Robust position-based routing in wireless ad hoc networks with irregular transmission ranges

Transcription

1 WIELESS COMMUNICATIONS AND MOBILE COMPUTING Wiel. Commun. Mob. Comput. 2003; 3: (DOI: /wcm.108) obust position-based outing in wieless ad hoc netwoks with iegula tansmission anges Lali Baièe Depatament de Matemàtica Aplicada i Telemàtica Univesitat Politècnica de Catalunya Bacelona, Spain Piee Faigniaud*, CNS, Laboatoie de echeche en Infomatique Univesité Pais-Sud Osay, Fance Lata Naayanan and Jaoslav Opatny Depatment of Compute Science Concodia Univesity Monteal, Canada Summay Seveal papes consideed the poblem of outing in ad hoc wieless netwoks using the positions of the mobile hosts. Peimete outing [1,2] gives an algoithm that guaantees delivey of messages in such netwoks without the use of flooding of contol packets. Howeve, this potocol is likely to fail if the tansmission anges of the mobile hosts vay because of natual o man-made obstacles. It may fail because eithe some connections ae not consideed, which effectively esults in a disconnection of the netwok, o because some cossing connections ae used, which could misdiect the message. In this pape, we descibe a obust outing potocol, a vaiant of peimete outing, which toleates up to 40% of vaiation in the tansmission anges of the mobile hosts. Moe pecisely, ou potocol guaantees message delivey in a connected ad hoc wieless netwok without the use of message flooding wheneve the atio of the maximum tansmission ange to the minimum tansmission ange is at most p 2. Copyight 2003 John Wiley & Sons, Ltd. KEY WODS wieless netwoks ad hoc netwoks MANET message outing Ł Coespondence to: Piee Faigniaud, CNS, Laboatoie de echeche en Infomatique, Univesité Pais-Sud, Osay, Fance. Piee@li.f A peliminay vesion of this pape appeaed in the poceedings of the 5th Int. Wokshop on Discete Algoithms and Methods fo Mobile Computing and Communications (DIALM), July Copyight 2003 John Wiley & Sons, Ltd.

2 142 L. BAIÈE ET AL. 1. Intoduction A wieless mobile ad hoc netwok, often abbeviated as MANET, is a netwok consisting of mobile hosts that is established as needed, typically without any assistance fom a fixed infastuctue. Using wieless boadcasts, each mobile host can communicate with othe hosts within its tansmission ange. Howeve, we allow the possibility that not all hosts ae within the tansmission ange of each othe. Thus, communication between two hosts in the netwok is in geneal achieved by multihop outing, in which intemediate nodes coopeate by fowading packets. Host mobility means that the topology of the netwok can change with time and is unstuctued. Futhemoe, no assumption can be made about the initial topology of the netwok. As a consequence, mobile hosts have to build and update thei outing tables automatically. In most cases, it is assumed that the mobile hosts have the same tansmission ange. The standad model fo a MANET epesents the mobile hosts spead out in some envionment by a set of points in the Euclidean plane. The tansmission ange specifies the (Euclidean) distance between two mobile hosts up to which they ae able to diectly communicate. A MANET can thus be epesented by a unit disk gaph: a host is epesented by a node in the plane detemined by the geometic coodinates of the host and two nodes ae adjacent if and only if they ae at an Euclidean distance of at most. See Figue 1 fo an example of a unit disk epesentation of a MANET in which the common tansmission ange is indicated by a cicle aound one of the hosts. In the last few yeas, a lage numbe of outing potocols have been poposed fo MANETs [3 6]. Since no assumption can be made about the location of the hosts in the netwok, most of these outing potocols use a type of flooding of the netwok by contol packets to obtain infomation about the netwok topology and to guaantee the delivey of messages. Howeve, flooding can use up significant amounts of scace wieless bandwidth, since, owing to the mobility of hosts, topology infomation must be continuously updated. To educe the amount of Fig. 1. A unit disk gaph epesentation of a MANET. contol taffic in MANETs, seveal authos [2,7 9] have ecently poposed the use of host location infomation. In this case, evey node is assumed to know its own location, fo example, by using the global positioning system (GPS), and each node knows the location of the destination host, that is, the host to which it is sending a message. We assume in this pape that each node is awae of its own position in the plane given by its x- andy-coodinates. Mobile hosts ae hencefoth identified by thei coodinates in the plane. Ko and Vaidya [9] essentially use the Dynamic Souce outing (DS) potocol [3], but suggest that a node fowads a packet to a node only if it is in a equest zone that is likely to contain a path to the desied destination. The distance outing effect fo mobility (DEAM) potocol [7] uses a limited flooding of data packets. Stojmenovic and Lin [10] poposed to guaantee delivey by following the geedy outing and flooding all neighbos by nodes that have no close neighbos to advance the message, followed by a subsequent ejection of these nodes to paticipate futhe in the delivey of the same message. They show that the flooding ate of this method is still lowe than that of a location-aided outing (LA) method [9] that also may ceate loops (like any othe diection-based appoach) and does not guaantee delivey. On the othe hand, puely geedy outing [11] completely stays away fom the flooding paadigm and at evey step, a node fowads a packet to exactly one of its neighbos. Fo instance, Geogaphic distance outing [12] constucts a sequence of mobile hosts w 0,w 1,...,w k whee w 0 D u and w k D v,and w ic1 is selected by w i as its neighbo closest to the destination v. Obviously, thee ae seveal scenaios in which geedy outing fails. Fo instance, if the mobile host w i is close to the destination than any of its neighbos, o if two adjacent mobile hosts w i 1 and w i ae equally close to the destination, and none of thei othe neighbos is close, then a livelock esults. An altenative to geedy outing is compass outing [13], also called diectional outing, fo which seveal impoved vaiants have been poposed in [7,14]. In compass outing, the next node w ic1 is chosen among w i s neighbos to minimize the angle between segments w i v and w i w ic1. As fo geedy outing, thee ae known configuations in MANETs in which compass outing fails. Combining compass and geedy outing fails in geneal, although thee ae lage classes of netwoks fo which this combination

3 OBUST POSITION-BASED OUTING 143 of the two outing stategies ensues message delivey [8,15]. To ovecome the failue of all these outing stategies, Bose et al. [1] descibed a outing algoithm that guaantees delivey of messages in MANETs. (The tansfomation of this algoithm into a potocol named Geedy Peimete Stateless outing (GPS) was late pesented by Kap and Kung [2].) In ode to ensue delivey, the algoithm uses peimete outing in the so-called Gabiel gaph of the netwok [16]. The gabiel gaph is a spanning subgaph of the oiginal netwok. It is defined as follows: given any two adjacent nodes u and v in the netwok (i.e. two nodes at a distance of at most ), the edge fu, vg belongs to the gabiel gaph if no othe node w of the netwok is located in the disk with the segment u, v as its diamete (see Figue 2a) and the edge is emoved othewise (see Figue 2b). It is known that the gabiel gaph is plana and connected if the undelying gaph is a unit disk gaph [17]. Once the gabiel gaph is extacted fom the netwok, outing is pefomed along its edges. Its planaity and connectivity ensue message delivey by outing along the faces of the gaph (e.g. by using the ight-hand ule) that intesect the line between the souce and the destination. A ough idea of peimete outing is given in Figue 3. To impove the efficiency of the algoithm in tems of outing pefomance, peimete outing can be used in combination with geedy outing [1]. outing of a message is in the geedy state as long as a mobile host can find a neighbo close than itself to the destination, othewise the outing is switched to the peimete state. A message in the peimete u (a) v Fig. 2. Gabiel gaph constuction. The edge fu, vg is etained in (a) but not in (b). state is outed aound the faces of the gabiel gaph. As fa as the message delivey is concened, one could leave the message in this state until it eaches its destination. Howeve, in most cases it is moe efficient to move the message back to the geedy state once it has eached the othe extemity of the face, if possible (see Figue 4). Although the algoithm in [1] is the fist-known distibuted potocol ensuing message delivey in MANETs without the use of flooding [9,12], it is likely to fail if thee is some iegulaity in the tansmission anges of the mobile hosts. By iegulaity of tansmission anges, we mean that the aea a mobile host can each is not necessaily a disk and the ange can vay in a nonunifom manne between D 1 ε and, ε>0 as shown in Figue 5 whee the actual tansmission ange of u is indicated by the shaded aea. In such cases of iegulaity, two mobile hosts at distance d, whee <d, may o may not be able to communicate diectly. This situation commonly occus if thee ae obstacles such as buildings u (b) w v Destination Souce Fig. 3. Peimete outing.

4 144 L. BAIÈE ET AL. Destination Geedy outing Peimete outing Geedy outing Souce Fig. 4. Altenation of geedy and peimete outing. in a netwok by equiing that a communication link fom the host u to the host v is valid only when an acknowledgment fom v is eceived by u. Even when only bidiectional communications ae assumed, the potocols in [1,2] may fail because of a slight vaiation of the anges. The main eason fo this is that the gabiel gaph that is used fo outing is not a obust stuctue. Fo instance, Figue 6 shows a case in which the vetices u and v might not agee on whethe to etain the edge fu, vg in the coesponding Gabiel gaph. Accoding to u, the host w is in the disk with the segment u, v as diamete and thus the edge fu, vg should not be in the Gabiel gaph. Howeve, since the distance between v and w is geate than, host v might not be able to communicate with w and thus it does not see w as being in the disk. Thus, v would etain the edge fu, vg in the Gabiel gaph. Since the outing decisions in MANETs ae done in a distibuted manne, peimete outing would not be consistent any moe and it might fail. emoving the edge fu, vg may disconnect the netwok, but keeping the edge may ceate a nonplana gaph (see Figue 7), and the planaity of the Gabiel gaph is cucial fo peimete outing. In this pape, we pesent a distibuted outing potocol fo a MANET with iegula tansmission anges and pove its coectness. Moe pecisely, ou potocol ensues message delivey in a connected u v Fig. 5. The tansmission ange of the mobile host u vaies between and. o mountains in the teain, which distub the adio tansmissions. Petubations can also be caused by bad weathe conditions (heavy ain, snow stom, etc.) o othe unelated adio tansmissions. Even a slight vaiation of the anges with ε abitaily small can cause subtle poblems fo outing. Fo example, the vaiation in tansmission ange may ceate some unidiectional communication links that would not exist if all hosts had the same unifom tansmission ange. Bidiectional communication links can be enfoced u Fig. 6. Nodes u and v do not agee whethe to etain the edge fu, vg in the gabiel gaph. w

5 OBUST POSITION-BASED OUTING 145 Section 4. Finally Section 5 contains some concluding emaks. w u v Fig. 7. Example of a possible failue in the Gabiel gaph constuction in a netwok with iegula anges. netwok wheneve the atio of the maximum tansmission ange to the minimum tansmission ange is at most p 2. This estiction guaantees that fo any two hosts u and v, if thee is anothe host w in the disk aound the segment u, v, then at least one of u and v will be awae of the existence of w. We note that the existence of a distibuted potocol that ensues message delivey if / is unbounded is unlikely since allowing abitay values of and foces the consideation of gaphs in which nodes ae not connected to othe nodes at a shot distance fom them while being connected to nodes futhe away (we discuss this poblem futhe in Section 4). Like the potocols in [1,2], ou potocol may fail in a totally asynchonous envionment, fo example, if a mobile host is abitaily slow while executing the potocol and makes its neighbos awae of its existence only vey late in the unning of the potocol. On the othe hand, ou potocol does not equie full synchonization between the mobile hosts, and a ealtime envionment in which evey atomic opeation has a bounded execution time is enough to ensue the validity of the potocol. The pape is oganized as follows. Section 2 defines the model of MANET we use, intoduces some notation and pesents ou outing potocol. Section 3 poves the coectness of the potocol. Discussion of the pefomance of the outing potocol is in w 2. The Netwok Model and the outing Scheme We model a MANET as follows. A set of mobile hosts spead out in an envionment is modeled by a Euclidean plane, each mobile host with x- and y-coodinates is epesented by point x, y. Alldistances ae Euclidean distances in the plane. We will use the following hypotheses and notation: ž Any mobile host knows the coodinates x, y of its position. ž The minimum tansmission ange is, that is,any two mobile hosts at distance d ae able to communicate diectly. ž The maximum tansmission ange is, that is, no pai of mobile hosts at distance d> can communicate diectly. ž Two mobile hosts at distance d, with <d, may o may not be able to communicate diectly. ž Communication links ae bidiectional, that is,if a mobile host u is able to eceive signals fom a mobile host v, thenv is also able to eceive signals fom u. (This can be enfoced as mentioned in the intoduction). The atio / of the maximum ange to the minimum ange influences the efficiency of ou potocol, but it is designed to wok fo all possible atios up to p 2. The netwok is epesented by a geometic undiected gaph, G D V, E, with vetices epesenting mobile hosts, and an edge connecting any pai of hosts that can communicate diectly. The set of vetices V is thus a set of points in the Euclidean plane. Let d u, v be the Euclidean distance between the points u and v in the plane. The set of edges E satisfies ffu, vg : u, v 2 V, d u, v g E and E ffu, vg : u, v 2 V, d u, v g,thatis,e contains all the pais of mobile hosts at a distance of at most, andsome subsets of the pais at a distance between and. We define the ball with cente u and adius, denoted B u,, to be the set of points in the plane at a distance of at most fom point u. Thedisk detemined by the two points u and v in the plane, denoted by D u,v, is defined to be the set of points in the cicle with the segment u, v as diamete. We assume that the tansmission conditions do not vay apidly with time compaed to the speed of electonic communications. This implies that the netwok

6 146 L. BAIÈE ET AL. may change, fo example, because of a change in the weathe conditions, but it is at a timescale that allows an easy esetting of the netwok. Thus, we assume fom now on that the connections between mobile hosts ae fixed. Howeve, the stuctue of these connections and the exact shape of the tansmission anges ae not known, and it is the ole of ou potocol to ensue message delivey in this unknown netwok. Initially, evey mobile host is tuned on and becomes awae of its position in the plane (e.g. by the use of GPS). The only knowledge initially given to a mobile host is its x- andy-coodinates in the plane. Ou outing potocol is a scheme consisting of thee phases. These phases ae called the completion phase, the extaction phase and the outing phase. The aim of the completion phase is to add edges, called vitual edges to the physical gaph G, and hence to etun a supegaph S G of G. This adding of vitual edges guaantees that the gaph esulting fom the extaction phase is connected and plana. Once the completion phase is done, the extaction phase is executed, and it etuns a connected and plana spanning subgaph of S G. Finally, the outing phase pefoms message delivey between mobile hosts. All computations in all phases ae done locally at hosts and do not equie any cental contolle. The completion phase is a new contibution, and is descibed in detail. The extaction phase is oughly the same as the gabiel Gaph constuction of [1], and theefoe its desciption is only sketched. Only the modifications induced by the vitual edges ae descibed in detail, as they ae oiginal. The outing phase is basically the same as the outing phase in [1,2] and it altenates geedy geogaphic distance outing (GEDI) [12] with peimete outing, that is, outing aound the faces of a plana gaph using the ight-hand ule. Thus, we include only a bief discussion of the outing phase Completion Phase Initially, evey mobile host u ceates the empty adjacency list L u of its neighbos and boadcasts a equest fo the coodinates of its neighbos, that is, the hosts with which u can communicate diectly. Evey mobile host v eceiving this equest immediately esponds to u with its own coodinates. The nodes coesponding to the mobile hosts, and the edges between a mobile host and its neighbos fom the gaph G. The mobile host u then adds to L u each of its neighbos in G. It maks any node added to L u as unpocessed. Fo evey unpocessed v 2 Ł u, u stats pocessing the edge fu, vg as follows. Pocessing of fu, vg by u. Fo evey w 2 L u,ifw 2 D u,v and w/2 B v, then 1 ž u sends the message (new,w) to v, togethe with the coodinates of w; ž u sends the message (new,v) to w, togethe with the coodinates of v; Then u maks v pocessed. End-Pocessing. In paallel with the pocessing of its incident edges, evey node is eady to eceive the new nodes coodinates fom its neighbos, and to update its adjacency list L. Updating the adjacency list of u. Upon eceipt of a message (new,w) fom a neighbo v, u checks whethe w 2 L u. If not, then u adds w to L u as an unpocessed vetex. The edge fu, wg is then set up as a vitual edge, andw is a vitual neighbo of u. Node u stoes the path ve u, w D u,v, w coesponding to the vitual edge fu, wg. End-Updating. Infomally, the edge fu, wg is set up as a vitual edge if w is a node in the disk D u,v fo some edge fu, vg of G o a vitual edge fu, vg of S G. Thus the Euclidian distance between w and u is less than the Euclidian distance between u and v. Since initially the adjacency list of any node contains edges of G of length at most, the Euclidian distance of w fom u is less than. Thus the edge fu, wg should exist in case the tansmission anges of u and w ae egula balls of diamete. The fact that any vitual edge of S G is connecting nodes at an Euclidian distance less than will be used late. Clealy, the pocessing of edges involves sending messages to vitual neighbos, that is, to destinations that ae not diectly in the tansmission ange of the souce. This equies a simple outing potocol called vitual outing to distinguish it fom the outing potocol esulting fom ou thee-phase scheme. We descibe heeafte the sending and the fowading of messages in the vitual outing potocol: Sending a message. 1 This condition is easy to check since, if u D u 1,u 2 and v D v 1, v 2,thenD u,v is the set of points x, y satisfying x 2 x u 1 C v 1 C u 1 v 1 C y 2 y u 2 C v 2 C u 2 v 2 0, and B v, is the set of points x, y satisfying x v 1 2 C y v

7 OBUST POSITION-BASED OUTING 147 If node u has to send a message M to a (possibly vitual) neighbo v, then u calls send v,m, which poceeds as follows: ž If v is a neighbo of u in G, then the message v,m is tansmitted to v; ž Othewise, v is a vitual neighbo of u, and, accoding to ve u, v D u, w, v, u pefoms send w, v,m. End-Sending. Note that if w tuns out to be a vitual neighbo of u, then u epeats the pocess, and actually sends M to w 0 whee ve u, w D u, w 0,w. Eventually, the message will be sent though a eal edge of G incident on u. Fowading a message. Upon eceipt of a message u, M fom a neighbo, node u poceeds as follows: ž If M D v,m 0, v 6D u, then u pefoms send v,m 0 ; ž Othewise u stoes M; End-Fowading. The gaph obtained in the completion phase by the above pocess of adding vitual edges to G is denoted by S G. At the end of the completion phase, evey node knows the coodinates of its neighbos in S G, and knows a path in G coesponding to each of its incident vitual edges. Note that S G fomally exists only afte all nodes have teminated the completion phase, that is, afte all nodes lean about all new (vitual) neighbos and afte the coesponding (vitual) edges have been pocessed. When a node u has completed the pocessing of all its edges (vitual o not), that is, when all nodes in L u ae maked as pocessed, it entes the extaction phase. Note that u may keep on switching between the extaction phase and the completion phase in the sense that u may lean about the existence of a new vitual neighbo v afte it has aleady enteed the extaction phase. In that case, u stops the extaction phase and comes back to the completion phase to pocess the edge fu, vg. We will pove late that this does not affect the coectness of ou potocol. One could ague that the knowledge of the coodinates of nodes within adius of a given node u could be sufficient and could be somehow obtained using GPS alone. Howeve, the knowledge of the existence of v in the neighbohood of u, without knowledge of a path between u and v is not sufficient fo ou puposes. Ceating vitual edges, in the manne descibed above, enables the constuction of these paths Extaction Phase The aim of the extaction phase is to extact a connected plana spanning subgaph of S G. This is pefomed by extacting the gabiel gaph of S G, denoted by GG S G. The vetex set of the gabiel gaph is V, that is, the set of nodes. An edge fu, vg of S G, possibly vitual, belongs to GG S G if and only if thee is no othe node w 2 D u,v. As stated ealie, if H is a unit gaph then GG H is a connected plana spanning subgaph of H. We show late that if G is connected then GG S G is a connected plana spanning subgaph of S G. Moe pecisely, in the extaction phase evey node u poceeds as follows. Let L 0 u be initially equal to L u, the list of neighbos and vitual neighbos of u in G. Fo evey v 2 L 0 u, u detemines whethe fu, vg belongs to GG S G as follows. Validating fu, vg by u. Node v is emoved fom the list L 0 u. Ifthee exists w 2 L u \ D u,v, then u deletes the edge fu, vg in the sense that v is not a neighbo of u (vitual o othewise) in GG S G. Othewise the edge fu, vg is kept in GG S G. End-Validating. Once L 0 u is empty, u has completed the extaction phase, and is eady to oute messages to othe hosts in the netwok (i.e. the messages that ae not used to set up the system duing the completion phase). It then entes the outing phase, which is in fact the steady state of the potocol. Howeve, if at any time u becomes awae of a new neighbo, it comes back to the completion phase, and fom thee to the extaction phase. We will show that this does not affect the coectness of the potocol outing Phase Once the completion and extaction phases ae completed and GG S G is set up, outing in G is pefomed accoding to the stategy descibed in [1], that is, a combination of the geedy geogaphic distance outing (GEDI) [12] in G and peimete outing in GG S G is applied. Of couse, peimete outing in GG S G could be used alone. Geedy outing geneally pefoms faste than peimete outing, but

8 148 L. BAIÈE ET AL. it may fail at some nodes wheeas peimete outing ensues message delivey. It is impotant to note that peimete outing equies outing though vitual edges. This outing is pefomed using the vitual outing potocol descibed in the completion phase. As pointed out befoe, the lack of synchonization between the nodes may cause a node u to ente the outing phase, and to lean late about the existence of a (vitual) neighbo v. Thenu comes back to the completion phase to pocess the edge fu, vg. Onceu completes the completion phase again, it will e-ente the extaction phase, and eventually the outing phase again. The discovey of new neighbos may happen seveal times, and u may go back and foth between the seveal phases until it eventually stabilizes in the outing phase (no new neighbo emains to be discoveed). If node u comes back to the completion phase, it caies on outing accoding to its possibly wong outing table. As we will see, this may cause a tempoay misdiection of a message, but the outing will eventually succeed. In fact, as will be discussed in Section 4, fequent altenations between the diffeent phases of ou potocol do not occu often, and can be caused only by some specific placements of mobile hosts that ae unlikely to occu in pactice. 3. Poof of Coectness The coectness of the outing potocol altenating geedy outing with peimete outing wheneve the undelying gaph used fo peimete outing is plana and connected has been shown in [1,2]. Theefoe, in this section, we pove the planaity and the connectivity of the gaph GG S G, and the temination of ou 3-phase scheme. Fist, we need to show that the constuction of S G teminates in a valid state. emak. We assumed bidiectional communication links as a pat of ou model specification. Thus, if u knows its neighbo v in the oiginal netwok G, thenv knows u as well. Lemma 1 If the vetex u leans about a vitual neighbo v, thenv eventually leans about the existence of u. Futhemoe, the vitual outing potocol ensues the coect delivey of messages though vitual edges. Poof: A node u leans about the existence of a new vitual neighbo v duing the pocessing of an edge e by a node diffeent fom u and v. Thee ae two cases coesponding to the two messages in the pocedue pocessing: eithe e Dfu, wg, v 2 D u,w, with w pocessing e, oe Dfv,wg, u 2 D v,w,with w pocessing e. In both cases, w will send to u the coodinates of v and to v the coodinates of u. It emains to be checked whethe these coodinates ae eventually eceived by u and v, that is, that the vitual outing potocol ensues delivey of these coodinates. By constuction, evey vitual edge of S G has two paths in the netwok G associated with it, one path fo evey end point. Indeed, the paths associated with a nonvitual edge fu, vg ae P u,v D u, v, and P v,u D v,u. Moeove, the paths associated with a vitual edge fu, vg ae ž P u,v D P u,w jp w,v whee w is the node fom which u leant about v, and j is the concatenation of two paths; and ž P v,u D P v,w 0jP w 0,u whee w 0 is the node fom which v leant about u. Note that P u,v and P v,u may not be symmetic as the actual composition of a path depends on who is the fist neighbo to infom u and v, espectively. Next, we show that the vitual outing potocol outes a message M fom u to v along P u,v. The poof is by induction on the length l of the path P u,v. If l D 1thenfu, vg is an edge in the oiginal gaph G, and the pocedue send tansmits the message M fom u to v along fu, vg. Assume now that l>1. The edge fu, vg is a vitual edge. The node u has stoed ve u, v D u, w, v, and calls send w, v,m. By the induction hypothesis, the message v,m will eventually each node w. Thee the pocedue fowading is executed, that is, w will pefom send v,m. Again, by the induction hypothesis, the message v,m will eventually each the node v in which it will be stoed. This completes the poof of coectness of the vitual outing potocol, and theefoe of the lemma. Lemma 1 has the following simple consequence. Lemma 2 The completion phase teminates, that is, all nodes eventually complete the completion phase, and neve come back to it. Poof: Evey node u pocesses at most k u edges whee k u is the numbe of nodes contained in B u,. The pocessing of an edge consists of sending a finite numbe of messages to a finite numbe of nodes, and

9 OBUST POSITION-BASED OUTING 149 will theefoe teminate. Lemma 1 showed that the outing of these messages will also teminate. Finally, once an edge has been pocessed, it does not need to be pocessed again. Lemma 1 shows that if a node u has to make a decision about whethe the (possibly vitual) edge fu, vg must be emoved fom S G, thenv eventually has to make a decision as well. The next lemma shows that u and v will make the same decision. Lemma 3 If u decides to emove the (possibly vitual) edge fu, vg duing the extaction phase, then v eventually emoves fu, vg as well. Poof: By definition of the gabiel gaph constuction, if u decides to emove the edge fu, vg, then thee is a neighbo w of u in D u,v.now,sinceu is cuently in the extaction phase, the edge fu, vg has aleady been pocessed duing the completion phase. Duing that pocessing, u sent the coodinates of w to v. Fom Lemma 1, these coodinates eventually each v, and hence v eventually decides to emove fu, vg fom S G. It is impotant to note that because of the asynchonous natue of the algoithm, it might happen that v peviously decided to keep fu, vg in S G because v was not yet awae of the existence of w when it consideed fu, vg. Howeve, ou potocol ensues that as soon as v becomes awae of w (and this will eventually occu) it stops the extaction phase (even possibly the outing phase) and etuns to the completion phase to pocess fv,wg. Late, v will eente the extaction phase, and will econside whethe the edge fu, vg should be emoved fom GG S G.Since w is known to v at that point, v will decide to emove the edge fu, vg fom GG S G. Lemma 4 The extaction phase teminates, that is, all nodes eventually complete the extaction phase, and neve come back to it. Poof: The extaction phase consists of taking a local decision about which edges should be deleted. This local decision pocess always teminates. We may ente the extaction phase seveal times because of the discovey of new neighbos. Howeve, fom Lemma 2, thee is a time at which evey node finishes the completion phase and will not come back to it any moe. The extaction phase is then pefomed fo the last time and is not enteed any moe. Lemma 5 If G is connected and / p 2, then the gaph obtained afte the temination of the extaction phase is plana and connected. In othe wods, GG S G is plana and connected. Poof: Assume that GG S G is not plana, and let fu, vg and fu 0, v 0 g be two cossing edges in GG S G. Conside the paallelogam fomed by the nodes u, u 0, v, v 0. Clealy, at least one of the angles in the paallelogam is at least /2. We may assume, without loss of geneality, that the angle u, u 0, v ½ /2. Then u 0 is in D u,v.ifd u, u 0 o d v,u 0 then u 0 would be a neighbo of at least one of the two nodes u and v, and theefoe, fom Lemma 3, the edge fu, vg would have been emoved fom GG S G. Theefoe d u, u 0 > and d v,u 0 >. Since u 0 2 D u,v, we get d u, v > p 2 Ð ½. This contadicts the fact that any edge of G o any vitual edge of S G constucted in the completion phase is of a length at most. Hence GG S G is plana. S G is connected since G is connected and S G is a supegaph of G. By Lemma 3, two neighboing nodes u and v make the same decision on whethe the edge fu, vg of S G is kept in GG S G. Theefoe, we can focus on the local connectivity of a node with its neighbos S G. InextactionofGG S G, an edge fu, vg is emoved fom S G by u if thee is a node w in D u,v such that u is awae of its pesence, that is, fu, wg 2E S G. Node u infomed v of w s coodinates when it pocessed fu, vg. Theefoe v is also awae of the pesence of w and u, w, v is an altenative path fom u to v. Now, two poblems may occu: (1) It may happen that w emoves the edge fw, vg fom GG S G because thee is a node w 0 in D w,v. Howeve, since w 2 D u,v, w 0 6D u, and hence the altenative path fo w, v does not use the edge fu, vg. (2) It may also happen that u emoves the edge fu, wg because of w 00 in D u,w. Howeve, since w 2 D u,v, w 00 6D v, and hence the altenative path fo u, w does not use the edge fu, vg. Theefoe, fo evey edge emoved fom S G, thee is an altenative path in GG S G joining its two endpoints. Thus, GG S G is connected wheneve S G is connected.

10 150 L. BAIÈE ET AL. Theoem 1 If / p 2 then the 3-phase outing scheme poduces a outing potocol that allows delivey of messages fom any souce to any destination in any connected wieless ad hoc netwok. Poof: Lemma 4 shows that all nodes eventually ente the outing phase and emain in it. Moeove, Lemma 5 shows that when nodes ae all in the outing phase, they agee on a gaph GG S G, which is plana and connected. Ou outing potocol applies peimete outing to GG S G when geedy outing fails. It is known that peimete outing ensues delivey of messages in a plana and connected gaph. By Lemma 1, message delivey is ensued along each edge of GG S G. Theefoe, once the outing phase is enteed by all nodes, ou whole outing potocol ensues delivey of messages between any pai of nodes in the netwok. 4. Discussion This section discusses the pefomance of ou 3-phase outing scheme. Ou fist esult concens the poblem of outing along vitual edges. It shows that the oute coesponding to a vitual edge cannot be abitaily lage, povided that thee is a lowe bound on the minimum distance between any two mobile hosts. Popety 1 [Good news] Let be the minimum Euclidean distance between any two mobile hosts. If / p 2, then the length of the oute in G coesponding to a vitual edge in S G is at most 1 C Poof: Fist we show that if fu, vg is a vitual edge in S G, then the length of the oute in G between u and v is at most 1 C 2 d 2 u, v / 2. This inequality diectly implies the uppe bound in Popety 1 since if fu, vg is a vitual edge, then d u, v >. The poof is by induction on the length l e of the oute. Let e Dfu, vg with l e D 2. We have e D u, w, v, fo some w, with eithe u 2 D v,w o v 2 D u,w. Assume u 2 D v,w.sinced u, w ½, we get that d 2 u, v d 2 v,w 2. Theefoe, 2 d 2 u, v / 2 ½ 1 as claimed. The case v 2 D u,w is simila. Let e Dfu, vg with l e D k>2. We have e D u, w, v, fosomew, with eithe u 2 D v,w o v 2 D u,w. Moeove, since / p 2, if u 2 D v,w then fu, wg is an edge of G and l fv,wg D k 1, othewise (i.e. v 2 D u,w ), fv,wg is an edge of G and l fu, wg D k 1. Assume that u 2 D v,w. By the induction hypothesis, we have k 1 1 C 2 d 2 v, w / 2. Moeove, d 2 v,w ½ d 2 u, v C 2. Theefoe k 1 C 2 d 2 u, v / 2 as claimed. The case v 2 D u,w is simila. The above popety implies that the values of, and can be used to find an uppe bound on the numbe of ounds a node might need to e-ente the completion phase. It is known that unit disk gaphs with a estiction on the minimum distance between two nodes have nice algoithmic popeties [18]. This also seems to be the case as fa as efficient communications in wieless ad hoc netwoks ae concened. Indeed, the following is a diect coollay of Popety 1. Coollay 1 If / D 1 C ε, 0 <ε p 2 1, then placing the mobile hosts so that no two mobile hosts ae at distance less than p ε Ð ensues that the length of the oute coesponding to a vitual edge is at most 4. Fo instance, if the vaiation in tansmission anges is at most 10%, then placing the mobile hosts at a mutual distance of at least /3 ensues that the length of the oute coesponding to a vitual edge is at most 4. Howeve, thee ae cases in which ou thee-phase scheme geneates a high volume of communication, and in which thee will be no benefit compaed to flooding. Such a bad case can be summaized as follows: Popety 2 [Bad news] Fo any k>0, and any atio / p 2, thee is a configuation of the mobile hosts fo which the path coesponding to a vitual edge is ½ k Poof: The bad configuation is depicted on Figue 8. Thee ae two sets of nodes, half of them on the left line and the othe half on the ight line. Each of the nodes on the left line is at distance lage than fom any of the nodes on the ight line, and they do not see each othe, except fo the nodes x and y, which ae connected by the eal edge a of G. The pocessing of a by x yields the vitual edge b. The pocessing of b by y yields the vitual edge c. The pocessing of the edge e yields a vitual edge f, which yields a vitual edge g, and so on. The pocess ends with the constuction of the vitual edge labeled

11 OBUST POSITION-BASED OUTING 151 i in Figue 8. Since the numbe of nodes on each side can be abitaily lage, the oute coesponding to the vitual edge i can also be made abitaily lage. Note that the impotance of the bad news should be modeated in view of the good news in the sense that the good news equies mobile hosts to be not too close, wheeas the bad example equies mobile hosts to be abitaily close. The eality is somewhat in between these two exteme cases. The bad case depicted in Figue 8 could coespond to a case whee a lage obstacle sepaates the nodes into two goups with only one eal edge between the two goups. Clealy, the shotest communication paths between nodes of the two goups ae long and the use of the vitual edge i only doubles the length of some of the communication paths. Notice that the efficiency of the constuction of S G could be impoved by applying heuistics. Fo example, edges can be pocessed in the ode of deceasing lengths. i We poved the coectness of ou outing potocol in case / p 2. Given a geometic gaph G in which / p 2, we can do a distibuted constuction of S G stating with the situation in which each node initially knows only its immediate neighbos. The need to stat the pocess of adding vitual edges is detected fom this local infomation. If the condition / p 2 is emoved, then fo some geometic gaphs, local infomation is insufficient to detect the pesence of edges that coss one anothe as shown below. Popety 3 If the atio / > p 2 then fo any intege k thee exists a geometic gaph G such that the existence of two cossing edges in G cannot be detected by any algoithm in which any host u knows only the positions of hosts at most k hops fom u in the gaph. Poof: Conside the netwok in Figue 9 in which the dots epesent a path of length of at least k. Since / > p 2, no host can detect the existence of hosts u and v as well as hosts x and y, if all hosts have infomation only about the locations of hosts at most k-hops away in G. Thus, the existence of a pai of cossing edges would emain undetected. 5. Conclusions g e h f We have poposed a obust position-based outing scheme allowing outing in ad hoc wieless netwoks in which tansmission anges may vay up to a atio of p 2. The value of p 2 is the lagest value that guaantees that fo any two hosts u and v, ifthee is anothe host w in the disk aound the segment u, v, then at least one of u and v will be awae x x c a d b y u v y Fig. 8. A configuation of the mobile hosts yielding long vitual edges. Fig. 9. A netwok with / > p 2 in which no host will detect both the edges fu, vg and fx, yg.

12 152 L. BAIÈE ET AL. of the existence of w. Ou scheme does not equie flooding the netwok to set up the outing potocol, but athe evey node pefoms a limited exploation of its locality. The degee of the locality of these exploations depends on the atio of the maximal ange to the minimum ange, and on the minimum distance between two mobile hosts. It is expected to be small in pactice. In this pape we focus on the algoithmic aspects of ou outing potocol and its coectness. We ae cuently investigating the stetch facto of the outes used in the outing phase, and expect to un some expeiments to get empiical evidence about the efficiency of the algoithm. Many poblems emain open, in paticula: 1. What can be done if the atio / > p 2? (In this case, ou scheme may fail because one cannot guaantee the planaity of GG S G.) 2. What can be done if the connections ae not bidiectional, that is, u can eceive signals fom v but v cannot eceive signals fom u? (In this case, ou scheme fails because the gabiel gaph constuction equies symmety between the two endpoints of an edge.) 3. Would it be possible to deive a outing scheme that ensues the temination of the setup of the outing befoe the outing stats? (Ou 3-phase scheme allows nodes to stat outing messages while the setup of the outing tables is not completed.) Acknowledgements This eseach was done while the fist two authos wee visiting Concodia Univesity in Montéal, the fist autho with suppot fom NATO and the second autho with suppot fom the NT pogam OM. The eseach of the last two authos is suppoted by NSEC gants. efeences 1. Bose P, Moin P, Stojmenovic I, Uutia J. outing with guaanteed delivey in ad hoc wieless netwoks. In 3d Wokshop on Discete Algoithms and Methods fo Mobile Computing and Communication (DIALM 99), 1999; pp (Jounal vesion appeaed in 19.) 2. Kap B, Kung H. GPS: geedy peimete stateless outing fo wieless netwoks. In 6th ACM Confeence on Mobile Computing and Netwoking (Mobicom 00), Boch J, Johnson D, Maltz D. The Dynamic Souce outing Potocol fo Mobile Ad Hoc Netwoks. Intenet-daft, daftietf-manet-ds-00.txt, Mach Pak V, Coson S. A highly adaptive distibuted outing algoithm fo mobile wieless netwoks. In Poceedings of INFOCOM 97, 1997; pp Pekins C. Ad Hoc on Demand Distance Vecto (AODV) outing. Intenet daft, daft-ietf-manet-aodv-00.txt, Novembe Pekins C, Bhagwat P. Highly dynamic destination-sequenced distance-vecto outing (DSDV) fo mobile computes. In Poceedings of SIGCOMM 94, August 1994; pp Basagni S, Chlamtac I, Syotiuk V Woodwad B. A distance outing effect fo mobility (DEAM). In 4th ACM/IEEE Confeence on Mobile Computing and Netwoking (Mobicom 98), 1998; pp Bose P, Moin P. Online outing in tiangulations. In 10th Annual Intenational Symposium on Algoithms and Computation (ISAAC 99), 1999; pp Ko Y-B, Vaidya N. Location-aided outing (LA) in mobile ad hoc netwoks. In 4th ACM/IEEE Confeence on Mobile Computing and Netwoking (Mobicom 98), 1998; pp Lin X, Stojmenović I. GEDI: loop-fee location based outing in wieless netwoks. In IASTED Int. Conf. on Paallel and Distibuted Computing and Systems (PDCS 99), 1999; pp Finn G. outing and Addessing Poblems in Lage Metopolitan-Scale Intenetwoks. ISI eseach epot ISU/ , Mach Stojmenovic I, Lin X. Loop-fee hybid single-path/flooding outing algoithms with guaanteed delivey fo wieless netwoks. IEEE Tansactions on Paallel and Distibuted Systems 2001; 12(10): Kanakis E, Singh H, Uutia J. Compass outing on geometic netwoks. In Canadian Confeence on Computational Geomety (CCCG 99), 1999; pp Ko Y-B, Vaidya N. Geocasting in mobile ad hoc netwoks: location-based multicast algoithms. In Wokshop on Mobile Computing Systems and Applications (WMCSA 99), Bose P, Bodnik A, Calsson S, Demaine E, Fleische, Jacobsen L, López-Otiz A, Moin P, Muno J. Online outing in convex subdivisions. In 11th Annual Intenational Symposium on Algoithms and Computation (ISAAC 00), Gabiel K and Sokal. A new statistical appoach to geogaphic vaiation analysis. Systematic Zoology 1969; 18: Okabe A, Boots B, Sugihaa S. Spatial Tessellations: Concepts and Applications of Voonoi Diagams. John Wiley & Sons: New Yok, Hunt III H, Maathe M, adhakishnan V, avi S, osenkantz D, Steans. NC-appoximation schemes fo NP- and PSPACE-had poblems fo geometic gaphs. Jounal of Algoithms 1998; 26(2): Bose P, Moin P, Stojmenovic I, Uutia J. outing with guaanteed delivey in ad hoc wieless netwoks. ACM/Kluwe Wieless Netwoks 2001; 7(6):

13 OBUST POSITION-BASED OUTING 153 Authos Biogaphies Lali Baiee obtained he licence in mathematics at the Univesity of Bacelona, in 1989, and he Ph.D. at the Technical Univesity of Catalunya, in She is an associate pofesso in the Applied Mathematics Depatment at the Technical Univesity of Catalunya. He eseach inteests include symmety in netwoks, outing algoithms, and mobile agents. Piee Faigniaud eceived the Fench Enginee degee fom the Institut National Polytechnique of Genoble (INPG), Fance, in 1987, and the Ph.D. degee in compute science fom the Ecole Nomale SupØieue of Lyon (ENS Lyon), Fance, in He is pesently Diecteu de echeches CNS, located at LI, UnivesitØ Pais Sud. His main eseach inteests include vaious aspects of communication in netwoks and of distibuted computing. gaph algoithms. Lata Naayanan is an associate pofesso in the Depatment of Compute Science at Concodia Univesity. She obtained he baccalaueate at the Bila Institute of Technology and Science, Pilani, India. She eceived he M.S. in 1989 and he Ph.D. in 1992 fom the Univesity of ocheste. He eseach inteests include mobile computing, outing algoithms, and Jaoslav Opatny eceived his fist degee in 1968 fom Chales Univesity in Pague, Czech epublic, and the Ph.D. in compute science fom the Univesity of Wateloo in Fom 1975 to 1977 he was an assistant pofesso in the Compute Science Depatment at the Univesity of Albeta, Canada. In 1977, he joined the Compute Science Depatment at Concodia Univesity, Monteal, Canada, whee he is pesently a pofesso. His main inteests ae inteconnection netwoks, communications in netwoks, and design of algoithms.