Routing with Guaranteed Delivery in ad hoc Wireless Networks*

Transcription

1 Routig with Guarateed Delivery i ad hoc Wireless Networks* Prosejit Bose School of Computer Sciece Carleto Uiversity jit@scs.carleto.ca Pat Mori School of Computer Sciece Carleto Uiversity mori@scs.carleto.ca Jorge Urrutia Computer Sciece, SITE Uiversity of Ottawa jorgeqsite.uottawa.ca Iva Stojmeovic Computer Sciece, SITE Uiversity of Ottawa iva@site.uottawa.ca Abstract We cosider routig problems i ad hoc wireless etworks modeled as uit graphs i which odes are poits i the plae ad two odes ca commuicate if the distace betwee them is less tha some fixed uit. We describe the first distributed algorithms for routig that do ot require duplicatio of packets or memory at the odes ad yet guaraty that a packet is delivered to its destiatio. These algorithms ca be exteded to yield algorithms for broadcastig ad geocastig that do ot require packet duplicatio. A byproduct of our results is a simple distributed protocol for extractig a plaar subgraph of a uit graph. We also preset simulatio results o the performace of our algorithms. 1 Itroductio Mobile ad hoc etworks (MANETS) cosist of wireless hosts that commuicate with each other i the absece of fixed ifrastructure. Two odes i a MANET ca commuicate if the distace betwee them is less tha the miimum of their two broadcast rages [l]. For health ad efficiecy reasos, it is geerally ot possible (or desirable) for all hosts i a MANET to be able to commuicate with each other directly. Thus, sedig messages betwee two hosts i a MANET may require routig the message through itermediate hosts. I may cases, MANETS are pieced together i a ucotrolled maer, chages i topology are frequet ad ustructured, ad hosts may ot kow the topology of the etire etwork. I this paper, we cosider routig i MANETS for which hosts kow othig about *This work was partly fuded by the Natural Scieces ad E& eerig Research Coucil of Caada. pcmissio to m&c digital or hard copies of all or part of this work for persoal or classroom USC is grated without fee provided that copies are ot made or distributed for profit or commercial advatage ad that copies bear this otice ad the full citatio o the tit Page. To copy otherwise, to republish, to post o servers or to redistribute to lists. requires prior specific permissio ad/or a fee. DIAL M 99 Seattle WA ES.4 Copyright ACM 1999 I /99/ m the etwork except their locatio ad the locatios of the hosts to which they ca commuicate directly. GPS (global positioig system) provides each host with its geographic locatio ad global timig [ll]. I particular, we cosider the case i which all hosts have the same broadcast rage. Let S be a set of poits i the plae. The the uit graph U(S) is a geometric graph that cotais a vertex for each elemet of S. A edge (u, v) is preset i U(S) if ad oly if dist(u, v) 5 1, where dist(s, y) deotes the Euclidea distace betwee x ad y. Uit graphs are a reasoable mathematical abstractio of wireless etworks i which all odes have equal broadcast rages. I this paper we describe algorithms for routig o uit graphs which do ot require global iformatio about U(S). Each vertex v E U(S) represets a trasmissio statio, ad has o iformatio about U(S) except the set of odes N(w) to which it is adjacet. A packet that is stored at vertex v ca be trasmitted to ay vertex i N(u). I accordace with other papers [l, 3, 7, 91, it is assumed that the source kows from the begiig the exact geographical positio of the destiatio. I the mutig problem, the source v,, ad destiatio v&t are poits of S ad v&t must receive a message origiatig at v,,. I the geocastig problem [6, 111 the source v,, is a poit i S while the destiatio r&t is a regio, ad all vertices i r&t must receive a message origiatig at v,,. I this work we take r&t to be a disk, but our algorithms geeralize to arbitrary covex regios. Broadcastig is the a special case of geocastig i which r&t is a disk with ifiite radius. Previous algorithms for olie routig i uit graphs ca be broadly classified ito two categories: Greedy algorithms apply some type of greedy pathfidig heuristic that does ot guaratee that a packet ultimately reaches (all of) its destiatio(s). These iclude the geographic distace routig (GEDIR) algorithm of Lm ad Stojmeovic 191, the directioal routig (DIR), a.k.a, compass routig algorithm of Basagi et al [l], 48

2 Ko ad Vaidya [6], ad Kraakis et al [8], the MFR algorithm of Takagi ad Kleirock [15], ad their 2-hop variats [9]. Floodig algorithms use some type of cotrolled packet duplicatio mechaism to esure that every destiatio receives at least oe copy of the origial packet. These are exemplified by the locatio-aided routig (LAR) protocols of Ko ad Vaidya [7, 61. I order for floodig algorithms to termiate, packets i the etwork must remember which packets they have previously see. I cotrast, our routig algorithms always guaratee that a packet will be delivered to (all of) its iteded recipiet(s) so log as the uit graph U(S) is static.ad coected. Our algorithms do ot make use of ay memory at the odes of U(S) ad require oly that a packet carry a small costat amout of iformatio i additio to its message. Our algorithms also ever require duplicatio of a packet, so that at ay poit i time there is exactly oe copy of each message i the etwork. Although the delivery is guarateed oly for fixed graphs, it may be possible to apply our algorithms to movig hosts, i cojuctio with locatio update techiques [l, 71. Our algorithms work by fidig a coected plaar subgraph of U(S) ad the applyig routig algorithms for plaar graphs o this subgraph. I Sectio 2 we show how to fid a coected plaar subgraph of U(S) i a olie distributed maer. I Sectio 3 we describe algorithms for routig, broadcastig, ad geocastig i plaar graphs. I Sectio 4 we describe simulatio results for our algorithm. Fially, i Sectio 5 we summarize ad coclude with ope problems i the area. 2 Extractig a Coected Plaar Subgraph I this sectio we describe a distributed algorithm for extractig a coected plaar subgraph from U(S). I order to ru the algorithm, the oly iformatio eeded at each ode is the positio of each of its eighbors i U(S). Our algorithm works by computig the itersectio of U(S) with a well-kow plaar graph. Let di&(u, u) be the disk with diameter (.u, v). The, the Gabriel graph [5] GG(S) is a geometric graph i which the edge (u, v) is preset if ad oly if disk(u, w) cotais o other poits of S. The followig lemma shows that the Gabriel graph is useful for extractig a coected subgraph from U(S). Lemma 1. If U(S) is coected the GG(S) U(S) is coected. Proof. It is well kow that a miimum spaig tree MST(S) is a subset of GG(S) [13]. Thus, we eed oly prove that MST(S) c U(S) if U(S) is coected. Assume for the sake of cotradictio that MST(S) co- tais a edge (21, w) whose legth is greater tha 1. Removig this edge from MST(S) produces a graph with two coected compoets, &(S) ad C,,(S). Sice U(S) is coected it cotais a edge (w,z) of legth ot greater tha 1 such that w E C,(S) ad z E C,(S). By replacig the edge (u,v) with (w,z) i MST(S) we obtai a coected graph o S with weight less tha MST(S), a cotradictio. 0 Let (u, w) be a edge of U(S) such that (u,v) $! GG(S). The, by the defiitio of GG(S) there exists a poit w that is cotaied i the disk with u ad u as diameter, ad this poit acts as a witess that (21, w) 4 GG(S). The followig lemma shows that every such edge ca be idetified ad elimiated by u ad w usig oly local iformatio. Lemma 2. Let u ad w be poits of U(S) such that (u, w) 4 GG(S) ad 1 e t w be a witess to this. The (u, w) E U(S) ad (w,w) E U(S). Proof. Let m be the midpoit of (.u, w). The dist(u, m) 5 l/2, dist(w,m) 5 l/2 ad dist(w,m) 5 l/2. Therefore, by the triagle iequality, dist(u, w) 5 1, dist(w, 20) < 1 ad (2~, w) ad (v, w) are i U(S). cl Thus, upo reachig a vertex w E S, a packet ca elimiate the edges icidet o w that are ot i U(S) fl GG(S) by simply elimiatig ay edge that is ot i GG(N(w) U {w}). This leads to the followig algorithm that is executed by each vertex w E S. Algorithm: GABRIEL 1: for each 2~ E N(v) do 2: if disk(u, w) (N(w) \ {u, w}) # 0 the 3: delete (u, w) 4: ed if 5: ed for Lemma 1 guaratees that if we apply this algorithm to each vertex of S the the resultig graph is coected. Sice GG(S) is plaar [12, 10, 51, the resultig graph is also plaar. As described above, the algorithm requires O(d2) time, where d is the degree of w. By usig efficiet algorithms for costructig the Vorooi diagram (VD) ad Delauay triagulatio (DT) [12,13] of N(v) U {v}, ad keepig edges of DT that itersect correspodig edges of VD [lo, 51, this ca be reduced to O(dlogd). Theorem 1. If U(S) as. coected the algorithm GABRIEL computes a coected plaar subgraph of U(S). The cost of the computatio performed at vertex w E S is O(d log d) where d is the degree of w. Remark: More realistically, the elimiatio of edges ot i GG(S) could be doe whe the etwork is iitialized or whe chages i etwork topology occur. 49

3 Figure 1: Routig from w,, to vdst usig FACE-~. Figure 2: Routig from w,,, to U&t usig FACE-2. 3 Routig i Plaar Graphs I this sectio we describe olie algorithms for routig, broadcastig, ad geocastig i a coected plaar graph G. Sice we have show that a coected plaar subgraph of U(S) is easily computable by a routig algorithm, these algorithms also apply to uit graphs. 3.1 Routig I this sectio we describe algorithms for routig i plaar graphs. The first algorithm, called FACE- 1, is due to Kraakis et al [8]. The secod algorithm, called FACE-Z, is a modificatio of their algorithm that performs better i practice. A coected plaar graph G partitios the plae ito faces that are bouded by polygoals made up of edges of G. Give a vertex v o a face F, the boudary of F ca be traversed i the couterclockwise (clockwise if F is the outer face) directio usig the well-kow right had rmle [2]. Treatig this face traversal techique as a subroutie, Kraakis et al [8] give the followig algorithm for routig a packet from v,, to U&t. Algorithm: FACE- 1 1: P+~src 2: repeat 3: let F be the face of G with p o its boudary that itersects lie segmet (p, 2)&t) 4: for each edge (u, V) of F do 5: if (u,v) itersects (p,v,jst) i a poit p ad dist(p, U&t) < dist(p, wdst) the 6: P + P 7: ed if 8: ed for 9: Traverse F util reachig the edge (u, v) cotai- ig P lo: util p = v&t The operatio of algorithm FACE-~ algorithm is illustrated i Figure 1. The followig theorem summarizes the performace of this algorithm. Theorem 2 (KSU 1999 [8]). Algorithm FACE-~ reaches u&t after at most 41Ej steps, where IEl is the umber of edges i G. Notice that this algorithm traverses the etire face F to determie the poit p, ad the must retur to the poit p. The boud 4lE( stated i the theorem ca be reduced to 3lEI by havig the retur trip to p.be alog the shorter of the two possible paths aroud F. However, i practice, as we will show i Sectio 4, the followig modified versio of FACE-~ works eve better. Algorithm: FACE-2 1: p+vsrc 2: repeat 3: let F be the face of G with p o its boudary that itersects (p, v&t) 4: traverse F util reachig a edge (u, w) that itersects (p, U&t) at some poit p # p 5: P + P 6: util p = V&t The operatio of FACE-:! is illustrated i Figure 2. Clearly this algorithm also termiates i a fiite umber of steps, sice the distace to V&t is decreasig durig each roud. However, i pathological cases it may visit fl(2) edges of G. Theorem 3. Algorithm FACE-2 reaches v&t i a fiite umber of steps. 3.2 Broadcastig De Berg et al [4] describe a algorithm for eumeratig all the faces, edges, ad vertices of a coected embedded plaar graph G. It requires o memory at the odes of the graph ad uses oly O(1) additioal memory i the packet that is travelig aroud the etwork. The algorithm works by defiig a spaig tree o the faces of G ad performig depth first search o this spaig tree i O( ) time, where is the umber of vertices of G.l This algorithm ca be made ito a routig algorithm that allows a sigle packet to visit every vertex i G. We refer to this algorithm as BROADCAST. Theorem 4. I at most O(2) steps algotithm BROAD- CAST termiates after havig visited every vertex of G. Actually, the algorithm has ruig time 0 (xi IFi I ) where IFi ( deotes umber of edges i the ith face of G. 50

4 3.3 Geocastig De Berg et al also exted their results to widow queries i which all the faces itersectig a rectagular or circular query regio r&t are to be visited. To start their algorithm, a vertex of a face that itersects r&t must be give as part of the iput. By applyig Algorithm FACE-~, such a face ca be foud i O() steps by settig the value of v&t to the ceter of the query regio. The algorithm should termiate whe it reaches a vertex v cotaied i T&t or whe it ca o loger make progress, i.e., it visits the same face twice. I the first case we apply the algorithm of de Berg et al to have the packet visit every vertex i the query regio, while i the secod case we ca quit, sice there is o vertex of G cotaied i the query regio. We call this algorithm GEOCAST. Theorem 5. I at most O(+ic2) steps algorithm GEO- CAST termiates after havig visited every vertex of G cotaied i T&t, where k is the complexity of all faces of G that itersect r&t. Remark: The delivery time for a message i the broadcastig ad geocastig algorithms ca be improved i practice by traversig subtrees of the spaig tree i parallel, at the cost of havig several copies of the same packet i the etwork simultaeously. 4 Experimetal Results I this sectio we measure the quality of the paths foud by our routig algorithms. Our test sets cosist of radomly costructed uit graphs. Test cases were geerated by uiformly selectig poits i the uit square as vertices, sortig all the ( - 1)/2 iterpoit distaces ad settig the value of a uit to achieve the desired average degree. Ay such radom graph that did ot result i a coected graph was rejected. For each graph geerated, routig was performed betwee all ( - 1) ordered pairs of vertices i the graph. Every data poit show i our graphs is the average of 200 idepedet trials coducted o 200 differet radomly geerated graphs. The results of these trials are give as 95% cofidece itervals i Appedix A. For compariso purposes the performace of our algorithms were measured agaist, ad i combiatio with, geographic distace routig (GEDIR) as described by Li ad StojmeoviC [9]. The GEDIR algorithm is a greedy algorithm that always moves the packet to the eighbour of the curret vertex whose distace to the destiatio is miimized. The algorithm fails whe the packet crosses the same edge twice i successio. The GEDIR algorithm was chose for compariso purposes because, of the three basic algorithms tested by Li ad StojmeoviC, GEDIR had the best performace i terms of delivery rate ad average dilatio (defied below). Figure 3: Delivery rates for the GEDIR algorithm. The experimets measured two quatities. Let X be the set of pairs of vertices (u, w) E G, u # u such that routig algorithm A succeeds i fidig a path from u to v ad let 1x1 deote the cardiality of X. The delivery rate of A is defied as D&(G) = IXl/(( - 1)). Note that, because our algorithms guaratee the delivery of a packet, they have a delivery rate of 1. The average dilatio of A is defied as ADA(G) = (l/lx/> c AP(%v)/SP(u,v), (%V)EX where AP(u, v) is the umber of edges i the path from u to v foud by A ad SP(u, v) is the umber of edges i the shortest path from u to o. Note that havig a low average dilatio is oly useful if the delivery rate is high sice a average dilatio of 1 is easily achieved by (for example) a algorithm that oly succeeds i routig betwee two odes if they axe directly adjacet. To illustrate the importace of havig guarateed delivery of messages, Figure 3 shows the delivery rate of GEDIR o graphs with varyig average degrees ad umber of odes. These results show that delivery failures are ot ucommo with the GEDIR algorithm, ad i very sparse graphs delivery rates ca be as low as 50%. I.e., there are some vertices from which half of the graph is ureachable. Figure 4 compares the FACE-~ algorithm with the FACE-~ algorithm i terms of average dilatio for varyig average degrees ad umber of odes. Not surprisigly, FACE-~ outperforms FACE- 1 due to the fact that it does ot require the packet to travel all the way aroud each face. What may be surprisig is that the average dilatio for both strategies seems to icrease as the average degree icreases. This ca be explaied by the fact that the subgraph GG(S) U(S) o which these algorithms operate is a plaar graph ad therefore has 51

5 Figure 4: Average dilatio algorithms. of the FACE-~ ad FACE-2 Figure 6: Average dilatio rithms. of the GEDIR ad GFG algo- Figure 5: Average dilatio of the GEDIR ad GEDIR+FACE-2 algorithms. average degree at most 6, but they are beig compared to the shortest path i U(S) whose average degree is icreasig. Thus, the algorithms are hadicapped from the start. Although these observatios may lead oe to believe that algorithms FACE-~ ad FACE-2 are ot very good o their ow, they may evertheless be useful i combiatio with aother algorithm. We tested two such combiatios ad compared their average dilatio with the average dilatio of GEDIR. Figure 5 shows the results of combiig the GEDIR algorithm with FACE-~ by applyig the GEDIR algorithm util it either failed or reached the destiatio. If the GEDIR algorithm failed, routig was the completed usig the FACE-2 algorithm. I this sceario FACE-2 ca be viewed as actig as a backup for the GEDIR algorithm. We refer to this algorithm as GEDIR+FACE-2. Figure 6 shows the results of applyig GEDIR util the packet reaches a ode v such that all of v s eighbours are further from the destiatio tha v is. The FACE-2 algorithm was the applied util the packet reached aother vertex u that was strictly closer to the destiatio tha Y at which poit the GEDIR algorithm was resumed. I this sceario, FACE-2 ca be see as a meas of overcomig local miima i the objective fuctio (distace to the destiatio). We refer to this algorithm as GFG. Both these hybrid algorithms exhibit similar performace with the GFG algorithm showig a slight advatage i very sparse graphs. These results show that the average dilatio of GEDIR is cosistetly low, but this comes at the price of low delivery rate i sparse graphs. O the other had, the combied algorithms sometime have high average dilatio, but this oly occurs whe the delivery rate of GEDIR is low ad the combied algorithms are ofte forced to apply the FACE-2 algorithm. The combied algorithm simultaeously ejoys the advatages guarateed delivery i sparse graphs ad low average dilatio i dese graphs. 5 Coclusios We have described algorithms for routig, broadcastig, ad geocastig i uit graphs. The algorithms do ot require duplicatio of packets or memory at the odes of the graph ad yet guaratee that a packet is always delivered to (all of) its destiatio(s). The empirical results for our routig algorithms suggest that although the FACE-~ ad FACE-2 algorithms are ot very efficiet o their ow, they ca be useful i cojuctio with simpler algorithms that do ot guaratee delivery. The BROADCAST ad GEOCAST algorithms are probably ot very applicable i practice due to their quadratic message cout ad delivery time behaviour. A iterestig ope problem that is curretly uder ivestigatio is whether or ot algorithms exist that do ot require memory at the odes of U(S) ad takes subquadratic time to visit all vertices of U(S). Results for static etworks like those i this ad

6 other papers [9] help i fidig the most promisig cadidates for the desig of routig protocols i mobile etworks. There are a umber of directios i which the work preseted here ca be exteded ad/or geeralized, icludig results for dyamically chagig etworks, etworks i three-dimesioal space, odes with uequal trasmissio rages, ad power-aware routig schemes [ 141. Refereces [l] S. Basagi, I. Chlamtac, V. R. Syrotiuk, ad B. A. Woodward. A distace routig effect algorithm for mobility (DREAM). I ACM/IEEE Iteratioal Coferece o Mobile Computig ad Networkig (Mobicom 98), pages 76-84, [2] J. A. Body ad U. S. R. Murty. Graph Theory with Applicatios. Elsevier North-Hollad, [3] D. Camara ad A. F. Loureiro. A ovel routig algorithm for ad hoc etworks. Mauscript, [4] M. de Berg, M. va Kreveld, R. va Oostrum, ad M. Overmars. Simple traversal of a subdivisio without extra storage. Iteratioal Joural of Geographic Iformatio Systems, 11: , [5] K. R. Gabriel ad R. R. Sokal. A ew statistical approach to geographic variatio aalysis. Systematic Zoology, 18: , [S] Y.-B. Ko ad N. H. Vaidya. Geocastig i mobile ad hoc etworks: Locatio-based multicast algorithms. Techical Report TR , Texas A&M Uiversity, September [7] Y.-B. Ko ad N. H. Vaidya. Locatio-aided routig (LAR) i mobile ad hoc etworks. I ACM/IEEE Iteratioal Coferece o Mobile Computig ad Networkig (Mobicom%), pages 66-75, [8] E. Kraakis, H. Sigh, ad J. Urrutia. Compass routig o geometric etworks. I Proceedigs of the 11th Caadia Coferece o Computatioal Geometry (CCCG 99), To appear. [9] X. Li ad I. Stojmeovic. Geographic distace routig i ad hoc wireless etworks. Techical Report TR-98-10, SITE, Uiversity of Ottawa, December [lo] D. W. Matula ad R. R. Sokal. Properties of Gabriel graphs relevat to geographic variatio research ad the clusterig of poits i the plae. Geographical Aalysis, 12:205, July [ll] J. C. Navas ad T. Imieliski. Geocast - geographic addressig ad routig. I ACM/IEEE Iteratioal Coferece o Mobile Computig ad Networkig (Mobicom 97), pages 66-76, PI P31 WI 1151 A A. Okabe, B. Boots, ad K. Sugihara. Spatial Tesselatios: Cocepts ad Applicatios of Vorooi Diagrams. Joh Wiley ad Sos, Frac0 P. Preparata ad Michael Ia Shamos. Computatioal Geometry. Spriger-Verlag, New York, I. StojmeoviC. Power-aware routig i ad hoc wireless etworks. Techical Report TR-98-11, SITE, Uiversity of Ottawa, December H. Takagi ad L. Kleirock. Optimal trasmissio rages for radomly distributed packet radio termials. IEEE l%asactios o Commuicatios, 32(3): , Simulatio Results This appedix presets the results of simulatios i tabular form. The variable d is the average degree of the graph ad the variable is the umber of vertices i the graph. 53

7 d f rt f f f f f f f l.oof f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f Table 1: 95% cofidece itervals for delivery rates of GEDIR. d f f f f f f f f f f f f f f f f f f f f III f f f f f f zt f f f f f f f f f f f f f f f f f Table 2: 95% cofidece itervals for average dilatio of GEDIR. d f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f Table 3: 95% cofidece itervals for average dilatio of FACE-~. 54

8 d f f f f f f f f f f f % f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f Table 4: 95% cofidece itervals for average dilatio of FACE-Z. d f f f f f f zt f f l.oof f f f f f f f f % f f f f f f f f f f f f f f f f f f f ho f ho f ho f Table 5: 95% cofidece itervals for average dilatio of GEDIR+FACE-2. d YO 1.22f f f f l.oofo.ooo f f f f f zk f f & f f f % f f f f z!z f f f f f f f f Ifr f f f f f f f f f f f ho f Table 6: 95% cofidece itervals for average dilatio of GFG. 55