JOURNAL OF SELECTED AREAS IN COMMUNICATIONS 1

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1 JOURNAL OF SELECTED AREAS IN COMMUNICATIONS On the Interdependence of Dstrbuted Topology Control and Geographcal Routng n Ad Hoc and Sensor Networks Tommaso Meloda, Student Member, IEEE, Daro Pompl, Student Member, IEEE, and Ian F. Akyldz, Fellow, IEEE Abstract Snce ad hoc and sensor networks can be composed of a very large number of devces, the scalablty of network protocols s a major desgn concern. Furthermore, network protocols must be desgned to prolong the battery lfetme of the devces. Typcal exstng routng technques for ad hoc networks are known not to scale well. On the other hand, the so-called geographcal routng algorthms are known to be scalable but ther energy effcency has never been extensvely and comparatvely studed. In a geographcal routng algorthm, data packets are forwarded by a node to ts neghbor based on ther respectve postons. The neghborhood of each node s consttuted by the that le wthn a certan rado range. Thus, from the perspectve of a node forwardng a packet, the next hop depends on the wdth of the neghborhood t perceves. The analytcal framework proposed n ths paper allows to analyze the relatonshp between the energy effcency of the routng tasks and the extenson of the range of the topology knowledge for each node. A wder topology knowledge may mprove the energy effcency of the routng tasks but ncreases the cost of topology nformaton due to sgnalng packets needed to acqure ths nformaton. The problem of determnng the Optmal Topology Knowledge Range for each node to make energy effcent geographcal routng decsons s tackled by Integer Lnear Programmng. It s shown that the problem s ntrnscally localzed,.e., a lmted topology knowledge s suffcent to make energy effcent forwardng decsons. The leadng forwardng rules for geographcal routng are compared n ths framework, and the energy effcency of each of them s studed. Moreover, a new forwardng scheme, Partal Topology Knowledge Forwardng (), s ntroduced, and shown to outperform other exstng schemes n typcal applcaton scenaros. A PRobe-bAsed Dstrbuted protocol for knowledge range adjustment (PRADA) s fnally ntroduced that allows each node to effcently select onlne ts topology knowledge range. PRADA s shown to rapdly converge to a near-optmal soluton. Index Terms Wreless Ad Hoc and Sensor Networks, Mathematcal Programmng/Optmzaton, Geographcal Routng, Topology Control. I. INTRODUCTION RECENT advances n wreless communcatons and electroncs are pavng the way for the deployment of lowcost, low-power, large scale ad hoc networks such as untethered and unattended networks of sensors and actuators. Sensor networks [] dffer from tradtonal ad hoc networks n many aspects. The number of n a sensor network can be several orders of magntude hgher than n ad hoc networks, and the deployment of s usually denser. Moreover, sensor are lmted n power, computatonal capactes and memory, and they may not have global dentfcaton (ID) because of the very large number of and the related overhead. The authors are wth the Broadband and Wreless Networkng Laboratory, Georga Insttute of Technology, Atlanta, GA. Due to the above constrants, sensor network protocols and algorthms must be endowed wth self-organzng capabltes,.e., sensors must be able to cooperate n order to effcently perform networkng tasks. The prmary desgn constrants of these algorthms are energy effcency, scalablty and localzaton. It has been recently ponted out [2] that energy effcency n moble systems can be mproved by desgnng protocols and algorthms wth a cross-layer approach,.e., by takng nto account nteractons among dfferent layers of the communcaton process so that the overall energy consumpton can be mnmzed. Hence, n ths paper we consder nterdependences between physcal and network layer functonaltes to mprove the energy effcency of the routng tasks. A prmary requrement of confguraton algorthms for large scale ad hoc networks, such as routng algorthms, s scalablty,.e., these algorthms should perform well for wreless networks wth an arbtrary number of. The noton of scalablty s strctly related to that of localzaton: n a scalable algorthm each node exchanges nformaton only wth ts neghbors (localzed nformaton exchange) [3]. In a localzed routng algorthm, each node selects the next hop based only on the poston of tself, of ts neghbors, and of the destnaton node. As a result, the local routng decson of each node strves to acheve a global network objectve such as mnmum latency or mnmum energy consumpton. Conversely, n a non-localzed routng algorthm a node mantans an accurate descrpton of the overall network topology to select the next hop. Ths way, the routng problem s equal to the shortest path problem f the hop count s used as the global performance metrc or the shortest weghted path f power [4] or cost [5][6] lnk metrcs are used. It has been shown [7][8] that routng protocols that do not use geographcal locaton nformaton are not scalable, e.g., AODV (Ad hoc On-demand Dstance Vector), DSDV (Destnaton Sequenced Dstance Vector) or DSR (Dynamc Source Routng). On the other hand, the recent avalablty of small, nexpensve and low-power GPS (Global Postonng System) recevers, together wth technques to deduce relatve sensor coordnates from sgnal strengths [9] encourage researchers to develop Geographcal Routng [] (also Poston Based Routng) algorthms, whch are deemed to be the most promsng solutons for crtcally power-constraned ad hoc networks. For these reasons, ths paper deals wth the nterdependences between topology control [] and energy effcent geographcal routng. The queston we try to answer s How extensve should be the Local Knowledge of the global topology n each node, so that energy effcent geographcal routng

2 JOURNAL OF SELECTED AREAS IN COMMUNICATIONS 2 decsons can be taken?. The answer to the queston s clearly related to the degree of localzaton of the routng scheme. If each node could hold a complete vson (knowledge) of the network topology, t could then compute the globally optmal next hop,.e., the neghborng node on the mnmum energy path. However, acqurng ncreasngly accurate topology nformaton has an ncreasng cost,.e., the energy requred to exchange the sgnalng traffc that convoys ths nformaton. Hence, we develop an analytcal framework to capture the trade-off between what we refer to as the topology nformaton cost, whch ncreases wth ncreasng range of knowledge of each node, and the communcaton cost, whch may decrease when the knowledge becomes more complete. We apply ths analytcal framework to dfferent poston based forwardng schemes [2][3][4][5][6] and show by Monte Carlo smulatons that a lmted knowledge s suffcent to make energy effcent routng decsons. Wth respect to the exstng lterature on geographcal routng, we try to better defne the terms localzed and neghbor. A neghbor for a certan node s another node whch falls nto ts Topology Knowledge Range, denoted as KR n the followng. The man contrbutons of ths paper are: We ntroduce an analytcal framework to evaluate the energy consumpton of geographcal routng algorthms for power constraned large scale ad hoc networks; We provde an Integer Lnear Programmng (ILP) formulaton of the topology Knowledge Range optmzaton problem; We conduct a detaled comparson of the leadng exstng forwardng schemes and ntroduce a new scheme called Partal Topology Knowledge Forwardng (); We ntroduce a PRobe-bAsed Dstrbuted protocol for knowledge range adjustment (PRADA), that allows each node to effcently select onlne ts topology knowledge range, and show that PRADA leads to near-optmal energy consumpton. The remander of the paper s organzed as follows. In Secton II we revew exstng forwardng schemes for geographcal routng and other related work. In Secton III we state the problem and n Secton IV we formulate t as an optmzaton problem. In Secton V we ntroduce PRADA, a dstrbuted protocol for onlne Knowledge Range adjustment. In Secton VI we show numercal performance results, whle n Secton VII we conclude the paper and draw the man conclusons. II. RELATED WORK In ths Secton, we descrbe the exstng poston based forwardng rules that wll be compared n the followng of the paper, and revew other exstng work on the topc whch consttutes the background of our work. A. Forwardng Rules In a localzed geographcal routng scheme, node S (Fg. ) whch currently holds the message, only knows the poston of ts neghbors,.e., the wthn ts Knowledge Range, and Fg.. Dfferent Forwardng Schemes of the destnaton node D. For the convenence of the reader, let us ntroduce the followng defntons: Defnton : Gven a sender node S, and a destnaton node D, the progress of a generc node X s the orthogonal projecton of the lne connectng S and X onto the lne connectng S and D. Defnton 2: Gven a sender node S and a destnaton node D, the advance of a generc node X s the dstance between S and D mnus the dstance between X and D. Takag and Klenrock proposed the frst geographcal routng scheme, based on the noton of progress. In ther Most Forward wthn Radus () scheme [2], the message s forwarded to the maxmum progress neghbor, e.g., node M n Fg., whose progress s Sm. Note that although node G s closer to the destnaton, ts progress Sg s smaller than Sm. Hou and L [3] dscuss the Nearest Forward Progress () method whch selects the mnmum progress neghbor wthn the topology Knowledge Range of S, e.g., node N n Fg., whose progress s Sn. Fnn [4] proposes the Greedy Routng Scheme (), whch s based on geographcal dstance: node S selects among ts neghbors the closest to the destnaton,.e., the node wth maxmum advance, e.g., G n Fg.. In the so-called Routng method [5], the message s forwarded to a neghbor, e.g., C n Fg., such that the angle CSD s mnmum,.e., the drecton SC s the closest to the drecton SD. The Random Progress Forwardng () method [6] selects a random next hop among the wthn the Knowledge Range. A suffcent condton for a geographcal routng scheme to be loop free s that only next hop wth postve advance can be selected. Accordng to Defnton 2, a generc node has a postve advance wth respect to a sender node f t s closer than the sender to the destnaton. When a routng scheme s constraned to select a node as next hop only f t has postve advance, then the overall path s guaranteed to be loop free. Conversely, a postve progress for each next hop s not a suffcent condton for a routng scheme to be loop free, as can be nferred from the counterexample n Fg. 2, where three, A, B and a destnaton node D are shown. A s a possble next hop for B and vce versa, snce both A and B have postve progress wth respect to each other

3 JOURNAL OF SELECTED AREAS IN COMMUNICATIONS 3 Fg. 2. Counterexample on the noton of progress Fg. 3. Neghborhood Dscovery Protocol (Ak >, Bh > ). However, ths does not avod loops as both could choose the other as next hop, although as shown n [7] the two can recognze the loop and stop t. Referrng agan to the example n Fg. 2, when a postve advance s a necessary condton for a node to be next hop, A s a feasble next hop for B, but not vce versa, snce A s closer than B to the destnaton (AD < BD). Snce postve advance s a stronger condton, and guarantees loop free paths, we take postve advance as a necessary condton for a node to be the next hop. In other words, a node must choose the next hop among the wthn ts Knowledge Range and wth postve advance wth respect to the destnaton node, for all the consdered forwardng schemes. B. Other Related Work An excellent survey on poston based routng technques for ad hoc networks s gven n [][8]. Locaton update technques,.e., methods to determne absolute and relatve coordnates for network, are revewed n [9]. Most of the pror research assumes that can ether work n a greedy mode or n a recovery mode. When n greedy mode, the node that currently holds the message tres to forward t towards the destnaton. The recovery mode s entered when a node fals to forward a message n the greedy mode, snce none of ts neghbors s a feasble next hop. Usually ths occurs because the node observes a vod regon between tself and the destnaton. Such a node s referred to as concave node. For example, the GFG algorthm [2] makes greedy forwardng decsons (as n Secton II.A). When a packet reaches a concave node, GFG tres to recover by routng around the permeter of the vod regon. Recovery mechansms, whch allow a packet to be forwarded to the destnaton when a concave node s reached, are out of the scope of ths paper. Here we assume that the packet s drectly forwarded to the destnaton whenever such a node s reached. The Trajectory Based Forwardng (TBF) algorthm s proposed n [2], where the packet s forwarded along a predefned parametrc curve encoded n the packet at the source. Several localzed algorthms for power, cost and power-cost effcent routng are proposed and ther effcency s analyzed n [22]. Scalablty propertes of dfferent ad hoc routng technques such as flat, herarchcal and geographcal routng are dscussed n [23]. The GAF topology control algorthm [24] dentfes that are equvalent from a routng perspectve based on poston nformaton, and adaptvely turns unnecessary off n order to mantan a constant level of performance. A taxonomy of locaton systems s gven n [9] for ubqutous computng applcatons ncludng locaton sensng technques and propertes as well as a survey of commercally avalable locaton systems. In [25] t s shown how to derve poston nformaton for all usng Angle of Arrval (AOA) capabltes, when only a fracton of the have postonng capabltes. Fnally, a dstrbuted locaton servce (GLS) s descrbed n [7], where a node sends ts poston updates to ts locaton servers wthout knowng ther actual denttes. Ths nformaton s then used by the other n the network to perform geographcal routng operatons. III. PROBLEM SETUP In ths secton we ntroduce the Topology Knowledge Range problem, whch s then formulated as an Integer Lnear Program (ILP) n Secton IV. Frst, we descrbe a neghborhood dscovery protocol whch allows each node to gather nformaton about ts neghborhood. Then, we ntroduce the network and energy models and defne some useful notons. Fnally, we present a new localzed forwardng scheme called Partal Topology Knowledge Forwardng (). Let us consder the followng neghborhood dscovery protocol. Wth reference to Fg. 3, node S perodcally sends a Neghborhood Dscovery packet, (ND-packet), to gather localzaton nformaton about ts neghbor, at a power level that allows the packet to be receved by all wthn ts chosen Knowledge Range (KR n Fg. 3). As a result, N, N 2 and N 3 receve the ND-packet whle farther do not. All that receve the NDpacket reply wth a Locaton Update packet (LU-packet), that contans the geographcal poston of the node. It s ntutve that ncreasng the KR may result n more effcent routng decsons. However, ths comes at the expense of a hgher energy consumpton needed to exchange sgnalng traffc. Hence, we are tryng to determne the Knowledge Range (KR) for each node so that the energy requred by the network to perform the routng tasks s mnmzed.

4 JOURNAL OF SELECTED AREAS IN COMMUNICATIONS 4 A. Network Model The network s represented as (V,E), where V = {v, v 2,.., v N } s a fnte set of n a fnte-dmenson terran, wth N = V, and where E s the matrx whose element (, j) contans the value of the dstance between v and v j. We assocate to each node v k ts Knowledge Range, r k, based on the neghborhood dscovery protocol as explaned above. Thus, the array R = [r, r 2,.., r N ] descrbes the KRs of all n the network. Let S be the set of traffc sources and D the set of destnaton. We defne P = {(s, d) : s S, d D} as the set of source-destnaton connectons. The nformaton rate of each connecton s descrbed by the traffc matrx P = [p j ], where p j represents the average nformaton rate (bt/s) between a source node S and a destnaton node j D. Let us ntroduce the followng defntons: Defnton 3: Gven a node v, ts KR r and a destnaton node v d, a loop-free Forwardng Rule F assocates the node v wth another node v k n V \ {v }, n such a way that the path {v, v k,.., v d } obtaned by applyng the rule from source to destnaton s composed of dstnct. We ndcate wth v k = l F v (v d, r ) that v k s the next hop of node v towards v d wth KR r, accordng to F. Note that for the sake of smplcty we wll also refer to a generc node v k as k, and omt the ndex F. Thus, l F v (v d, r ) s referred to as l (d, r ). Gven the array R of the KRs of all, the rule F nduces paths among any possble source-destnaton par n the network. Thus, F : R x sd j (R) () where x sd j (R)= ff the lnk between node and node j s part of the path between node s and node d wth the gven choce R of ranges, when we apply the forwardng rule F. B. Energy Model An accurate model for the energy consumpton per bt at the physcal layer s E = E trans elec + βd α + E rec elec (2) where Eelec trans s the dstance-ndependent amount of energy consumed by the transmtter electroncs (PLLs, VCOs, bas currents, etc.) and dgtal processng, Eelec rec s the energy utlzed by recever electroncs, whle βd α accounts for the radated power necessary to transmt over a dstance d between source and destnaton. As n [26], we assume that E trans elec = E rec elec = E elec (3) Hence, the overall expresson for E n (2), whch we refer to as lnk metrc hereafter, smplfes to E = 2 E elec + βd α (4) Accordng to ths lnk metrc, the topology nformaton cost for node v s expressed as C INF (r ) = [L D βr α + (N (r ) + ) L D E elec + + m ζ (r ) L U βd α m + 2N (r ) L U E elec ] T M (5) where: α s the path loss (2 α 5); β s a constant [Joule/(bt m α )]; L D s the length of an ND-packet [bt]; L U s the length of an LU-packet [bt]; E elec s the energy needed by the transcever crcutry to transmt or receve one bt [Joule/bt]; N (r ) s the number of neghbors of node when ts Knowledge Range s r ; ζ (r ) s the set contanng the ndexes of the n range r of node ; T M s the perod between two consecutve neghborhood dscovery messages [s]; The expresson βr α represents the energy needed to transmt one bt at dstance r ; thus L D E elec +L D βr α s the energy needed for node to transmt the ND-packet to all n ts Knowledge Range, where as each of the N (r ) n ts KR spends L D E elec to receve the ND-packet. By addng these two components we obtan the frst lne of (5). Then, each of the N (r ) transmts an LU-packet. The energy expendture has agan a constant factor, L U E elec, plus a factor, L U βd α m, whch depends on the dstance between the transmttng node v m and node v. Moreover, v spends L U E elec to receve each of the N (r ) LU-packets. By addng all these components, and dvdng by T M, whch s the nverse of the locaton update frequency, we obtan the fnal expresson for C INF. In other words, C INF s the average energy needed for node v to obtan topology nformaton wthn range r. The communcaton cost for node v can be expressed as C COM (R) = [βd α l (d,r ) + 2 E elec] p sd (6) wth (s,d) Π (R) Π (R) = {(s, d) s.t. x sd j = for at least one j} (7) The set Π (R) contans all source-destnaton pars whose path ncludes v as a transt node, as well as those for whch v s the source. Thus, n (6) we sum over all the connectons where v s a transmttng node. Note that each term has a dstance-ndependent component 2 E elec (the energy needed to transmt and receve one bt), and a dstance dependent component, d α l (d,r ), whch represents the α-th power of the dstance between v and v l (d,r ), ths last beng the next hop of v towards v d when ts KR s r. Every term s then multpled by the average bt rate of the communcaton p sd. (R) represents the average energy expendture for all the communcatons node v s nvolved n. We can now state the total cost for node v as Thus, C COM C T OT (R) = C COM (R) + C INF (r ), V (8) Note that whle the nformaton cost of each node only depends on ts own KR, the communcaton cost depends on the KRs of all nvolved n the communcaton process.

5 JOURNAL OF SELECTED AREAS IN COMMUNICATIONS 5 IV. INTEGER LINEAR PROGRAMMING (ILP) FORMULATION As stated above, our objectve s selectng the vector R of Knowledge Ranges (KR) whch mnmzes the energy expendture of the overall network, gven the set of connectons P and a Forwardng Rule F: Fg. 4. Partal Topology Knowledge Forwardng () mn R CT OT = (C COM (R) + C INF (r )) (9) V C. Partal Topology Knowledge Forwardng () We now ntroduce a novel forwardng scheme, called Partal Topology Knowledge Forwardng (). can be classfed as a localzed shortest weghted path routng scheme wth a power lnk metrc, where routes are calculated based only on a lmted local knowledge of the overall topology. Let us refer to Fg. 4 and consder a node that s holdng a message (S) and s n charge of forwardng t to a gven destnaton node (D). If S had a complete topologcal vew, t could calculate the optmal path towards the destnaton,.e., the shortest weghted path that concdes wth the mnmum energy path when we consder a lnk metrc of 2 E elec +βd α, accordng to (4). Ths s shown on the left box of Fg. 4 where the topologcal vew of node S s consttuted by all of the network wth postve advance (see Def. 2) wth respect to the destnaton (grey ). Conversely, n we assume that, gven a lmted KR, S only knows the poston of all nsde ths range and the poston of the destnaton node. The topologcal vew of S s consttuted by node D and by all the n the KR wth postve advance wth respect to D (see rght box n Fg. 4). In ths case, the mnmum energy path towards the destnaton s calculated only based on ths lmted topologcal vew,.e., the shortest weghted path only takes nto account n the KR and the destnaton, as the other are unknown to S. It s assumed that on the border of the KR can reach the destnaton node drectly n one hop, e.g., node N 2 n the rght box of Fg. 4 can drectly reach D. Hence, we consder a fne graned topology close to the node holdng the message (wthn the KR), and an extremely coarser graned topology outsde (only the poston of the destnaton node s consdered). Thus, S wll forward the message to the frst node N on the mnmum energy path calculated n ths way. In ts turn, N calculates the path towards the destnaton D, but ths tme accordng to ts own KR. Ths can actually result n a very dfferent path beng chosen by N as compared to the path calculated by S. Note that, unlke the forwardng schemes descrbed n Secton II-A, s not a greedy scheme. Ths scheme becomes more localzed when the KR of each node gets smaller. We wll show n the followng that small KRs are chosen when energy effcency s the major concern. We refer to ths problem as Optmal Topology Knowledge Range problem and formulate t as an ILP. We consder dscrete values of the Knowledge Ranges. The granularty of ths quantzaton can be whatever, but obvously fner-graned transmsson ranges ncrease the sze of the space of possble solutons, thus makng t harder to fnd the optmal values. Each varable r, r r max assumes one out of k max dscrete, equdstant values n the set {r, r,.., r kmax }, wth r k r k = r, k s.t. k k max, wth r = and r max = r kmax. We refer to the set of ndces {,,.., k max } as R. We ntroduce the followng notatons and varables: r(k) s the k-th Knowledge Range; r α (k) s the α-th power of the k-th KR; N (k) s the number of neghbors for node v when t selects the k-th KR; f j dk = ff, accordng to F, node v j s the next hop for node v, when v d s the destnaton, and the k-th Range s chosen; a k j = ff node v j s n the k-th KR of node v ; d α j s the α-th power of the dstance between v and v j. We ntroduce the followng routng varables: x sd j = ff lnk (, j) s part of the path between v s and v d. The assgnment varables are: y k = ff node v uses k-th Knowledge Range. We refer to the varables y k as Knowledge Range ndces. We can now express the problem as: Optmal Topology Knowledge Range Problem: Mnmze: Subject to: j V j V C T OT = V (C COM + C INF ) () y k =, V () k R (x sd sj x sd js) =, s S, d D s.t. s d; (2) (x sd dj x sd jd) =, s S, d D s.t. s d; (3) j V (x sd j x sd j ) =, s S, d D, V s.t. s d, s, d; (4) x sd j (y k f j dk ), s S, d D,, j V; (5) k R

6 JOURNAL OF SELECTED AREAS IN COMMUNICATIONS 6 x sd sj = (ys k f sj dk ), s S, d D, j V s.t. s d (6) k R C INF = (L N β k R(y k r α (k) + ( k R (y k N (k)) + ) L N E elec + m V(L U β d α m + 2 L U E elec ) (y k a m (k))), V. (7) T M C COM k R = s S (x sd j p sd (2 E elec d D j V +β d α j)), V. (8) Constrant () mposes the exstence of a sngle Knowledge Range ndex dfferent from zero for each node. Constrants (2)(3)(4) express conservaton of flows [27], whle constrants (5)(6) mpose that paths are bult accordng to the forwardng rule defned by the nput parameters f j dk. Fnally, constrants (7)(8) express the nformaton and communcaton cost wth the Knowledge Range ndex notaton, respectvely. Note that gven a forwardng rule F, expressed by the f j dk parameters, the assgnment of the routng (xsd j ) varables s completely dependent on the choce of Knowledge Ranges (y k varables). Once the values of the yk varables have been selected, the set X = {x sd j } defnes the path from source to destnaton for any connecton n P. V. PRADA: A DISTRIBUTED PROTOCOL FOR TOPOLOGY KNOWLEDGE RANGE ADJUSTMENT The soluton of the ILP problem s not feasble n a practcal settng due to ts complexty and centralzed nature. Hence, we ntroduce the PRobe-bAsed Dstrbuted protocol for knowledge range adjustment (PRADA), whch determnes the KRs onlne n a dstrbuted way. The objectve of PRADA s to allow network to select stable and effcent topology Knowledge Ranges (KRs). Ths global target s acheved through dstrbuted decsons and by means of probe packets exchanged among the. The man dea behnd PRADA s to allow each node to adjust ts KR accordng to the feedback nformaton t receves from neghborng nvolved n the same mult-hop connectons. In Secton VI, we wll show that PRADA quckly converges to a near-optmal soluton. To trade-off between the topology nformaton cost and the communcaton cost, each node whch s part of the path of a partcular connecton (as a source or a transt node), perodcally probes ts possble KRs. The node s thus able to assocate an ncrease/decrease n the overall energy expendture to each KR. To clearly understand the ratonale behnd PRADA we pont out that whle the nformaton cost of each node only depends on ts KR, the communcaton cost depends on the KRs of all nvolved n the communcaton process. Thus, the communcaton cost must be montored wth probe packets. PRADA s executed at each node v that has an actve role n the network as a source or a transt node. We ndcate as P the set of connectons where v has an actve role. Perodcally, each actve node selects a certan KR to be probed, dfferent from the current one, n the dscrete set of possble KRs. We refer to the selected KR as r probe and to the current KR as r current. For each connecton p P, v selects the next hop lv F (v p d, r probe), where v p d s the destnaton node of the connecton p P, accordng to the forwardng rule F and to ts current KR. The node calculates C T OT (r probe ) = C INF (r probe ) + c p (r probe) (9) p P where c p (r probe) s the cost of the transmssons along the path from v to the destnaton of the connecton p, wth KR r probe. Ths accounts for the cost of transmttng data from the node tself to all the destnatons, plus the cost of nformaton assocated to the new KR r probe. If C T OT (r probe ) < C T OT (r current ), the value of the KR s updated (r current = r probe ). A probe packet has fve data felds. The frst two felds contan the geographcal coordnates of the source and the destnaton. The thrd contans a parameter called cumulatve communcaton cost and the fourth contans the value r probe of KR. The last feld s a one-bt flag, whch s equal to f the packet s on the forward path towards the destnaton, or equal to f t s on the reverse path. The cumulatve communcaton cost feld, ntalzed to when the packet s created, s updated hop-by-hop by addng the ncremental communcaton cost,.e., the communcaton cost necessary to reach the next hop, to the communcaton cost stored n the packet. Ths way, partal cumulatve communcaton costs are computed hop-byhop along the path from the sender to the destnaton. Algorthm PRADA begn randomly select r probe r current for each p P do v lv F (v p d, r probe): probe packet end for wat for return packets C T OT f (C T OT (r probe ) = C INF (r probe ) < C T OT r current = r probe end f end (r probe ) + p P c p (r probe) (r current )) then After choosng a KR r probe, for each connecton n P the node sends a probe packet to the relevant next hop and wats for ts return. When a node receves a probe packet on the forward path, t looks nto a cost record table to check f t already knows the ncremental communcaton cost needed to reach ths destnaton. If t does, there s no need to forward the probe packet to the destnaton. The probe packet s sent back wth the updated nformaton and the path bt s set to reverse. If t does not, the packet s forwarded to the next hop towards the destnaton n order to evaluate the communcaton cost. The packet s forwarded untl a node wth nformaton for that destnaton or the destnaton tself s reached. When a node has gathered all the cost nformaton assocated to a certan r probe, t calculates the cost assocated to r probe as n (9).

7 JOURNAL OF SELECTED AREAS IN COMMUNICATIONS 7 Scenaro Scenaro 2 Scenaro 3 Sze (mxm) (mxm) (5mx5m) KRs (, 2, 4, 6, 8)m (, 2, 4, 6, 8)m (, 4,.., 2)m α 4 3 vares L D 28bts 28bts 28bts L U 28bts 28bts 28bts T M s vares.s E elec vares 5pJ/bt 5nJ/bt β pj/bt/m α pj/bt/m α pj/bt/m α Rates kbt/s kbt/s kbt/s TABLE I.5 PARAMETERS OF THE MODEL USED FOR SIMULATIONS Algorthm descrbes the operatons performed by a node v whch executes PRADA. In order to reach stablty, the KR s updated only f the movng average of the communcaton cost for the last N probe values gathered s lower than the cost of the current range. In the experments we assume that all the KRs are probed wth the same probablty. More sophstcated strateges can also be mplemented n order to selectvely scan the KRs, amed at savng transmsson power, e.g., by avodng values of KR that are not lkely to brng any beneft. VI. PERFORMANCE EVALUATION We mplemented the forwardng schemes descrbed n Secton II-A, gven n Secton III-C and PRADA, gven n Secton V n a smulator. We further mplemented the ILP problem n AMPL [28] and solved t wth CPLEX [29]. We are partcularly nterested n scenaros where the densty of s hgh, such as those encountered n sensor network applcatons. However, due to the computatonal complexty of the problem, and to the large amount of the nput data, a state-of-the-art workstaton can fnd the optmal soluton wth CPLEX for networks wth at most. Thus, we consder small geographcal areas n order to take nto account the effects of hgh node denstes on the problem. The model depends on several nput parameters, and on the approprate choce of these parameters whch are hghly dependent on the technology and on the target applcatons. Our choce for these parameters was motvated by the model presented n [26]. However, we also vary these parameters n order to study ther relevant effects on the network performance. We present smulaton results for the scenaros llustrated n Table I. In Scenaro, are randomly deployed n a mxm terran. All are sources wth kbt/s flows drected towards a sngle snk node. In Fg. 5 we show the optmal cost (the mnmum of the objectve functon of the Optmal Knowledge Range problem, stated n ()), wth ncreasng number of for all the mplemented forwardng schemes (Sectons II-A and III-C). The value chosen for the parameter E elec s 5 9 J/bt [26]. Note that confdence ntervals are not shown for the sake of clarty. Snce the area of the terran s very lmted, mult-hop paths are often not Fg. 5. Scenaro - for the mplemented forwardng schemes, E elec = 5 9 J/bt Protocol Cost Fg. 6. Scenaro - Cost wth PRADA for the mplemented forwardng schemes, E elec = 5 9 J/bt energy effcent, whch leads source to drectly transmt to the destnaton wthout relyng on ntermedate forwardng. For ths reason, dfferent forwardng schemes show smlar performance. In Fg. 6 we show the total cost for all the mplemented forwardng schemes n Scenaro, obtaned by applyng PRADA wth N probe = 3. In Fg. 7 we compare the optmal cost obtaned for wth three dfferent approaches for the soluton of the optmzaton problem, wth 95% confdence ntervals. The problem s solved wth CPLEX (optmal soluton), wth a greedy local search heurstc, and by applyng the PRADA dstrbuted protocol. CPLEX fnds the optmal soluton for mxed nteger problems by usng a branch and bound algorthm. The greedy local search heurstc scans the one after another and selects for each of them the KR whch mnmzes the cost; the process s repeated perodcally untl stablty s reached. The PRADA curve s very close to the CPLEX and the greedy local search heurstc curves. Ths behavor, as wll

8 JOURNAL OF SELECTED AREAS IN COMMUNICATIONS Greedy Local Search Cost PRADA Cost Confdence Interval (Optmal) Confdence Interval (Greedy Local Search) Confdence Interval (PRADA) Fg. 7. Scenaro - Comparson of for wth dfferent approaches, E elec = 5 9 J/bt Fg. 9. Scenaro - for the mplemented forwardng schemes, E elec = 5 J/bt N=.5 Dstrbuton of Kowledge Ranges x 3 9 N= N=5.5 6 N= Knowledge Range [meters] Fg. 8. Scenaro - Dstrbuton of values of Knowledge Range, E elec = 5 9 J/bt Fg.. Scenaro - for the mplemented forwardng schemes, E elec = 5 J/bt be shown, becomes more evdent when the problem becomes more localzed,.e., when mult-hop paths are more energy effcent. In Fg. 8 we show the dstrbuton of the values of the KRs n Scenaro, wth N =, 3, 5 and 7. In ths scenaro the average KR s below.5 meters, and as can be seen most ether have a KR equal to (.e., they prefer to know nothng about ther neghborhood and drectly transmt to destnaton) or they try to know far (4, 6 meters) to use them as ntermedate relays. As a result, t s ether effcent to drectly transmt to destnaton or use at most one ntermedate node as relay. By decreasng the E elec parameter, we decrease the weght of the component n energy expendture (lnk metrc n (4)) whch s ndependent of the dstance. Hence, t becomes more energy effcent to select mult-hop paths, snce the overall dstance ndependent part of the energy expendture ncreases wth the number of hops. We would obtan the same effect by ncreasng the area of the terran, but we would have a less dense deployment. It can be nferred by comparng Fgures 5, 9, and that the more mult-hop paths are energy effcent (low values for E elec ), the more (Secton III-C) outperforms the other schemes. When long paths are energy effcent, takes a better advantage of the local knowledge of the neghborhood. In the above fgures, the values for E elec are 5 9, 5, 5 and 5 2 J/bt, respectvely. For E elec = 5 2 J/bt, the cost obtaned wth PRADA s optmal, as can be seen from Fg. 2. When the dstance ndependent term E elec n (4) becomes small as compared to the area of the terran, mult-hop paths become more energy effcent. When ths occurs, by selectng KRs whch are optmal only locally, as PRADA does, we obtan globally optmal solutons, because the problem becomes more localzed when E elec decreases. In Fg. 3 we show that t s more energy effcent to select close as next hop (KRs are 2 meters),

9 JOURNAL OF SELECTED AREAS IN COMMUNICATIONS x Fg.. Scenaro - for the mplemented forwardng schemes, E elec = 5 2 J/bt Fg. 4. Scenaro 2 - for the mplemented forwardng schemes, T M =.s x Protocol Cost Greedy Local Search Cost PRADA Cost Confdence Interval (Optmal) Confdence Interval (Greedy Local Search) Confdence Interval (PRADA) Fg. 2. Scenaro - Comparson of for wth dfferent approaches, E elec = 5 2 J/bt Fg. 5. Scenaro 2 - Cost wth PRADA for the mplemented forwardng schemes, T M =.s N= N=3 N=5 N= Dstrbuton of Knowledge Ranges Knowledge Ranges [meters] Fg. 3. Scenaro - Dstrbuton of values of Knowledge Range, E elec = 5 2 J/bt as E elec decreases. Ths s partcularly true when the densty ncreases. In Scenaro 2, all are sources wth kbt/s flows drected towards a sngle snk node. In Fg. 4 we report optmal costs wth ncreasng number of for all the mplemented forwardng schemes (Secton II-A). Agan, performs better than the other forwardng schemes. More greedy schemes such as Nearest Forward Progress () and Most Forward wthn Radus () are shown to consume more energy. Fgure 5 shows the total cost n Scenaro 2 for all the mplemented forwardng schemes, obtaned by applyng PRADA wth N probe = 3. Fgures 4 and 5 are almost dentcal, whch s explctly shown by Fg. 6 where we compare the results obtaned for wth the three dfferent optmzaton approaches (CPLEX, greedy local search, PRADA). In Fg. 7 we depct the nformaton cost (7) and the communcaton cost (8) for, agan wth the three dfferent approaches. The communcaton cost s shown to hghly exceed the n-

10 JOURNAL OF SELECTED AREAS IN COMMUNICATIONS Convergence of Total Cost Average Cost wth PRADA after n steps Greedy Local Search Cost PRADA Cost Confdence Interval (Optmal) Confdence Interval (Greedy Local Search) Confdence Interval (PRADA) Fg. 6. Scenaro 2 - Comparson of for wth dfferent approaches, T M =.s steps Fg. 9. Scenaro 2 - Convergence of PRADA wth, 7, T M =.s.5..9 Convergence of Total Cost Average Cost wth PRADA after n steps Cost of Communcaton (Optmal) Cost of Informaton (Optmal) Cost of Communcaton (Greedy Local Search) Cost of Informaton (Greedy Local Search) Cost of Communcaton (PRADA) Cost of Informaton (PRADA) Fg. 7. Scenaro 2 - Informaton Cost and Communcaton Cost for, T M =.s steps Fg. 2. Scenaro 2 - Convergence of PRADA wth, 4, T M =.s KR [meters] Average Optmal Knowledge Range Fg. 8. Scenaro 2 - Average KR wth dfferent forwardng schemes, T M =.s formaton cost when relatvely hgh data rate flows must be supported. In Fg. 8 we show the average value of the Knowledge Range wth ncreasng number of for all the proposed schemes. It s shown that a very lmted knowledge of the topology s needed n average, less than 2 meters. In Fg. 9 and 2 we show the average convergence dynamcs of PRADA to the optmal soluton wth 7 and 4, respectvely. At each step, one node randomly selects and probes one of ts KRs. For 7, after 3 steps we obtan a near-optmal soluton. In Fg. 2 we assume a lower locaton update frequency (hgher T M ). Thus, we set T M =. As can be seen n Fg. 2, for lower locaton update frequences even more evdently outperforms the other schemes. A more extended local topology knowledge brngs benefts n terms of energy to the scheme whch best explots ths nformaton. Ths s confrmed by Fg. 22 that shows how the average KRs ncrease n general, and partcularly for whch s able by ts nature to better take advantage of a more extended knowledge. Stll, the extenson of local knowledge

11 JOURNAL OF SELECTED AREAS IN COMMUNICATIONS watts Fg. 2. Scenaro 2 - for the mplemented forwardng schemes, T M = s Fg. 23. α = 3 Scenaro 3 - for the mplemented forwardng schemes, Average Optmal Knowledge Range KR [meters] 2. watts Fg. 22. Scenaro 2 - Average KR wth dfferent forwardng schemes, T M = s Fg. 24. α = Scenaro 3 - for the mplemented forwardng schemes, of the topology s very lmted compared to the dmensons of the terran. In Scenaro 3, we consder traffc patterns that are more lkely encountered n an ad hoc network. In ths case, 25% of the deployed generate a kbt/s traffc flow, each drected towards another randomly selected node. Fgures 23 and 24 report optmal cost wth ncreasng number of for all the mplemented forwardng schemes, wth α = 3 and α = 4, respectvely. For hgh values of α the optmal cost decreases as the node densty ncreases. Conversely, for low values of α the ncreased traffc overcomes the postve effect of a hgher node densty. As the number of becomes hgher, the cost of nformaton and the optmal KRs ncrease wth the overall effect of decreasng the optmal cost. Agan, n all the experments performed n Scenaro 3, s shown to perform better than any other scheme, whle more greedy schemes, such as and, are shown to lead to hgher energy consumptons. VII. CONCLUSIONS We dscussed how to determne optmal local topology knowledge for energy effcent geographcal routng n ad hoc and sensor networks. We provded an Integer Lnear Programmng Formulaton of the problem whch consttutes a framework for the analyss of the energy effcency of dfferent forwardng schemes. We ntroduced a new localzed forwardng scheme for geographcal routng,, and a dstrbuted protocol for onlne knowledge range adjustment, PRADA. s shown to outperform exstng greedy forwardng schemes and PRADA s shown to lead to near-optmal energy consumpton. Furthermore, we demonstrated that only a lmted local topology knowledge s needed to take energy effcent routng decsons. Future research wll nclude the extenson of the model, prmarly to nclude features such as battery and bandwdth constrants for the.

12 JOURNAL OF SELECTED AREAS IN COMMUNICATIONS 2 REFERENCES [] I. Akyldz, W. Su, Y. Sankarasubramanam, E. Cayrc, Wreless Sensor Networks: A Survey, Computer Networks (Elsever) Journal, pp , March 22. [2] Z.J. Haas, Desgn Methodologes for Adaptve and Multmeda Networks, IEEE Communcatons Magazne, Vol. 39, no., November 2, pp 6-7. [3] D. Estrn, R. Govndan, J. Hedemann, S. Kumar, Next Century Challenges: Scalable Coordnaton n Sensor Networks, Proc. IEEE/ACM Mobcom 999, Seattle, pp [4] V. Rodoplu, T. Meng, Mnmum Energy Moble Wreless Networks, IEEE Journal of Selected Aresas In Communcatons, Vol. 7, no. 8, August 999, pp [5] S.M. Woo, S. Sngh, C. S. Raghavendra, Power-Aware Routng n Moble Ad Hoc Networks, Proc. of IEEE/ACM Mobcom 998, pp [6] J. Chang, L. Tassulas, Energy Conservng Routng n Wreless Ad-hoc Networks,, Proc. of IEEE Infocom 2, pp [7] J. L, J. Jannott, D. De Couto, D. Karger, R. Morrs, A Scalable Locaton Servce for Geographc Ad Hoc Routng, Proc. IEEE/ACM Mobcom 2, pp [8] R. 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Urruta, routng on geometrc networks, Proceedngs of the th Canadan Conference on Computatonal Geometry, Vancouver, Canada, August 999. [6] R. Nelson, L. Klenrock, The spatal capacty of a slotted ALOHA multhop packet rado network wth capture, IEEE Transactons on Communcatons, Vol. 32, no. 6, 984, pp [7] I. Stojmenovc, X. Ln, Loop-Free Hybrd Sngle-Path/Floodng Routng Algorthms wth Guaranteed Delvery for Wreless Networks, IEEE Transactons on Parallel and Dstrbuted Systems, Vol. 2, no., 2, pp [8] S. Gordano, I. Stojmenovc, L. Blazevc, Poston Based Routng Algorthms for Ad Hoc Networks: A Taxonomy, Ad Hoc Wreless Networkng, Kluwer, 23 [9] I. Stojmenovc, Locaton Updates for Effcent Routng n ad hoc network, Handbook of Wreless Networks and Moble Computng, Wley, 22, [2] P. Bose, P. Morn, I. Stojmenovc, J. Urruta, Routng wth guaranteed delvery n ad hoc wreless networks, ACM Wreless Networks, 7,6, November 2, pp [2] D. Nculescu, B. Nath, Trajectory based forwardng and ts applcatons, Tech. Rep. 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W. Kernghan AMPL: A Modelng Language for Mathematcal Programmng, Duxbury Press / Brooks/Cole Publshng Company, 22. [29] Tommaso Meloda receved the Laurea degree n Telecommuncatons Engneerng from the Unversty La Sapenza, Rome, Italy, n 2. He then worked on a natonal research project on Moble Networkng and Wreless Personal Area Networks at the same Unversty. He s currently pursung hs Ph.D. and workng as a research assstant at the Broadband and Wreless Networkng Laboratory, Georga Insttute of Technology, Atlanta. Hs man research nterests are n Wreless Ad Hoc and Sensor Networks, Wreless Sensor and Actor Networks, Underwater Acoustc Sensor Networks, Personal and Moble Communcatons. Daro Pompl receved the Laurea degree n Telecommuncatons Engneerng n 2, magna cum laude, from the Unversty of Rome La Sapenza. From June 2 he has been workng at the same unversty on the European Unon IST Brahms and Satp6 projects. In 23 he worked on Sensor Networks at the Broadband and Wreless Networkng Laboratory, Georga Insttute of Technology, Atlanta, as a vstng researcher. Currently he s pursung the Ph.D. degree n Electrcal Engneerng at the Georga Insttute of Technology. Hs man research nterests are n Wreless Sensor Networks and Underwater Acoustc Sensor Networks. Ian F. Akyldz s the Ken Byers Dstngushed Char Professor wth the School of Electrcal and Computer Engneerng, Georga Insttute of Technology and Drector of Broadband and Wreless Networkng Laboratory. He s the Edtor-n-Chef of Computer Networks (Elsever) and Ad Hoc Networks (Elsever) Journal. Dr. Akyldz s an IEEE FELLOW (995), an ACM FELLOW (996). He served as a Natonal Lecturer for ACM from 989 untl 998 and receved the ACM Outstandng Dstngushed Lecturer Award for 994. Dr. Akyldz receved the 997 IEEE Leonard G. Abraham Prze award (IEEE Communcatons Socety) for hs paper enttled Multmeda Group Synchronzaton Protocols for Integrated Servces Archtectures publshed n the IEEE Journal of Selected Areas n Communcatons (JSAC) n January 996. Dr. Akyldz receved the 22 IEEE Harry M. Goode Memoral award (IEEE Computer Socety) wth the ctaton for sgnfcant and poneerng contrbutons to advanced archtectures and protocols for wreless and satellte networkng. Dr. Akyldz receved the 23 IEEE Best Tutoral Award (IEEE Communcaton Socety) for hs paper enttled A Survey on Sensor Networks, publshed n IEEE Communcaton Magazne, n August 22. Dr. Akyldz receved the 23 ACM SIGMOBILE award for hs sgnfcant contrbutons to moble computng and wreless networkng. Hs current research nterests are n Sensor Networks, InterPlaNetary Internet, Wreless Networks and Satellte Networks.