Optimal Geographic Routing for Wireless Networks with Near-Arbitrary Holes and Traffic

Transcription

1 Optimal Geographic Routig for Wireless Networks with Near-Arbitrary Holes ad Traffic Sudar Subramaia ad Sajay Shakkottai Departmet of ECE The Uiversity of Texas at Austi {ssubrama, Piyush Gupta Complex Systems Aalysis & Optimizatio Departmet Bell Labs, Alcatel-Lucet Abstract We cosider the problem of throughput-optimal routig over large-scale wireless ad-hoc etworks. Gupta ad Kumar (2000) showed that a throughput capacity (a uiform 1 rate over all source-destiatio pairs) of Θ( log ) is achievable i radom plaar etworks, ad the capacity is achieved by straight-lie routes. I reality, both the etwork model ad the traffic demads are likely to be highly o-uiform. I this paper, we first propose a radomized forwardig strategy based o geographic routig that achieves ear-optimal throughput over radom plaar etworks with a arbitrary umber of routig holes (regios devoid of odes) of varyig sizes. Next, we study a radom plaar etwork with arbitrary source-destiatio pairs with arbitrary traffic demads. For such etworks, we demostrate a radomized local load-balacig algorithm that supports ay traffic load that is withi a poly-logarithmic factor of the throughput regio. Our algorithms are based o geographic routig ad hece iherit their advatageous properties of lowcomplexity, robustess ad stability. I. INTRODUCTION We study the problem of throughput-optimal routig i large wireless etworks such as ad-hoc ad sesor etworks. I such large etworks, there is a eed for scalable, low-complexity ad distributed routig algorithms that ca provide good data rates for the traffic flows. The work i [8], [6] has show that a throughput-capacity of Θ( ) is achievable i uiform etworks with uiform traffic demads, ad that the capacity achievig routes are straight-lie paths. I may practical etworks, both the etwork ad the traffic distributio may be highly o uiform. No-uiformities may arise due to factors such as etwork holes (regios devoid of ay livig odes), the arbitrary locatios of source-destiatio pairs or due to variatios i the required data rates. I recet studies [9], [10], [17], [11], [5], geographic forwardig based protocols have bee suggested as a stable routig (providig fixed routes that do ot flip) techique over large o-uiform etworks as they are scalable, lowcomplexity ad highly distributed. However, i recet work [19], it was demostrated that etwork o-uiformities ca cause sigificat losses i throughput (rates could be as low as Θ(1/)) while employig such schemes. A critical issue is that covetioal shortest-path (such as straight-lie) routes are oblivious to the distributio of other routes (betwee other source-destiatio pairs) ad may cause heavy losses i throughput due to spatial cogestio. 1 Typically, throughput optimal routig schemes over ouiform etworks ad traffic demads are based o (i) solvig a global optimizatio problem [1] (settig up routes such that the traffic balaced over the wireless liks) or (ii) adaptive schemes [21], [18] that coverge to a optimal set of routes (or a per-packet route) over time. While global optimizatio requires co-ordiatio ad heavy computatio by the odes, adaptive schemes may take a log time to coverge to good paths ad also have issues of stability. I this paper, we are iterested i developig distributed routig algorithms that are ear-optimal (close to the rates obtaied by a global optimizatio) over o-uiform etworks with arbitrary traffic demads, but are still low-complexity, distributed ad stable. A. Mai Cotributios We cosider a radom plaar etwork with odes arbitrarily distributed over a uit regio, with each ode havig a log uiform circular radio rage of M() = C, for ay C > 1 π. This scalig esures that the resultat graph is coected [7]. (a) We first cosider o-uiform etworks with large umber of routig holes ad /2 uiformly radomly distributed source-destiatio pairs. I cotrast to earlier work [19], with fiite umber of holes of costat area, we allow for a arbitrary umber of holes of varyig sizes. Over such etworks, we demostrate that 1 a ear-optimal throughput capacity of Θ( ) is achievable (up to poly-logarithmic factors) by our algorithm RadHT(). Ulike the RANDOMWAY algorithm [19], the ew algorithm does ot overload the etwork with icreasig umber of holes, ad is also oblivious to the umber of holes i the etwork. (b) Next, we cosider etworks with a arbitrary umber of source-destiatio pairs with arbitrary locatios ad varyig rate requiremets. We assume that the etwork however has o routig holes. Covetioally, cutset bouds (amout of traffic that ca eter/leave the boudary of ay sub-regio of the etwork) have bee used to characterize upper bouds o etwork capacity [14], [12]. However, whe sources ad siks ca be arbitrarily close or far, ad with widely varyig traffic

2 requiremets, cut-set bouds aloe are isufficiet to characterize etwork loads. This is because traffic flows that ever leave the regio are uaccouted by such cuts. I this paper, we joitly utilize trasport capacity bouds (bouds that arise due to the iterferece ature of the chael) alog with cut-set based bouds to characterize the allowable set of source-destiatio pairs. That is, we demostrate that all routig scheme have a local coservatio property, where usig less amout of the local cut capacity requires it to use more of the limited local trasport-capacity, ad vice-versa. Usig this key property, we demostrate that locally load balacig the traffic by meas of a low-complexity radomized algorithm (RadLLB) is optimal up to a poly-logarithmic factor. Ulike RANDOMSPREAD [19], this algorithm does ot assume that the source-destiatio pairs are Θ(1) away from each other, ad distributes the traffic over a appropriate area rather tha over the whole etwork as performed i RANDOMSPREAD. Fially, we discuss some cosideratios whe implemetig these algorithms i practical deploymets. These iclude combiig RadHT ad RadLLB algorithms over mixed etworks ad icorporatig GPSR-like algorithms [9], [5] ito RadHT i order to guaratee (low-rate) coectivity i worst-case etwork topologies. B. Related Work Routig i large wireless etworks has bee widely studied i the past decade (see [15] for a overview). May of these algorithms are derived from Iteret routig protocols, ad do ot scale well (i terms of route setup, routig table complexity) i large etworks. Recetly, geography based routig algorithms [9], [10], [17], [11], [5] have bee ivestigated for providig low-complexity routig protocols that are scalable ad stable. I these schemes, packets are greedily routed towards the locatio of the destiatio ode ad if the greedy routes are trapped i a routig local miima, techiques such as plaarizatio ad face-traversal are used to route aroud these holes. However, i recet work [19], we demostrated that traditioal shortest path schemes (such as DSDV or AODV) ad greedy geographic schemes (with facetraversal) ca cause heavy throughput losses i the presece of etwork o-uiformities or ubalaced traffic demads. As source-destiatio pairs setup routes without kowledge of other flows i the etwork, greedy or shortest path routig ca cause spatial cogestio. Certai radomized strategies to route aroud holes were suggested i [3]. However, such schemes may fail i etworks with typical hole cofiguratios, ad eve whe workig, may provide low throughput. Traditioally, throughput-optimal schemes have either bee based o a global optimizatio [1] (the routes are cetrally chose to balace the flows over the etwork) or o adaptive schemes where the packets are routed accordig to curret queue/traffic states (e.g., back-pressure algorithms). Two mai drawbacks of global schemes are the eed for etwork-wide coordiatio ad the high complexity of solvig the optimizatio. O the other had, while adaptive schemes that use oly local coordiatio have bee developed [21] (ad more recet follow-ups [18], [13]), these algorithms have to coted with issues of stability, slow covergece to optimality (especially i large-scale etworks) ad log packet delays. Radomized approximatios to throughput optimal routig have bee studied sice the classical paper of [22], where the traffic flow is distributed equally over the etire etwork ad recombied at the destiatio. We used a extesio of this idea i [19] to provide ear-optimal throughput i etworks where the source-destiatio pairs were radomly chose. Further, i [19] we assumed etworks with a fiite umber of routig holes (whose size was comparable to the etwork size), ad radomly distributed source-destiatio pairs (thus the typical distace betwee a source ad its destiatio was comparable to the etwork diameter). For such etworks, we proposed scalable ad distributed algorithms based o geographic schemes that were ear throughput-optimal. I this paper, we allow for (i) a more complex etwork topology ad (ii) a arbitrary umber of arbitrarily located source-destiatio pairs with variable traffic requiremets. We ote that Valiat-like schemes (as used earlier i [19]) for distributig traffic over the whole etwork are provably suboptimal as they require packets to be uecessarily trasported over log distaces. Such etworks require ew routig algorithms as well as differet proof techiques to demostrate optimality, as we shall show i the rest of this paper. II. SYSTEM DESCRIPTION We iitially cosider a radom plaar etwork where odes are radomly ad uiformly throw over a uit torus (a square regio with wrap-aroud at the edges). We allow for a uiform circular radio rage, M() = C log /, to esure coectivity ad a o-zero umber of odes i ay tile of size M() M(). A. Networks with Routig Holes I the first part of the paper, we cosider a radom traffic patter where /2 source-destiatio pairs are chose uiformly radomly from the torus. Further, we allow for a arbitrary umber of holes to occur o the etwork. We igore the traffic geerated by ay source or destiatio ode that are removed by the occurrece of a hole. We assume the followig coditios o the holes. (After the occurrece of holes with these assumptios, the coectivity ad the o-zero odes i ay tile M() M() outside the hole is preserved.) Assumptio 2.1: Hole placemets: Let δ r be the side of the smallest uique axis-parallel square that cotais the hole r, ad ɛ r = δ r (1 + ) be the side of a larger cocetric square aroud the hole. The, o other hole t ca be placed such that its ɛ t outer square ca itersect with that of hole r. This esures that each hole is separated from ay other hole by a distace proportioal to its diameter. Assumptio 2.2: Hole shapes: Cosider the tilig of the uit regio by square tiles of dimesio p p for some small

3 The aular regio is reachable by a straight path No iterferig Holes to illustrate our proof method clearly. We ca exted it to ay rate model by assumig a o-itegral umber of sources of the basic rate that are collocated, ad our proof method ca be used to show this result. Fig. 1. δ r ε r Assumptios o holes i wireless etworks. p > γ 1/2 for some 0 < γ < 1/2. The the holes are composed by the uio of cotiguous tiles. Further, ay ode A i the iterior of the δ-square ca reach ay poit i the aular regio betwee the ɛ ad the δ-squares by straight lie ot itersectig the hole. For a illustratio, see figure 1. Thus, ay survivig odes iside the δ-square are easily reachable from outside the hole. This assumptio essetially disallows the formatio of holes with complex topologies that house survivig odes that are extremely hard to reach by local search methods. The holes ca still be cocave. Holes ca be chose arbitrarily to occur over the etwork subject to the above assumptios. These assumptios are similar to [19] - however, we ow allow for a large umber of holes of varyig sizes to occur o the etwork. Thus, the hole sizes are decoupled from the etwork size. We ote that this allows for a sigificatly larger class of o-uiform etwork topologies. We also ote that after the removal of odes due to holes, the umber of survivig source-destiatio pairs are Θ() (w.h.p). B. Networks with Arbitrary Traffic I the secod part, we are cocered with the issue of traffic o-uiformity i etworks. Here, the radom plaar etwork is without ay routig holes, but with arbitrarily chose source ad destiatio pairs from the etwork. Formally, we allow for H source-destiatio pairs, with 0 H 2 ad with l-th source-destiatio pair (l {1, 2,, H}) at a distace αl 1/2 away from each other, for 0 < α α l 1 2. The algorithms ad proofs described immediately exted to ay costat scalig of the distace model described above. However for otatioal simplicity, we keep the costat as uity. The rate required by ay flow is assumed to be from a fiite set R = { 1 γ 1,, 1 }, γ R with 0 γ i <. Thus, for a give source-destiatio cofiguratio, a rate vector r = [r 1,, r H ], r l R describes the traffic demad. I other words, the S-D pairs may be arbitrarily close to each other (compared to etwork diameter). A α = 1 2 sigifies source-destiatios that are a uit distace 1 away from each other, ad a α = 0 a distace of, the average distace betwee earest eighbors i a radom plaar etwork. Further, we assume the fiite-level rate model oly C. Iterferece Model ad Stadard Defiitios Defiitio 2.1: The throughput capacity T() of a etwork is defied as the maximum data-rate that is simultaeously achievable by all survivig source-destiatio pairs. Also, we assume the followig to model the iterferece effects of simultaeously trasmittig odes which are withi each other s radio rage. Defiitio 2.2 (Protocol Model, [8]): A trasmissio betwee a ode A ad its receivig ode B is assumed to be successful if d(a, B) M() ad d(c, B) > (1 + d)m(), for some d > 0, for all other trasmittig odes C A. This successful trasmissio occurs at rate 1 WLOG. We defie the packet delay D() as the maximum time take by the routig algorithm to travel from the source to its destiatio over all source-destiatio pairs. We defie f() = Θ(g()) if f() = O(g()(log ) k ) ad g() = O(f()(log ) k1 ) for some k, k 1 <, ad thus, a throughput T() is ear-optimal if it achieves Θ(T ()), where T () is the optimal throughput. III. ROUTING WITH NETWORK HOLES I this sectio, we cosider the problem of routig over a etwork with a large umber of holes - i particular, we cosider etworks i which the umber of holes may be comparable to the umber of odes i the etwork. A importat questio is to determie if geographic forwardig based schemes ca provide routig strategies that are throughput ad delay optimal. Geography based routig schemes are preferred for routig over large etworks predomiatly for two reasos. Firstly, the routig iformatio is scalable, i.e., the amout of routig iformatio that a ode eeds to remember is proportioal to the umber of its eighbors ad does ot icrease sigificatly with the etwork size. Secodly, the routig strategy is stable, low complexity ad scalable - the routes are chose i a greedy geographic maer, ad hece the routes are easily computed ad do ot flip/switch due to the loss or the additio of a few extra odes. I earlier work [19], we studied a etwork with a fiite umber of holes, ad demostrated that pure greedy forwardig strategies such as GPSR ca cause the throughput capacity of the etwork to be cosiderably reduced. We also proposed a radomized forwardig algorithm (RANDOMWAY) that was throughput optimal (while iheritig the icer properties of geographic routig schemes) for etworks with a fiite umber of costat area holes. While the routig scheme was oblivious to the actual locatio of the holes, a drawback of the proposed scheme was (i) a expoetial drop i throughput with icreasig umber of etwork holes, (ii) the algorithm required a kowledge of the umber of holes i the etwork.

4 Field Name TOPOLOGY or DATA TOPOLOGY DATA SRC-LOC STAGE NEXT-DEST SEC-DEST FINAL-DEST DATA Fuctioality Toggle bit - Topology iformatio or Data Packet. Iformatio about Hole locatio ad dimesio The ID ad locatio of source The stage of routig Locatio of the ext waypoit Locatio of ext+1 waypoit Locatio ad ID of the origial destiatio Message to the destiatio ode TABLE I FIELDS IN THE HEADER OF THE PACKET. A Stage 2 Stage 0 Box 2 C Box 1 A Stage 1 Regio (Stage 3) Stage 3 B Box 3 Stage 4 D Box 4 B Stage 0 Fig. 2. RadHT algorithm - Routig aroud a hole. I this sectio, we propose a radomized routig algorithm based o greedy forwardig that provides ear-optimal throughput ad delay eve i the presece of a more complicated etwork topology (a arbitrary umber of etwork holes), ad operates without the kowledge of the umber of holes. We also characterize the scalig laws for its throughput, delay ad routig iformatio at each ode. The etwork model is as described i Sectio II-A. A. The RadHT() Algorithm We first defie a packet structure to provide a commo commuicatio scheme betwee odes. See Table I. The source ode while sedig out a data packet sets the data flag bit, ad sets its SRC-LOC ad FINAL-DEST. It sets STAGE = 0, NEXT-DEST = FINAL-DEST ad other fields to a NULL symbol. We shall iitially assume that the odes that are o the boudary of a hole h kow the dimesios ad locatio of the smallest (up to a order) axis-parallel square that cotais the hole h, i.e., they kow the pair {xmi(h), ymax(h)} which are the ed poits of the diagoal of the cotaiig square. We deote this square as Sq(h). We will shortly describe a update scheme by which the odes o the hole boudary ca obtai this data. The radomized hole traversig algorithm (RadHT()) is defied as follows (See Figure 2.) Algorithm RadHT(): A ode o receivig a packet with the data flag set (i.e., sigifyig that it is a data packet) checks if the FINAL-DEST id is idetical to its ow. If yes, it accepts the packet. Else it checks if it is o the boudary of a hole. If the ode is ot o the boudary of a hole, it first checks if its ode locatio matches (withi a radio-rage hop) the NEXT-DEST. If that does ot match its ow locatio, it forwards the packet greedily towards NEXT-DEST. If it is the NEXT-DEST, 1) The ode checks the STAGE to see what stage of routig the packet is i. If STAGE = 0, NEXT-DEST is always FINAL-DEST. The ode would have already accepted the packet. 2) If STAGE = 1, it updates STAGE = 2, ad sets NEXT- DEST = SEC-DEST ad clears SEC-DEST to ull, ad forwards the packet to eighbor closest to the ew NEXT-DEST. 3) If STAGE = 2, it picks a radom locatio B from the 2 Sq(h) Box 3 ad sets NEXT-DEST = B, STAGE = 3 ad forwards packet greedily towards B. 4) If STAGE = 3, it picks a radom locatio B from the 2 Sq(h) Box 4 ad sets NEXT-DEST = the itersectio of B B ad the lie joiig source ad destiatio, sets STAGE = 4, ad greedily forwards towards NEXT- DEST. 5) If STAGE = 4, it sets NEXT-DEST = FINAL-DEST, STAGE = 0, ad greedily forwards towards NEXT- DEST. If it does lie o a hole, 1) it geerates two radom poits A ad A from 2 Sq(h) boxes 1 ad 2 respectively, ad sets NEXT-DEST =the itersectio of AA ad the lie joiig source ad destiatio, sets STAGE = 1, SEC-DEST = A ad greedily forwards towards NEXT-DEST. 2) It also updates the TOPOLOGY DATA field to provide the xmi(h), ymax(h) of the hole h that it is borderig. This provides the odes the iformatio about the holes dimesios to compute radom poits from appropriate boxes. Note that the SEC-DEST is modified oly by a ode that is o the boudary of a hole. Ed of Algorithm Calculatio of the Hole s dimesios: A ode o the hole perimeter (at locatio (x, y) receivig a packet with the topology flag set (i.e., sigifyig that it is a topology packet) computes xmi(h) = mi(xmi(h), x), ymax = max(ymax(h), y) ad passes it to the clock-wise closest eighbor that is o the hole boudary. A periodic update of such messages, alog with their respective timeout mechaisms ca be used to geerate a kowledge of hole dimesios at the boudaries. More iformally, our algorithm costructs a radom path (as show i Figure 2) i the aular regio aroud the hole, ad the cotiues o i its straight-lie path oce it leaves the (1+ )Sq(h) regio aroud the hole. I our algorithm, we choose the hole traversal algorithm to go above the hole for aalytical simplicity. I practice oe ca radomize this choice (to go above or below) to perform better load balacig although the results would be order-wise the same. We ote that the above algorithm ca be either used to iitialize static routes that ca be remembered, or each packet ca be idepedetly routed.

5 For the followig aalysis, we assume that RadHT is ru to setup static routes. B. Aalysis of RadHT() Algorithm I this sectio, we provide a quatitative aalysis of the throughput-capacity achievable i etworks with holes (as defied i Sectio II-A) ad radom source-destiatio pairs. Before we begi our aalysis, we show the followig upper boud o the best achievable throughput capacity. I this sectio, we skip the proofs of the claims ad refer to [20] for details. Claim 1: I etworks with holes ad a radom distributio of source-destiatio pairs the best achievable throughputcapacity T() = O( 1 ). Theorem 3.1: Cosider etworks as defied i Sectio II-A. The simultaeously achievable throughput capacity T() = Θ( 1 ). Further, the delay D() = Θ(T()). Thus, we show that our routig scheme achieves earoptimal throughput & delay (at the maximum capacity), ad the routig iformatio at odes does ot grow sigificatly. Proof: We shall make use of the followig result (whose proof is similar to Lemma 4.13 of [8] ad is skipped for brevity). Result 3.1: Cosider a torus of dimesios γ 1/2, with 0 < γ < 1/2. We pick R γ log radom source destiatio pairs ad coect them with straight-lies. The, i each tile of size M() M(), there are O( log ) lies through ay tile, with high probability. We begi by cosiderig a tilig of the uit torus by square tiles of the size M() M() ad showig that the umber of lies through ay arbitrary tile chose from the tilig is Θ( ) w.h.p. Note that a route may pass through the same tile more tha oce - each time usig a differet straight-lie path. The, based o stadard colorig argumets i [8], [2], we ca show that the costat badwidth available at a tile ca be uiformly split amog all lies passig through it to provide a throughput T() = Θ( 1 ) for all routes. There are three kids of tiles: (i) Tiles that lie outside the (1+ )Sq(h) of all holes h, (ii) Tiles that lie withi Sq(h) for some h, (iii) Tiles that lie i the aular regio (1 + )Sq(h) Sq(h) of some h. We show the above boud for each of these possibilities. CASE 1: If a tile is outside the aular regio, the tile is exactly equivalet to a tile i a etwork without holes where Θ() radom source-destiatio pairs are chose. This is due to the fact that outside the aular regios, the umber of lies that go through a tile is uchaged if the routig were accordig to our scheme or my a direct straight-lie path - i.e., our stage 0 routes ad the straight-lie paths from source to destiatio are exactly the same o regios outside the regio (1 + )Sq(h) of ay h. From stadard results o throwig /2 radom lies due to radom choice of source-destiatio pairs o a uit torus (Lemma 4.13 of [8] or Claim 2 of [19]), we kow that the maximally loaded tile is at most Θ( ) with probability CASE 2: If a tile is iside the square regio Sq(h), for some hole regio h, it is clear that packets of oly 2 stages pass through it. The stage 0 lies may pass through a tile i this regio if a radomly chose destiatio is o the other side of the hole. If a stage 0 packet hits a hole, it leaves the regio by usig the reverse path (Stage 1 packet) to a radom poit C i the aular regio (Figure 2). Thus, for every stage 0 packet through a tile, there is at most oe stage 1 packet passig through it. Sice the stage 0 of ay route is a exact subset of the straight-lie betwee the radom source-destiatio pair (stage 1 is a subset as well, but with flows i the opposite directio), the total load o a tile i the Sq(h) regio is agai upper bouded by Θ( ). CASE 3: Note that all stages of packets may pass through the aular regio. But as the traffic due to stages 0 ad 1 have bee show to be Θ( ), w.h.p, we restrict our attetio to Stage 3 of ay route. This is because, stage 2 routes are subsets of AA ad stage 4 routes are subsets of BB ad both AA ad BB are symmetric to AB (i the sese that their distributio is idetical to AB over the correspodig rectagular arm - Regio (Stage 3)). Thus, if we show the load due to stage 3 of routes is o more tha Θ( ) with probability 1 1, our claim o the achievable throughput 2 capacity follows. First, we show a boud o the umber of stage 3 routes that are geerated for ay hole h, ad let h be the side of the smallest square cotaiig the hole. Claim 2: The umber of stage 3 routes aroud ay hole h is O( h log ). Now, we cosider a tile i the rectagular regio where stage 3 routes are active (See Figure 2). The distributio of stage 3 routes over this regio is ot uiform for stadard bouds to apply. We upper boud this system by the followig uiform system. Cosider a toroidal regio T boud of side 2(1 + ) h. I this regio we throw 2(1+ ) 2 h K(log ) 2 (we choose a sufficietly large K) radom source-destiatio pairs. Noticig that this etwork is a smaller aalog of the uiform etwork cosidered i the proof of Lemma 4.13 of [8], we apply our stadard bouds o uiform etworks to show the followig claim. Claim 3: The umber of lies through ay tile is o more tha Θ( (log ) 2 ) with probability at least I this toroidal regio, we pick two boxes B 2, B 3 of size ( h ) 2 that are a distace h apart from each other, i.e., a regio similar to Regio(stage 3) i Figure 2. We show that the umber of source destiatio pairs such that the source lies i box B 2 ad destiatio i box B 3 is greater tha the umber of stage 3 routes of the origial etwork, ad further as each of these lies are idepedetly ad idetically distributed as the lie segmet AB. Now, as we throw 2(1+ ) 2 h K log over (1 + 1/ ) 2 tiles of size ( h ) 2, the umber of sources over box 2 is at least Θ( h (log ) 2 ) with probability Further, each of these sources picks a radom destiatio. We cout the umber of destiatios that would fall i box 3. As we throw Θ( h (log ) 2 ) over (1 + 1/ ) 2 boxes, there exist at least Θ( h (log )) source-destiatio pairs that have a radom source i box B 2 ad a radom destiatio i box B 3 (this is with probability at least 1 1 ). Let L be the umber 2

6 of lies over a tile of size M() M() i the Regio (Stage 3), ad let L be the umber of lies passig through ay M() M() tile i toroidal regio T boud. The, we ca show that P(L > Θ( (log ) 2 )) 2/ 2. By our schedulig algorithm where each tile of size M() M() ca be allocated a costat fractio of a time-slot for collisio-free trasmissios (the iterferece graph is a fiite degree graph that ca be colored with fiite colors [2]) it follows that every lie through a tile ca be provided a equal rate of Θ( 1 ) thus providig the same throughput to all routes i the etwork. Further, the delay D() is the sum of the time spet by a packet i each hop. Note that the umber of hops is at most 3 dist(s D), ad the delay at each hop due to schedulig is o more tha Θ( ). Thus, delays are o more 3 tha M() Θ( ) = Θ(T()). Note that this lies o the optimal throughput delay curve [4], [16]. C. Scalig of Routig Iformatio A mai motivatio of geographic routig schemes is the miimal amout of routig iformatio that each ode has to store. Here, we discuss the scalig of routig iformatio of our algorithm. The RadHT() algorithm ca be used i two ways: (i) The route for each packet to its destiatio was setup idepedetly ad radomly accordig to RadHT, or (ii) the RadHT algorithm is ru oce iitially to setup static routes (i.e. all packets from a S D pair follow the same route). I case (i), the oly routig iformatio eeded at ay ode is the locatios of the eighborig odes, which grows as Θ(log ). This is due to the fact that the waypoit odes are ot required to remember the ext waypoit, but geerate it radomly, from the iformatio available i the packet. I case (ii), the waypoit odes are required to remember the ext waypoit so that the packets are routed alog the static routes. However, we ote that a maximum of 1/2 γ holes ca occur o the path betwee a source ad its destiatio, ad thus each path may have at least 3 1/2 γ waypoits, ad with routes, this implies that each ode is a waypoit for 1/2 γ routes o a average. We ote that while our aalysis for the throughput assumed static-routes for tractability, we strogly believe that the throughput achieved by per-packet routes would be orderwise uchaged. IV. NETWORKS WITH ARBITRARY TRAFFIC PATTERNS I this sectio, we cosider the problem of routig betwee arbitrarily chose source-destiatio pairs, with arbitrary traffic demads. Thus, we cosider a fairly geeral etwork ad traffic model (cf. Sectio II-B for a descriptio of the model). A critical issue is to determie if some form of radomized geographic routig ca provide ear-optimal throughput. Such a routig scheme would provide highly distributed etworks (with low computatioal capabilities) to achieve high data rates without ay route setup overheads. Also, geographic routig would coverge immediately to the ear-optimal routes. I previous work [19], we studied etworks with radomly chose source-destiatio pairs (such that the sourcedestiatio pairs are Θ(1) distace away from each other, which correspods to α = 1/2) with a two-level traffic demad, ad demostrated a radomized routig algorithm RANDOMSPREAD that was ear-optimal. Here, we geeralize the model to allow arbitrary locatios of source ad destiatio (cf. Sectio II-B). Typically, upper bouds o etwork capacity have utilized cut-set ideas to limit the traffic that ca leave ay set [12]. Essetially, if we cosider ay closed regio of space, the amout of traffic that ca eter or leave this area is bouded by the amout of radio resource alog the boudary of the set. However, whe sources ad siks ca be arbitrarily close or far, ad with widely varyig traffic requiremets, the cutset boud aloe is ot sufficiet to characterize etwork load distributios. (The traffic flows that ever leave the regio are uaccouted by such cuts.) I this paper, we joitly utilize trasport capacity bouds (bouds that arise due the iterferece ature of the chael) alog with cut-set based bouds to characterize the allowable set of source-destiatio pairs. The joit approach is based o the followig reasoig. Every sub-regio of the geographic regio cotais two resources : (i) the trasport capacity of the sub-regio, ad (ii) the amout of traffic that ca eter/leave the sub-regio through its boudary (the perimeter cut-capacity). For each S-D pair, ay routig algorithm uses up some amout of each of the two resources. For istace, if the S-D pair lies completely withi a sub-regio, straight lie routig uses up oly the trasport capacity withi the regio. O the other had, if the S-D pair decides to route by spreadig the load over the etire geographic regio, it will use the perimeter cut-capacity of the sub-regio alog with some amout of the trasport capacity of the sub-regio. We demostrate that ay routig scheme has a local coservatio property betwee these two resources, amely, that usig less amout of the local trasport capacity resource, requires it to use more of the local cut-capacity resource, ad vice-versa. We propose a algorithm RadLLB (Radomized Local Load Balacig) ad demostrate usig the above property that it is ear-optimal for arbitrary traffic demads (a fiitelevel traffic model is cosidered for aalytical tractability - this ca be readily exteded to a arbitrary traffic model). We describe the algorithm below. Algorithm RadLLB() Cosider a source-destiatio pair l {1,, H} demadig a rate r R, ad whose destiatio is α 1/2 away from its source 1 (with 0 < α < α 1/2). 1) The source ode chooses α locatios at radom from withi a circle of radius α 1/2 about the source locatio for its first waypoit S (i), for 1 i α. 2) The source ode the chooses α locatios at radom from withi a circle of radius α 1/2 about the destiatio locatio for its secod waypoit D (i). 1 For otatioal simplicity, we suppress the source-destiatio idex l i α (i.e. α l ) with the uderstadig i the proof that each source-destiatio pair has potetially a differet α l.

7 Box B i 1 S (i) D (i) Box B i S D M() M() Circle 1 Circle 2 s A source i the aular regio Cotaiig Torus for aalysis Ay give tile T Fig. 3. Local load balacig with three-hop routig. Fig. 4. The traffic load through ay arbitrary tile. 3) Thus, each source costructs α paths from itself to the destiatio, ad radomly distributes the traffic load over a regio that is proportioal to the square of the distace betwee the source ad destiatio. 4) The source splits its rate uiformly over the α multipath routes. For a illustratio of this process, see Figure 3. Ed of Algorithm A. Aalysis of RadLLB() Algorithm We show that the above algorithm spreads the local traffic load i a appropriate maer such that the traffic through ay tile is maageable for ay cofiguratio of source-destiatio pairs ad their traffic demads if they are achievable by ay other scheme. That is, we show the followig theorem. Theorem 4.1: Cosider a arbitrary distributio of sourcedestiatio pairs over a radom plaar etwork (see Sectio II-B). Let a rate vector Λ = [λ s,d R] be achievable by ay scheme. The, the RadLLB() achieves a throughput rate of Θ(Λ), i.e, the algorithm is throughput-optimal up to poly-logarithmic factors. Proof: The mai steps i the proof are as follows: (i) We assume all the sources require a rate r R. (ii) We develop two basic spatial costraits (cut-set ad trasport capacity based bouds) o the positios of the S-D pairs that are ecessary for all routig schemes. (iii) We costruct a boud o the total traffic that may pass through ay give tile, give the costraits o the positios of S-D pairs. (iv) We show that the traffic through ay tile (of the size of the radio rage) is o more tha Θ(log()) for ay achievable S-D pair cofiguratio ad sice each tile of the radio rage is capable of supportig a costat traffic (to its eighborig tiles), we ca scale dow the throughput of all sources by a log-factor to achieve earoptimal throughput. (v) We repeat the argumet for each traffic level i the fiite set R. 2 2 Alterately, we ca sharpe this boud by cosiderig all rate requiremets that are a multiple of a basic rate ˆr. By showig achievability for this basic rate ˆr with arbitrary S- D pairs, we ca immediately geeralize to rate requiremets that are a multiple ˆr. This is because ay multiple of ˆr ca be viewed as a group of sources (destiatios) that are co-located i the same tile. However, we skip a formal proof of this due to space costraits. Thus, we first assume that all S-D pairs require a rate r. Note that there are three types of traffic - the iitial outward-star traffic, i.e., the routes betwee S ad S i, (1 < i < α ) ( traffic), the traffic betwee the 1st ad the 2d waypoits of each route (routes betwee S i ad D i for 1 < i < α ) or (#-traffic), ad the fial iward-star traffic. By symmetry, the traffic load see due to the iward-star traffic is same as the -traffic. Cosider ay give tile (as i Figure 4) of size M() M(), ad costruct cocetric squares B i 1) Necessary coditios o source-destiatio pairs: The followig are ecessary coditios for ay routig scheme: Coditio 4.1: (i) Traffic boud I (trasport capacity): Sice ay tile ca at most support a rate 1, the total traffic supported iside ay box B i is at most (2i 1) 2. (ii) Traffic Boud II (perimeter or cut-set): The total traffic leavig (or eterig) box B i is at most 4 (2i 1). Now, cosider a source whose destiatio is α 1/2 away, or equivaletly, for a M() M() tilig of the space, α / log boxes away. Let this source be i the square regio B i. The the followig holds: (i) Dist(S D) < i 3 boxes: I this case, we ca show that a upper boud o the umber of sources that ca affect tile T withi Box B i ad with distace to destiatio less tha i/3 caot be more tha 32i/r (details i [20]). (ii) Dist(S D) > 2 2i boxes: The destiatio lies outside the box B i, ad hece it uses up r uits of capacity from the perimeter boud (allowable 4 (2i 1)). Thus the total umber of such sources is upper bouded by 4(2i 1) 1 r. (iii) Dist(S D) is betwee i 3 ad 2 2i boxes: I this case, the destiatio ca either lie iside or outside the box. If the destiatio was outside, the source uses up r uits from the allowable perimeter capacity of 4(2i 1). If the destiatio was iside, a arbitrary part r i is supported completely iside the box, ad r r i leaves the box. Note that this is true for every routig scheme. The the rate iside the box uses up at least r i i 3 of the allowable trasport capacity (2i 1) 2. The rate outside the box uses up r r i of the perimeter boud 4 (2i 1). Thus the total umber of such sources is upper bouded by ( (2i 1) 2 i/3 + 4(2i 1) ) 1 r.

8 Thus, a uiform boud o the umber of sources i regio B i that ca affect tile T is give by 32 r (2i 1), which is greater tha ( (2i 1) 2 i/3 + 4(2i 1) ) 1 r + 32 i r. 2) Traffic through a tile due to a source: We ow provide a boud o the traffic through tile T due to a source s i B i B i 1. The bouds arise from two kids of traffic: The -traffic: Note that the outward-star traffic is geerated by choosig α poits at radom from a circle of radius α 1/2 cetered at the source ode. If the source-destiatio separatio was less tha i/3 boxes, the -traffic does ot touch tile T. Else, the umber of lies that ca go through tile T (say lies(t, s)), lies(t, s) K log 2i 1 α with probability 1 1 for some K <. (The above 4 boud ca be obtaied from stadard results o throwig α lies radomly at 4(2i 1) boxes, whe 4(2i 1) = O( α )). Thus, the traffic through a tile due to oe source is r. α lies(t, s) = r K log 2i 1 with probability The #-traffic: Agai, if the source-destiatio separatio was less tha i/3 boxes, the #-traffic does ot touch tile T. If greater, ote that the traffic is geerated by pickig α radom lies that have a source i Circle 1 ad destiatio i Circle 2 (of Figure 3). Claim 4: The umber of # lies through ay give tile is O(log ) with probability at least Thus, the traffic through ay touchable tile is (umber of lies) (traffic through each lie). Sice the rate of r was split uiformly amog α routes, the # traffic through ay tile is upper bouded by K 1 log r α. As α is at-least i/3 log for ay source whose # traffic ca touch tile T, the total traffic K 1 log r α K 1 r log 1 2i 1. We ote that the iward-star traffic is symmetric to the - traffic. 3) Maximizig the traffic through ay tile: Previously, we characterized the ecessary coditios o the umber of sources i ay B i B i 1 aular regio, ad also provided a upper boud (that holds with high probability) o the load see o a tile due to ay source i B i B i 1. Thus, to demostrate that our algorithm does ot overload ay tile, we maximize the traffic o ay tile give the costraits o the source-destiatio pairs, ad show that the maximum traffic is Θ(log ), i.e., is ear-optimal. Let a i be the umber of sources i regio B i B i 1 that ca affect tile T. The, i l=1 a l 32 r (2i 1) for all 1 i / log. Hece, the maximum traffic through a tile is upper bouded by the solutio to the followig optimizatio problem. Let K 2 = K 1 + K. i l=1 max /log i=1 a l 32 (2i 1), r {1 10 r rk 2 log ( r log ) ai K 2 2i 1 i / log}. s.t (1) Claim 5: a = 32 r [1, 2, 2,, 2] maximizes the above problem. The proof is available i [20]. The traffic through ay tile is at most / log i=1 2 2i 1 = Θ(log2 ()). (2) Thus, we show that for ay allowable source-destiatio cofiguratio, the traffic through ay tile is at most Θ(log 2 ()). By meas of a fiite-colorig scheme for the tiles, we ca provide a costat throughput for each tile, ad hece, by reducig the throughput of each source by a poly-logarithmic factor, we ca support Θ(Λ). We showed the above proof for a give rate r - the method ca be similarly used for other rates r from R. V. DISCUSSION AND CONCLUSIONS I the previous sectios, we formally demostrated that radomized geographic schemes ca obtai ear-optimal throughput performace, with low complexity ad very little coordiatio. Here, we try to address some issues that may arise i practical etworks. A. Networks with Traffic ad Node No-uiformity A key property of the RadLLB algorithm that allows it to achieve optimality is that a source-destiatio pair that is separated by a distace d spreads its traffic oly over tiles that are of the same distace from either of them. However, i etworks with holes, it is possible that the shortest path betwee a source ad its destiatio is much larger tha the Euclidea distace betwee them. I such situatios, combiig the RadLLB algorithm with the RadHT algorithm may be sub-optimal, as the hole traversig algorithm itroduces a traffic demad of rate r over a regio of much larger size tha the source-destiatio separatio. Cosider etworks with the followig property (i additio to Sectio II). Coditio 5.1: Let d(x, y) be the Euclidea distace betwee odes x ad y. The, the shortest-distace path betwee odes x ad y i the etwork = d N (x, y) K 3 d(x, y) (x, y). For such etworks, we propose the followig scheme. 1) Each source x performs RadLLB() o its traffic to its destiatio y, spreadig over a area (2K 3 d(x, y)) 2. 2) A packet o hittig a hole h s boudary checks if the h is greater tha d(src-loc, FINAL-DEST). 3) If greater, the packet is dropped at the hole boudary. 4) If the hole is smaller, it performs a RadHT() to traverse the hole.

9 Note that the above scheme has the followig properties: (i) A source-destiatio pair (x, y) oly loads tiles that are withi Θ(d(x, y)),ad (ii) The shortest path has a tubular regio of width at least h /2 aroud it that is ot affected by holes. We expect that the above scheme optimally combies the two algorithms proposed i the paper, amely, RadHT ad RadLLB. We pla to provide a formal proof of the above claim i a future work. B. Networks with Arbitrarily Coected Graphs While i may practical scearios of ad-hoc wireless etworks we may model the o-uiformity of the etwork as occurrece of holes, the actual etwork topology ca be fairly complex ad ot satisfy the hole assumptio i Sectio II-A. I such cases, costructig greedy routes i a throughputoptimal maer may require much more complex algorithms. Moreover, algorithms such as RadHT() may fail to costruct a path to the destiatio i such complex etworks. Although GPSR-like algorithms provide low throughput eve with miimal etwork o-uiformity, they are capable of costructig a path to the destiatio if oe exists (however, with poor load-balacig). To overcome such pathological etworks, practical algorithms could combie the radomized algorithms proposed i this paper alog with determiistic GPSR-like algorithms to provide worst-case performace guaratees. For example, they could be combied i the followig maer: 1) Each source tries to costruct both a GPSR based route (gree) ad Radomized route (red) to the destiatio. 2) I each tile of size M() M(), the chael access time available at each tile is divided ito a fractio β for radomized schemes ad a fractio 1 β for GPSR-like schemes. 3) Based o the fractio of red ad gree packets received at the destiatio, the β-factor ca be updated (by some gossip mechaism) to utilize the more efficiet of the two schemes. This assures that if the etwork is complex, GPSR-like schemes guaratee a path to the destiatio, while if the etwork has maageable holes, the radomized algorithms provide much better throughput. C. Practical Issues We wish to emphasize here that the focus of these algorithms is o providig a ear-optimal performace - there are some implemetatio issues that may arise i practical protocols: 1) Idetificatio of hole perimeter - I our algorithms, we had assumed that the odes have kowledge of whether they are o the hole boudary or ot. I practice, techiques explored i [5] may be used by the odes to lear of their membership o a hole-perimeter. 2) Stability of topology updates - With a chagig topology, where odes may move i ad out of holes ad hole shapes could chage sigificatly over time, a importat issue is if the hole update mechaisms ca still provide a good path to the destiatio. As a part of future work, we will ivestigate such effects of ode mobility. VI. ACKNOWLEDGMENTS This research was supported i part by NSF Grats CNS , CNS , CNS , CCR , ad Darpa CBMANET ad ITMANET programs. REFERENCES [1] D. Bertsekas ad R. Gallager. Data Networks. Pretice Hall, Eglewood Cliffs, NJ, [2] J.A. Body ad U.S.R. Murty. Graph Theory with Applicatios. Macmilliam Press, [3] S. Douglas, J. De Couto, ad R. Morris. Locatio proxies ad itermediate ode forwardig for practical geographic forwardig. Techical Report MIT-LCS-TR-824, MIT Laboratory for Computer Sciece, [4] A. El Gamal, J. Mamme, B. Prabhakar, ad D. Shah. Throughput delay trade-off i wireless etworks. I Proc. of IEEE Ifocom, March [5] Q. Fag, J. Gao, L. J. Guibas, V. desilva, ad L. Zhag. Glider: Gradiet ladmark-based distributed routig for sesor etworks. I Proc. of IEEE Ifocom, March, [6] M. Fraceschetti, O. Dousse, D. Tse, ad P. Thira. Closig the gap i the capacity of radom wireless etworks. I Proceedigs of Itl Symp. o Ifo. Theory, [7] P. Gupta ad P. R. Kumar. 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