Improved Algorithms for Data-Gathering Time in Sensor Networks II: Ring, Tree and Grid Topologies

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1 Improved Algorthms for Data-Gatherng Tme n Sensor Networs II: Rng, Tree and Grd Topologes Yoram Revah and Mchael Segal Communcaton Systems Engneerng Department Ben-Guron Unversty of the Negev, Beer-Sheva, Israel 8405 Abstract We address the problem of gatherng nformaton n sensor webs consstng of sensors nodes, where n a round of communcaton sensor nodes have messages to be sent to a dstant central node (called the base staton) over shortest path. There s a wde range of data gatherng applcatons le: target and hazard detecton, envronmental montorng, battlefeld survellance, etc. Consequently, effcent data collecton solutons are needed to mprove the performance of the networ. In ths paper, we tae nto account the fact that nterference can occurs at the recepton of a message at the recever sensor. In order to save redundant retransmssons and energy, we assume a nown dstrbuton of sources (each node wants to transmt at most one pacet) and one common destnaton. We provde a number of schedulng algorthms ontly mnmzng both the completon tme and the average pacet delvery tme. We defne our networ model usng drectonal antennas and consder Rng, Tree, and Grd Networ (and ts generalty) topologes. All our algorthms run n low-polynomal tme. Key words: Schedulng algorthms, Optmzaton problems, Half-duplex One-port model. Introducton Recent advances n commercal IC (Integrated Crcuts) fabrcaton technology have made t possble to ntegrate sgnal processng and sensng n one ntegrated crcut. These devces are popularly nown as * The prelmnary verson of ths paper has been appeared by ICNS'07 (Intl. Conf. on Networng and Servces, 2007). The wor by Yoram Revah has been partally supported by the Kretman fellowshp for excellence and Israel Mnstry of Scence fellowshp for PhD studes. The wor by Mchael Segal has been partally supported by Israel Mnstry of Trade (consortum REMON for 4G communcatons) and INTEL.

2 wreless ntegrated networ sensors (WINS) and nclude mcro-electromechancal systems (MEMS) technology components such as sensors, actuators, RF component and CMOSS buldng blocs. WINS combnes mcro-sensor technology and low power computng and wreless networng n a compact system. Sensor nodes are dspersed over the area of nterest and are capable of RF (rado frequency) communcaton, and contan sgnal processng (DSPs) engnes to manage the communcaton protocol and data processng before transmsson. The ndvdual nodes have a lmted capablty, but are capable of achevng a large tas through coordnated effort n a networ that typcally conssts of hundreds to thousands of nodes. Networs of such devces can autonomously perform varous sensng tass such as envronmental (sesmc, meteorologcal) montorng and mltary survellance, enemy tracng, target detecton, dstrbuton of tmng and poston nformaton and mult-hop communcaton []. The sensor could be sensng temperature, pressure, ol lea, radaton, etc. Generally, these networs are referred to as wreless ad-hoc sensor networs or smply sensor networs. It can also be a collecton of moble sensor nodes that dynamcally form a temporary networ wthout the use of any exstng networ nfrastructure or centralzed admnstraton. In other words, the prmary applcaton of such networs has been n dsaster relef operatons, mltary use, conferencng and envronment sensng. A typcal applcaton n web s gatherng of sensed data at dstant central processng system named the base staton (BS) (or the root node n the networ graph). Ths rood node s assumed to be wth greater computatonal, storage, and transmsson capabltes than the rest of the nodes n the networ. The root node typcally serves as an entry pont to the sensor networ, ntegratng the sensor networ wth wred networ. In each round of ths data gatherng applcaton, all the data from all nodes need to be collected and transmtted to the BS, where the end-user can access the data. In some sensor networ applcatons, data collecton may be needed only from a regon and, therefore, a subset of nodes wll be used. A smple approach to accomplshng ths data gatherng tas s for each node to transmt ts data drectly to the BS. Snce the BS s typcally located far away, the energy cost to transmt to the BS from any node s qute hgh. Therefore, an mproved approach s to use few and mult-hop transmssons as possble to the BS. In contrast, n many 2

3 emergng and envsoned applcatons, sensor networs wll be both dstrbuted and wreless (n terms of communcaton and power) [2]. Dstrbuton s necessary for mprovng sensng qualty: when the precse locaton of a sgnal s unnown, then dstrbuted sensors wll allow sensng to tae the place closer to the event of nterest than by any sgnal sensor. Dstrbuton also mproves robustness to envronmental obstacles, whch s especally crucal n stuatons where sensng requres lne-of-sght. The sensor unts, thus, need to rely on fnte, local energy sources and wreless communcatons channels. Fnally, shorter range communcaton s generally much cheaper than longer range communcaton because the rado-sgnal power can drop off wth a quadratc power of dstance [3]. As a result, t s much cheaper to transmt nformaton usng mult- hoppng among sensor unts. One way to reduce the amount of data that must be transmtted (and reduce energy) n rado networs s schedulng forwarded nformaton gathered by sensor nodes. The schedulng process s ntended to prevent collsons that mght arse from mproper or neffcent use of the networ resources by random messagng across the networ wthout tang nto account the networ model. Then, we am to solve the problem for a gven certan topology of rado networ and a networ model, ntal nformaton (messages) located at some nodes and a sngle desgnated destnaton. We consder the Rng and the Tree networs and gve optmal schedulng solutons that acheve a mnmum completon tme as well as a mnmum average delvery tme. For the Grd networ topology (and ts extensons) we propose an approxmaton algorthm to our problem provdng.5 approxmaton rato for maxmum completon tme. We present a low-polynomal tme solutons for our problem for above mentoned networ topologes and we also provde some useful nsghts. Our research can be practcally mplemented n those networs: for example, whenever a node has a pacet to transmt, t sends a very short message (to save battery energy) called a Schedule Request to a central computer (BS) that serves as the only destnaton n the networ. The requests can be sent over an upstream control channel (or multple upstream control channels) usng, for example, ALOHA or CDMA randomaccess schemes. The base staton s assumed to have full nformaton about the nput and the networ topology. It produces a schedule and perodcally transmts the schedule requests (called a MAP message) to 3

4 the nodes that requested to send. Ths s done over a separate downstream control channel. Note that the separaton n channels s between the downstream and upstream control channels and the channel n whch the data messages are transmtted. In our prevous wor [4], we consder lnear, two-branch, and star (or mult-branch) networ topologes. For each topology we provded an optmal schedule for routng all the messages to the base staton, ontly mnmzng both the completon tme and the average pacet delvery tme, whle all our algorthms run n polynomal tme. It should be noted that the presented (optmal) data gatherng algorthms are centralzed and requre cooperaton between nodes whch s not necessarly compatble wth the requrements of sensor networs. Therefore for stronger requrements, these algorthms may no longer be practcal. However, they contnue to provde a lower bound on data gatherng tme of any gven collecton schedule. We focused our analyss on systems equpped wth drectonal antenna snce from comparson results (wth respect to completon tme) between drectonal antenna systems to omn-drectonal antenna systems obtaned by Florens and McElece [44] t follows that former outperforms the later by 50% on Lnear Networ. The dea of usng drectonal antenna n wreless communcaton s not new. It has been already extensvely used n base staton of cellular networs for frequency reuse, to reduce nterference, and to ncrease the capacty of allowable users wthn a cell. However, the applcatons of drectonal antenna to wreless ad-hoc or sensor networ to reduce the transmt power of each node to acheve power-effcency n routng problem s relatvely new. Our problem was partly addressed over the past few decades. A number of wors (see [5 2]) dscuss rado networs under a smlar networ model, but wth a dfferent target functon that leads to maxmzng the number of transmssons n one hop wthout referrng to specfc sources and destnatons across the networs. Ths problem, and ts varatons are nown to be NP-hard, and the suggested solutons are heurstc approxmaton algorthms. Other wors (see, e.g. [22 29]) dealt wth our problem consderng Grd and Tree topology, but under other (weaer) networ models. For example, authors n [22] used the same target functon as we suggest, but the dscusson s based on several varatons of hot potato routng. In ths model 4

5 each node can successfully receve and transmt more than one message smultaneously. Ths s a completely dfferent model from the classcal rado networ model, whch we chose to apply n our analyss. Furthermore, papers that are concerned wth hot potato routng offer upper and lower bounds on performance n terms of order of magntude, whle n our wor, we produce exact results or tght bounds to our problems. Smlarly, other papers (see [30-34]) assume the same target functon to mnmze the completon tme and the model under whch each node can successfully receve and transmt more than one message smultaneously s explored n Rng, Tree, Grd networs and general graphs. Y. Cho et al. [35] presents a protocol for routng data messages from any sensor to the base staton n a sensor networ twodmensonal grd topology, by usng and mantanng a spannng tree wth root servng as the base staton completely gnorng the nterferences. Srdhran and Krshnamachar [36] presented some problem of converge-castng flows wth rate control from nodes to the root of the gven routng tree of the networ. Lau and Zhang [37] and Krumme et al. [38] also study the gosspng problem of communcatng a unque tem from each node n a graph to every other node under two-dmensonal grd networ topology. They have suggested that the gosspng problem can be studed under four dfferent communcaton models, whch have dfferent restrctons on the use of the lns, as well as the ablty of a node n handlng ts ncdent lns. The four models beng consdered are: () the full-duplex, all port model, (2) the full-duplex, one-port model, (3) the half-duplex, all-port model, and (4) the half-duplex, one port model, whch can be dentfed by the labels F*, F, H*, and H respectvely. In ther [37,38] notatons, we assume a networ model denoted H or called The half duplex one port model, snce ths model of communcaton maes the weaest assumptons about both hardware and software capabltes. Gronvst [39] assumes a stochastc model for the general networ topology problem and presents a number of results under ths model. Some other papers (see [40-43]) transformed a networ nto an undrected graph G(V,E) wth V as the set of nodes and E as the set of edges and modeled the transmsson area and the nterference area as balls n graph by ntroducng two parameters: d T, the transmsson radus and d I the nterference radus wth d d. They deal wth gatherng nformaton n I T 5

6 specfc rado networs: Lne, Grd, wth the same target functon of mnmum completon tme, gnorng the requrement of mnmzng average delvery tme and usng omn-antenna. They also show [43] that n the case of general networ the problem s NP hard. Fnally, Florens and McElece [44] consder exactly our problem under a crteron of mnmum completon tme, gnorng the requrement of mnmzng average delvery tme. In fact, ther schedulng strategy does not consderng the dle tme of the messages and produces unnecessary dependences among messages. Ths, consequently, causes unnecessary delays for messages. For example, t s unreasonable not to transmt a message toward the destnaton f t can be transmtted wthout any delay. They [44] also do not provde any tme-complexty analyss of ther algorthms. Ths paper s organzed as follows: Frst we explan n detals the networ and channel model wth a precse defnton of our problem. Next, we address the Rng Networ case. After that, we consder the Tree Networ problem. The optmal schedulng strategy under both target functons for Rng networ and Tree Networ s explaned n Sectons 3.3 and 3.4, respectvely. Fnally, we consder a Grd Networ and ts extensons. We propose an approxmaton algorthm to our problem provdng.5 approxmaton rato for maxmum completon tme, where the approxmaton bound holds for any BS locaton n the Grd Networ. We conclude the paper wth drectons for further research. 2 General Networ and Channel Model A sensor or ad-hoc networ s modeled as a drected graph G(V,E) wth N nodes ( n the case of Grd networ N N nodes), V s a set of nodes, each of them representng a communcaton devce, where each node v V s a sensor that can transmt and receve data; E s a set of edges connectng nodes. There s an edge between node v and w f and only f v can hear w s transmssons when v ponts ts drectonal transmsson antenna towards w. The networ has a specal node v 0, the Base Staton (BS), that serves as a destnaton for all messages. Ths node s assumed to be the root of the graph wth large computatonal, storage, and transmsson capabltes. The root node typcally serves an entry pont to the sensor networ, ntegratng the sensor networ wth an external wred networ. 6

7 We assume that at tme t=t 0, each node v V has at most one message to transmt to the destnaton. Ths s referred to as a legal nput. We assume that all the nformaton about the nput and topology of the networ s avalable at the BS and there are separate, collson free, control channels between the BS and the other nodes. We also assume that every node n the networ ncludng the BS has the same transmsson power r and that a node can not transmt and receve message smultaneously. In our model we assume that the capacty of each node's buffer s one message. We also assume the use of drectonal antennas. The sgnal from node v to node w propagates n a straght lne n the drecton of node w wthout dspersng to other drectons. We also assume that f a message arrves successfully to the recever, the recever can send an acnowledgment to the sender usng drectonal antenna on a separate channel. Based on the above, the condtons for a successful transmsson are: v, wv a message from node v that s transmtted to node w, arrves successfully at node w f for all smultaneous transmssons from u V, u v, w usng drectonal antennas ponted n the drecton of v the followng relatons hold: r, 0 v w r, u w (here v stands for the locaton of node v). We also assume n our model that tme s slotted and one hop transmsson consumes one tme slot (TS). As we mentoned above node can ether transmt or receve n one tme slot. Ths model of channel s called n the lterature S-TDMA channel model. In summary, we model our networ by a rooted graph, where the root represents the BS and an edge represents an exstng wreless connecton (a ln) between two statons. A necessary condton for connecton exstence between v and w s the fact that v and w are at dstance between them of at most r. We denote by d v, w the dstance, measured n number of hops, between node v and node w. 3 Problem Statement and Our Performance Measure In ths secton we carefully defne our problem. We are nterested n solvng the problem for varous networ topologes: Rng, Bnary Tree and Grd networs. 7

8 3. Problem Statement Assumng some networ topology wth N nodes (or N N nodes n the case of Grd networ), M of whch have messages to be send to BS (each node has to transmt at most one message), and assumng our networ model wth the fact that BS s recevng the requests for transmsson from the nodes that have a message to send to the BS on separate, collson free channels, the purpose s to fnd an algorthm that schedules and routes all the messages to the BS n a mnmum tme (prmary crteron) and also mnmzes the average message-delvery tme (a secondary crteron, whch s equvalent to mnmzaton of the sum of the message dle tmes). 3.2 General Target Functons We wsh to fnd a schedulng and routng soluton for every possble nput set of messages to destnaton. We denote by tme t taes for message sum of delays that Tend mn dle tmes for all messages. Thus, the mnmum tme for all messages to reach the destnaton, and by T the m to reach the destnaton. The delay tme or dle tme of a message m s a total m ncurs startng at t 0 untl arrvng to the destnaton. Denote by S the mnmum sum of Tend mn max T mn m () M S mn (2) 3.3 Analyss Rng Networ Frst, we nvestgate a Rng Networ topology, wth each sensor playng a role of node, see Fgure. 8

9 V - V V + V 6 V 5 V 4 V 3 V N-2 V 2 (BS) V V N- V 0 Fgure : The Rng Networ. The Rng consst of N nodes ncludng the BS and each node denoted by, 0,.. N. The networ has M, M N messages to transmt to the BS. In ths topology, t s obvous that the transmssons towards the BS can go two ways: clocwse or counterclocwse,.e. node v can transmt ether to node v or to node v. We also assume that the dstance between any two adacent nodes s less than or equal to r. Followng our problem defnton we would le to prove the exstence of an optmal schedulng algorthm that can handle any type of a legal nput (at t 0 any sensor eeps at most one message). The optmalty of the algorthm s measured n terms of T end mn and S. We develop the proof n stages, by provng the exstence of such algorthm that schedules and routes all the messages to the BS n a mnmum completon tme (prmary crteron) and also mnmzes the average message-delvery tme. We denote by T ( T ) the mnmum tme t v taes for message m to reach the destnaton BS usng clocwse (counterclocwse) path, and by ( ) the total sum of delays that (counterclocwse) drecton. m ncurs startng at t 0 untl arrvng to the destnaton usng clocwse 9

10 We compute, for each message m the values T ( T ) and ( ) by applyng Lnear Networ Algorthm [4] fxng a drecton of message movement to be clocwse (counterclocwse) wth respect to BS. Thus, we obtan for each message m the couples T T, and,. Algorthm Rng: If T T, send the message to the drecton (clocwse or counterclocwse path) that s defned by mn T T,. If T T then the drecton of message s defned by values,. If, send the message to the drecton (clocwse or counterclocwse path) that s defned mn(, ). If, we can choose any drecton. After the drecton for each message s determned, we apply Two Branch Networ Algorthm [4]. Theorem : The schedulng produced by Rng Networ Algorthm schedules and routes all the messages to the BS n a mnmum completon tme (prmary crteron) and also mnmzes the average message-delvery tme. Proof: Accordng our model, the capacty of each node's buffer s one message. It s obvous that two dfferent messages can not be sent to the BS n overlappng paths n dfferent drectons snce otherwse there must me a node wth a buffer capacty 2. As a consequence of t we wll fnd a pvot node message on the Rng. The pvot node message s the message such that all the messages that are located closer to the BS (ncludng ths message) wll be transmtted to BS n the same drecton and the remanng messages, (f any) wll be transmtted n the other drecton. In other words, n the general case there are at most two groups of the messages (as we wll see latter), one part wll be transmtted n clocwse drecton, whle the other part wll be transmtted counterclocwse. Lets us denote by T pnm clocwse and counterclocwse, respectvely. and T pnm the arrval tme of the pvot node message 0

11 After we apply the Lnear Networ Algorthm ndependently to the same nput n both drectons there are two cases: Case : For all the messages we havet T. Ths means that T pnm Tpnm. If for all ether T T or T T we obtan an nstance of Lnear Networ and our problem s solved optmally. Otherwse, we have two adacent pvot node messages (not necessary adacent nodes). We obtan a stuaton wth one pvot node message wth a group of messages n counterclocwse drecton towards BS and the second pvot node message wth a group of messages n clocwse drecton towards BS. In ths case, our Rng Networ algorthm s equvalent to Two Branch Networ algorthm [4], snce we have two separate groups of messages that have to be routed to the BS over two dfferent paths (lnes) [4]. By applyng Two Branch Networ Algorthm [4], we acheve both () and (2) crtera. Then, T end mn max( T pnm T ) f T pnm Tpnm. If, pnm T pnm T pnm, then accordng Two Branch Networ Algorthm [4], Tend mn max( T tpm decde to send two pvot node messages at the same drecton we obtan Case 2: For one message holdst pnm T. pnm Tend T pnm In ths case t s easy to see that applyng Two Branch Networ algorthm [4] gvest However, n order to mnmze the second crtera, we chec the, T pnm )+, snce f we +2 > max( T pnm, T pnm )+. end mn T pnm or T pnm, values and send the pvot node message n the path that guarantees mnmum dle tme snce Two Branch Networ algorthm n ether case wll postpone the schedulng of mnmum group of dependent messages n the case of collson but the pvot node message wll suffer mn(, ) delays. pnm pnm. Theorem 2: Gven a rng networ, mnmzng the sum of dle tmes for all messages does not lead to mnmzng the total completon tme. Proof: Lets we loo at more specfc example (case, Theorem, for all : T T ) wth only two messages and the last message depends on the frst message n the counterclocwse drecton, thus havng dle tme.

12 But n the clocwse drecton the message has a zero dle tme despte the fact that completon tme grows up. The runnng tme of ths algorthm s domnated by the runnng tme of Two branch Networ algorthm whch 2 s O (N ). 3.4 Analyss Bnary Tree Networ The base staton (BS) plays a role of the root of a bnary tree graph and each sensor play a role of nodes. In the Bnary Tree Networ (BTN) G(V,E,L), root s connected to the (possbly empty) left and the rght subtrees that are also BTNs. Every such connecton s a Lne Networ (see [4]), where we call the endponts of a connecton man nodes. In other words, man nodes n BTN are connected by lne networs wth sensors that serve as edges, see Fgure 2. The number of the lne networs n the graph denoted by L and we numerate them from the left to the rght as depcted n Fgure 2. We denote by v 0 the man node that s connected n the end of lne networs, and by v 0 when ust one lne s connected to ths man node. Notce, that n ths notaton BS s represented by ether v 2 0 or v 0. In general, any man node can eep an arbtrary number of messages, however as we wll see later t s enough to assume that the capacty of each man node's buffer s one message. However, the man node can not transmt and receve message at the same tme: n the same tme slot t can receve at most one message from nodes of lne networs connected to t from chldren drecton or transmt the message to the node belongng to the lne networ n parent drecton. The man nodes, n some sense, act as relays to transfer messages towards BS. Let V v v..., v, 2 N nodes (sensors) n the lne networ numbered (n short, -lne networ) n the BTN. be a set of N 2

13 V 0 BS Sensor 2 V Man node V V 2 V V 4 V 3 3 V V V V 4 V 3 3 V V0 V0 V 0 V 56 0 V V 0 + Fgure 2: The BTN Networ In ths topology, we defne a legal nput x X as a collecton of sensor nodes (exclude the man nodes and the BS) that have at most one message to transmt to the base staton at t=t 0. The goal s to transmt M messages to the BS, M L N, where each lne networ wth a legal nput x, has M messages to transmt to the BS, whle M N. Let us denote by or dle tme T to be an arrval tme of message of a message m s a total sum of delays that m from -lne networ at destnaton BS. The delay tme m underwent untl arrvng to the destnaton 3

14 (BS). Thus, usng our target functon, we are nterested n optmal schedulng algorthm that brngs to L M and S m Lne,,2... L mnmum the followng crtera: T max T end Agan, n order to do that, we develop the proof n stages. Algorthm BTN : Every node n lne networs behaves as normal node n lne networ algorthm,.e. f some node contans a message that can be sent to ts (rght) neghbor (wth no message), then we send t. Every man node acts as followng: f t has a message, then the message s transmtted to t's parent. If the man node does not have a message to send and only one node of two possble nodes from chldren lne networ connectons has a message to transmt, the man node receves t. Moreover, once man node decdes a drecton from whch t receves messages t contnue to receve them untl there s no message from ths drecton. The man node do not serve the other lne networ connected to t from chldren drecton as long as the messages from the drecton that has been started to be serve by man node contnue to arrve. In the case when the man node has been dle and now there are two messages from two dfferent drectons to arrve at ths node, t arbtrary starts to serve one of the lnes. The BS apples Two Branch Networ algorthm [4]. Theorem 3 The BTN Algorthm optmzes both crtera. Proof: Notce that the maxmum transmsson rate of any man node s /2. Ths s due to the fact that accordng our model the man node can not transmt and receve message at the same tme. The concluson from the above observaton s that n order to acheve maxmum effcency t s suffcent to supply the messages to all man nodes at rate /2. We begn our explanaton for one arbtrary sub-tree, say left sub-tree. We would le to "create" groups of messages of maxmal length [4] that can be transmtted at the maxmum rate (/2). By usng the algorthm above, accordng to Two Branch Networ algorthm and Lnear Networ algorthm [4] at t 0, we have at every. 4

15 -lne a number, say z, of ndependent groups of maxmal length of frst order. We denote them as Two Branch Networ algorthm [4], these groups can be transmtted to BS at maxmal rate /2. w z. In In addton, n the BTN, we denote by w x a set of ndependent groups of messages of maxmal length at the left sub-tree that can be transmtted to BS at maxmal rate /2. Our algorthm produces them n the followng fashon. If we have w g (g groups n the -lne networ) then t s easy to see that g w g =w, and we denote by new T end the tme t taes the last message n wg to reach the BS whle applyng Lnear Networ Algorthm, assumng we have only the left sub tree. Afterwards, we loo for the next consecutve maxmal group w (e.g., ths group s located n lne ) closest to w. The closest means that g the frst message from curr w s at mnmal dstance ( d mn ) from BS. Accordng to Lnear Networ algorthm, curr new end f d mn T 2 then the groups are merged to a frst order group w w g w and smultaneously updated new end new end curr T = T + 2 n, whle n curr s the number of the messages n w. By ths way we gong down on the tree to loo for the next closest group, checng agan the condton curr new end d mn T 2 and contnue q producng new w wg w... wu, where w denote the next closest group wth curr new end d mn T 2. When we fnsh ths process, we start producng w 2, set w x of mnmal q u w 2, and so on, untl we buld a. In the case that, we do not have any groups n the -lne networ, we start the process wth a group curr w 2 d mn. The correspondng set for rght sub-tree z s bult n analogous fashon. In addton, t s clear that f two any two groups of messages arrve to a man node smultaneously, the tme that t taes for two groups to pass through the man node s ndependent on the group that the man node starts wth. Fnally, we obtan two logcal lnes wth w x 2 w and z nputs that t s equvalent to the case of Two Branch Networ wth those nputs. If we apply Two Branch Networ algorthm [4] to w x and 2 w z nputs, we mnmze both crtera as proved n [4]. 5

16 Remar : Notce that man nodes may have messages at tme t 0. The algorthm remans the same. Remar 2: The above scheme can be generalzed to deal wth -ary trees. The only dfference s that we construct ndependent groups of messages of maxmal length for sub-trees and BS apples a strategy of - Star Networ algorthm. The runnng tme of the above scheme s domnated by the runnng tme of Two Branch Networ algorthm, 2 whch s O ( N ), wth N standng for number of nodes. 3.5 Analyss Grd Networ In ths secton we wll address the problem of a grd topology. The networ has NxN nodes. In our model defnton each node can do at most one operaton at a gven tme slot, meanng that t can not transmt and receve a message at the same tme; t can at most transmt one message or at most receve one message. There are two nds of node as depcted n Fgure 3: Sensor and Relay nodes. The sensor nodes are located n the sensng zone. The sensor node has the same functon as n the prevously dscussed networs: t senses the nformaton whch s transformed to a one message to be transmtted to the BS. Relay nodes, are located outsde the sensng zone on the rght and the down border of the Grd Networ. D 2 D 6 D D 0 P ˆ x, y D 5 D 4 D 9 D 3 D 8 D 2 D 7 BS D D 0 D 6 D 5 D 4 D 3 D 2 D D 0 Relay zone Sensng zone Fgure 3: Grd Networ 6

17 The relay node acts as a messages delver, t ust receves the messages from the sensng zone and transmts them to the BS (the relay node operates smlarly to sensor node n sense that t performs only one operaton per one tme slot). The coordnate of node,0, N v, s denoted by P x, y, and the dstance between any two adacent nodes s the same and equals. For now, we wll place the destnaton (BS) at node x P (the relay nodes are coordnated at P x, 0,0 0, y0 0, 0 y and P, 0 x, y0, =,..,n, =,..,n). In our model a message can advance to the destnaton only on the shortest path from the source to destnaton. Therefore, a message from pont P ( x, y ) can move to the destnaton only n rght or down drecton as, mared n Fgure 3. Therefore, f the length of shortest path from the source to destnaton (BS) s d x y steps (hops) long, then n order to reach a destnaton, we need to move x steps n the x drecton and y steps n the y drecton. Thus, the number L ~ of shortest paths from source d destnaton s equal to L ~ x d d x d y P, to. Notce that the nodes on the same dagonal have the same dstance to the destnaton. We also can conclude that, f there s more than one message on the same dagonal then the messages n the nodes located at ths dagonal are dependent. Recall from [4] that message to be dependent on message m f m s not transmtted n the current tme slot because we need to m s sad transmt m. For example f we have only two messages located n the same dagonal then at least one of the messages has to wat at least one dle unt tme n ts way to the BS, because the other message s beng transmtted. Note, that the base staton can have a maxmum throughput of, snce t can receve messages n each tme slot from ether the relay node above t or the relay node to the left of t, alternately. Therefore, t s not possble to transfer a number of ndependent messages more than a number of dagonals n the grd. In what follows we propose an approxmaton algorthm to our problem provdng.5 approxmaton raton for maxmum completon tme. Before we ntroduce our heurstc and prove ts performance and bounds we wll present some defntons, observatons and lemmas that wll assst us n the analyss of our algorthm. Let us defne by D D ) the dagonal that contans at least one message at the maxmum (mnmum) dstance max ( mn 7

18 ~ from the BS and d d ) be the correspondng dstances, respectvely. Denote by d ( ) the dstance max ( mn between any node located on dagonal D to the BS ( hops). D Observaton 5 Gven x X, so that any two adacent nodes v,, v, l satsfy ( x y ) ( x yl ) 2, t s possble to send all the messages to BS wth T end d and S=0. mn max Observaton 6 Gven a Grd Networ of sze NxN wth M messages to be sent at t 0 to the BS, t always happens thatt end d M. mn mn It s very mportant to menton that for achevng ths lower bound (as we wll see later n the example of lemma 7), the nput must enable a partton nto two groups of messages wth the messages belongng to one group havng no affect on the movement of messages from other We denote by S [ a, b] a set of all messages wth a dstance from BS beng between values a and b. Lemma 7 Assumng a Grd networ ncludng M messages to be sent at t 0 to the BS, any schedulng algorthm fulflls the lower bound ( ) M d mn, f t fulfls the necessary condton that: S T end, S 2, S 3, S M. [ d, dmn ] [ dmn, dmn ] [ dmn, dmn 2] [ dmn, dmn M ] Proof: mn, Accordng to d defnton, t always happens that S. It s clear, that n order to reach the mn [ dmn, dmn ] completon tme bound of M d mn, the BS should receve messages contnuously n every tme slot durng the next M tme slots. It means that the dstance from the next consecutve message to be sent can ncrease by at most. The proof follows. 8

19 Below, we show an example that the condton descrbed n Lemma 7 s not suffcent. Suppose we have the followng nput: four messages coordnated at (3, 2), (4, 2), (5, 2), (6, 2) respectvely n the Grd Networ. It s easy to see that ths nput fulfls the necessary condton of lemma 6, but accordng to the networ model we wll never succeed to acheve ths bound, snce n any scenaro at least two close messages (at dstance at most ) wll compete at the same tme slot to be sent to BS. We call ths stuaton a collson. ˆ be a node havng a message to transmt (n short,, P, Let P x, y ˆ ), and let l P x, y ˆ, be a node coordnated at node x, y whose message s ntended to be sent to destnaton BS towards axs l ( l x, y),.e. f l=x t means that the message s to be sent to BS frst y steps towards the x axs followng x steps towards BS. Let C be the arrval tme of message at BS and In our heurstc, the dle tme equals C d( Pˆ,, BS). denotes total dle tme of message at BS. The algorthm numerates dagonals D q, q 2N, that have messages to send to BS. We obtan the sorted group H= D q, where N s the number the messages n D D, and G denotes a collecton of nodes wth messages located n dagonal D. Assume that message s located at hops from BS. Then ~ ' C d ( D ) C. The functon max( D [x]) (max( D [y])) determnes the node wth a message that s located at D and has the maxmal x coordnate (maxmal y coordnate). The functon N um( O, y) produces the number of nodes wth the messages n the group O G that the y-coordnate of each node s less or equal to y value. Let N um( O, y)= N - N um( O D, y). Our algorthm treats two possble types of dagonals from H. The frst type of dagonals ncludes dagonals that contan at least two messages. In ths case, we treat ths dagonal D by schedulng the messages startng at the message max( D [y]) routed to the BS va va P ˆ y, P ˆ x,. The algorthm contnues to schedules the rest of messages from. Then schedule the message max( D [x]) to the BS D n the smlar fashon, 9

20 alternately, completng the entre schedule of D n tme slot t. After we complete to schedule all the messages from D, we chec whether there s any message from D that can be sent n a consecutve tme slot t+. If the answer s postve, then we wat one tme slot before schedulng the messages from D ' ',.e. we start schedulng messages from ' D startng at tme slot t+2. Ths s done n order to prevent future undesrable collsons. Otherwse (the answer s negatve), algorthm contnues (wthout any delay) the schedulng process. After we fnsh schedulng messages from D ', we repeat the same process of nsertng an artfcal delay as explaned above,.e. we chec whether the next message can be sent n a consecutve tme slot. The second type of dagonals ncludes dagonals from H wth one message to send. Suppose we are dealng wth dagonal l D and the message located at node P x, y ˆ from D l s to be scheduled at tme slot t, ( x y ). In order to determne the drecton of ths message towards BS, the algorthm checs whether there s a message from D ' l that can be sent n a consecutve tme slot t+. If the answer s negatve, we can choose any drecton we le. If the answer s postve, then the algorthm schedule the messages from both dagonals l D and l ' D consequentally, startng wth the message of D. If N um( O ', y ) > l l N um( O l ', y) then the algorthm starts routng the message located at y D dagonal va P x, y l ˆ,. Routng of messages from l ' D dagonal starts from the message max( D [x]) towards x-axs followng routng of l ' l max( D ' [y]) towards y-axs and contnues at the same fashon, alternately. If N um( O ', y ) l N um( O l ', y), the algorthm starts routng the message located at x D dagonal va P x, y l ˆ, and the messages from l ' D dagonal are routed startng from max( D [y]) towards y-axs followng routng of l ' l ' max( D [x]) towards x-axs and proceedng n alternate fashon. 20

21 After we fnsh schedulng messages from D l ', we repeat the same process of nsertng an artfcal delay as explaned above,.e. we chec whether the next message can be sent n a consecutve tme slot. Lemma 8: Grd algorthm explaned above satsfes crtera () and (2) when he nput fulflls condton of Observaton 5. Proof: Snce the dstance between any two adacent messages s at least 2, the algorthm does not nsert any addtonal delay. Lemma 9 The Grd Algorthm schedules the messages wthout collsons. Proof: The algorthm schedules two types of dagonals. The frst type concerns about the dagonals contanng more than one message. It s easy to see that the messages from any dagonal of such nd are dvded nto two groups: those to be routed to the BS towards x-axs and those to be routed to the BS towards y-axs. Thus, the collsons between the messages from two groups are mpossble. (For a dagonal contanng only one message we even do not have groups.) In addton, no collson occurs at the BS staton snce the algorthm gves to the messages successve arrval tme (every group arrves at BS wth /2 rate. The second type of dagonals ncludes dagonals contanng exactly one message. The method based on values N um ensures that no collson s possble between messages from ths and followng dagonal snce all the messages that wll be routed towards, say y-axs, have ther nodes y coordnate greater (smaller or equal) than the messages that wll routed towards x-axs. As a result, those two groups of messages are routed through dsont paths to the BS. It remans to show that no collsons are possble between the messages of dfferent dagonals. Ths follows mmedately due to the fact that we nsert an addtonal delay tme between transmssons at consecutve tme slots from adacent dagonals accordng to algorthm. Theorem 0 Gven opt x X wth n messages when T end ( x), T ( x) mn Grd end stand for the completon tme as functons of nput x for an optmal algorthm and Algorthm Grd, respectvely we obtan: 2

22 T max xx T Grd end opt end mn ( x) ( x).5 Proof: Accordng to lemma 7, T opt end mn n. Notce that our algorthm nserts addtonal delay tme slots per at least 2 messages that are scheduled, and therefore, the total number of delays s at most n/2. Thus, T Grd end Grd opt T T opt end T end mn n / 2 n / 2 n / 2 end mn n / 2. We obtan that. 5. opt opt opt T end T end T end n mn mn mn Proposton The second constrant condton s unbounded n Algorthm Grd Proof: We construct an example when we have to schedule the specal nput wth all dagonals n the grd contanng one message to send. The messages are located n the followng manner: each message from odd numbered dagonal D s located at node,, 3 2( N ), and each message from even numbered dagonal D s located at node,, 4 2 2( N ). It s clear that optmal schedule that s appled to ths nput leads to T end d and S=0. However, our proposed algorthm adds a delay after every par of messages. mn max The proof follows. The runnng tme of the proposed algorthm s affected by effcent mantenance of maxmal and mnmal coordnates n nodes wth messages at each dagonal and computng values N um( O, y). All ths can be done n total O( M log M ) tme by usng mnmal/maxmal order heaps and balanced bnary search trees, where M stands for the total number of messages n the entre grd. Remar 3: The algorthm above can be generalzed to wor wth graphs that represent connected (partal) pavng of plane by grds of the same sze, see for example Fgure 4. Moreover, the base staton can be located anywhere n the graph. The basc dea s to dvde (logcally) the graph nto 4 quarters and determne the 22

23 quarter wth a closest message to BS. Then we contnue to send messages to BS untl we have a tme slot wth no message. In that case, we loo for the next closest message to BS that has not been sent yet. In order to avod collson between the last sent message and the new one we send them n a dfferent drectons (we always can do t). BS BS (a) cube networ (b) logcal spreadng nto 4 quarters Fgure 4: Cube and ts planar representaton 4 Conclusons Ths paper contnues to nvestgate the problem that we began at [4]: how to mnmze the tme completon and the sum of delays needed for gatherng nformaton at base staton, from the sensor networ nodes. We assume half duplex one port model equpped wth drectonal antennas. Polynomal tme algorthms for Rng Networ, Tree Networ and Grd Networ topologes have been desgned and analyzed. We provde optmal solutons for Rng and Tree networs and for Grd Networ, we present.5 approxmate soluton. A possble future research ncludes examnaton of more general topology models such as general graph problem, as well as tght analyss of other target functons, e.g. ncludng the cost of transmsson dependng on battery energy. 23

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