Least-Latency Routing over Time-Dependent Wireless Sensor Networks

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1 1 Least-Latency Routng over Tme-Dependent Wreless Sensor Networks Shouwen La, Member, IEEE, and Bnoy Ravndran, Senor Member, IEEE Abstract We consder the problem of least-latency end-to-end routng over adaptvely duty-cycled wreless sensor networks. Such networks exhbt a tme-dependent feature, where the lnk cost and transmsson latency from one node to other nodes vary constantly n dfferent dscrete tme moments. We model the problem as the tme-dependent Bellman-Ford problem. We show that such networks satsfy the FIFO property, whch makes the tme-dependent Bellman-Ford problem solvable n polynomal-tme. Usng the β-synchronzer, we propose a fast dstrbuted algorthm to construct all-to-one shortest paths wth polynomal message complexty and tme complexty. The algorthm determnes the shortest paths for all dscrete tmes n a sngle executon, n contrast wth multple executons needed by prevous solutons. We further propose an effcent dstrbuted algorthm for tme-dependent shortest path mantenance. The proposed algorthm s loop-free wth low message complexty and low space complexty of O(maxdeg), where maxdeg s the maxmum degree for all nodes. We dscuss a sub-optmal mplementaton of our proposed algorthms that reduces ther memory requrement. The performance of our algorthms are expermentally evaluated under dverse network confguratons. The results reveal that our algorthms are more effcent than prevous solutons n terms of message cost and space cost. Index Terms Tme-dependent, shortest path, wreless sensor networks, routng, routng mantenance, least-latency. 1 INTRODUCTION Multhop data routng over wreless sensor networks (WSNs) has attracted extensve attenton n the recent years. Snce there s no nfrastructure n sensor networks, the routng problem s dfferent from the one n tradtonal wred networks or the Internet. Some routng protocols [2], [3] over WSNs presented n the lterature are extended from the related approaches over wred/wreless ad-hoc networks. They usually fnd a path wth the mnmum hop count to the destnaton, whch s based on the assumpton that the lnk cost (or one-hop transmsson latency) s relatvely statc for all wred/wreless lnks. However, for duty-cycled WSNs [4] [6], that assumpton may not always be vald. Duty-cycled WSNs nclude sleep-wakeup mechansms, whch can volate the assumpton of statc lnk costs. Currently, many MAC protocols support WSNs operatng wth low duty cycle, e.g., B-MAC [5], X-MAC [6]. In such protocols, sensor nodes operate n low power lstenng (or LPL) mode. In the LPL mode, a node perodcally swtches between the actve and sleep states. The tme duraton of an actve state and an mmedately followng sleep state s called the LPL checkng nterval, whch can be dentcal, or can be adaptvely vared for dfferent nodes, referred to as ALPL [7]. The duty-cycled mechansm has been shown to acheve excellent dle energy savngs, scalablty, and easness n mplementaton. However, they suffer from tme-varyng neghbor dscovery latences (the tme between data arrval and dscovery of S. La s wth Qualcomm, Inc, 5775 Morehouse Drve, San Dego, CA, E-mal: shouwenl@qualcomm.com B. Ravndran s wth the Department of Electrcal and Computer Engneerng, Vrgna Tech, Blacksburg, VA, E-mal: bnoy@vt.edu The prelmnary result was presented at IEEE INFOCOM 2010 [1]. the adjacent recever), whch s also ponted out by Ye et.al. [8]. As shown n Fgure 1, the neghbor dscovery latency between two neghbors s varyng wth dfferent departure tmes. Even wth synchronzed duty-cyclng, the neghbor dscovery latency s varyng at dfferent tme moments due to adaptve duty cycle settng as shown n Fgure 1. To formally defne the problem, we frst defne the lnk cost as the tme delay between data dspatchng tme, whch s the earlest tme when a sender wakes up for data transmsson, and the data arrval tme at the recever. The lnk cost s tme-varyng n adaptvely dutycycled WSNs due to varyng neghbor dscovery latences, even though the physcal propagaton condton does not change wth tme. The dspatchng tme s the tme moment when the data s ready for transmsson at the sender sde. Thus, ths rases a non-trval problem: wth tme-varyng lnk costs, how to fnd optmal paths wth least nodes-to-snk latency for all nodes at all dscrete dspatchng tme moments? A smlar problem has been modeled n prevous works as the tme-dependent shortest path problem (or TDSP) [9], [10] n the feld of traffc networks [11], tmedependent graphs [12], and GPS navgaton [13]. The general tme-dependent shortest path problem s at least NP-Hard, snce t may be used to solve a varety of NP-Hard optmzaton problems such as the knapsack problem. However, dependng on how one defnes the problem, t may not be n NP, snce ts output s not polynomally bounded. Moreover, there are even contnuoustme nstances of the TDSP problem n whch shortest paths consst of an nfnte sequence of arcs, as shown by Orda and Rom [14]. In ths paper, we study a specal case where the networks are known as FIFO networks, n whch commodtes travel along lnks n a Frst-In- Frst-Out manner. Under the FIFO condton, the tme-

2 2 dependent shortest path problem s solvable n polynomal tme. The TDSP problem has also been studed wth a dstrbuted approach. The only prevous dstrbuted soluton [10] computes the shortest paths for a specfc departure tme n each executon. If the whole tme perod has M dscrete ntervals (M s for nfnte tme ntervals), we have to execute the algorthm n [10] M tmes, whch s neffcent n terms of message complexty and tme complexty, gven the lmted power and rado resource n WSNs. Therefore, the frst motvaton of our work s to desgn a fast dstrbuted algorthm for the problem, whch can effcently enumerate all optmal paths wth least end-to-snk latency for nfnte tme ntervals. The second motvaton of our work s to propose an algorthm whch can dynamcally and dstrbutvely mantan tme-dependent least-latency paths. In WSNs, a node may update ts duty-cycle confguraton (e.g., based on ts resdual energy), or jon or leave the network, thereby changng the network topology. In such stuatons, the duty-cycle updatng node or the jonng/leavng node may change the cost of all the lnks wth ts neghbors, whch means that a sngle node update can cause multple lnk updates. Prevous efforts on ths problem [15], [16] are effcent n handlng sngle lnk updates. Applyng such solutons for multple lnk updates would mply that multple dstrbuted updates execute concurrently for a sngle node update, whch s not effcent n terms of message cost and memory cost for resource-lmted WSNs. The thrd motvaton s to address practcal mplementaton ssues. Our work requres schedule-awareness n neghborhood. One scenaro for such requrement s that all nodes have none nformaton from each other after ntal deployment n feld. In ths scenaro, we would lke to get schedule-awareness wthout global tme synchronzaton. Local synchronzaton and lght nformaton exchange s desrable to get duty-cyclng nformaton and tme dfference from neghbors for a sensor node. One canddate to acheve ths s by neghbor dscovery protocol, just smlar as the lnk layer neghbor dscovery protocol n the Internet. Nodes wthout schedules of neghbors would stay awake and broadcast neghbor dscovery message perodcally (.e., once every multple predefned tme slots) and neghbors can response back ther schedule nformaton, whch s a knd of local level synchronzaton. Once reachng a stable stage, any node would get ts neghbor s schedule and the tme dfference from ts neghbors, and swtch back to duty-cyclng state. The update of nodes schedule can be undergong n background ether probatvely or reactvely. Because our work s not lmted to any specfed MACprotocol, we dscuss dfferent methods to acheve schedule awareness over several underlyng mechansms, such as B-MAC, S-MAC, and quorum-based wakeup schedulng n Secton 7.1. Regardng to another practcal mplementaton ssues, we also need to understand how to smplfy the vector presentaton so that only smaller vector szes are requred, gven the lmted memory resource of sensor nodes. We present a sub-optmal mplementaton, whch acheves a trade-off between latency and memory usage. Fnally, we dscuss the complextes of our algorthms n some specal scenaros, lke statc lnk costs and multple snk nodes. In ths paper, we frst propose a dstrbuted algorthm to compute the tme-dependent paths wth least-latency for all nodes n a duty-cycled WSN. The algorthm has low message and space complextes. The algorthm s based on the observaton that the tme-varyng lnk cost functon s perodc, and hence by dervaton, the tme-varyng dstance functon for each node s also perodc. We show that the lnk cost functon satsfes the FIFO property [9]. Therefore, the tme-dependent shortest path problem s not an NP-hard problem, and thus s solvable n polynomal tme. We also propose dstrbuted algorthms for mantanng the shortest paths. The proposed algorthms re-compute the routng paths based on prevous path nformaton. The message complexty of our algorthms s O(δ 2 ) per node update, where δ s the number of nodes that change ether the dstance or the parents n ther shortest paths to the snk as a consequence of the correspondng nodes update. The algorthms space complexty s O(maxdeg). Fnally, we propose a sub-optmal mplementaton, whch requres vectors wth smaller szes to represent lnk cost functons and the dstance functon. The contrbutons of the paper are as follows: 1) We model adaptvely duty-cycled WSNs as tme-dependent networks. We show that such networks satsfy the FIFO condton and the trangular path condton 2) We present dstrbuted algorthms for fndng the tme-dependent shortest paths to the snk node for all nodes. When compared to the prevous soluton [10], our algorthms fnd the shortest paths n a sngle executon for nfnte tme ntervals 3) We present dstrbuted shortest path mantenance algorthms wth low message complexty and space complexty 4) We propose sub-optmal mplementaton wth vector compresson. To the best of our knowledge, we are not aware of any other efforts that consder duty-cycled WSNs as tmedependent networks and solve the problem of fndng or updatng the shortest paths wth effcent message and space costs. The rest of the paper s organzed as follows: We survey past and related works n Secton 2, and outlne our assumptons and defne the problem n Secton 3. We formally model the lnk cost functon and the dstance functon n Secton 4. The algorthms for route constructon and route mantenance are descrbed n Sectons 5 and 6, respectvely. We dscuss practcal mplementaton ssues n Secton 7. Smulaton results are reported n Secton 8. We conclude n Secton 9. 2 RELATED WORK We summarze the lterature on LPL schedulng and the tme-dependent shortest path problem as follows. LPL/ALPL n WSNs. LPL means that a node only wakes up and lstens the channel state for a short tme

3 3 perod. Examples nclude B-MAC [5], whch s a CSMAbased technque that utlzes low power lstenng and an extended preamble to acheve low power communcaton. In B-MAC, nodes have an awake and a sleep perod, and an ndependent sleep schedule. If a node wshes to transmt, t precedes the data packet wth a preamble that s slghtly longer than the sleep perod of the recever. Durng the awake perod, a node samples the medum, and f a preamble s detected, t remans awake to receve the data. Wth the extended preamble, a sender s assured that at some pont durng the preamble, the recever wll wake up, detect the preamble, and reman awake n order to receve the data. The desgners of B-MAC show that B- MAC surpasses exstng protocols n terms of throughput, latency, and for most cases, energy consumpton. Whle B- MAC performs qute well, t suffers from the overhearng problem, and the long preamble domnates the energy usage. To overcome some of B-MAC s dsadvantages, XMAC [6] and DPS-MAC [17] were proposed. In X- MAC or DPS-MAC, short preamble was proposed to replace the long preamble n B-MAC. Also, recever nformaton s embedded n the short preamble to avod the overhearng problem. The man dsadvantage of B-MAC, X-MAC, and DPS-MAC s that t s dffcult to reconfgure the protocols after deployment, thus lackng n flexblty. X-MAC [6] and DPS-MAC [17] are compatble wth LPL mechansms. However, they do not explctly support adaptve duty cyclng, where nodes choose ther duty cycle dependng on ther resdual energy. Jurdak et. al. [7] and Vgorto et. al. [4] present adaptve low power lstenng (ALPL) mode based on nodes resdual energy. These works provde the applcaton spaces for our work. In ALPL, snce nodes have heterogenous duty cycle settng, t s more dffcult for neghbor dscovery snce a node cannot dfferentate whether a neghbor s sleepng or falng when t does not receve feed-back from the neghbor. ALPL also ncurs tme-dependent lnkcost and end-to-end latency as llustrated n Secton 4. Recently, B-MAC [5] was also extended to support the ALPL mode n TnyOS. Delay-effcent routng over adaptvely duty-cycled WSNs. Over adaptvely duty-cycled WSNs, routng becomes more dffcult due to two reasons: ntermttent connecton between two neghbor nodes and changes n the transmsson latency at dfferent tmes. Some works have studed the delay-effcent routng problem over adaptvely duty-cycled WSNs n recent years. Lu [18] et al. proposed two methods to solve routng over ntermttently connected WSNs due to duty cyclng. One s by an on-demand approach that uses probe messages to determne the least latency route. The other one s a proactve method, where all least latency routes at dfferent departure tmes are computed at the begnnng. The frst method does not work well for frequent data delveres. The second method s a centralzed approach, and s not flexble for dstrbuted constructon. Our algorthms also follow the proactve approach, but are dstrbuted. Yu [19] et al. consdered the problem wth a dfferent perspectve. They studed how to consume a mnmum amount of energy whle satsfyng an end-to-end delay bound specfed by the applcaton. In [20], they studed how to guarantee the end-to-end latency by adjustng duty cyclng n ndvdual nodes. There are also some other works that have studed the energy-delay tradeoff for duty-cycled WSNs, such as [21] and [22]. These efforts are smlar to our work, but there s a fundamental dfference: we study the least routng latency, gven the duty-cycle or energy confguraton on each node. Tme-Dependent Shortest Path Problem. Ths problem was frst proposed by Cooke and Halsey [9]. It has been well studed n the feld of traffc networks [11], tmedependent graphs [12], and GPS navgaton [13]. Prevous solutons for ths problem mostly work offlne usng a centralzed approach [12]. Although these solutons can provde nspratons, they cannot be appled to WSNs where the global network topology s not known by a centralzed node, gven the large-scale sze of a WSN. For the dstrbuted tme-dependent shortest path problem, the only prevous work [10] computes the shortest paths for a specfc departure tme n each executon, whch s not tme-effcent. If the whole tme perod has M dscrete ntervals (M s for nfnte tme ntervals), we have to execute the algorthm n [10] M tmes, whch s neffcent n terms of message complexty and tme complexty. For multple executons, the algorthm n [10] suffers from hgh message cost, whch s undesrable for resource-lmted WSNs. The work n [10] dscusses two polces for the tmedependent shortest path problem: watng and nonwatng. Watng does not mean watng n the buffer, but means watng for some tme after the data has been delvered (.e., the recever s awake). Non-watng means that a sender wll mmedately send the data once the recever s awake. We do not consder the watng polcy n our work, snce the end-to-end latency does not beneft from watng. Dynamc Shortest-Path mantenance. Many works [15], [16], [23] exst for handlng lnk decreases and ncreases, and node deletons and nsertons n statc networks. In [24], an algorthm s gven for computng all-pars shortest paths, whch requres O(n 2 ) messages when the network sze s n. In [25], an effcent ncremental soluton has been proposed for the dstrbuted all-pars shortest paths problem, requrng O(nlog(nW )) amortzed number of messages over a sequence of edge nsertons and edge weght decreases. Here, W s the largest postve nteger edge weght. In [26], Awerbuch et al. propose a general technque that allows to update the all-pars shortest paths n a dstrbuted network n O(n) amortzed number of messages and O(n) tme, by usng O(n 2 ) space per node. In [23], Ramarao and Venkatesan gve a soluton for updatng all-pars shortest paths that requres O(n 3 ) messages, O(n 3 ) tme, and O(n) space. They also show that, n the worst case, the problem of updatng shortest

4 4 paths s as dffcult as computng shortest paths. They suggest two possble drectons toward devsng effcent fully dynamc algorthms for updatng all-pars shortest paths: 1) explore the trade-off between the message, tme and space complexty for each knd of dynamc change 2) devse algorthms that are effcent n dfferent complexty models (wth respect to worst case and amortzed analyss). However, the algorthms n [15], [23] need O(n) space at each node, whch s mpractcal for sensor nodes wth lmted memory capacty. In addton, none of the prevous works are effcent for shortest path mantenance n tme-dependent networks. β-synchronzer [27]. As descrbed n [27], the synchronzer s a methodology for desgnng effcent dstrbuted algorthms n asynchronous networks. Researchers have used synchronzers to reduce message complexty of some asynchronous algorthms, such as Bellman-Ford. A synchronzer works as follows. A synchronzer generates sequences of clock-pulses at each node of a network. At each node, a new pulse s generated only after t receves all the messages whch were sent to that node by ts neghbors at the prevous pulse. Thus, a synchronzer runs n a phase-by-phase manner. A β-synchronzer s a specal type of synchronzer, whch has an ntalzaton phase, n whch a leader s s chosen n the network and a spannng tree rooted at s s constructed (e.g., by a Breadth-Frst-Search). After the executon of one phase, the leader s wll eventually learn that all the nodes n the network are safe. At that tme, t broadcasts a message along the spannng tree, notfyng all the nodes that they may generate a new pulse. The communcaton pattern for recevng all acknowledgments s just lke convergecast. Therefore, wth a β-synchronzer, whenever a node learns that t s safe and all ts descendants n the tree are safe, t sends an acknowledgment to ts parent. In each phase, a β-synchronzer ncurs low message complexty, whch s O( V ) [27], where V s the network sze, but at a hgher tme cost. For a dstrbuted shortest path algorthm armed wth a β-synchronzer, f the number of phases are lmted, the total message complexty wll be bounded. 3 ASSUMPTIONS AND PROBLEM DEFINITION We model a WSN as a drected graph G = (V, E, C), wth V nodes and E lnks. C = {τ,j (t) (, j) E} s a set of tme-dependent lnk delays,.e., τ,j (t) s a strctly postve functon of tme defned for [0, ), descrbng the delay of a message over lnk (, j) at tme t. Each node n only knows the dentty of the nodes n ts neghbor set, defned as N. We assume that tme axs are arranged as consecutve numbered tme slots. We denote the duraton of one tme slot for node n as T. It s possble that T T j (ALPL) for two nodes n and n j. The tme expanson of each node n s modeled as dscrete and nfnte, where T = {t 0, t1, t2,, tm }, M s +, and t k tk 1 = T. We use the terms checkng nterval and tme slot nterchangeably. The wakeup schedule depends on the underlyng MAC protocol. We frst assume that a node can be operated n the LPL mode: a node wakes up at the begnnng of a tme slot to check the channel state. If there s no actvty, the node goes back to sleep; otherwse, t stays awake. Then, we relax the assumpton and dscuss how our work can be appled to other wakeup schedules, such as quorum schedules [28]. We consder the non-watng polcy (.e., the sender mmedately delvers data once the recever s awake) at each node, snce the node-to-snk delay wll not beneft from watng. Thus, once the data arrves at an ntermedate node, the node wll attempt to dspatch the data mmedately. Dspatchng tme represents the earlest tme when a sendor s awake for data transmsson. Thus, dspatchng tmes are not the same as the data departure tmes, as the data may stll be buffered n the sender s memory. For smplcty n modelng and desgn, a node dspatches the receved data at t k T. A nonnegatve travel tme τ,j (t k ) s assocated wth each lnk (, j) wth the followng meanng: f t k s the data dspatchng tme from node n along the lnk (, j), then t k + τ,j(t k ) s the data arrval tme at node n j. The general problem of determnng the shortest paths wth the least latency n tme-dependent WSNs can be defned as follows: Fnd the least-tme paths from all nodes to the snk node n s correspondng to the mnmum achevable delay d, n V and t k T, where: d (t k ) = mn n j N {τ,j (t k ) + d j ( t k + τ,j (t k ) ) } (1) Equaton 1 s an extenson of Bellman s equatons [29] for the tme-dependent network and s referred to as TD- Bellman s equaton hereafter. We also assume that a message arrves correctly n fnte tme from a sender to a recever, whch can be acheved by any relable MAC-layer transmsson mechansm. We do not assume that the entre network s tmesynchronzed,.e., all nodes are not equpped wth GPS devces or are not operated by global tmesynchronzaton protocols [30]. However, due to the proactve routng nature n our proposed algorthms, we need to know the lnk costs at the begnnng of route constructon. Therefore, we assume awareness of wakeup schedules of the neghborhood for each node. Such schedule awareness can be acheved by neghbor dscovery protocols such as perodc neghbor detecton mechansms. We further assume that all nodes have the same tme frequency, and ther clocks drft at relatvely slow speeds. 4 MODELING ADAPTIVELY DUTY-CYCLED WSNS In ths secton, we model adaptvely duty-cycled WSNs as tme-dependent networks. The algorthms we propose are based on Equaton 1. Bascally, t k s nfntely dscrete. Note that sensor nodes have lmted memory and exchangng messages s expensve. Thus, n order to mplement Equaton 1 n practce, we must make τ,j (t k )

5 5 and d j ( t k + τ,j (t k )) (where k [0, ]) fnte, so that the tme-dependent lnk cost and dstance can be represented by vectors. We wll now show that the lnk cost functon s perodc and establsh that the tme-varyng dstance functon s also perodc. Havng done so, we wll show how TD- Bellman s Equaton can be mplemented by vector representatons of lnk costs and dstances. 4.1 Lnk Cost Functon Wthout loss of generalty, suppose there are two adjacent nodes n and n j, where n s the sender and n j s the recever. 4.2 Dstance to Snk We refer to the node-to-snk delay as dstance, for compatble representaton wth that n the statc Bellman-Ford algorthm [29]. Consder node n and ts neghbor n j, where T T j. For dspatchng tme t 0 at n, let us suppose that the data arrvng tme at n j s t j0 j. Then, for the dspatchng tme t k at n, wth the same tme frequency for the two nodes, the correspondng tme nstant at n j s t j0 j t0,j + k T. Hence, based on the neghbor dscovery mechansm (e.g., B-MAC [5], X-MAC [6]), n j wll be dscovered by n at the tme nstant t j0 j + k T t,j T j T j, wth respect to n j s tme clock. Therefore, the functon of dstance from n to n s through the path through n j s: 0 d (t k ) = τ,j (t k ) + d j (t k +j 0 j ) (5) Fg. 1. Varyng neghbor dscovery latency n heterogenous LPL mode Suppose at tme t 0, the neghbor dscovery latency s t0,j. Then at tme tk = t0 + k T, the neghbor dscovery latency can be expressed as: tk,j = T j (k T t0,j )mod T j (2) In an actual deployment, for example wth the X-MAC protocol, we can measure t0,j n the followng way: at the begnnng of the tme slot whch starts at t 0, node n sends out a preamble whch contans the node ID of n j. n j mmedately feeds-back an ACK contanng the value of T j once t receves the preamble. After recevng the ACK by n, the one-hop round trp delay from t 0 to the tme at whch the ACK s receved s set to t0,j. Once we have measured t0,j, tk,j (k 0) can be computed by Equaton 2. For data transmsson wth fxed length data packets, we defne the data propagaton tme as τ data. Now, for a drected lnk (, j), we can set the lnk cost functon as, for t k T, τ,j (t k ) = T j (k T t0,j )mod T j + τ data (3) If τ data s relatvely small when compared wth T and T j, we can set τ data = 0. Ths s especally true for some WSN applcatons wth small nformaton reports, such as target trackng and envronment montorng. Theorem 1: For every lnk (, j), the tme-varyng lnk cost functon s perodc. The mnmum perod for the functon regardng k s, where τ,j (t k )=τ,j(t common multple. P (τ,j ) = LCM(T, T j ) T (4) k+p (τ,j) ) (k 0) and LCM s the least for t k T, where k = (k T t0,j )/T j for t k j T j, and j 0 s the data arrvng tme slot at n j wth respect to dspatchng tme t 0. Theorem 2: For a path n n 1 n 1 n s, the dstance functon d (t k ), tk T, s a perodc functon, where the mnmum perod for the functon regardng k s: P (d ) = LCM(T 0, T 1,, T ) (6) T k+p (d) for d (t k ) = d (t ), where T 0, T 1,, T are duratons of the LPL checkng ntervals of nodes n s, n 1,, n, respectvely. Proof: The proof s by nducton. For = 1, d 1 (t k 1) = τ 1,s (t k 1) + d s (t k 1 ). Snce d s 0, d 1 (t) = τ 1,s (t k 1). Accordng to Theorem 1, d 1 (t k 1) s perodc and ts perod s LCM(T 0, T 1 )/T 1. Assume that the clam s true for node n 1. Now, for node n, d (t k ) = τ, 1(t k ) + d 1(t k +j 0 1 ), where k = (k T t0,j )/T j and j 0 s the data arrval tme slot at n 1 wth respect to t k, based on Equaton 5. Let us defne f 1 = τ, 1 (t k ) and f 2 = d 1 (t k +j 0 1 ). The mnmum perod for functon f 1 s P (f 1 ) = LCM(T, T 1 )/T. Let = LCM(T 0, T 1,, T 1 ). Based on the nducton step, for dstance functon d 1 (t k 1 ), the perod s /T 1. The perod for functon f 2 s P (f 2 ) = /gcd(, T ). Snce gcd(, T ) = T LCM(,T, we have P (f ) 2) = LCM(,T ) T. Therefore, the mnmum perod for d (t k ) s: P (d ) = LCM[P (f 1 ), P (f 2 )] = LCM(T 0, T 1,, T )/T. Gven a WSN wth dfferent LPL checkng ntervals, the perod for the dstance functon of any node s bounded by LCM(T 0, T 1,, T n )/ mn{t }, from Equaton 6. In practcal mplementatons, t s recommended that LCM(T 0, T 1,, T n )/ mn{t } s not arbtrarly large. Thus, our mechansm for fndng the shortest paths at the routng layer should be based on cross-layer desgn. For example, {100ms, 200ms, 500ms, 1000ms} s a good confguraton set, where there s a bounded perod LCM(100, 200, 500, 1000)/mn{100, 200, 500, 1000} = 10 for the dstance functon. It means that for any node, ts

6 6 dstance to the snk wll repeat at most every 10 checkng ntervals. 4.3 Implementaton va Vectors We mplement the dscrete, perodc, and nfnte lnk cost functons and the dstance functons as vectors, and mplement the TD-Bellman s equaton (Secton 3) by vector operatons. Our goal s to use vectors wth lmted szes n order to mplement our algorthm wth lmted memory and same message sze over the ar, although the tme axs s nfnte. We mplement the lnk cost functon τ,j (t k ) (tk T ) wth a vector τ,j, where τ,j = LCM(T, T j )/T and τ,j [k] represents a set of numbers as follows: τ,j τ,j [k] = {τ,j (t k ), τ,j (t k+ ), τ,j (t k+2 τ,j ), } (7) For the node n, ts dstance functon d (t k ) (t k T ) can be mplemented by d, where d = P (d (t k )). d [k] represents a set of numbers as follows: d [k] = {d (t k ), d (t k+ d ), d (t k+2 d ), } (8) However, there are two dffcultes for the mplementaton of the TD-Bellman s equaton by vector operatons. The frst one s vector mappng. To mplement Equaton 5, even f we know τ,j and d j, we cannot add up the two vectors drectly. We defne a new vector d j as: d j[k] = d j [(k + j 0 ) mod d j ] (9) where k = (k T t0,j )/T j and j 0 s the correspondng tme slot for τ,j [0] at n j (.e., t 0 + t0,j = tj0 j ). Only after mappng d j [k] to d j [k], we can add τ,j[k] to d j [k]. By vector mappng, the sze of the new vector d j s: d j = d j T j gcd( d j T j, T ) (10) The second dffculty comes from the varous szes of vectors for lnk cost and dstance. Suppose d (j) s the vector representng the dstance of n from a path through n j n dscrete tme ntervals. To mplement Equaton 5, f τ,j and d j have the same sze, we can drectly add them up for computng d (j). Otherwse, we need to expand the two vectors to be of the same sze, whch means expandng τ,j by LCM( τ,j, d j )/ τ,j tmes and d j by LCM( τ,j, d j )/ d j tmes. After the expanson, we can drectly add up the expanded vectors. We call such an operaton, vector expanson. Vector mappng and expanson do not change the value of the dscrete functons τ,j and d j. They just change the representaton of values of the two dscrete functons. The vector expanson s vald snce the tme expanson s nfnte. We defne the followng functons for mplementaton: Defnton 1: For a vector v: ror( v, ofs): output v where k [0.. v 1], v [k] = v[(k + ofs) mod v ]; rol( v, ofs): output v where k [0.. v 1], v [k] = v[( v + k ofs) mod v ]; map( v, a, b,, ofs): output v where k [0.. v 1], v [k] = v[( a k b + ofs) mod v ]; exp( v, e) = v v... v (e tmes) ( presents catenatng operaton) We utlze these functons to mplement the TD- Bellman s equaton. Suppose that n has receved the dstance vector d j of node n j. Suppose τ,j [0] s assocated wth the tme slot l,j 0 at node n, and the data arrvng tme slot for τ,j [0] s l 0 j at n j. Then, d (j), the dstance vector of n to the snk from the path through n j, can be calculated as: d j d (j) d (j) = map[ror( d j, l 0 j ), T, T j, τ,j [0], 0]; = exp( τ,j, e 1 ) + exp( d j, e2 ); = rol[ d (j), l,j 0 ] = vec add( τ,j, d j, l,j 0, l0 j ) (11) where d j s defned n Equaton 10 and by defnng A = LCM( τ,j, d j ), e 1 = A/ τ,j and e 2 = A/ d j. We update d and the correspondng parent vector p n the followng way. Suppose the orgnal d [0] s assocated wth the tme slot 0 at n (.e., d [0] = d (t 0 )). Now, ( d, p ) = vmn{exp(d, e 1), exp( d (j),, e 2]} (12) where B = LCM( d (j), d ), e 1 = B/ d, and e 2 = B/ d (j). The functon ( d, p ) = vmn( v 1, v 2 ) compares the correspondng elements n the two vectors v 1 and v 2, and copes the smaller element of each par nto the correspondng element n d and the correspondng vector ID nto p. In addton, we defne an operator < for comparng two vectors v 1 and v 2. Let C = LCM( v 1, v 2 ). If k [0..C 1] exp( v 1, C/ v 1 )[k] exp( v 2, C/ v 2 )[k], then v 1 < v 2. For example, [1, 3] <[2, 3]. Example. We now gve an example to llustrate the vector mplementaton. In Fgure 2, there are four nodes n 0, n 1, n 2, and n 3. n 0 s the snk. By the measurement method ntroduced n Secton 4.1 and Equaton 2, we have τ 1,0 = [155, 5], τ 2,0 = [50, 200], τ 3,1 = [20, 120, 70], and τ 3,2 = [175, 275, 75]. By Equaton 5, we drectly have d 1 = [155, 5] and d 2 = [50, 200], whch means that the shortest latences from node n 1 to node n 0 wll be repeatedly 155 and 5 every two tme slots, startng from t 0 1. Smlarly, ths apples for node n 2. Then node n 1 and n 2 send ther dstance vectors d 1 and d 2 to n 3 by messages. After n 3 receves the messages, by Equaton 11, t wll compute the dstance vector d 3 (1), whch s the latency of routng through node n 1, and the dstance vector d 3 (2), whch s the latency of routng through node n 2. We now have d 1 = [155, 5, 155] (the extenson of d 1). Thus, d 3 (1) = d 1 + τ 3,1 = [20, 120, 70] + [155, 5, 155] = [175, 125, 225]. Smlarly, d 2 = [50, 200, 50] (the extenson of d 1 ), and d 3 (2) = d 2 +τ 3,2 = [50, 200, 50]+[175, 275, 75] = [225, 475, 125].

7 7 n0 d0=0 nj nj n1 n2 n3 T0 = 300ms T1 = T2 = 150ms T3 = 100ms 1,0=[155,5] d1=[155,5] t t² 3,1=[20,120,70] d =[175,125, 225] p =[n1,n1, n1] t 2,0=[50,200] d2=[50,200] t t¹ 3,2=[175,275,75] n (a) nk Fg. 3. Trangular path condton: the drect path n n j always acheves the least latency among all paths from n to n j n (b) nk Fg. 2. Example vector mplementaton Therefore, d3 = vmn{[175, 125, 225], [225, 475, 125]} = [175, 125, 125], and correspondngly p 3 = [n 1 n 1 n 2 ]), whch means that the shortest dstance wll be repeatedly 175, 125, and 125 every three tme slots, startng from t 0 3. The correspondng parents are repeatedly n 1, n 1, and n 2 every three tme slots, startng from t Propertes The FIFO condton [9] means that a packet whch was delvered earler wll always arrve at a drect neghbor earler. We wll prove that an adaptvely duty-cycled WSN satsfes the FIFO condton. Theorem 3: FIFO condton: The lnk cost functon τ,j (t k ) satsfes the FIFO property, whch means, for any t k1 t k2, t k1 < + τ,j (t k1 ) t k2 + τ,j (t k2 ) (13) Proof: From Equaton 3, we have t k2 + τ,j (t k2 ) t k1 τ,j (t k1 ) = (k 2 k 1 ) T +(k 1 T t0,j ) mod T j (k 2 T t0,j ) mod T j = (k 2 k 1 ) T + [(k 1 k 2 ) T ] mod T j = (k 2 k 1 ) T [(k 2 k 1 ) T ] mod T j 0. We now gve an ntutve explanaton for Theorem 3. In a statc network, τ,j (t k2 ) τ,j (t k1 ) = 0, because the lnk cost s constant. In adaptvely duty cycled WSNs, the lnk cost τ,j (t) captures the tme dfference between when node n wakes up at t and when node n j wakes up. Snce node n, whch wakes up at t k1, can always detect the awake neghbor n j earler (or at least at the same tme) than the case n whch n wakes up at t k2, we have t k1 + τ,j (t k1 ) t k2 + τ,j (t k2 ). By Theorem 3, the tme-dependent shortest path problem n adaptvely duty-cycled WSNs s not NP-hard and s solvable n polynomal-tme [9]. Theorem 4: Suppose node n has two neghbors n j and n k, whch are one-hop away from each other. Then, at a tme nstant t, we have the trangular property: τ,j (t ) τ,k (t ) + τ k,j [t + τ,k (t )] (14) Proof: Suppose at tme t, the data arrvng tme slot at n j s t j, and the data arrvng tme at n k s t k. If t k t j, whch means that the data arrvng tme at n k s earler than the data arrvng tme at n j, τ,k (t ) + τ k,j [t + τ,k (t )] = t k t + t j t k = t j t = τ,j (t ). If t k > t j, whch means that the data arrvng tme at n k s later than the data arrvng tme at n j, we have: τ,k (t )+τ k,j [t +τ,k (t )] = t k t +t j t k > t j t +t j t k > t j t > τ,j (t ). The theorem follows. Theorem 4, as llustrated n Fgure 3(a), llustrates that node n wll always drectly arrve at ts neghbor n j wthout gong through other nodes. We now have the followng clam: Lemma 1: Trangular Path Condton: For a node n and ts neghbor n j, at any dspatchng tme, the one-hop path n n j always has the least tme delay for data transmsson. Proof: We prove ths by nducton. Suppose there are multple nodes along the path from n to n j. We defne these nodes as n k, n k+1,, n k+. If = 0, based on Theorem 4, Lemma 1 s true. Now, assume that = k and Lemma 1 s true. We prove that Lemma 1 s true when = k+1. Suppose the adjacent node to n along the path s n k. Then the drect path n k n j has shorter latency than the path along n k+1,, n k+. Also, based on Theorem 4, n n j has shorter latency than the path (n > n k > n j ). Therefore, the drect path n n j has shorter latency than the path along n k, n k+1,, n k+. The lemma follows. An llustraton s gven n Fgure 3(b). Note that the trangular path condton does not exst n statc networks. 5 ALGORITHM FOR INITIAL ROUTE CON- STRUCTION We now present an algorthm for ntal tme-dependent shortest path route constructon n duty-cycled WSNs, where the dstances from all nodes to the snk node are ntally nfnte. The proposed algorthm, referred to as the algorthm, for Fast Tme-Dependent Shortest Path algorthm, s nspred by the work n [10]. As descrbed n Secton 4, although the tme axs s nfnte, the tme-varyng lnk costs and dstance can be mplemented by vectors. Therefore, our algorthm whch mplements Equaton 1 s bascally smlar to the dstrbuted Bellman-Ford algorthm. The dfference s that our algorthm s exchangng vectors (.e., for tme-varyng lnk costs and dstances), rather than sngle values of statc lnk cost and dstance. s adapted from the dstrbuted Bellman-Ford algorthm and s augmented wth a β-synchronzer [27]. We choose a β-synchronzer n order to avod exponental message complexty, as dscussed n Secton 2. Wth a β- synchronzer, the tme cost and message cost are bounded and there s no gan n the best case. However, due to the lmted resource n WSNs, we beleve t s more mportant to avod the exponental message complexty of the tradtonal Bellman-Ford algorthm. Equpped wth the vector mplementaton, computes the shortest paths for nfnte dscrete tme ntervals

8 8 n one executon, n contrast wth the soluton n [10], whch only calculates the shortest paths for a specfed dscrete tme. Let D m denote the dameter of the longest shortest path for all nodes. We show that the message complexty of s O( D m E ) and the tme complexty s O( D m V ). does not suffer from exponental message complexty, lke that n prevous works for the statc shortest path problem over asynchronous networks (e.g., [31]). 5.1 Dstrbuted Algorthm Descrpton There are essentally two steps n our algorthm for constructng routng paths. In the frst step, we buld a spannng tree. In the second step, we calculate the shortest paths and send back the acknowledgment to the snk for nodes n the network wth a layer-by-layer approach. In our algorthm, we combne the two steps and mplement them through teratons. In the frst teraton, computes the shortest paths for nodes n the nearest layer to the snk. In the followng teratons, goes beyond one layer each tme, untl t reaches the last layer. We present the data structures and message formats n : d : vector of dstance from n to n s, defned n Equaton 8; ntally all elements are ; τ,j : lnk delay from n to ts neghbor n j, defned n Equaton 7; τ,j [0] s obtaned by measurement; p : vector of parents for n n the shortest path n n s for nfnte tme ntervals; ntally all elements are node n ; MSG(ID src, ID from, d, updated): control message; ID src s the node ID of snk node, or update source node (see Secton 6); ID from s the sender s node ID; d s the dstance vector of the sender; updated ndcates whether there s an update n the current teraton, whch wll be explaned later; ACK[j]: boolean ndcatng whether a node receves a control message from ts neghbor n j We assume that n knows the duraton of the checkng nterval T j of all ts neghbors n j N after measurng lnk delays. Intally, a drected spannng tree rooted at n s s bult upon the network. We assume that n knows ts parent st p n the spannng tree. We also assume that s nvoked by hgher-level protocols that create START mpetuses at n s. The frst teraton of begns when node n s receves the START mpetus. Subsequent ones begn whenever n s completes an teraton and determnes whether another one s necessary by checkng whether there s a node whose dstance was mnmzed n the last teraton. Each teraton begns at node n s by sendng a control message to all ts neghbors. When reples from all ts neghbors have been receved, node n s concludes that an teraton s completed. Every other node,.e., n (n n s ), begns an teraton upon recevng a control message from ts parent st p n the spannng tree, upon whch t sends control messages to all ts neghbors except st p. When reples are receved from all these neghbors, a control Algorthm 1: Algorthm for snk node n s n : Intalzaton: n j N s, ACK[j] = 0, updated[j] = true; On recevng START: send MSG(s, s, 0, false) to n j N s; On recevng MSG(j, d j, l j, bchanged): ACK[j] = 1; updated[j] = bchanged; f ( n j N s, ACK[j] == 1) then ACK[j] = 0; f ( n j N s, updated[j] == false) then STOP; else send MSG(s, s, 0, false) to nj N s; message s sent to the parent, thereby completng the current teraton at node n. The control message from node n j contans the dstance vector d j (known thus far durng the prevous teratons) between n j and n s. When such a message s receved at node n, node n checks whether the new nformaton decreases the value of any element n the current dstance vector. It does so by consderng the path that goes through n j, takng nto account the most recent nformaton from n j. We descrbe the procedures n Algorthms 1 and 2. Algorthm 2: Algorthm for node n ( s) n : Intalzaton: st p = NULL; for n j N do ACK[j] = 0; updated[j] = true; Measure lnk delay 0,j at the begnnng of any tme slot; l,j 0 = the tme slot number for measurement; l j = the data arrvng tme slot number at n j ; for k [0..LCM(T, T j )/T j ] do τ,j [k] = T j (k T 0,j )mod T j; On recevng MSG(s, j, dj, bchanged) from n j : ACK[j] = 1; updated[j] = bchanged; d prev = d ; f n j == st p then send MSG(s,, d, false) to n j N except st p ; d (j) = vec add( d, d j, l,j 0, l j); /* Equaton 11*/ ( d, p ) = vmn( d, d (j)); /* Equaton 12*/ f ( d < d prev ) then updated[j] = true; f ( n j N, ACK[j] == 1) then f ( n j N, updated[j] == true) then bupdated = true; else then bupdated = false; send MSG(s,, d, bupdated) to st p ; n j N, ACK[j] = 0; The functons vec add( ), vmn( ), and operator < are defned n Secton Correctness and Complexty Let P AT H(, t k ) be a path obtaned by node n, whch s startng at tme t k and movng along ts parent p [k]. Let d [k] m denote the value of d [k] after the m th teraton n. We have the followng propertes: Theorem 5: 1) After termnaton, P AT H(, t k ) s loopfree and concatenated. 2) In the m th (m 0) teraton, a node n whose shortest path s at most m-hop away from the snk node wll be determned, for all dscrete tme ntervals t k T.

9 9 Proof: For part 1, t k T, after termnaton, suppose p [k] s set to the node n j for the shortest path wth respect to n s. Snce n j s the parent of n at the tme slot t k, P AT H(, t k ) s a path composed of P AT H(j, tk +τ,j(t k )), whch s appended to node n t k T. Thus, for any node n and t k T, P AT H(, t k ) s concatenated. We prove that P AT H(, t k ) s loop-free by contradcton. Wthout loss of generalty, assume that there s a loop: (n 0 n 1 n 2 n +k n 0 ). Ths means that, there s a shortest path (n 0 n 1,, n +k ), where n +k s one-hop away from n 0. Accordng to the trangular path condton n Lemma 1, such a shortest path cannot exst because n 0 n +k s always the shortest one among all paths from n 0 to n +k, contradctng the assumpton. We prove part 2 by nducton on m. It s easy to fnd that the clam s true for m = 0. Now, assume that the clam s true for m 1 (.e., the nductve hypothess). We now prove for m by nducton. Consder a specfc tme t k and a node n such that there s a shortest path wth at most m hops between n and n s. Let SP (, s, t k, m) be the shortest path, whch s at most m hops from n to n s at tme t k. Let n j be n s parent on SP (, s, t k, m) at tk. Ths means that, there s a path wth at most m 1 hops between n j and n s. By the nductve hypothess, n j determned ts shortest dstance at the (m 1) th teraton. In the m-th teraton, node n j sends ts mnmzed dstance vector d j m 1 to node n. Thus, SP (, s, t k, m) s determned after recevng the vector d j m 1 n the m th teraton, whch completes the nductve step. Snce t k s chosen arbtrarly, ths holds for all values of t k T. Part 2 n Theorem 5 mples that, for m D m, all nodes determne ther mnmum delay and the correspondng parents for all tme ntervals, snce all shortest paths contan at most D m nodes. Theorem 6: The message and tme complexty of s O(D m E ) and O(D m V ), respectvely. Proof: Based on the mplementaton of the β- synchronzer n [32], n each teraton, there s exactly one message traversng each lnk n the spannng tree, totally E messages exchanged. By Theorem 5, the number of teratons s upper bounded by the longest shortest path s length D m. Thus, the message complexty s O(D m E ). Suppose the largest delay for transmttng a message n the spannng tree s a constant, denoted by C. In each teraton, the tme consumed s at most V C. Snce there are at most D m teratons, the tme complexty s O(D m V ). 6 ALGORITHM FOR DYNAMIC ROUTE MAINTE- NANCE When compared wth statc networks, lnk changes and node changes are more frequent n duty-cycled WSNs. If a node changes ts duty-cycle confguraton, or dynamcally jons or leaves the network, the lnks connectng wth all ts neghbors wll be changed at multple tme ntervals. In such a stuaton, a sngle node update usually causes multple lnk updates. Some prevous works n statc networks (e.g., [15]) have proposed solutons that effcently deal wth sngle lnk updates. They are neffcent for multple lnk updates caused by a sngle node update. The algorthms n [15] are also memory-neffcent, snce each node stores the route entres for all other nodes, ncurrng the space complexty of O( V ). Unlke prevous works [15], where each node stores the route nformaton for all other nodes, we propose a soluton n whch a node only stores the route to the snk, whch s more practcal n WSNs due to ther memory constrants. In our proposed algorthm, when one node s updated (denoted as the source node), the algorthm does not update the shortest path for the whole network from scratch, but only updates necessary nodes. Thus, the man dea s frst to dentfy whch nodes need to be updated. After that, the algorthm updates the shortest path for these dentfed nodes. The updatng process s smlar to the route constructon descrbed n Secton 5. But the startng pont s the source node, rather than from the snk node, and the updatng scope s just a subset of the whole network. The proposed dstrbuted algorthms for path mantenance are also equpped wth the β synchronzer. Agan, we choose the β-synchronzer n order to avod exponental message complexty as dscussed n Secton 2. The proposed algorthms, referred to as -M ( M meanng mantenance), focus on per-node update and can be easly extended to node nserton and deleton. If there are multple node updates, the algorthms wll run concurrently at multple nodes. 6.1 Overvew and Ratonal of the Dstrbuted Algorthms Suppose the source update node s n u and the correspondng nput change s σ. We dvde σ nto two parts: σ nc and σ dec, where σ nc ncludes the ncreasng lnks for t k u T u, and σ dec ncludes the decreasng lnks for t k u T u. Let δ(σ nc ) be the set of nodes that change ether the dstance or the parents for all nfnte dscrete tme ntervals, as a consequence of σ nc. Smlarly, let δ(σ dec ) be the set of nodes affected by σ dec. Apparently, δ(σ) = δ(σ nc ) δ(σ dec ). We dentfy the nput change δ(σ) as δ(σ nc ) and δ(σ dec ), because δ(σ dec ) s easy to handle gven the suffcent loopfree condton clamed n [33], whch s referred to as the dstance ncrease condton (or DIC). Wth DIC, at tme t, f node n detects a lnk-cost decrease or a decrease n the dstance reported by a neghbor, node n s free to choose ts new parents. Therefore, the node n δ(σ dec ) can safely select a new parent. However, thngs become more complcated for nodes n δ(σ nc ), snce a loop can be formed f they drectly choose a new parent [33]. In order to address ths ssue, we adopt two phases. In the frst phase, we ncrease the nodeto-snk dstance for all nodes, whch are descendants of n u. In the second phase, those nodes wll re-select ther parents based on the Bellman-Ford approach. We use the β-synchronzer n phase 1 and phase 2, n order to bound the message complexty, though t wll

10 10 ntroduce addtonal tme cost. For WSNs, whch have lmted energy resource, less communcaton s usually more mportant for some applcatons, whch s the motvaton for usng the β-synchronzer. 6.2 Algorthm Descrptons We use smlar data structures and message formats as that n Secton M conssts of two phases for node n u and all nodes n δ(σ). We descrbe our algorthms n Algorthms 3 and 5. In phase 1, an ntal spannng tree s bult up gradually to contan all nodes n δ(σ nc ). The purpose of phase 1 s to let all nodes n δ(σ nc ) ncrease ther dstances to the snk node as a consequence of σ nc, along the tme-expanded shortest path trees rooted at n u. After the termnaton of phase 1, all nodes n δ(σ) wll never ncrease ther dstance agan. In each teraton, node n u sends a control message to all neghbors. Every other node,.e., n (n n u ) wll send control messages to all ts neghbors f t s n the spannng tree and receves a message from ts parent. If n s not n the spannng tree n the current teraton, t checks whether n j s ts parent n ts shortest path after recevng a control message from n j, whch can be done by checkng whether n j p. If true, n v wll jon the spannng tree and set newsp = n j. By dong so, the spannng tree wll ncrease at most one level n each teraton. Algorthm 3: Operatons at update node n u n - M: Intalzaton: Same ntalzaton as n Algorthm 2; f p u null then for n j N u do ( d j,lj 0) = get dst(n j); /* retreve d j, detaled mplementaton s omtted */ d j (u)=vec add( τ u,j, d j,lu,j 0,l0 j ); /* Equaton 11*/ nc update( d u, p u, n j, dj (u)); Phase 1: On recevng START: send MSG(u,u, d u,false) to n j N u; Phase 1: On recevng MSG(u, j, dj, bchanged): ACK[j] = 1; updated[j] = bchanged; f ( n j N u, ACK[j] == 1) then f ( n j N u, updated[j] == false) then Begnnng Phase 2; else send MSG(u,u, d u,false) to n j N u; n j N u, ACK[j] = 0; Phase 2: On Begnnng Phase 2: send MSG(u,u, d u,false) to n j N u; Phase 2: On recevng MSG(0, j, dj, bchanged) from n j : ACK[j] = 1; updated[j] = bchanged; d prev u = d u; d u(j) = vec add( d u, d j, l,j 0, l j, ); /* Equaton 11 */ ( d u, p u) = vmn( d u, d (j)); /* Equaton 12 */ f ( d u < d prev u ) updated[j] = true; f ( n j N u, ACK[j] == 1) then f ( n j N u, updated[j] == false) then STOP; else send MSG(u,u, d u,false) to n j N u; n j N u, ACK[j] = 0; Algorthm 4: Functon: nc update( d 1, p 1,n 2, d 2 ) d 1 = exp( d 1,( d 1 d 2 )/ d 1 ); d 2 = exp( d 2,( d 1 d 2 )/ d 2 ); p 1 = exp( p 1,( d 1 d 2 )/ d 1 ); flag = false; for k = 0 to d 1 d 2 1 do f p 1 [k] == n 2 && d 1 [k] < d 2 [k] then d 1 [k] = d 2 [k]; flag = true; return flag; A control message wll traverse from the root (n u ) to all other nodes n the spannng tree just lke that n. When node n receves a control message from n j, t only updates the element to be ncreased n ts dstance vector, as llustrated n Algorthm 4. When reples from all ts neghbors have been receved, node n u concludes that an teraton s completed. When reples are receved from all neghbors by a node, a control message s sent to the parent, thereby completng the teraton at the node. When there s no dstance ncrease for all nodes n δ(σ nc ), phase 1 wll be termnated and node n u wll start phase 2. In phase 2, the ntal spannng tree bult up n phase 1 s contnuously growng untl t contans all nodes n δ(σ). Phase 2 s also runnng by teratons. In each teraton, when a node n not n the spannng tree receves a control message from n j, f the value of any elements n ts dstance vector s decreased, t wll jon the spannng tree by settng ts parent newsp to n j. The dstance update and message traversng n phase 2 of -M are just smlar to that n. 6.3 Correctness and Complexty In phase 1, all nodes n δ(σ nc ) do not change ther parents, but ncrease ther dstances as a consequence of σ nc. Thus, there s no loop n phase 1. In phase 2, all nodes n δ(σ) wll never ncrease ther dstances, thereby satsfyng the dstance ncrease condton [33]. All paths are therefore loop-free. Theorem 7: In phase 1, each node n δ(σ nc ) wth at most m hops away from n u along the tme-dependent shortest path wll not ncrease ts dstance after m teratons. Proof: The proof s by nducton on m. It s easy to fnd that the clam s true for m = 0. Now, assume that the clam s true for m 1 (.e., the nductve hypothess). We now prove for m by nducton. Consder a specfc tme t k and a node n δ(σ nc ) such that there s a shortest path wth at most m 1 hops between n and n s after node update. Let SP (, u, t k, m 1) be the shortest path whch s at most m 1 hops from n to n u at tme t k. Let n j be n s neghbor whch s not updated yet due to σ nc. Then n j s at most m hops away from n u. By the nductve hypothess, n wll not ncrease ts dstance after (m 1) th teraton. In the m-th teraton, node n sends ts updated dstance vector d j m 1 to node n j, and n j wll determne ts updated dstance n the teraton f n j δ(σ nc ). Thus, SP (j, u, t k, m) s determned after recevng the vector d m 1 n the m th teraton, whch completes the nductve step. Snce t k s chosen arbtrarly, ths holds for all values of t k T.

11 11 Algorthm 5: Operatons n node n (n n u ) n - M: Intalzaton: newsp = null; n j N,updated[j] = false; Phase 1: On recevng MSG(u, j, dj, bchanged) from n j : f n j N u then re-measure τ,u at the begnnng of one tme slot; reset l 0,j and l0 u ; d (j)=vec add( τ,j, d j,l,j 0,l0 j ); f newsp == null then f (nc update( d, p,n j, d (j))) then newsp = n j ; send MSG(u,, d j,true) to n j ; else send MSG(u,, dj, false) to n j ; else FORWARD(u); /* u means the MACRO s executed n phase 1 */ ACK[j] = 1; updated[j] = nc update( d, p,n j, d (j)); ACK REPLY(u); Phase 2: On recevng MSG(0, j, dj, bchanged): f newsp == null then f (updated[j]) then newsp = n j ; send MSG(0,, dj, true) to n j ; else send MSG(0,, dj, false) to n j ; else FORWARD(0); /* 0 means the MACRO s executed n phase 2 */ ACK[j] = 1; updated[j] = bchanged; d prev = d ; d u(j) = vec add( d u, d j, l,j 0, l j); /* Equaton 11*/ ( d u, p u) = vmn( d u, d (j)); /* Equaton 12*/ f ( d u < d prev u ) updated[j] = true; ACK REPLY(0); FORWARD(nt dct): code macro f (newsp == n j ) then send MSG(dct,, d,false) to n j N except newsp ; ACK REPLY(nt dct): code macro f ( n j N, ACK[j] == 1) then f ( n j N, updated[j] == true) then bupdated = true; else then bupdated = false; send MSG(dct,, d,bupdated) to newsp ; n j N, ACK[j] = 1; bupdated = false; By Theorem 7, after δ(σ nc ) teratons, all nodes n δ(σ nc ) wll not ncrease ther dstance anymore. Defnton 2: Updated-subpath: for any node n δ, the updated-subpath s from n to the frst node n e not n δ along the shortest path from n to n u. Theorem 8: In phase 2, all generated updated-subpaths are loop-free, and updated-subpaths wth at most m hops long are determned n the m th teraton. Proof: After phase 1, the DIC loop-free condton [33] s satsfed. Thus, all updated-subpaths are loop free n phase 2. The proof for at most m hops-long updated-subpaths beng determned n the m th teraton can be done by nducton. It s easy to fnd that the clam s true for m = 0. Now, assume that the clam s true for m 1 (.e., the nductve hypothess). We now prove for m by nducton. Consder a specfc tme t k and a node n δ such that there s an updated-subpath whch contans m hops between n and n u. Let UP (, u, t k, m) be the updatedsubpath whch s at most m hops long at tme t k. Let n j be n s parent on UP (, u, t k, m) at tk. Ths means that there s an updated-subpath wth at most m 1 hops between n j and n s. By the nductve hypothess, the updated subpath from n j to n u s determned at the (m 1) th teraton. In the m- th teraton, node n j sends ts mnmzed dstance vector d j m 1 to node n. Thus, UP (, u, t k, m) s determned after recevng the vector d j m 1 n the m th teraton, whch completes the nductve step. Snce we choose t k arbtrarly, the theorem holds for t k T. Theorem 9: The message complexty for per node update wth δ(σ) output change s O( δ(σ) 2 maxdeg). The tme complexty s O( δ(σ) 2 ), and space complexty s O(maxdeg). Proof: In phase 1, the number of teratons s δ(σ nc ) δ(σ). In phase 2, there are at most δ(σ) teratons before all updated-subpaths are decded. Thus totally, there are O( δ(σ) 2 maxdeg) messages. In each teraton, the consumed tme s at most δ(σ) C (C s the largest transmsson delay for all lnks). Thus, the message complexty s O( δ(σ) 2 ). Snce a node only stores the nformaton of all ts neghbors, the space complexty s O(maxdeg). 7 PRACTICAL IMPLEMENTATION We now dscuss some practcal mplementaton ssues. In partcular, we dscuss how to acheve awareness of schedules of neghborhood and how to reduce the vector sze gven a large LCM n Equaton Schedule Awareness and -M requre awareness of wakeup schedules of the neghborhood. Achevng ths requrement depends on the specfc underlyng MAC protocol. We dscuss three scenaros for achevng schedule awareness over dfferent MAC protocols. Actve Neghbor Dscovery: Ths means that a node needs to probe the schedules of ts neghbors actvely. We consder two scenaros whch need actve neghbor dscovery. One s the LPL mode as adopted by B-MAC [5] and X-MAC [6]. The other s the low duty-cyclng mode, where tme axs are arranged as consecutve short tme slots, and all slots have the same duraton. For ether scenaro, we assume that beacon messages are sent out at the begnnng of wakeup slots, smlar to [28], [34]. In order to dscover neghbors, a node has to stay awake n order to detect the beacon message of ts neghborhood. The node should wat untl beacons are receved from ts neghbors. The frequency wth whch a node should detect ts neghbors schedule depends on mplementaton consderatons. Quorum-based Duty-Cyclng: Actve neghbor dscovery mechansms requres a node to stay awake actvely. Now we ntroduce quorum-based duty-cyclng whch does not have that requrement. Here, the wakeup schedule follows a quorum system desgn [35]. In quorumbased duty cyclng, two neghbor nodes can hear each other at least once wthn bounded tme slots va the nonempty ntersecton property. We choose cyclc quorum systems [28] for presentaton. We use the followng defntons for brefly revewng quorum systems (used for wakeup schedulng). Consder a cycle length n and U = {0,, n 1}.

12 12 Defnton 3: A quorum system Q under U s a superset of non-empty subsets of U, each called a quorum, whch satsfes the ntersecton property: G, H Q : G H. If G, H Q, {0, 1,...n 1}: G (H + ), where H + = {(x + ) mod n : x H}, Q s sad to have the rotaton closure property. Cyclc quorum systems (or cqs) satsfy the rotaton closure property, and are denoted as C(A, n), where A s a quorum and n s the cycle length. For example, the cqs {{1, 2, 4}, {2, 3, 5}, {7, 1, 3}} can be denoted as C({1, 2, 4}, 7). Gven two dfferent cyclc quorum systems C(A 1, n 1 ) and C(A 2, n 2 ), f two quorums from them, respectvely, have non-empty ntersectons, then, even wth clock drft, they can be used for heterogenous wakeup schedulng n WSNs. For example, gven C({1, 2, 4}, 7) and C({1, 2, 4, 10}, 13), two quorums from them, respectvely, have non-empty ntersecton for every 13 tme slots. Therefore, two nodes that wake up wth schedules complyng wth any two quorums from these two cyclc quorum systems can hear each other. By embeddng the wakeup schedule nformaton n the beacon message, a node can always detect the wakeup schedules of ts neghborhood through the non-empty ntersecton property of the quorum desgn. Synchronzaton: MAC protocols wth synchronzaton requre that all neghborng nodes wake up at the same tme. The smplest method for dong ths s to use a fully synchronzed pattern, lke that n the S-MAC protocol [36]. In ths case, all nodes n the network wakeup at the same tme accordng to a perodc pattern. A further mprovement can be acheved by allowng nodes to swtch off ther rado when no actvty s detected for a tmeout value, lke that n the TMAC protocol [37]. In ths scheme, neghborng nodes form vrtual clusters to set up a common sleep schedule. The man dsadvantages of such scheduled rendezvous schemes are the complexty of mplementaton and the overhead for synchronzaton. Through synchronzaton, a node can convenently know the schedules of ts neghbors. Schedule awareness can be acheved by perodc message exchange between a node and ts neghbors. 7.2 Sub-optmal Implementaton wth Vector Compresson The key mplementaton aspect of our proposed algorthms s the vector representaton of lnk cost functons and dstance functons. However, f the vector sze s too large (.e., the LCM n Equaton 6 s too large), the proposed algorthms, and -M, may not be feasble gven the lmted memory resource of embedded sensor nodes, and neffcent due to dstrbuted message exchangng. Based on Theorem 2, the vector sze s dependng on LCM(T 0, T 1,, T )/T. The worst case s that all nodes have dfferent cycles, and the sze of the dstance vector can be very large. In a real mplementaton, to avod arbtrarly long vectors, there are two possble solutons: 1) Use a predefned LCM(T0,T1,,T) duty cycle set, so that the T can be bounded by carefully selectng a duty cycle set, as dscussed n Sectons 4.1 and 4.2; 2) Adopt vector compresson to acheve a trade-off,.e., adopt a low-accurate dstance vector, whch takes less memory space, to represent the end-to-end latency. Hence, the output path s sub-optmal n terms of latency. The frst soluton can be appled to small-scale networks, where the node number s not large and predefnng a duty cycle set s not dffcult. For a large-scale network, we mght need the second soluton n whch a LCM(T0,T1,,T) bounded, global T s not necessary. The basc dea of vector compresson n the second soluton s to smooth all values n a vector and represent the vector wth less nformaton. For example, for a vector [ ] wth 6 elements, we can approxmately represent the vectors by a vector wth 2 elements, such as [(2, 3), (5, 3)] (where 2 = (1+2+3)/ and 5 = (4+5+6)/3). Each tuple (v, s) n the vector represents the average value of s elements n the orgnal vector. The formal descrpton of vector compresson s as follows: Vector compresson: Suppose the source vector s v s = [v 1, v 2,, v n ] and the target vector sze s m (n > m). We compress v s by: v t = [( v 1 + v v len, len), len ( v 1+len + v 2+len + + v 2 len, len),, len ( v 1+(m 1) len + v 2+(m 1) len + + v n, n (m 1) len)] n (m 1) len where len = n m. We choose the average value v 1+ len+v 2+len + +v (+1) len len as the value of v t [], because the expected devaton can be mnmzed by P = len j=1 1 len v t[] v[j]. In addton to the averagng flter used n the above equaton, other flters, such as the Wavelet transform flter [38] can also be appled for vector compresson, as typcally used n mage compresson. 7.3 Remarks The algorthm descrbed n Secton 5 s a proactve routng protocol. Although ts tme complexty s O(D m V ) for ntal route constructon, t s affordable n the ntal stage of WSNs. The low space complexty (O( maxdeg )) for route mantenance makes the algorthm scalable for large-scale WSNs. Note that and -M target the ALPL mode [7] wth varous checkng ntervals. When all nodes have homogenous LPL checkng ntervals (lke that n the standard B-MAC), accordng to Equatons 3 and 5, the lnk cost functon and the dstance functon wll become constants. In such a case, our algorthms wll default to the statc shortest path algorthm. However, and -M wll yeld the same message complextes and tme complextes for the statc stuaton. Our work focuses on the scenaro of a sngle snk node. However, t can be extended to multple destnatons. In WSNs, there s usually no end-to-end communcaton between two arbtrary nodes. We only consder the generalzaton of communcaton between one node and multple

13 13 snk nodes, rather than the communcaton between two arbtrary nodes. 8 EXPERIMENTAL RESULTS We evaluated the performance of, -M, and the sub-optmal mplementaton through extensve smulatons usng the OMNET++ dscrete event smulator [39]. We compared our algorthms wth other related algorthms for the TDSP problem, ncludng the dstrbuted Bellman-Ford algorthm [31] adapted to the tme-dependent model (Secton 4), referred to here as TD-Bellman, and DSPP1 [10]. The followng three major metrcs were measured n the evaluaton: 1) message count, 2) tme cost, 3) average memory cost. We examned two man factors that affect the performance of our algorthms, ncludng network sze and the underlyng duty cycle settng. Our expermental settngs were compatble wth typcal confguratons, as n [3], [7], [8]. The wreless communcaton range n our smulaton was set to 10m. We adopt the wreless loss model used n [40], whch consders the oscllaton of rado lnks. We generated 8 network sze sets wth varyng szes, G 1,,G 8, whch are lsted n Table 1. For each network sze, we randomly generated 10 topologes. Each data pont presented n our smulaton results n ths secton s the average of 10 topologes, wth 10 runs on each topology. whch ndcates that the largest dstance vector sze s 10 by Equaton 6, and vared the network sze. Wth the number of nodes ncreasng from 50 to 2000 n G 1,,G 8, the average tme consumed and the message count are shown n Fgure 4. The average executon tme of DSPP1 s about 10 tmes larger than that of, snce DSPP1 has to be executed 10 tmes to compute the shortest paths for all tme ntervals. FD-Bellman s better than when the network sze s small, snce FD-Bellman does not have a dstrbuted synchronzer n ts executon. When the network sze becomes large (.e., 1K), outperforms FD-Bellman due to the exponental worst case message complexty of the Bellman-Ford algorthm. We observed smlar trends for tme cost for the three algorthms, as shown n Fgure 4. Tme(sec) 100K 10K 1K 100 DSSP1 TD Bellman 10 Network Sze Message Count 10M 1M 100K 10K 1K DSSP1 TD Bellman 100 Network Sze Fg. 4. Comparson of tme effcency and message effcency by varyng V TABLE 1 Network Sze Sets G 1 G 2 G 3 G 4 V G 5 G 6 G 7 G 8 V 600 1K 1.5K 2K TABLE 2 Tme Slot Sets C 1 (ms) {100, 100, 100, 100} C 2 (ms) {100, 200, 300, 600} C 3 (ms) {100, 200, 400, 800} C 4 (ms) {100, 200, 500, 1000} Tme(sec) DSSP1 TD Bellman C1 C2 C3 C4 Duty Cycle Sets Message Count 800K 600K 400K 200K 0 DSSP1 TD Bellman C1 C2 C3 C4 Duty Cycle Sets We chose two MAC protocols: ALPL (adaptve low power lstenng) and quorum-based duty-cyclng. In the ALPL mode, a node just wakes up for a short tme durng a checkng nterval to check the channel actvtes. The duraton of the checkng nterval vares for dfferent nodes. We changed the duraton of the checkng nterval n our smulaton experments wth 4 sets, C 1, C 2, C 3, and C 4, as lsted n Table 2. Wth each set, we randomly chose one element as the value of the LPL checkng nterval for each node. Wth dfferent tme slot sets, the sze of a message (changed wth vector sze) s changng. Thus, we used a flexble packet sze n our smulaton. Each element n a vector occuped 1 byte n all experments. For quorum-based duty-cyclng, we choose the (7, 3, 1), and (21, 5, 1) dfference sets for the heterogenous wakeup schedule settngs. The duraton of one tme slot was set to 100ms n quorum-based duty-cyclng. Snce, - M, TD-Bellman, and DSPP1 are ndependent of wakeup schedulng, we argue that the comparson s far even when we choose quorum-based duty-cyclng. 8.1 Least-latency Path Constructon In the frst set of smulaton experments, we measured the ALPL mode and chose C 4 as the tme slot settng, Fg. 5. Comparson of tme effcency and message effcency by varyng tme slots n ALPL mode Tme(sec) 100K 10K 1K 100 DSSP1 TD Bellman 10 Network Sze Message Count 10M 1M 100K 10K 1K DSSP1 TD Bellman 100 Network Sze Fg. 6. Comparson of tme effcency and message effcency for quorum-based duty-cycle settng We also vared the tme slot sets wth a fxed network sze of G 6, whch represents medum-szed WSNs. The results are shown n Fgure 5. We observe that and FD- Bellman do not change ther message count sgnfcantly snce they only depend on the network sze. The tme cost of all algorthms become worse when the average value of all elements n the selected tme slot set becomes larger, snce the average lnk delay s correspondngly ncreasng. Fnally, we measured the performance for quorumbased duty-cyclng by fxng the network sze of G 6. Each node randomly chose the (7, 3, 1) and (21, 5, 1) dfference

14 14 sets for ts heterogenous schedule settngs. As shown n Fgure 6, we observe smlar trends for executon tme and message count. The average executon tme of DSPP1 s about 3 tmes larger than that of, snce DSPP1 has to be executed 3 tmes to compute the shortest paths for all tme ntervals gven the (7, 3, 1) and (21, 5, 1) dfference sets. The tme costs for all algorthms become worse for larger szed networks, whch s consstent wth the concluson n Theorem Least-latency Path Mantenance For evaluatng the path mantenance performance of -M, we return to statc networks by selectng the tme slot set of C 1 n Table 2 for the ALPL mode. We do so for the purpose of a far comparson wth the prevous work n [15], referred to here as Full-Dynamc and DSDV [41] wth uncast support. We frst evaluate the effect of nput changes on all algorthms for medum-szed networks by choosng G 6. As shown n Fgure 7, -M acheves the medan message cost and tme cost when nput changes become large. The reason s that, DSDV suffers from exponental message complexty. In addton, -M uses the synchronzer, whch consumes addtonal tme and messages, whch s not as effcent as that n Full-Dynamc. We also measured the average memory cost by varyng the network sze for the two underlyng duty-cyclng mechansms. We frst chose the ALPL mode and set the tme slot set of C 1 n Table 2 for all nodes. The results shown n Fgure 8 ndcate that -M acheves the best memory cost, whch does not depend on the network sze. The memory costs of DSDV and Full-Dynamc ncrease wth the network sze, snce each node stores an entry for all other nodes. Tme(sec) M Full Dynamc DSDV [1 10] [40 60] [80 100] [ ] [ ] #Nodes n nput change Message Count 100K 10K 1K 100 M Full Dynamc DSDV 10 [1 10] [40 60] [80 100] [ ] [ ] #Nodes n nput change Fg. 7. Performance comparson for route mantenance by varyng nput change: ALPL mode Memory (#entres) n ALPL Mode M Full Dynamc DSDV Network sze Memory (#entres) n Quorum based Duty Cyclng Mode M Full Dynamc DSDV Network sze Fg. 8. Performance comparson for route mantenance on memory requred n each node Fnally, we measured the average memory cost for the quorum-based duty-cyclng mechansm. Each nodes randomly chose the (7, 3, 1) dfference sets for the heterogenous schedule settngs. As shown n Fgure 8, we observe smlar trends for the quorum-based duty-cyclng mechansm. The only dfference s that wth (7, 3, 1) dfference sets, each node mantans 3 entres n the routng table for all neghbors. 8.3 Performance of Sub-Optmal Implementaton For sub-optmal mplementaton wth vector compresson, the performance s a trade-off between path latency and message sze. We evaluated ths tradeoff for both ntal route constructon and route mantenance. We frst fxed the duty cycle settng by choosng C 4 and compared the performance between and ts sub-optmal mplementaton. Snce the message count does not rely on vector mplementaton, we compared the vector sze (n terms of the number of elements n a vector). Snce each element took one byte n a message packet, the packet sze (n bytes) represents the vector sze. Max Mn Latency (ms) Sub Optmal 0 Network Sze Vector Sze Sub Optmal 0 Network Sze Fg. 9. Performance comparson for route constructon: latency and vector sze (n bytes) over ALPL mode Max Mn Latency (ms) Sub Optmal 0 Network Sze Vector Sze Sub Optmal 4 Network Sze Fg. 10. Performance comparson for route constructon: latency and message sze over quorum-based duty-cyclng mode Fgure 9 shows the results. We observe that the suboptmal mplementaton acheves less message sze wth vector compresson. To understand the end-to-end latency of the two technques, we compared the maxmum value of the least latency acheved by all nodes (defned as max-mn latency). As shown n Fgure 9, the sub-optmal mplementaton has hgher max-mn latency than. However, the sub-optmal mplementaton has a smaller average message sze than. We then fxed the duty cycle settng by choosng the (7, 3, 1) and (21, 5, 1) dfference sets as the wakeup schedule for all nodes. We observed smlar trends, as shown n Fgure 10. We observe that the sub-optmal mplementaton has hgher max-mn latency than, but has a smaller vector sze. The smulaton results show the performance tradeoff after ntroducng the suboptmal mplementaton, whch has smaller message sze but hgher latency.

15 15 9 CONCLUSIONS In ths paper, we addressed the dstrbuted shortest path routng problem n duty-cycled WSNs. Our contrbutons are four-fold. Frst, we modeled duty-cycled WSNs as tme-dependent networks, whch satsfy the FIFO condton. Second, we presented the algorthm for fndng shortest paths n such networks. has polynomal message complexty and s more tme-effcent than prevous solutons. Thrd, we presented -M for dstrbuted route mantenance wth node nserton, updatng, and deleton. -M s memory effcent and has polynomal message complexty. Fnally, we proposed a sub-optmal mplementaton on vector representatons to reduce memory requrements. The vector sze of the sub-optmal soluton does not depend on the largest LCM value as shown n Equaton 6. Smulaton results valdated the effectveness and effcency of our solutons. We envson several drectons for future work. One s to nvestgate the tme-dependent mnmum spannng tree problem, whch s NP-Hard n duty-cycled WSNs. Another drecton s to study tme-dependent multcast routng n duty-cycled WSNs, whch s a requred servce for many applcatons and s the reverse drecton of allto-one least-latency routng. REFERENCES [1] Shouwen La and Bnoy Ravndran, On dstrbuted tmedependent shortest paths over duty-cycled wreless sensor networks, n IEEE Conference on. Computer Communcatons (INFO- COM), [2] C. Schurgers and M.B. Srvastava, Energy effcent routng n wreless sensor networks, n Mltary Communcatons Conference (MILCOM 2001), 2001, vol. 1, pp [3] A. Woo, T. Tong, and D. Culler, Tamng the underlyng challenges of relable multhop routng n sensor networks, n Proceedngs of the 1st nternatonal conference on Embedded networked sensor systems (Sensys), 2003, pp [4] C.M. Vgorto, D. Ganesan, and A.G. Barto, Adaptve control of duty cyclng n energy-harvestng wreless sensor networks, n IEEE Communcatons Socety Conference on Sensor, Mesh and Ad Hoc Communcatons and Networks (SECON), June 2007, pp [5] J. Polastre, J. Hll, and D. Culler, Versatle low power meda access for wreless sensor networks, n Internatonal conference on Embedded networked sensor systems (Sensys 04), 2004, pp [6] Mchael Buettner, Gary V. Yee, Erc Anderson, and Rchard Han, X-mac: a short preamble mac protocol for duty-cycled wreless sensor networks, n ACM Sensys, New York, NY, USA, 2006, pp [7] Raja J., Perre B., and Crstna V., Adaptve low power lstenng for wreless sensor networks, IEEE Transactons on Moble Computng, vol. 6, no. 8, pp , [8] Y. Gu and T. He, Data forwardng n extremely low dutycycle sensor networks wth unrelable communcaton lnks, n Proceedngs of the 6th ACM conference on Embedded network sensor systems (Sensys), 2007, pp [9] K. L. Cooke and E. Halsey, The shortest route through a network wth tme-dependent nternodal transt tmes, J. Math. Anal. Appl., vol. 14, pp , [10] Arel Orda and Raphael Rom, Dstrbuted shortest-path protocols for tme-dependent networks, Dstrbuted Computng, vol. 10, no. 1, pp , [11] I. Chabn, Dscrete dynamc shortest path problems n transportaton applcatons: Complexty and algorthms wth optmal run tme, Transportaton Research Records, vol. 1645, pp , [12] Boln Dng, Jeffrey Xu Yu, and Lu Qn, Fndng tme-dependent shortest paths over large graphs, n Proceedngs of Extendng database technology (EDBT 08), 2008, pp [13] H. Chon, D. Agrawa, and A. Abbad., Fates: Fndng a tme dependent shortest path, Moble Data Management, vol. 2574, pp , [14] Arel Orda and Raphael Rom, Mnmum weght paths n tmedependent networks, Networks, vol. 21, pp , [15] Serafno C., Gabrele D., Danele F., and Umberto N., A fully dynamc algorthm for dstrbuted shortest paths, Theoretcal Computer Scence, vol. 297, no. 1-3, pp , [16] G. D Angelo, S. Ccerone, G. D Stefano, and D. Frgon, Partally dynamc concurrent update of dstrbuted shortest paths, n Internatonal Conference on Computng: Theory and Applcatons, 2007, pp [17] H. Wang, X. Zhang, F. Abdesselam, and A. Khokhar, Dps-mac: An asynchronous mac protocol for wreless sensor networks, June 2007, vol. 7, pp [18] Lu Su, Changle Lu, Hu Song, and Guohong Cao, Routng n ntermttently connected sensor networks, 2008, pp [19] Tan He Yu Gu, Boundng communcaton delay n energy harvestng sensor networks, 2010, pp [20] Yu Gu, Tan He, Mngen Ln, and Jnhu Xu, Spatotemporal delay control for low-duty-cycle sensor networks, n Proceedngs of the th IEEE Real-Tme Systems Symposum (RTSS), 2009, pp [21] Xue Yang and N.H. Vadya, A wakeup scheme for sensor networks: achevng balance between energy savng and end-toend delay, [22] G. Lu, N. Sadagopan, B. Krshnamachar, and A. Goel, Delay effcent sleep schedulng n wreless sensor networks, 2005, pp [23] K. V.S. Ramarao and S. Venkatesan, On fndng and updatng shortest paths dstrbutvely, J. Algorthms, vol. 13, no. 2, pp , [24] S. Haldar, An all pars shortest paths dstrbuted algorthm usng 2n2 messages, J. Algorthms, vol. 24, no. 1, pp , [25] Guseppe F. Italano, Dstrbuted algorthms for updatng shortest paths (extended abstract), n Proceedngs of the 5th Internatonal Workshop on Dstrbuted Algorthms (WDAG), London, UK, 1992, pp , Sprnger-Verlag. [26] B. Awerbuch, I. Cdon, and S. Kutten, Communcaton-optmal mantenance of replcated nformaton, n Proceedngs of the 31st Annual Symposum on Foundatons of Computer Scence (SFCS), Washngton, DC, USA, 1990, pp vol.2, IEEE Computer Socety. [27] Baruch Awerbuch, Complexty of network synchronzaton, Journal of the ACM (JACM), vol. 32, no. 4, pp , [28] S. La, B. Zhang, B. Ravndran, and H. Cho, Cqs-par: Cyclc quorum system par for wakeup schedulng n wreless sensor networks., n Internatonal Conference on Prncples of Dstrbuted Systems (OPODIS). 2008, vol. 5401, pp , Sprnger. [29] Rchard Bellman, On a routng problem, Quarterly of Appled Mathematcs, vol. 16, no. 1, pp , [30] Phlpp Sommer and Roger Wattenhofer, Gradent clock synchronzaton n wreless sensor networks, n ACM IPSN, 2009, pp [31] K. Man Chandy and J. Msra, Dstrbuted computaton on graphs: shortest path algorthms, Communcatons of the ACM, vol. 25, no. 11, [32] A. Segall, Dstrbuted network protocols, IEEE Transactons on Informaton Theory, vol. 29, no. 1, pp , Jan [33] J. J. Garca-Lunes-Aceves, Loop-free routng usng dffusng computatons, IEEE/ACM Trans. Netw.,, no. 1, pp , [34] Yanjun Sun, Omer Gurewtz, and Davd B. Johnson, R-mac: a recever-ntated asynchronous duty cycle mac protocol for dynamc traffc loads n wreless sensor networks, n Proceedngs of the ACM conference on Embedded network sensor systems (Sensys), 2008, pp [35] W.S. Luk and T.T. Huang, Two new quorum based algorthms for dstrbuted mutual excluson, n Proceedngs of the Internatonal Conference on Dstrbuted Computng Systems (ICDCS), 1997, pp [36] W. Ye, J. Hedemann, and D. Estrn, Medum access control wth coordnated adaptve sleepng for wreless sensor networks, IEEE/ACM Transactons on Networkng (TON), vol. 12, pp , [37] T.V. Dam and K. Langendoen, An adaptve energy-effcent mac protocol for wreless sensor networks, n The Frst ACM Conference on Embedded Networked Sensor Systems (Sensys), [38] Stphane Mallat, A Wavelet Tour of Sgnal Processng, Thrd Edton: The Sparse Way, Academc Press, [39] OMNET++,, [40] M. Zunga and B. Krshnamachar, Analyzng the transtonal regon n low power wreless lnks, n IEEE Communcatons Socety Conference on Sensor, Mesh and Ad Hoc Communcatons and Networks (SECON), 2004, pp [41] Charles E. Perkns and Pravn Bhagwat, Hghly dynamc destnaton-sequenced dstance-vector routng (dsdv) for moble computers, SIGCOMM Comput. Commun. Rev., vol. 24, no. 4, pp , 1994.

16 16 Shouwen La s a senor engneer n Qualcomm Inc where he s workng on CDMA software development. He receved hs Ph.D degree n the Department of Electrcal and Computer Engneerng n Vrgna Polytechnc Insttute and State Unversty (Vrgna Tech) n Hs research topcs nclude wreless sensor networks, real-tme systems and moble computng. He s also nterested n system development and mplementatons for wreless networkng, embedded systems and dstrbuted systems. Before comng to Vrgna Tech, he worked for two years as a research engneer n Htach (Chna) R & D corporaton where he conducted research and development works for network moblty, securty provson, multmeda applcatons over 3G and WLAN networks. Bnoy Ravndran s an Assocate Professor n the ECE Department at Vrgna Tech. Hs research nterests nclude real-tme, embedded, and networked systems, wth a partcular focus on resource management at varous levels of abstracton from OSes to vrtual machnes to runtmes to mddleware. He and hs students have publshed more than 170 papers n ths space, and some of hs group s results have been transtoned to US DoD programs. Dr. Ravndran s an US Offce of Naval Research Senor Faculty Fellow, an ACM Dstngushed Speaker, and an Assocate Edtor of ACM Transactons on Embedded Computng Systems.

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