Distributed Construction of a Planar Spanner and Routing for Ad Hoc Wireless Networks

Transcription

1 Disribed Consrcion of a Planar Spanner and Roing for Ad Hoc Wireless Neorks Xiang-Yang Li Gria Calinesc Peng-Jn Wan Absrac Seeral localied roing proocols [1] garanee he delier of he packes hen he nderling neork opolog is he Delana rianglaion of all ireless nodes. Hoeer, i is expensie o consrc he Delana rianglaion in a disribed manner. Gien a se of ireless nodes, e more accrael model he neork as a ni-disk graph UDG, in hich a link in beeen o nodes exis onl if he disance in beeen hem is a mos he maximm ransmission range. Gien a graph H, a spanning sbgraph G of H is a -spanner if he lengh of he shores pah connecing an o poins in G is no more han imes he lengh of he shores pah connecing he o poins in H. In his paper, e presen a noel localied neorking proocol ha consrcs a planar.5-spanner of UDG, called he localied Delana rianglaion, as neork opolog. I conains all edges ha are boh in he ni-disk graph and he Delana rianglaion of all ireless nodes. Or experimens sho ha he delier raes of exising localied roing proocols are increased hen localied Delana rianglaion is sed insead of seeral preiosl proposed opologies. The oal commnicaion cos of or neorking proocol is O(n log n) bis. Moreoer, he compaion cos of each node is O(d log d ), here d is he nmber of 1-hop neighbors of in UDG. I. INTRODUCTION In a ireless ad hoc neork (or sensor neork), assme ha all ireless nodes hae disincie ideniies and each saic ireless node knos is posiion informaion, eiher hrogh a lo-poer Global Posiion Ssem (GPS) receier or hrogh some oher a. For simplici, e also assme ha all ireless nodes hae he same maximm ransmission range and e normalie i o one ni. B a simple broadcasing, each node can gaher he locaion informaion of all nodes ihin he ransmission range of. Conseqenl, all ireless nodes S ogeher define a ni-disk graph UDG(S), hich has an edge if and onl if he Eclidean disance beeen and is less han one ni. One of he cenral challenges in he design of ad hoc neorks is he deelopmen of dnamic roing proocols ha can efficienl find roes beeen o commnicaion nodes. In recen ears, a arie of roing proocols [], [3], [4], [5], [6], [7], [8] argeed specificall for ad hoc enironmen hae been deeloped. For he reie of he sae of he ar roing proocols, see sres b E. Roer and C. Toh [9] and b S. Ramanahan and M. Seensrp [10]. Seeral researchers proposed anoher se of roing proocols, namel he localied roing, hich selec he nex node o forard he packes based on he informaion in he packe header, and he posiion of is local neighbors. Bose and Morin [1] shoed ha seeral localied roing proocols garanee o delier he packes if he nderling neork opolog is he Deparmen of Comper Science, Illinois Insie of Technolog, Chicago, IL {xli, calinesc, an}@cs.ii.ed. The research of he second ahor as performed in par hile isiing he Deparmen of Combinaorics and Opimiaion of Uniersi of Waerloo, and pariall sppored b NSERC research grans. Delana rianglaion of all ireless nodes. The also gae a localied roing proocol based on he Delana rianglaion sch ha he oal disance raeled b he packe is no more han a small consan facor of he disance beeen he sorce and he desinaion. Hoeer, i is expensie o consrc he Delana rianglaion in a disribed manner, and roing based on i migh no be possible since he Delana rianglaion can conain links longer han one ni. Then, seeral researchers proposed o se some planar neork opologies ha can be consrced efficienl in a disribed manner. Lin e al. [11], Bose a al.[1] and Karp e al. [13] proposed o se he Gabriel graph. Roing according o he righ hand rle, hich garanees delier in planar graphs [1], is sed hen simple greed-based roing herisics fail. Gien a graph H, a spanning sbgraph G of H is a -spanner if he lengh of he shores pah connecing an o poins in G is no more han imes he lengh of he shores pah connecing he o poins in H. In his paper, e design a localied algorihm ha consrcs a planar -spanner for he ni-disk graph UDG(S), sch ha some of he localied roing proocols can be applied on i. We obain a ale of approximael.5 for he consan. Gien a se of poins S, le UDel(S), he ni Delana rianglaion, be he graph obained b remoing all edges of Del(S) ha are longer han one ni. We firs proe ha UDel(S) is a -spanner of he ni-disk graph UDG(S). We hen gie a localied algorihm ha consrcs a graph, called localied Delana graph LDel (1) (S). We proe ha LDel (1) (S) is a -spanner b shoing ha i is also a spergraph of UDel(S). We hen sho ho o make he graph LDel (1) (S) planar efficienl. The oal commnicaion cos of or approach is O(n log n) bis, hich is opimal ihin a consan facor. Bose e al. [1] and Karp e al. [13] proposed similar algorihms ha roe he packes sing he Gabriel graph o garanee he delier. Appling he roing mehods proposed in [1], [13] on he planaried localied Delana graph LDel (1) (S), a beer performance is expeced becase he localied Delana rianglaion is denser compared o he Gabriel graph, b sill ih O(n) edges. Or simlaions sho ha he delier raes of seeral localied roing proocols are increased hen he localied Delana rianglaion is sed. In or experimens, seeral simple local roing herisics, applied on he localied Delana rianglaion, hae alas sccessfll deliered he packes, hile oher herisics ere sccessfl in oer 90% of he random insances. Moreoer, becase he consrced opolog is planar, a localied roing algorihm sing he righ hand rle garanees he delier of he packes from sorce node o he desinaion hen simple herisics fail. The experimens also sho ha seeral localied roing algorihms (noabl, com /0/$17.00 (c) 00 IEEE. 168 IEEE INFOCOM 00 Ahoried licensed se limied o: CiU. 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2 pass roing [14] and greed roing) also resl in a pah hose lengh is ihin a small consan facor of he shores pah; e alread kno sch a pah exiss since he localied Delana rianglaion is a -spanner. The remaining of he paper is organied as follos. In Secion II, e reie some srcres ha are ofen sed o consrc he opolog for ireless neorks. In Secion III, e sho ha he ni Delana rianglaion UDel is a -spanner, here = 1+ 5 π. We also claim ha can be redced o π.4. We define localied Delana rianglaions LDel (k) (S) and sd heir properies in Secion IV. Secion V presens he firs localied efficien algorihm ha consrcs a planar graph, PLDel(S), hich conains UDel(S) as a sbgraph. Ths, PLDel(S) is a planar -spanner. The correcness of or algorihm is jsified in he Appendix. We demonsrae he effecieness of he localied Delana rianglaion in Secion VI b sding he performance of arios roing proocols on i. We conclde or paper and discss possible fre research direcions in Secion VII. II. PRELIMINARIES A. Voronoi Diagram and Delana Trianglaion We begin ih definiions of he Voronoi diagram and he Delana rianglaion [15]. We assme ha all ireless nodes are gien as a se S of n nodes in a o dimensional space. Each node has some compaional poer. We also assme ha here are no for nodes of S ha are co-circlar. A rianglaion of S is a Delana rianglaion, denoed b Del(S), if he circmcircle of each of is riangles does no conain an oher nodes of S in is inerior. A riangle is called he Delana riangle if is circmcircle is emp of nodes of S. The Voronoi region, denoed b Vor (p), of a node p in S is he collecion of o dimensional poins sch ha eer poin is closer o p han o an oher node of S. The Voronoi diagram for S is he nion of all Voronoi regions Vor (p), here p S. The Delana rianglaion Del(S) is also he dal of he Voronoi diagram: o nodes p and q are conneced in Del(S) if and onl if Vor (p) and Vor (q) share a common bondar. The shared bondar of o Voronoi regions Vor (p) and Vor (q) is on he perpendiclar bisecor line of segmen pq. The bondar segmen of a Voronoi region is called he Voronoi edge. The inersecion poin of o Voronoi edge is called he Voronoi erex. Each Voronoi erex is he circmcener of some Delana riangle. B. Spanner Consrcing a spanner of a graph has been ell sdied. Le Π G (, ) be he shores pah connecing and in a eighed graph G, and Π G (, ) be he lengh of Π G (, ). Then a graph G is a -spanner of a graph H if V (G) =V (H) and, for an o nodes and of V (H), Π H (, ) Π G (, ) Π H (, ). Wih H ndersood, e also call he lengh srech facor of he spanner G. There are seeral geomerical srcres hich are proed o be -spanners for he Eclidean complee graph K(S) of a poin se S. For example, he Yao graph [16] and he θ-graph [17] hae been shon o be -spanners. Hoeer, boh hese o geomerical srcres are no garaneed o be planar in o dimensions. Gien a se of poins S, i is ell-knon ha he Delana rianglaion Del(S) is a planar -spanner of he compleed Eclidean graph K(S). This is firs proed b Dobkin, Friedman and Spoi [18] ih pper bond 1+ 5 π 5.08 on. Then Kein and Gin [19], [17] improed he pper bond on o π 3cos π 6 be π.4. The bes knon loer bond on is π/, hich is de o Che [0]. = C. Proximi Graphs Le S be a se of n ireless nodes disribed in a odimensional plane. These nodes indce a ni-disk graph UDG(S) in hich here is an edge if and onl if 1. Varios proximi sbgraphs of he ni-disk graph can be defined [1], [], [3], [4], [16]. For conenience, le disk(, ) be he closed disk ih diameer, le disk(,, ) be he circmcircle defined b he riangle, and le B(, r) be he circle cenered a ih radis r. Le x() and () be he ale of he x-coordinae and -coordinae of a node respeciel. The consrained relaie neighborhood graph, denoed b RNG(S), consiss of all edges sch ha 1 and here is no poin S sch ha <, and <. The consrained Gabriel graph, denoed b GG(S), consiss of all edges sch ha 1 and disk(, ) does no conain an node from S. The consrained Yao graph ih an ineger parameer k 6, denoed b YGk (S), is defined as follos. A each node, an k eqal-separaed ras originaed a define k cones. In each cone, choose he closes node o ih disance a mos one, if here is an, and add a direced link. Ties are broken arbiraril. Le YG k (S) be he ndireced graph obained b ignoring he direcion of each link in YGk (S). Bose e al. [5] shoed ha he lengh srech facor of RNG(V ) is a mos n 1 and he lengh srech facor of GG(V ) is a mos 4π n 4 3. Seeral papers [6], [7], [1] shoed ha he Yao graph YG k (V ) has lengh srech facor a 1 mos 1 sin. Hoeer, he Yao graph is no garaneed o be π k planar. The relaie neighborhood graph and he Gabriel graph are planar graphs, b he are no a spanner for he ni-disk graph. In his paper, e are ineresed in locall consrcing a planar graph ha is a spanner of he ni-disk graph. In or experimens, roing packes sing seeral simple localied roing algorihms sch as compass roing on his localied Delana rianglaion as alas or almos alas sccessfl, improing on roing on he Gabriel graph or he relaie neighborhood graph. D. Localied Roing Algorihms Le N k () be he se of nodes of S ha are ihin k hops disance of in he ni-disk graph UDG(S). A node N k () is called he k-neighbor of he node. Usall, here he consan k is 1 or, hich ill be omied if i is clear from he conex. In his paper, e alas assme ha each node of S knos is locaion and ideni. Then, afer one broadcas b eer node, /0/$17.00 (c) 00 IEEE. 169 IEEE INFOCOM 00 Ahoried licensed se limied o: CiU. Donloaded on Ma,0 a 08:10:13 UTC from IEEE Xplore. Resricions appl.

3 each node of S knos he locaion and ideni informaion of all nodes in N 1 (). The oal commnicaion cos of all nodes o do so is O(n log n) bis. A disribed algorihm is a localied algorihm if i ses onl he informaion of all k-local nodes of each node pls he informaion of a consan nmber of addiional nodes. In his paper, e concenrae on he case k =1. Tha is, a node ses onl he informaion of he 1-hop neighbors. A graph G can be consrced locall in he ad hoc ireless enironmen if each ireless node can compe he edges of G inciden on b sing onl he locaion informaion of all is k-local nodes. In his paper, e design a localied algorihm ha consrcs a planar -spanner for he ni-disk graph UDG(S) sch ha some localied roing proocols can be applied on i. Assme a packe is crrenl a node, and he desinaion node is. Seeral localied roing algorihms ha js se he local informaion of o roe packes (i.e., find he nex node of ) ere deeloped. Kranakis e al. [14] proposed o se he compass roing, hich basicall finds he nex rela node sch ha he angle is he smalles among all neighbors of in a gien opolog. Lin e al. [11], Bose e al. [1], and Karp e al. [13] proposed similar greed roing mehods, in hich node forards he packe o is neighbor inagien opolog hich is closes o. Recenl, Bose a al.[8], [1], [1] proposed seeral localied roing algorihm ha roe a packe from a sorce node s o a desinaion node. Specificall, Bose and Morin [1] proposed a localied roing mehod based on he Delana rianglaion. The shoed ha he disance raeled b he packe is ihin a small consan facor of he disance beeen s and. The also proed ha he compass roing and he greed roing mehod garanee o delier he packe if he Delana rianglaion is sed. III. GRAPH UDel(S) IS A SPANNER In his secion, e proe ha UDel(S) is a spanner ih srech facor = 1+ 5 π. We claim he sronger resl ha UDel(S) is a π-spanner, b omi he proof de o space limiaions. Dobkin, Friedman and Spoi proed ha, for an o poins and of a poin se S, he shores pah connecing and in he Delana rianglaion Del(S) has lengh no more han 1+ 5 π. Hoeer, i is no appropriae o reqire he consrcion of he Delana rianglaion in he ireless commnicaion enironmen becase of he possible massie commnicaions i reqires. Therefore, e consider he folloing sbse of he Delana rianglaion. Le UDel(S) be he graph b remoing all edges of Del(S) ha are longer han one ni, i.e., UDel(S) =Del(S) UDG(S). Call UDel(S) he ni Delana rianglaion. For he remainder of his secion, e ill proe ha UDel(S) is a -spanner of he ni-disk graph UDG(S). Or proof is based on he remarkable proof b Dobkin e al.[18]. The proed ha he Delana rianglaion is a - spanner b consrcing a pah Π df s (, ) in Del(S) ih lengh no more 1+ 5 π. The consrced pah consiss of a mos o pars: one is some direc DT pahs, he oher is some shorc sbpahs. Gien o nodes and, le b 0 =, b 1, b,, b m 1, b m = be he nodes corresponding o he seqence of Voronoi regions raersed b alking from o along he segmen. See Figre 1 for an illsraion. If a Voronoi edge or a Voronoi erex happens o lie on he segmen, hen choose he Voronoi region ling aboe. Assme ha he line is he x-axis. The seqence of nodes b i, 0 i m, defines a pah from o. In general, he [18] refer o he pah consrced his a beeen some nodes and as he direc DT pah from o. Then Dobkin e al. proed he folloing lemma. b1 b b 3 b4 b i b j 1 3 b i b j Fig. 1. Lef: The direc DT pah b 1 b b 3 b 4 beeen and shon b dashed lines; Righ: The shor c from node b i o node b j. Lemma 1: For all i, 0 i m, b i is conained ihin or on he bondar of disk(, ). A sronger resl is ha all nodes b i, 0 i m, are on he bondar of he nion of all circles C i, 1 i m, here C i = B(p i, p i b i ) and p i is he poin on he x-axis ha also lies on he bondar beeen he Voronoi regions Vor (b i 1 ) and Vor (b i ). The bondar of he nion of all circles C i has lengh a mos π ; For deails, see [18]. This implies ha if a direc DT pah alas lies aboe (or belo), hen is lengh is a mos π. If he direc DT pah connecing and is ling enirel aboe or enirel belo he segmen, iis called one-sided; see [18]. The Lemma 1 also implies ha he disance b i b j beeen an o nodes b i and b j is a mos. Conseqenl, e hae he folloing corollar. Corollar : All edges of he direc DT pah connecing o nodes s and hae lengh a mos s. The pah consrced b Dobkin e al. ses he direc DT pah as long as i is aboe he x-axis. Assme ha he pah consrced so far has brogh s o some node b i sch ha (b i ) 0, b i, and (b i+1 ) < 0. Le j be he leas ineger larger han i sch ha (b j ) 0. Noice ha here j exiss becase (b m )=0b assming ha is he x-axis. Then he pah consrced b Dobkin e al. ses eiher he direc DT pah o b j or akes a shorc as follos 1. Consrc he loer conex hll 0 = b i, 1,, l 1, l = b j of he folloing se of nodes: {q S x(b i ) x(q) x(b j ) and (q) 0 and q lies nder b i b j }. 1 See [18] for more deail abo he condiion hen o choose he direc DT pah from b i o b j and hen o choose he shorc pah from b i o b j /0/$17.00 (c) 00 IEEE. 170 IEEE INFOCOM 00 Ahoried licensed se limied o: CiU. Donloaded on Ma,0 a 08:10:13 UTC from IEEE Xplore. Resricions appl.

4 Noice ha excep 0 and l, all nodes 1,, l 1 do no belong o {b 1, b,, b m 1, b m } and he edges of he conex hll are no on he direc DT pah from o. The shorc pah consiss of aking he direc DT pah from k o k+1 for each 0 k l 1, hich is shon o be on one side of line k k+1 if he shorc pah is chosen. Dobkin e al. hen proed ha he lengh of he pah raersed from o has lengh a mos 1+ 5 π. Similar o he direc DT pah, e proe he folloing lemma. Lemma 3: All edges of he shorc pah connecing o nodes b i and b j hae lengh a mos. Proof: Figre 1 gies iniion on he proof ha follos. Le b i, b j be he projecion poins of nodes b i and b j on he x- axis (segmen ), respeciel. Then from he definiion of 0, 1,, l 1, l, e kno ha k, 0 k l lies inside or on he bondar of he rapeoid b i b j b j b i, hich lies inside he disk(, ). Conseqenl, edge k k+1, for each 0 k l 1 has lengh a mos. From Corollar, e kno ha all edges of he direc DT pah from k o k+1 hae lengh a mos k k+1. Then he lemma follos. Conseqenl, e hae he folloing lemma. Lemma 4: Le Π df s (, ) be he pah consrced b Dobkin e al. from o in he Delana rianglaion. All edges in Π df s (, ) hae lengh a mos. Then he folloing heorem is sraighforard. Theorem 5: For an o nodes and of S, Π UDel(S) (, ) 1+ 5 π Π UDG(S) (, ). Proof: Assme Π UDG(S) (, ) = 0 1 h 1 h, here = 0 and = h, is he shores pah connecing and in UDG(S). Then for each link i i+1, 0 i h 1, here is a pah Π Del(S) ( i, i+1 ) in he Delana rianglaion (consrced sing he mehod proposed in [18]) Del(S) ih lengh a mos 1+ 5 π i i+1. Noice ha i i+1 1and all edges in Π Del(S) ( i, i+1 ) hae lengh a mos i i+1. Therefore each pah Π Del(S) ( i, i+1 ), 0 i h 1, is also in he graph UDel(S). Then he pah formed b concaenaing all pahs Π Del(S) ( i, i+1 ), i =0,,h 1 has lengh a mos 1+ 5 π Π UDG(S) (, ). The heorem follos. Kein and Gin [19], [17] shoed ha he Delana rianglaion is a -spanner for a consan = π 3cos = 4 3 π 6 9 π.4. The proed his sing indcion on he order of he lenghs of all pair of nodes (from he shores o he longes). We can sho ha he pah connecing nodes and consrced b he mehod gien in [19], [17] also saisfies ha all edges of ha pah is shorer han. De o space limiaions, e omi he proof. Conseqenl, e hae: Theorem 6: UDel(S) is a π-spanner of UDG. IV. LOCAL DELAUNAY TRIANGULATION In his secion, e ill define a ne opolog, called local Delana rianglaion, hich can be consrced in a localied manner. We firs inrodce some geomeric srcres and noaions o be sed in his secion. All angles are measred in radians and ake ales in he range [0,π]. For an hree poins p 1, p, and p 3, he angle beeen he o ras p 1 p and p 1 p 3 is denoed b p 3 p 1 p or p p 1 p 3. The closed infinie area inside he angle p 3 p 1 p, also referred o as a secor, is denoed b p 3 p 1 p. The riangle deermined b p 1, p, and p 3 is denoed b p 1 p p 3. An edge is called Gabriel edge if 1 and he open disk sing as diameer does no conain an node from S. I is ell knon [15] ha he consrained Gabriel graph is a sbgraph of he Delana rianglaion. Recall ha a riangle belongs o he Delana rianglaion Del(S) if is circmcircle disk(,, ) does no conain an oher node of S in is inerior. Here e ofen assme ha here are no for nodes of S co-circmcircle. I is eas o sho ha nodes, and ogeher can no decide if he can form a riangle in Del(S) b sing onl heir local informaion. We sa a node x can see anoher node if x 1. The folloing definiion is one of he ke ingrediens of or localied algorihm. Definiion 1: A riangle saisfies k-localied Delana proper if he inerior of disk(,, ) does no conain an node of S ha is a k-neighbor of,, or; and all edges of he riangle hae lengh no more han one ni. Triangle is called a k-localied Delana riangle. Triangle is called localied Delana if i is a k- localied Delana riangle for some ineger k 1. Definiion : The k-localied Delana graph oer a node se S, denoed b LDel (k) (S), has exacl all Gabriel edges and edges of all k-localied Delana riangles. When i is clear from he conex, e ill omi he ineger k in or noaion of LDel (k) (S). Or original conjecre as ha LDel (1) (S) is a planar graph and hs e can easil consrc a planar -spanner of UDG(S) b sing a localied approach. Unfornael, as e ill sho laer, he edges of he graph LDel (1) (S) ma inersec. While LDel (1) (S) is a -spanner, is consrcion is a lile bi more complicaed han some oher non-planar -spanners, sch as he Yao srcre [16] and he θ- graph [17]. B e can make LDel (1) (S) planar efficienl, a resl e describe laer in his paper. Noice ha he k-localied Delana graph LDel (k) (S) oer a node se S saisfies a monoone proper: LDel (k+1) (S) is alas a sbgraph of LDel (k) (S) for an posiie ineger k. A. LDel (1) (S) ma be non-planar The definiion of he 1-localied Delana riangle does no preen o riangles from inersecing or preen a Gabriel edge from inersecing a riangle. Figre gies sch an example ih 6 nodes {,,, x,, } ha LDel (1) (S) is no a planar graph. Here =1. Triangle is a 1-localied Delana riangle. If he node does no exis, edge x is an Gabriel edge. The riangle inersecs he Gabriel edge x if does no exis, oherise i inersecs he 1-localied Delana riangle x /0/$17.00 (c) 00 IEEE. 171 IEEE INFOCOM 00 Ahoried licensed se limied o: CiU. Donloaded on Ma,0 a 08:10:13 UTC from IEEE Xplore. Resricions appl.

5 11 0 x B. LDel (k) (S) is a -spanner Fig.. LDel (1) (S) is no planar. Theorem 7: Graph UDel(S) is a sbgraph of he k-localied Delana graph LDel (k) (S). Proof: We proe he heorem b shoing ha each edge of he ni Delana rianglaion graph UDel(S) appears in he localied Delana graph LDel (k) (S). For each edge of UDel(S), he folloing fie cases are possible (see Figre 3 for illsraions). Case 1 Case.1 Case. Case Case 4 Case 5.1 Case 5. Fig. 3. The neighborhood configraion of edge. Dashed lines (solid lines) denoe edges ih lengh > 1( 1). Case 1: here is a riangle inciden on sch ha all edges of hae lengh a mos one ni. Becase he circmcircle disk(,, ) is emp of nodes of S, riangle saisfies he k-localied Delana proper and hs edge belongs o LDel (k) (S). Case : each of he o riangles inciden on has onl one edge ih lengh larger han one ni. Case 3: one riangle inciden on has onl one edge ih lengh larger han one ni and he oher riangle has o edges ih lengh larger han one ni. Case 4: each of he o riangles inciden on has o edges ih lengh larger han one ni. We proe he cases, 3, and 4 ogeher. Assme he o riangles are and. Le H, be he half-plane ha is diided b and conains node. Then edge is no he longes edge in riangle and hs he angle < π ; for an illsraion, see Figre 4. This implies ha he circmcircle disk(,, ) conains disk(, ) H,. Similarl, he oher half of disk(, ) is conained inside he circmcircle disk(,, ). Noice ha boh disk(,, ) and disk(,, ) do no conain an node of S inside. I implies ha disk(, ) is emp, i.e., edge is a Gabriel edge. Conseqenl, edge ill be insered o LDel (k) (S). Case 5: here is onl one riangle inciden on and i has a leas one edge ih lengh larger han one ni. Similar o cases Fig. 4. Gabriel edges. -4, e can sho ha disk(, ) is emp and herefore edge ill be insered o LDel (k) (S) as a Gabriel edge. C. LDel (k) (S), k, is planar The aboe proof implies ha each edge of UDel(S) is eiher a Gabriel edge or forms a 1-localied Delana riangle ih some edges from UDel(S). An o edges in UDel(S) do no inersec. Ths, each possible inersecion in LDel (k) (S) is cased b a leas one localied Delana riangle. We begin he proof ha LDel (k) (S), k, is planar b giing some simple facs and lemmas. Lemma 8: If an edge x inersecs a localied Delana riangle, hen x and can no be boh inside he circmcircle disk(,, ). Proof: For he sake of conradicion, assme ha x and are boh inside disk(,, ). Noice ha disk(,, ) is diided ino for regions b he riangle. Le û,, and ŵ be he hree fan regions defined b edges,, and respeciel. Firs of all, neiher x nor can be inside he riangle. Assme ha x is inside he region û and is inside he region. Then one of he angles and is less han π, hich implies ha one of he angle x and is larger han π. Ths, eiher < 1 or x < 1. In oher ords, he disk(,, ) conains a node from N 1 (). This conradics ha is a k-localied Delana riangle. Lemma 9: If a Gabriel edge x inersecs a localied Delana riangle, hen x and can no be boh oside he circmcircle disk(,, ). Proof: Le c be he circmcener of he riangle. Then a leas one of he,, and ms be on he differen side of line x ih he cener c; Le s sa. If boh x and are oside, hen x > π. Ths, is inside disk(x, ), hich conradics ha x is a Gabriel edge. Theorem 10: Assme o riangles and x inrodced o LDel (k) (S), k 1, inersec, hen eiher disk(,, ) conains a leas one of he nodes of {x,, } or disk(x,, ) conains a leas one of he nodes of {,, }. See he appendix for he proof. The aboe heorem garanees ha if o k-localied Delana riangles and x inersec, hen eiher disk(,, ) or disk(x,, ) iolaes he Delana proper b js considering he nodes {,,, x,, }. We hen sho ha LDel () (S) is a planar graph. Theorem 11: LDel () (S) is a planar graph. Proof: Noice ha o Gabriel edges do no inersec. Then eer inersecion ms inoles a localied Delana ri /0/$17.00 (c) 00 IEEE. 17 IEEE INFOCOM 00 Ahoried licensed se limied o: CiU. Donloaded on Ma,0 a 08:10:13 UTC from IEEE Xplore. Resricions appl.

6 angle. Assme ha an edge x of LDel () (S) inersecs a localied Delana riangle. Edge x is eiher a Gabriel edge or an edge of a localied Delana riangle, sa x. If x is a Gabriel edge, hen Lemma 9 implies ha eiher x or is inside he disk(,, ), sa. If x is an edge of a localied Delana riangle x, hen Theorem 10 implies ha eiher x or is inside he disk(,, ), sa. The riangle ineqali implies ha x + < x +. The exisence of he -localied Delana riangle implies ha / N 1 () N 1 () N 1 (). Ths, > 1, hich implies ha x < 1. In oher ords, x N 1 (). Conseqenl, N () becase of he pah x in he nidisk graph UDG(S), hich conradics o he exisence of - localied Delana riangle. The heorem follos. We defined a seqence of localied Delana graphs LDel (k) (S), here 1 k n. All graphs are -spanner of he ni-disk graph ih he folloing properies: UDel(S) LDel (k) (S), for all 1 k n; LDel (k+1) (S) LDel (k) (S), for all 1 k n; LDel (k) (S) are planar graphs for all k n; LDel (1) (S) is no alas planar. D. LDel (1) (S) has hickness In his sbsecion, e claim ha LDel (1) (S) has hickness o, or in oher ords, is edges can be pariioned in o planar graphs. From Eler s formla, i follos ha a simple planar graph ih n nodes has a mos 3n 6 edges, and herefore LDel (1) (S) has a mos 6n edges. De o space limiaions, e omi he proof. Theorem 1: Graph LDel (1) (S) has hickness. V. LOCALIZED ALGORITHM In his secion, e sd ho o locall consrc a planar -spanner of UDG(S). We assme ha he ideni of a node can be represened b O(log n) bis and is locaion can be represened b O(1) bis. Alhogh he graph UDel(S) is a -spanner for UDG(S), e do no kno ho o consrc i locall. We can consrc LDel () (S), hich is garaneed o be a planar spanner of UDel(S), b ih a oal commnicaion cos of his approach is O(m log n) bis, here m is he nmber of edges in UDG(S) and cold be as large as O(n ). In order o redce he oal commnicaion cos o O(n log n) bis, e do no consrc LDel () (S), and insead e exrac a planar graph PLDel(S) o of LDel (1) (S). A. Algorihm Recall ha LDel (1) (S) is no garaneed o be a planar graph. We propose an algorihm ha consrcs LDel (1) (S) and hen makes i a planar graph efficienl. The final graph sill conains UDel(S) as a sbgraph. Ths, i is a -spanner of he ni-disk graph UDG(S). In he folloing, he order of hree nodes in a riangle is immaerial. Algorihm 1: Localied Uni Delana Trianglaion 1. Each ireless node broadcass is ideni and locaion and lisens o he messages from oher nodes.. Assme ha node gahered he locaion informaion of N 1 (). I compes he Delana rianglaion Del(N 1 ()) of is 1-neighbors N 1 (), inclding iself. 3. For each edge of Del(N 1 ()), le and be o riangles inciden on. Edge is a Gabriel edge if boh angles and are less han π/. Node marks all Gabriel edges, hich ill neer be deleed. 4. Each node finds all riangles from Del(N 1 ()) sch ha all hree edges of hae lengh a mos one ni. If angle π 3, node broadcass a message proposal(,, ) o form a 1-localied Delana riangle in LDel (1) (V ), and lisens o he messages from oher nodes. 5. When a node receies a message proposal(,, ), acceps he proposal of consrcing if belongs o he Delana rianglaion Del(N 1 ()) b broadcasing message accep(,, ); oherise, i rejecs he proposal b broadcasing message rejec(,, ). 6. A node adds he edges and o is se of inciden edges if he riangle is in he Delana rianglaion Del(N 1 ()) and boh and hae sen eiher accep(,, ) or proposal(,, ). We firs claim ha he graph consrced b he aboe algorihm is LDel (1) (S). Indeed, for each riangle of LDel (1) (S), one of is inerior angle is a leas π/3 and is in Del(N 1 ()), Del(N 1 ()) and Del(N 1 ()). So one of he nodes amongs {,, } ill broadcas he message proposal(,, ) o form a 1-localied Delana riangle. As Del(N 1 ()) is a planar graph, and a proposal is made onl if π 3, node broadcass a mos 6 proposals. And each proposal is replied b a mos o nodes. Therefore, he oal commnicaion cos is O(n log n) bis. The aboe algorihm also shos ha LDel (1) (S) has O(n) edges, hich e kno from Theorem 1. Ping ogeher he argmens aboe, e hae: Theorem 13: Algorihm 1 consrcs LDel (1) (S) ih oal commnicaion cos O(n log n). We hen propose an algorihm o exrac from LDel (1) (S) a planar sbgraph. Algorihm : Planarie LDel (1) (S) 1. Each ireless node broadcass he Gabriel edges inciden on and he riangles of LDel (1) (S) and lisens o he messages from oher nodes.. Assme node gahered he Gabriel edge and 1-local Delana riangles informaion of all nodes from N 1 (). For o inerseced riangles and x knon b, node remoes he riangle if is circmcircle conains a node from {x,, }. 3. Each ireless node broadcass all he riangles inciden on hich i has no remoed in he preios sep, and lisens o he broadcasing b oher nodes. 4. Node keeps he edge in is se of inciden edges if i is a Gabriel edge, or if here is a riangle sch ha,, and hae all annonced he hae no remoed he riangle in Sep /0/$17.00 (c) 00 IEEE. 173 IEEE INFOCOM 00 Ahoried licensed se limied o: CiU. Donloaded on Ma,0 a 08:10:13 UTC from IEEE Xplore. Resricions appl.

7 We denoe he graph exraced b he algorihm aboe b PLDel(S). Noe ha an riangle of LDel (1) (S) no kep in he las sep of he Planariaion Algorihm is no a riangle of LDel () (S), and herefore PLDel(S) is a spergraph of LDel () (S). Ths, b sing Theorem 7, e hae: 1 Compass Random Compass Greed UDel(S) LDel () (S) PLDel(S) LDel (1) (S) α α Similar o he proof ha LDel () (S) is a planar graph, e can sho ha or algorihm does generae a planar graph PLDel(S). De o space limiaion, e omi he proof. The oal commnicaion cos o consrc he graph PLDel(S) is a O(log n) imes he nmber of edges of he graph LDel (1) (S), hich b Theorem 1 is O(n). Ping ogeher all he argmens aboe and Theorem 6, e hae: Theorem 14: PLDel(S) is planar π-spanner of UDG(S), and can be consrced ih oal commnicaion cos O(n log n). VI. ROUTING We discss ho o roe packes on he consrced graph. Recenl, Bose and Morin [1] firs proposed a localied roing algorihm ha roes a packe from a sorce node s o a desinaion node. Here a roing algorihm is localied if each relaing node decides o hich node o forard he packe onl based on he folloing informaion: he sorce node s, he desinaion node, he crren node and all nodes of N k (). We onl se k =1. Someimes, he algorihm ma se a mos a consan nmber of bis of addiional informaion. Their algorihm is based on he remarkable proof of Dobkin e al. [18] ha he Delana rianglaion is a -spanner of he complee Eclidean graph. Bose and Morin [1] shoed ho o find anoher pah locall ih lengh no more han Π df s (, ). Hoeer heir algorihm has a major deficienc b reqiring he consrcion of he Delana rianglaion and he Voronoi diagram of all ireless nodes, hich cold be er expensie in disribed comping. Bose e al. [1] proposed anoher algorihm ha roes he packes sing he Gabriel graph o garanee he delier. Noice ha he Gabriel graph is a sbgraph of PLDel(S). Ths, if e appl he roing mehod proposed in [1] on he nel proposed planar graph PLDel(S), e expec o achiee beer performance becase PLDel(S) is denser han he Gabriel graph (b sill ih O(n) edges). The consrced local Delana rianglaion no onl garanees ha he lengh of he shores pah connecing an o ireless nodes is a mos a consan facor of he minimm in he ni-disk graph, i also garanees ha he energ consmed b he pah is also minimm, as i incldes he Gabriel graph (see [9], [1]). Moreoer, becase he consrced opolog is planar, hen a localied roing algorihm sing he righ hand rle garanees he delier of he packes from sorce node o he desinaion node. We sd he folloing roing algorihms on he graphs proposed in his paper. Compass Roing Le be he desinaion node. Crren node finds he nex rela node sch ha he angle is he Mos Forarding Neares Neighbor Farhes Neighbor Fig. 5. Shaded area is emp and is nex node. smalles among all neighbors of in a gien opolog. See[14]. Random Compass Roing Le be he crren node and be he desinaion node. Le 1 be he node on he aboe of line sch ha 1 is he smalles among all sch neighbors of. Similarl, e define o be nodes belo line ha minimies he angle. Then node randoml choose 1 or o forard he packe. See[14]. Greed Roing Le be he desinaion node. Crren node finds he nex rela node sch ha he disance is he smalles among all neighbors of in a gien opolog. See [1]. Mos Forarding Roing (MFR) Crren node finds he nex rela node sch ha is he smalles among all neighbors of in a gien opolog, here is he projecion of on segmen. See [11]. Neares Neighbor Roing (NN) Gien a parameer angle α, node finds he neares node as forarding node among all neighbors of in a gien opolog sch ha α. Farhes Neighbor Roing (FN) Gien a parameer angle α, node finds he farhes node as forarding node among all neighbors of in a gien opolog sch ha α. Noice ha i is shon in [1], [14] ha he compass roing, random compass roing and he greed roing garanee o delier he packes from he sorce o he desinaion if Delana rianglaion is sed as neork opolog. The proed his b shoing ha he disance from he seleced forarding node o he desinaion node is less han he disance from crren node o. Hoeer, he same proof canno be carried oer hen he neork opolog is Yao graph, Gabriel graph, relaie neighborhood graph, and he localied Delana rianglaion. When he nderling neork opolog is a planar graph, he righ hand rle is ofen sed o garanee he packe delier afer simple localied roing herisics fail [1], [11], [13]. We presen or experimenal resls of arios roing mehods on differen neork opologies. Figre 6 illsraes some neork opologies discssed in his paper. Recall ha Gabriel graph, relaie neighborhood graph, Delana rianglaion, LDel () (S), and PLDel(S) are alas planar graphs. The Yao srcre, Delana rianglaion, LDel () (S), and PLDel(S) are alas a -spanner of he ni-disk graph. We se ineger parameer k =8in consrcing he Yao graph. In he experimenal resls presened here, e choose oal n =50ireless nodes /0/$17.00 (c) 00 IEEE. 174 IEEE INFOCOM 00 Ahoried licensed se limied o: CiU. Donloaded on Ma,0 a 08:10:13 UTC from IEEE Xplore. Resricions appl.

8 TABLE II THE MAXIMUM SPANNING RATIO OF DIFFERENT LOCALIZED ROUTING METHODS ON VARIOUS NETWORK TOPOLOGIES. GG RNG Yao Yao RNG GG Del LDel () PLDel NN FN MFR Compass RndCmp Greed Del LDel () PLDel Fig. 6. Varios planar neork opologies (excep Yao). hich are disribed randoml in a sqare area ih side lengh 100 meers. Each node are specified b a random x-coordinae ale and a random -coordinae ale. The ransmission radis of each ireless node is se as 30 meers. We randoml selec 10% of nodes as sorce nodes; and for eer sorce node, e randoml choose 10% of nodes as desinaion nodes. The saisics are comped oer 10 differen node configraions. Ineresingl, e fond ha hen he nderling neork opolog is Yao graph, LDel () (S), orpldel(s), he compass roing, random compass roing and he greed roing deliered he packes in all or experimens. Table I illsraes he delier raes of differen localied roing proocols on arios neork opologies. For neares neighbor roing and farhes neighbor roing, e choose he angle α = π/3. The LDel () (S) and PLDel(S) graphs are preferred oer he Yao graph becase e can appl he righ hand rle hen preios simple herisic localied roing fails. Boh [1] and [13] se he greed roing on Gabriel graph and se he righ hand rle hen greed fails. Table II illsraes he maximm raios of Π(s, ) / s, here Π(s, ) is he pah raersed b he packe sing differen localied roing proocols on arios neork opologies from sorce s o desinaion. In or experimen, e fond ha he raios Π(s, ) / s are small. TABLE I THE DELIVERY RATE OF DIFFERENT LOCALIZED ROUTING METHODS ON VARIOUS NETWORK TOPOLOGIES. Yao RNG GG Del LDel () PLDel NN FN MFR Compass RndCmp Greed VII. CONCLUSION I is ell-knon ha Delana rianglaion Del(S) is a - spanner of he compleed graph K(S). In his paper, e firs proed ha he UDel(S) is a -spanner of he ni-disk graph UDG(S). We hen gae a localied algorihm ha consrcs a graph, namel PLDel(S). We proed ha PLDel(S) is a planar graph and i is a -spanner b shoing ha UDel(S) is a sbgraph of PLDel(S). The oal commnicaion cos of all nodes of or algorihm is O(n log n) bis. The compaion cos of each node is O(d log d ), here d is he nmber of 1-hop neighbors of in UDG. Or experimens shoed ha he delier raes of exising localied roing proocols are increased hen localied Delana rianglaion is sed insead of seeral preiosl proposed planar opologies. We proed ha he shores pah in PLDel(S) connecing an o nodes and is a mos a consan facor of he shores pah connecing and in UDG. I remain open designing a localied algorihm sch ha he pah raersed b a packe from o has lengh ihin a consan of he shores pah connecing and in UDG. VIII. ACKNOWLEDGMENT The firs ahor is graefl o Xiang-Min Jiao of UIUC for sending him copied reference papers [17], [18]. REFERENCES [1] P. Bose and P. Morin, Online roing in rianglaions, in Proc. of he 10 h Annal In. Smp. on Algorihms and Compaion ISAAC, [] Daid B Johnson and Daid A Mal, Dnamic sorce roing in ad hoc ireless neorks, in Mobile Comping, Imielinski and Korh, Eds., ol Kler Academic Pblishers, [3] J. Broch, D. Johnson, and D. Mal, The dnamic sorce roing proocol for mobile ad hoc neorks, [4] S. Mrh and J. Garcia-Lna-Acees, An efficien roing proocol for ireless neorks, ACM Mobile Neorks and Applicaions Jornal, Special isse on Roing in Mobile Commnicaion Neorks, ol. 1, no., [5] V. Park and M. Corson, A highl adapie disribed roing algorihm for mobile ireless neorks, in IEEE Infocom, [6] C. Perkins, Ad-hoc on-demand disance ecor roing, in MILCOM 97, No [7] C. Perkins and P. Bhaga, Highl dnamic desinaion-seqenced disance-ecor roing, in In Proc. of he ACM SIGCOMM, Ocober, [8] P. Sinha, R. Siakmar, and V. Bharghaan, Cedar: Core exracion disribed ad hoc roing, in Proc. of IEEE INFOCOM, [9] E. Roer and C. Toh, A reie of crren roing proocols for ad-hoc mobile ireless neorks, IEEE Personal Commnicaions, Apr [10] S. Ramanahan and M. Seensrp, A sre of roing echniqes for mobile commnicaion neorks, ACM/Baler Mobile Neorks and Applicaions, pp , [11] X Lin and Ian Sojmenoic, GPS based disribed roing algorihms for ireless neorks, 000. [1] P. Bose, P. Morin, I. Sojmenoic, and J. Urria, Roing ih garaneed delier in ad hoc ireless neorks, in 3rd in. Workshop on Discree Algorihms and mehods for mobile comping and commnicaions, /0/$17.00 (c) 00 IEEE. 175 IEEE INFOCOM 00 Ahoried licensed se limied o: CiU. Donloaded on Ma,0 a 08:10:13 UTC from IEEE Xplore. Resricions appl.

9 [13] B. Karp and H. T. Kng, Gpsr: Greed perimeer saeless roing for ireless neorks, in ACM/IEEE Inernaional Conference on Mobile Comping and Neorking, 000. [14] E. Kranakis, H. Singh, and J. Urria, Compass roing on geomeric neorks, in Proc. 11 h Canadian Conference on Compaional Geomer, 1999, pp [15] Franco P. Preparaa and Michael Ian Shamos, Compaional Geomer: an Inrodcion, Springer-Verlag, [16] A. C.-C. Yao, On consrcing minimm spanning rees in k-dimensional spaces and relaed problems, SIAM J. Comping, ol. 11, pp , 198. [17] J. M. Keil and C. A. Gin, Classes of graphs hich approximae he complee eclidean graph, Discree Compaional Geomer, ol. 7, 199. [18] D.P. Dobkin, S.J. Friedman, and K.J. Spoi, Delana graphs are almos as good as complee graphs, Discree Compaional Geomer, [19] J.M. Keil and C.A. Gin, The Delana rianglaion closel approximaes he complee eclidean graph, in Proc. 1s Workshop Algorihms Daa Srcre (LNCS 38), [0] P.L. 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[6] Tamas Lkoski, Ne Resls on Geomeric Spanners and Their Applicaions, Ph.D. hesis, Uniersi of Paderborn, [7] Mahias Fischer, Tamas Lkoski, and Marin Ziegler, Pariioned neighborhood spanners of minimal odegree, Tech. Rep., Hein Nixdore Insie, German, [8] P. Bose, A. Brodnik, S Carlsson, E. D. Demaine, R. Fleischer, A. Lope- Ori, P. Morin, and J. I. Mnro, Online roing in conex sbdiisions, in Inernaional Smposim on Algorihms and Compaion, 000, pp [9] Xiang-Yang Li, Peng-Jn Wan, Y Wang, and Ophir Frieder, Coerage in ireless ad-hoc sensor neorks, in ICC, 00, To appear. IX. APPENDIX Lemma 15: If an edge x inersecs a localied Delana riangle, hen i inersecs o edges of. Proof: If i inersecs one edge of, hen eiher x or ms be inside he riangle, sa x. Then x < max(, ) 1, hich conradics ha is a localied Delana riangle. Then e presen he proof of Theorem 10. Proof: There are hree cases: riangles and x share o nodes (i.e., one edge), one node or do no share an node. Case 1: riangles and x share one edge. We proe ha his case is impossible. For he sake of conradicion, assme ha i is possible and he share an edge. In oher ords, e hae o localied Delana riangles and ha inersec. Noice ha and can no be eqal becase e assme ha no for nodes are co-circle. Assme ha <. Then he circmcircle disk(,, ) conains node inside. Noice ha node N 1 (). Ths, riangle does no saisf he localied Delana proper. I is a conradicion o he exisence of riangle in LDel (1) (S) (x) Case 1 Sbcase.1 Fig. 7. To inerseced riangles share an edge or a node. Case : riangles and x share one node. We also proe ha his case is impossible. For he sake of conradicion, assme ha i is possible and = x. Then he exisence of he riangle implies ha and ms be oside of disk(,, ) becase boh and are from N 1 (). Then here are hree sbcases abo he locaions of he segmen x and x. (x) (x) 0 1 Sbcase. Sbcase.3 Fig. 8. To inerseced riangles share a node. Sbcase.1: none of he segmens x and x inersecs he riangle. Then segmen ms inersec boh and. I can no inersec segmen ; oherise, eiher or is inside he riangle x. The righ figre in Figre 7 illsraes he proof ha follos. Le be he inersecion poin of segmen ih disk(,, ), hich is close o. Le be he oher inersecion poin of ih he circmcircle disk(,, ). Then x + > x + = π. I implies ha node is inside he circmcircle disk(x,, ). Noice ha x = 1. Therefore here exiss a node from N 1 (x) ha is inside disk(x,, ), hich conradics ha x is a localied Delana riangle. Sbcase.: onl one edge of x and x ha inersecs he riangle. Le s sa x. Then segmen ms inersec boh edges and. Oherise is inside he riangle x, hich conradics he exisence of riangle x. The lef figre in Figre 7 illsraes he proof ha follos. Le be anoher inersecion poin of segmen ih disk(,, ). Le be he inersecion poin of segmen ih he circmcircle disk(,, ), hich is close o. Then x + > x + > x + = π. I implies ha node is inside he circmcircle disk(x,, ). Noice ha x = 1. Therefore here exiss a node from N 1 (x) ha is inside disk(x,, ), hich conradics ha x is a localied Delana riangle. Sbcase.3: Boh segmens x and x inersec he riangle. The righ figre in Figre 7 illsraes he proof ha follos. Le be anoher inersecion poin of segmen ih disk(,, ). Le be anoher inersecion poin of segmen ih he circmcircle disk(,, ). Then /0/$17.00 (c) 00 IEEE. 176 IEEE INFOCOM 00 Ahoried licensed se limied o: CiU. Donloaded on Ma,0 a 08:10:13 UTC from IEEE Xplore. Resricions appl.

10 < + + = + + = π. I implies ha ( x + x)+( x + x) = 3π ( + + ) > π. Then from he pigeonhole principle, e hae eiher x+ x > π or x+ x > π. Conseqenl, he circmcircle disk(x,, ) of he riangle x conains eiher or in is inerior. This conradics o ha x is a localied Delana riangle. From he aboe analsis of case, o inerseced riangles and x can no share one common node, sa = x, becase in all hree cases, or ms be in he inerior of he circmcircle of and N 1 () or N 1 (). Case 3: riangles and x do no share an node. Wiho loss of generali, assme ha none of he nodes of x is conained inside he circmcircle disk(,, ). I is no difficl o sho ha here are onl o possible sbcases as illsraed b Figre 9. We hen proe ha disk(x,, ) conains a leas one of he nodes of,, and. x x Sbcase 3.1 Sbcase 3. Fig. 9. All or for edges of o riangles inersec. Sbcase 3.1: all edges of x and are inerseced b some edges of he oher riangle. Assme ha he nodes hae he order as illsraed b he lef figre in Figre 9. Then i is eas o sho ha all angles x, x,,,, x are less han π. Noice ha x + < π becase x is no inside he circmcircle disk(,, ). Similarl + < π and + < π. Therefore x+ + < 3π ( + + ) =π. Noice ha x+ + + x + + x = 4π. I implies ha x + + x > π. Then e kno ha a leas one of he nodes of,, and is conained inside he circmcircle disk(x,, ) (oherise b smmer, similarl e old hae x + + x < π). We hen proe ha sbcase 3.1 is impossible. For he sake of conradicion, assme ha i is possible. Then from he proof of he sbcase 3.1, eiher disk(,, ) conains one of he nodes of x, and ; or disk(x,, ) conains a leas one of he nodes of,, and. Wiho loss of generali, assme ha node x is conained in he inerior of disk(,, ). Then Lemma 8 implies ha boh and are oside of disk(,, ). The folloing Figre 10 illsraes he proof ha follos. The exisence of riangle implies ha x > 1, x > 1, and x > 1. Noice ha x 1 and x 1. Le c be he circmcener of he riangle. Here c can no be x becase x > 1, x 1 and is oside of he circle. Noice ha he angle x < π 3 becase ms be he shores edge of riangle x. Consider he folloing fie segmens ling in he inerior of he edge x: x, x, x, x, and x. From he pigeonhole principle, here are a leas hree sch segmens ling on he same side of he line xc. x x x c Fig. 10. Sbcase 3.1 is impossible. More precisel, e hae eiher x, x and x are on he same side of xc or x, x and x are on he same side of xc. Wiho loss of generali, assme ha he firs scenario happens. Then i is eas o proe ha x > min(x, x) > 1.This conradics o x 1. The righ figre of Figre 10 illsraes he proof sing ha x = xc + c xc c cos( xc),and c = c = c. Therefore, he assmpion ha sbcase 3.1 is possible does no hold. Sbcase 3.: one edge of each riangle is no inerseced b he edges of he oher riangle. We hen proe ha disk(x,, ) conains a leas one of he nodes of,, and. The righ figre of Figre 9 illsraes he proof ha follos. Le x be he inersecion poin of segmen x ih he circmcircle disk(,, ), hich is close o x. Le be he inersecion poin of segmen ih he circmcircle disk(,, ). Le x and be he o inersecion poins of segmen x ih he circmcircle disk(,, ), here x is close o x and is close o. Then x < x = x < x,and x < x = x < x.noice ha + x + x + x + = 3π.Then ( x + x)+( x + x) =3π ( x + x + x) > 3π ( x + x + x) =π. So eiher x + x > π or x + x > π from he pigeonhole principle. Conseqenl, disk(x,, ) conains eiher node or node. x c /0/$17.00 (c) 00 IEEE. 177 IEEE INFOCOM 00 Ahoried licensed se limied o: CiU. Donloaded on Ma,0 a 08:10:13 UTC from IEEE Xplore. Resricions appl.