worse than the one above n the man text 9 Only the expermental results of the TEMP trace are reported n ths secton to show the effect of m and α The results of the SOLAR trace have smlar trends 7 9 allocaton computed by Algorthm s always the same as the current allocaton Snce the experments started wth unformly allocated error bounds, Adaptve-PA degenerates to Unform-PA at m = The flexblty of precson allocaton ncreases wth m As seen from Fgure 6, an m value of 3 mproves network lfetme sgnfcantly compared to m = by factors of 34 and 7 n sngle-hop and mult-hop networks respectvely The network lfetme s generally nsenstve to m when m exceeds 5 Snce the largest allowable m for a canddate report message to ft nto the default message sze s 7, m was set at 7 n the remanng experments Recall that Adaptve-PA lmts the energy overhead of adjustments at the most energy-consumng node by a porton α of ts energy budget The settng of α reflects a tradeoff between overhead and adaptvty, both of whch ncrease wth α Fgure 7 shows the network lfetme for dfferent α values As expected, the curve of network lfetme s convex for most system confguratons tested In general, the performance of Adaptve-PA s not very senstve to the α value from % to 5% Therefore, we shall report only the expermental results for the default α = 2% n the remander of ths paper C Performance Comparson n Sngle-Hop Networks Fgure 8 shows the network lfetme as a functon of the desgnated error bound E on data aggregaton for dfferent precson allocaton schemes n the sngle-hop network of Fgure 3 Note that an error bound E = mples exact data aggregaton the leftmost ponts n Fgure 8 Wth exact data aggregaton, all sensor nodes must be allocated error bounds of Therefore, n ths case, the four precson allocaton schemes have smlar performance As seen from Fgure 8, the network lfetme ncreases wth error bound When E >, the proposed Adaptve-PA scheme sgnfcantly outperforms the other schemes for both traces tested Even f the readngs at all sensor nodes follow smlar changng patterns, t s not desrable to allocate the same error bound to all nodes because they are geographcally dstrbuted In a sngle-hop network, a node farther away from the base staton consumes more energy n sendng a data update than a node closer to the base staton Among the four precson allocaton schemes examned, Unform-PA and PGan-PA do not take ths heterogenety nto consderaton Thus, as shown n Fgure 8, Adaptve-PA mproves network lfetme by factors up to 34 and 26 compared to Unform-PA and PGan-PA respectvely To show the mportance of balancng energy consumpton n extendng network lfetme, we plot n Fgure 9 the total energy consumed by each sensor node by the tme when the frst node runs out of energy e, the network lfetme elapsed Under Adaptve-PA, most nodes were close to exhaustng ther energy when the network lfetme elapsed However, under Unform-PA and PGan-PA, the nodes close to the base staton e, nodes 3 and 8 n Fgure 3 consumed as low as 5% 2% of the energy budget only Burden-PA consders the heterogenety n communcaton cost due to transmsson dstance However, the objectve of Burden-PA s to mnmze the total communcaton cost Fgure 8 shows that our Adaptve-PA scheme extends network