Competitive Routing in the Half-θ 6 -Graph

Transcription

1 Compeiie Roing in he Half-θ 6 -Graph Proenji Boe Rolf Fagererg André an Renen Sander Verdoncho Arac We preen a deerminiic local roing cheme ha i garaneed o find a pah eeen any pair of erice in a halfθ 6-graph hoe lengh i a mo 5/ 3 = ime he Eclidean diance eeen he pair of erice. The half-θ 6- graph i idenical o he Delanay rianglaion here he empy region i an eqilaeral riangle. Moreoer, e ho ha no local roing cheme can achiee a eer compeiie panning raio herey implying ha or roing cheme i opimal. Thi i omeha rpriing ecae he panning raio of he half-θ 6-graph i 2. Since eery rianglaion can e emedded in he plane a a half-θ 6-graph ing O(log n) i per erex coordinae ia Schnyder emedding cheme (SODA 1990), or rel proide a compeiie local roing cheme for eery ch emedded rianglaion. 1 Inrodcion A fndamenal prolem in neorking i he roing of a meage from one erex o anoher in a graph. Wha make roing more challenging i ha ofen in a neork he roing raegy m e local. Informally, a roing raegy i local, hen he roing algorihm m decide hich erex o forard a meage o aed olely on knoledge of he orce and deinaion erex, he crren erex and all erice direcly conneced o he crren erex. Roing algorihm are conidered geomeric hen he nderlying graph i emedded in he plane. Edge are egmen connecing pair of poin and are eighed y he Eclidean diance eeen heir endpoin. Geomeric roing algorihm are imporan in irele enor neork (ee [13] and [15] for rey of he area) ince hey offer roing raegie ha e he coordinae of he erice o help gide he earch a oppoed o ing he more radiional roing ale. Papadimirio and Raajczak [14] poed a analizing qeion in hi area ha led o a flrry of aciiy: Doe eery 3-conneced planar graph hae a raigh- Reearch ppored in par y NSERC and he Danih Concil for Independen Reearch, Naral Science. School of Comper Science, Carleon Unieriy. ji@c.carleon.ca, arene@connec.carleon.ca, erdon@connec.carleon.ca. Deparmen of Mahemaic and Comper Science, Unieriy of Sohern Denmark. rolf@imada.d.dk. line emedding in he plane ha admi a local roing raegy ch a greedy 1 roing? They proided a parial aner y hoing ha 3-conneced planar graph can alay e emedded in R 3 ch ha hey admi a greedy roing raegy. They alo hoed ha he cla of complee iparie graph, K k,6k+1 for all k 1 canno e emedded ch ha greedy roing alay cceed ince eery emedding ha a lea one erex ha i no conneced o i neare neighor. Boe and Morin [4] hoed ha greedy roing alay cceed on Delanay rianglaion. In fac, a lighly rericed greedy roing raegy knon a greedy-compa i he fir local roing raegy hon o cceed on all rianglaion [3]. Dhandapani [6] proed he exience of an emedding ha admi greedy roing for eery rianglaion and Angelini e al.[1] proided a conrcie proof. Leighon and Moira [12] eled Papadimirio and Raajczak qeion y hoing ha eery 3-conneced planar graph can e emedded in he plane ch ha greedy roing cceed. One draack of hee emedding algorihm i ha he coordinae reqire Ω(n log n) i per erex. To addre hi, He and Zhang [9] and Goodrich and Srah [8] gae ccinc emedding ing only O(log n) i per erex. Recenly, He and Zhang [10] hoed ha eery 3-conneced plane graph admi a ccinc conex emedding 2 on hich a lighly modified greedy roing raegy alay cceed. In ligh of hee recen ccee, i i rpriing o noe ha he aoe roing raegie hae olely concenraed on finding an emedding ha garanee a local roing raegy ill cceed. No aenion i paid o he qaliy of he reling pah relaie o he hore pah. None of he aoe roing raegie hae een hon o e compeiie 3. Boe and Morin 1 A roing raegy i greedy hen a meage i alay forarded o he erex hoe diance o he deinaion i he malle among all erice in he neighorhood of he crren erex inclding he crren erex. 2 An emedding of a planar graph i conex hen eery face i conex. 3 A roing raegy i compeiie if he pah fond y he roing raegy i no more han a conan (he compeiie panning raio) ime he hore pah. The compeiie panning raio of a graph i defined a he minimm compeiie panning raio oer all roing raegie.

2 [4] ho ha many local roing raegie are no compeiie ho ho o roe compeiiely on he Delanay rianglaion. Hoeer, Dillencor [7] hoed ha no all rianglaion can e emedded in he plane a Delanay rianglaion. Thi raie he folloing qeion: can eery rianglaion e emedded in he plane ch ha i admi a compeiie local roing raegy? We aner hi qeion in he affirmaie. The half-θ 6 -graph a inrodced y Bonichon e al. [2] ho hoed ha i i idenical o he Delanay rianglaion here he empy region i an eqilaeral riangle. Or main rel i a deerminiic local roing cheme ha i garaneed o find a pah eeen any pair of erice in a half-θ 6 -graph hoe lengh i a mo 5/ 3 = ime he Eclidean diance eeen he pair of erice. On he ay o proing or main rel, e ncoer ome local properie of panning pah in he half-θ 6 -graph. Since Schnyder [16] hoed ha eery rianglaion can e emedded in he plane a a half-θ 6 -graph ing O(log n) i per erex coordinae, or main rel implie ha a compeiie local roing cheme exi for eery rianglaion emedded a ch. Moreoer, e ho ha no local roing cheme can achiee a eer compeiie panning raio on half-θ 6 -graph, implying ha or roing cheme i opimal. Thi i omeha rpriing ecae Che [5] hoed ha he panning raio of he halfθ 6 -graph i a mo 2. Th, or loer ond proide a eparaion eeen he panning raio of he half-θ 6 - graph and he compeiie panning raio of any local roing cheme on he half-θ 6 -graph. Finally, e conclde y highlighing ome imilariie and difference eeen he half-θ 6 -graph and he fll-θ 6 -graph [11]. 2 Preliminarie In a eighed graph G, le he diance d G (, ) eeen o erice and e he lengh of he hore pah eeen and in G. A graph H of G i a -panner of G if for all pair of erice and, d H (, ) d G (, ), 1. I panning raio i he malle for hich i i a -panner. The graph G i referred o a he nderlying graph. We conider he iaion here he nderlying graph G i a raighline emedding of he complee graph on a e of n poin in he plane denoed y K n, ih he eigh of an edge (, ) eing he Eclidean diance eeen and. A panner of ch a graph i called a geomeric panner. In hi paper, e ho ho o roe compeiiely on he geomeric panner called he half-θ 6 -graph [2]. To define hi graph, e need he folloing erminology. Le a cone C e he region in he plane eeen o ray originaing from he ame poin (referred o a he apex of he cone). For each erex of K n conider he ix ray originaing from ih angle o he poiie x-axi eing mliple of π/3. Each pair of conecie ray define a cone. Le C 1, C 0, C 2, C 1, C 0, C 2 e he eqence of cone in conerclockie order aring from he poiie x-axi, a depiced in Figre 1, and call C 0, C 1, and C 2 poiie cone, and C 0, C 1, and C 2 negaie cone. The name are choen ch ha ing modlo 3 arihmeic on he indice, a poiie cone C i ha he negaie cone C i+1 (C i 1 ) a i clockie (conerclockie) neighor. An analogo aemen hold for a negaie cone C i. When he apex i no clear from he conex, e e Ci o denoe cone C i ih apex. C 2 C 0 C 1 C 1 C 0 C 2 Figre 1: The cone haing apex. The half-θ 6 -graph [2] i conrced a follo: for each of he hree poiie cone of each erex, add an edge from o he cloe erex in ha cone, here diance i meared along he iecor of he cone. More formally, e add an edge eeen o erice and if Ci and for all poin Ci ( ),, here and denoe he orhogonal projecion of and, repeciely, on he iecor of Ci. For eae of expoiion, e only conider poin e in general poiion: no o poin lie on a line parallel o one of he ray ha define he cone. Thi implie ha each erex add a mo one edge per poiie cone o he graph, and hence here are a mo 3n edge in oal. Gien a erex in a poiie cone Ci of erex, e define he canonical eqilaeral riangle T o e he riangle defined y he order of Ci and he line hrogh perpendiclar o he iecor of Ci. Figre 2 gie an example ih C0. Noe ha for any pair of erice and, eiher lie in a poiie cone of, or lie in a poiie cone of, o here i exacly one canonical eqilaeral riangle (eiher T or T ) for he pair.

3 m α ame.l.o.g. ha lie in C0. We proe he heorem y indcion on he area of T (formally, indcion on he rank, hen ordered y area, of he riangle T xy for all pair of poin x and y). Le a and e he pper lef and righ corner of T, and le A = T C1 and B = T C2 (ee Figre 3). Figre 2: The canonical eqilaeral riangle T of and, i iecor m, and he angle α eeen and he iecor. a A B 3 Spanning Raio of he Half-θ 6 -Graph Bonichon e al. [2] hoed ha he half-θ 6 -graph i eqialen o he Delanay rianglaion aed on empy eqilaeral riangle, hich i knon o hae panning raio 2 [5]. In hi ecion, e proide an alernaie proof of he panning raio of he half-θ 6 -graph. Or proof ho ha eeen any pair of poin,, here alay exi a pah ih panning raio 2 ha lie in he canonical riangle. Thi i a key propery ed y or roing algorihm. For a pair of erice and, or ond i expreed in erm of he angle α eeen he line from o and he iecor of heir canonical eqilaeral riangle. See Figre 2. Theorem 3.1. Le and e erice ih in a poiie cone of. Le m e he midpoin of he ide of T oppoing, and le α e he nigned angle eeen he line and m. There exi a pah in he halfθ 6 -graph of lengh a mo ( 3 co α + in α) here all erice on hi pah lie in T. The expreion 3 co α + in α i increaing for α [0, π/6]. Inering he exreme ale π/6 for α, e arrie a he folloing. Corollary 3.1. The panning raio of he half-θ 6 - graph i 2. We noe ha he ond of Theorem 3.1 and Corollary 3.1 are igh: for all ale of α [0, π/6] here exi a poin e for hich he hore pah in he halfθ 6 -graph for ome pair of erice and ha lengh arirarily cloe o ( 3 co α + in α). A imple example appear laer in he proof of Theorem 4.1. Proof of Theorem 3.1. Gien o erice and, e Figre 3: The corner a and, and he area A and B. Or indcie hypohei i he folloing, here δ(, ) denoe he lengh of he hore pah from o in he half-θ 6 -graph ih all erice lying in T : If A i empy, hen δ(, ) +. If B i empy, hen δ(, ) a + a. If neiher A nor B i empy, hen δ(, ) max{ a + a, + }. We fir noe ha hi indcion hypohei implie Theorem 3.1: ing he ide of T a he ni of lengh, e hae m = in α and 3/2 = m = co α (ee Figre 2), hence he indcion hypohei gie ha δ(, ) i a mo 1+1/2+ m = 3 ( 3/2) + m = ( 3 co α + in α). Bae cae: T ha rank 1. Since he riangle i a malle riangle, i he cloe erex o in a poiie cone of. Hence he edge (, ) i in he half-θ 6 -graph, and δ(, ) =. From he riangle ineqaliy, e hae min{ a + a, + }, o he indcion hypohei hold. Indcion ep: We ame ha he indcion hypohei hold for all pair of poin ih canonical riangle of rank p o k. Le T e a canonical riangle of rank k + 1. If (, ) i an edge in he half-θ 6 -graph, he indcion hypohei follo y he ame argmen a in he ae cae. If here i no edge eeen and, le e he erex cloe o in he poiie cone C0, and le a and e he pper lef and righ corner of T, repeciely. See Figre 5. By definiion, δ(, ) + δ(, ), and y he riangle ineqaliy, min{ a + a, + }.

4 d d a a a Figre 4: Vializaion of he pah ineqaliie. Thick, dark gray line ignify pah occrring in he ineqaliie, and ligh gray area indicae empine. a a A c d C D B (a) a a a A E () Figre 5: The o cae: (a) lie in neiher A nor B, () lie in A. We perform a cae analyi aed on he locaion of : (a) lie neiher in A nor in B, () lie inide A, and (c) lie inide B. Cae (c) i analogo o Cae (), o e only dic he fir o cae. Cae (a): Le c and d e he pper lef and righ corner of T, repeciely, and le C = T C1 and D = T C2. See Figre 5. Since T ha maller area han T, e apply he indcie hypohei on T. Or ak i o proe all hree aemen of he indcie hypohei for T. 1. If A i empy, hen C i alo empy, o y indcion δ(, ) d + d. Since, d,, and form a parallelogram, e hae: (3.1) (3.2) (3.3) δ(, ) + δ(, ) + + d + d = +, hich proe he fir aemen of he indcion hypohei. Thi argmen i illraed in Figre 4, lef, here hick, dark grey line ignify pah occrring in he ineqaliie aoe, and ligh gray area indicae empine. B 2. If B i empy, an analogo argmen proe he econd aemen of he indcion hypohei. 3. If neiher A nor B i empy, y indcion e hae δ(, ) max{ c + c, d + d }. Ame, iho lo of generaliy, ha he maximm of he righ hand ide i aained y i econd argmen d + d (he oher cae i analogo). Since erice, d,, and form a parallelogram, e hae ha: (3.4) (3.5) (3.6) (3.7) δ(, ) + δ(, ) + + d + d + max{ a + a, + }, hich proe he hird aemen of he indcion hypohei. Thi argmen i illraed in Figre 4, middle. Cae (): Le E = T T, and le a e he pper lef corner of T. See Figre 5. Since i he cloe erex o in he poiie cone C0, T i empy. Hence, E i empy. Since T i maller han T, e can apply indcion on i. A E i empy, he fir aemen of he indcion hypohei for T gie δ(, ) a + a. Since a + a and, a, a, and a form a parallelogram, e hae ha δ(, ) a + a, proing he econd and hird aemen in he indcion hypohei for T. Thi argmen i illraed in Figre 4, righ. Since lie in A, he fir aemen in he indcion hypohei for T i acoly re. 4 Roing in he Half-θ 6 -Graph In hi ecion, e gie maching pper and loer ond for he compeiie roing raio on he half-θ 6 - graph. We egin y defining or model. A deerminiic k-local roing cheme defined on a graph G i a fncion f(,, N k ()) pecifying he erex he meage

5 m m m α α α Inance 1 Inance 2a Inance 2 Figre 6: Loer ond inance. hold e forarded o gien ha i he crren erex, i he arge or deinaion erex, and N k () i he k-neighorhood of. The k-neighorhood of a erex i he e of erice in he graph ha can e reached from y folloing a mo k edge. When k = 1, e drop he ale and refer o he cheme a a local roing cheme. We noe ha in he lierare, many arian of hi model hae een died here he roing cheme kno hich node en he meage o or he cheme ha ome memory. Hoeer, e dy one of he eake model and ho ha i i ill poile o roe compeiiely. Or pper ond hold for k = 1 and or loer ond hold for any fixed k. Since or graph are geomeric, he idenifier for a erex i i coordinae in he plane. The compeiie roing raio of a roing cheme i defined analogoly o he panning raio a he malle 1 for hich no roe comped y he roing cheme eeen any pair of erice i longer han ime he Eclidean diance eeen ha pair. Or ond are expreed in erm of he angle α eeen he line from he orce o he deinaion poin and he iecor of heir canonical eqilaeral riangle. See Figre 2. Theorem 4.1. Le and e erice ih in a poiie cone of. Le m e he midpoin of he ide of T oppoing, and le α e he nigned angle eeen he line and m. There i a local roing cheme on he half-θ 6 -graph for hich eery pah folloed ha lengh a mo i) ( 3 co α + in α) hen roing from o, ii) (5/ 3 co α in α) hen roing from o, and hi i e poile for deerminiic k-local roing cheme. The fir expreion i increaing for α [0, π/6], hile he econd expreion i decreaing. Inering he exreme ale π/6 and 0 for α, e ge he folloing or cae erion of Theorem 4.1. Corollary 4.1. Le and e o erice ih in a poiie cone of. There i a local roing cheme on he half-θ 6 -graph ih roing raio i) 2 hen roing from o, ii) 5/ 3 = hen roing from o, and hi i e poile for deerminiic k-local roing cheme. Since he panning raio of he half-θ 6 -graph i 2, he econd loer ond ho a eparaion eeen he panning raio and he compeiie roing raio in he half-θ 6 -graph. Since eery rianglaion can e emedded in he plane a a half-θ 6 -graph ing O(log n) i per erex ia Schnyder emedding cheme [16], an imporan implicaion of Theorem 4.1 i he folloing. Corollary 4.2. Eery n-erex rianglaion can e emedded in he plane ing O(log n) i per coordinae ch ha he emedded rianglaion admi a deerminiic local roing cheme ih compeiie roing raio a mo 5/ 3. In he remainder of hi ecion, e proe Theorem 4.1. We fir proe he loer ond, hen decrie he roing cheme, and finally proe he aed pper ond. Loer ond. Le he ide of T e he ni of lengh. From Figre 2 e hae m = in α and 3/2 = m = co α. From Inance 1 in Figre 6, he panning raio of he half-θ 6 -graph i a lea 1 + 1/2 + m = 3 ( 3/2) + m = ( 3 co α + in α) (ince he poin in he pper lef corner of T can e moed arirarily cloe o he corner). Thi i a loer ond for any roing cheme, and proe or aemen on roing from o. For roing from o, conider inance 2a and 2 in Figre 6. Any deerminiic 1-local roing cheme only ha informaion ao direc neighor,

6 hence canno diingih eeen he o inance hen roing o of. So a deerminiic algorihm m roe o he ame neighor of in oh inance, and eiher choice of neighor lead o a non-opimal roe in one of he o inance. The malle lo occr hen he choice i oard he cloe corner of T (making inance 2a he hard inance), hich gie a loer ond of (1/2 m ) = 5/2 m = (5/ 3 co α in α) (ince he poin in he corner of T can e moed arirarily cloe o he corner hile keeping heir relaie poiion). In fac, hi i preciely here he eparaion occr eeen he panning raio of 2 and he compeiie roing raio. Noe ha y exending inance 2a and 2 of Figre 6 o hae Ω(k) poin cloe o he corner ch ha i no in he k-neighorhood of, he loer ond hold for any deerminiic k-local roing cheme. Roing cheme. We le denoe he crren erex, and he fixed deinaion. The roing cheme need o deermine hich edge (, ) o follo nex. We ay e are crrenly roing poiiely (negaiely) hen i in a poiie (negaie) cone of. For eae of decripion, e ame.l.o.g. ha i in cone C 0 of hen roing poiiely, and in cone C 0 of hen roing negaiely. When roing poiiely, T inerec only C 0 among he cone of. When roing negaiely, T inerec C 0 a ell a he o poiie cone C 1 and C 2 of. We le X 0 = C 0 T, X 1 = C 1 T, and X 2 = C 2 T. We le a e he corner of T conained in X 1 and he corner of T conained in X 2. Thee definiion are illraed in Figre 7. C 0 Roing poiiely a C 1 X 1 X 0 X 2 C 0 Roing negaiely Figre 7: Roing erminology. The roing cheme ill only follo edge (, ) here lie in he canonical eqilaeral riangle of and. Roing poiiely i raighforard ince here i exacly one edge (, ) ih T, y he conrcion of he half-θ 6 -graph. The challenge i o roe negaiely. When roing negaiely, a lea one edge (, ) ih T exi, ince and are conneced y a pah in T, according o Theorem 3.1. The core of or roing cheme i ho o chooe hich C 2 edge o follo hen here are more han one. Iniiely, hen roing negaiely, or cheme rie o elec an edge ha make mearale progre oard he deinaion. When no ch edge exi, e are forced o ake an edge ha doe no make mearale progre, hoeer e are ale o hen dedce ha cerain region ihin he canonical riangle are empy. Thi allo o proe ha e ake ch an edge a mo once. In eence, e proe ha e can go in he rong direcion only once. Thi can e een in he inance 2a and 2 in Figre 6 here an aderary force a roing cheme o go in he rong direcion once. We proide a formal decripion of or roing cheme elo. Or roing algorihm can e in one of for ae. We call he iaion hen roing poiiely ae A, and diide he iaion hen roing negaiely ino hree ae B, C, and D, a follo: By conrcion of he half-θ 6 -graph, here i a mo one edge (, ) ih X 1 and he ame applie o X 2 ince hey are oh poiie cone of. Le ae B e he cae here oh X 1 and X 2 conain an edge, ae C he cae here exacly one conain an edge, and ae D he cae here oh are empy. A he ar of a roing ep, e are in exacly one of he ae A, B, C, or D. Roing in ae A i raighforard. We no decrie roing in ae B, C, and D. In ae B, he roing cheme fir rie o follo an edge (, ) ih X 0. If eeral ch edge exi, an arirary one of hee i folloed. If no ch edge exi, he roing cheme follo an edge in X 1 or X 2 : if a, i follo he ingle edge (, ) ih X 1 ; if a >, i follo he ingle edge (, ) ih X 2. In hor, he roing cheme faor moing oard he cloe corner of T hen i i no ale o moe direcly oard. Noe ha hi choice i made pecifically o enre e can ond he oal diance raelled hen faced ih inance imilar o 2a and 2 in Figre 6. In ae C and D, he roing cheme fir rie o follo an edge (, ) ih X 0. If eeral ch edge exi, i chooe a pecific one aed on ha e call he projeced lengh on a neighoring cone. Le e 1 ( e 2 ) e a ni ecor in he direcion of he ray from coniing he order of C 0 and C 1 (C 2 ). Since e 1 and e 2 are linearly independen, he ecor can e rien niqely a l 1 e 1 +l 2 e 2. We define he projeced lengh of he edge (, ) on he neighoring cone C 1 (C 2 ) o e l 1 (l 2 ). Figre 9 illrae he projeced lengh on C 1. In ae C, exacly one of X 1 or X 2 i empy. If here exi edge (, ) ih X 0, he roing cheme ill follo one of hee, chooing among hem in he folloing ay: If X 1 i empy, i chooe he edge ih large projeced lengh on C 1. Ele X 2 i empy, and i chooe he edge ih large projeced lengh on C 2.

7 a a a a Sae A Sae B Sae C Sae D Figre 8: The poenial φ in each of he ae. C 1 Figre 9: Projeced lengh on C 1. In hor, he roing cheme faor aying cloe o he empy ide of T. If no edge (, ) ih X 0 exi, he roing cheme follo he ingle edge (, ) ih in X 1 X 2. In ae D, oh X 1 and X 2 are empy, o here m exi edge (, ) ih X 0, a and are conneced y a pah in T. If a, he roing cheme follo he edge ih large projeced lengh on C 1 ; if a <, i follo he edge ih large projeced lengh on C 2. In hor, hen oh ide of T are empy, he roing cheme faor aying cloe o he large empy ide of T. Upper ond. The proof of he pper ond e a poenial fncion φ, defined a follo for each of he ae A, B, C, and D, here a and are he corner of T (T ) differen from (), and x {a, } in ae C i he corner conained in he non-empy one of he o area X 1 and X 2. Sae A: Sae B: Sae C: Sae D: φ = a + max( a, ) φ = a + a + min( a, ) φ = a + x φ = a + min( a, ) Thi definiion i illraed in Figre 8, here arline deignae poenial and gray area are empy. We ill refer o he fir erm of φ (i.e., a in ae A, a in ae B, C, and D) a he erical par of φ and o he re a he horizonal par. Or aim i o proe ha for any roing ep, he redcion in φ i a lea a large a he lengh of he edge folloed. Since φ i alay non-negaie, hi ill imply ha no pah folloed can e longer han he iniial ale of φ. A all edge hae ricly poiie lengh, he roing cheme m erminae. Frhermore, he iniial ale of φ in ae A and ae B are exacly he pah lengh appearing in he calclaion leading o he loer ond of inance 1 and 2a/2, repeciely, o he ame ond apply, no a pper ond. The iniial ale of φ in ae C and D are maller han he iniial ale in ae B, and hence can only lead o eer roing ond. Thi gie he pper ond aed in Theorem 4.1. Wha remain i o proe he aemen ha for any roing ep, he redcion in φ i a lea a large a he lengh of he edge folloed. We do hi y cae analyi of he poile roing ep. One imple oeraion ha i repeaedly ed in or analyi i mmarized in he fac elo. Fac 4.1. In an eqilaeral riangle, he ide lengh i he diameer, i.e. he longe diance defined y any o poin in he riangle. Sae A. For a roing ep aring in ae A, i eiher in a negaie or a poiie cone of. The fir iaion lead o ae A again. The econd lead o ae C or D, ince he area of T eeen and m e empy y conrcion of he half-θ 6 -graph. Thee iaion are illraed in Figre 10. For he cae ending in ae A, he redcion of he erical par of φ i a lea a large a y Fac 4.1. The horizonal par of φ can only decreae dring he ep. Hence he aemen hold for hi cae. The ame ype of analyi proe he aemen for he cae ending in ae C. For he cae ending in ae D, he final ale of φ can only e maller han for he cae ending in ae C, o again he aemen hold. Sae B. A roing ep aring in ae B canno lead o ae A, a he ep ay ihin T, i may

8 or Figre 10: Roing in ae A. Figre 11: Roing in ae B. lead o ae B, C, and D. There are o cae, depending on heher edge (, ) ih X 0 exi or no. Thee cae are illraed in Figre 11 for he iaion leading o ae B. In he fir cae, he redcion of he erical par of φ i a lea a large a, y Fac 4.1. The horizonal par of φ can only decreae. In he econd cae, he redcion of he horizonal par of φ i a lea a large a, y Fac 4.1. The erical par of φ can only decreae. In oh cae, he aemen i proen. If he ep inead lead o ae C or D (no illraed), he ale of φ afer he ep can only e maller han aoe, o he aemen alo hold here. Sae C. A roing ep aring in ae C canno lead o ae A, a he ep ay ihin T. We fir proe ha i canno lead o ae B, eiher. There are o cae, depending on heher edge (, ) ih X 0 exi or no. For he cae here edge (, ) ih X 0 do exi, he iaion a he ar of he ep i illraed lef of he arro in he lef half of Figre 12. By he conrcion of he half-θ 6 -graph, he exience of he edge (, ) implie ha he horizonally hached area m e empy. From hi i follo ha he erically hached area m alo e empy: if no, he opmo poin in i old hae an edge o, hile haing larger projeced lengh on he neighoring cone, conradicing he choice of in he roing algorihm. For he cae here edge (, ) ih X 0 do no exi, he iaion a he ar of he ep i illraed lef of he arro in he righ half of Figre 12. The horizonally hached area m e empy y conrcion of he half-θ 6 -graph. From hi i follo ha he erically hached area m alo e empy: if no, he opmo poin in i old hae an edge o, conradicing ha edge (, ) ih X 0

9 Figre 12: Roing in ae C. do no exi. Th, in oh cae he roing ep can only lead o ae C or D. The iaion hen ending p in ae C are illraed righ of he arro of Figre 12. In he lef half of Figre 12, he redcion of he erical par of φ i a lea a large a (y Fac 4.1) and he horizonal par can only decreae. In he righ half of Figre 12, he redcion of he horizonal par of φ i a lea a large a (y Fac 4.1) and he erical par can only decreae. In oh iaion, he aemen i proen. If he ep inead lead o ae D (no illraed), he ale of φ afer he ep can only e maller han hen leading o ae C, o he aemen alo hold here. Sae D. A roing ep aring in ae D (no illraed) ha exacly he ame analyi a he fir cae of a roing ep aring in ae C. Th, he aemen i proen in all cae, hich complee he proof of Theorem Conclding Remark The fll-θ 6 -graph, inrodced y Keil and Gin [11], i imilar o he half-θ 6 -graph excep ha all 6 cone are poiie cone. Th, he fll-θ 6 -graph i he nion of o copie of he half-θ 6 -graph, here one half-θ 6 - graph i roaed y π/3 radian. The half-θ 6 -graph and he fll-θ 6 -graph oh hae a panning raio of 2, ih imple loer ond example hoing ha i i igh for oh graph. Thi i rpriing ince he fll-θ 6 - graph can hae dole he nmer of edge of he halfθ 6 -graph. Or rel ho ha alhogh he panning raio are he ame, he compeiie roing raio differ. Since he fll-θ 6 -graph ha no negaie cone, i ha a compeiie roing raio of 2, hich i eqal o he panning raio and herefore i opimal. Noe ha ince he fll-θ 6 -graph coni of o roaed copie of he half-θ 6 -graph, one qeion ha come o mind i ha i he e panning raio if one i o conrc a graph a o roaed copie of he half-θ 6 -graph? Can one do eer han a panning raio of 2? Conider he folloing conrcion. Bild o half-θ 6 -graph a decried in Secion 2, roae each cone of he econd graph y π/6 radian. By Theorem 3.1, he panning raio for each of hee graph i 3 co α + in α, here α i he angle eeen he line connecing he erice in qeion and he cloe iecor. Since hi fncion i increaing, he panning raio i defined y he maximm poile angle o he cloe iecor, hich in hi conrcion i only π/12 radian, giing a panning raio of roghly By ing more copie, e improe he panning raio een frher: if each i roaed y π/(3k) radian, e ge a panning raio of 3 co π/(6k) + in π/(6k). Thi i eer han he knon pper ond for he fll θ 3k -graph [11] for k 4 and eer han he Yao 3k - graph [17] for k 5. Reference [1] P. Angelini, F. Frai, and L. Grilli. An algorihm o conrc greedy draing of rianglaion. J. Graph Algorihm Appl., 14(1):19 51, [2] N. Bonichon, C. Gaoille, N. Hane, and D. Ilcinka. Connecion eeen hea-graph, Delanay rianglaion, and orhogonal rface. In WG, page , [3] P. Boe, A. Brodnik, S. Carlon, E. D. Demaine, R. Fleicher, A. López-Oriz, P. Morin, and J. I. Mnro. Online roing in conex diiion. In. J. Comp. Geomery Appl., 12(4): , [4] P. Boe and P. Morin. Online roing in rianglaion. SIAM J. Comp., 33(4): , [5] P. Che. There are planar graph almo a good a he complee graph. J. Comp. Sy. Sci., 39(2): , [6] R. Dhandapani. Greedy draing of rianglaion. Dicree & Compaional Geomery, 43(2): , [7] M. B. Dillencor. Realizailiy of Delanay rianglaion. Inf. Proce. Le., 33(6): , [8] M. T. Goodrich and D. Srah. Sccinc greedy

10 geomeric roing in R 2. In ISAAC, page , [9] X. He and H. Zhang. Schnyder greedy roing algorihm. In TAMC, page , [10] X. He and H. Zhang. On ccinc conex greedy draing of 3-conneced plane graph. In SODA, page , [11] J. M. Keil and C. A. Gin. Clae of graph hich approximae he complee Eclidean graph. Dicree & Compaional Geomery, 7:13 28, [12] T. Leighon and A. Moira. Some rel on greedy emedding in meric pace. Dicree & Compaional Geomery, 44(3): , [13] S. Mihra, I. Wongang, and S. Chandra, edior. Gide o Wirele Senor Neork. Springer, [14] C. H. Papadimirio and D. Raajczak. On a conjecre relaed o geomeric roing. Theor. Comp. Sci., 344(1):3 14, [15] H. Räcke. Srey on oliio roing raegie. In CiE, page , [16] W. Schnyder. Emedding planar graph on he grid. In SODA, page , [17] A. C.-C. Yao. On conrcing minimm panning ree in k-dimenional pace and relaed prolem. SIAM J. Comp., 11(4): , 1982.