COMPETITIVE LOCAL ROUTING WITH CONSTRAINTS

Transcription

1 COMPETITIVE LOCAL ROUTING WITH CONSTRAINTS Proenji Boe, Rolf Fagerberg, André van Renen, and Sander Verdoncho Abrac. Le P be a e of n verice in he plane and S a e of non-croing line egmen beween verice in P, called conrain. Two verice are viible if he raigh line egmen connecing hem doe no properly inerec any conrain. The conrained Θ m -graph i conrced by pariioning he plane arond each verex ino m dijoin cone, each wih aperre θ = 2π/m, and adding an edge o he cloe viible verex in each cone. We conider how o roe on he conrained Θ 6 -graph. We fir how ha no deerminiic 1-local roing algorihm i o( n)-compeiive on all pair of verice of he conrained Θ 6 - graph. Afer ha, we how how o roe beween any wo viible verice of he conrained Θ 6 -graph ing only 1-local informaion. Or roing algorihm garanee ha he rerned pah i 2-compeiive. Addiionally, we provide a 1-local 18-compeiive roing algorihm for viible verice in he conrained half-θ 6 -graph, a bgraph of he conrained Θ 6 -graph ha i eqivalen o he Delanay graph where he empy region i an eqilaeral riangle. To he be of or knowledge, hee are he fir local roing algorihm in he conrained eing wih garanee on he lengh of he rerned pah. 1 Inrodcion A fndamenal problem in any graph i he qeion of how o roe a meage from one verex o anoher. Wha make hi more challenging i ha ofen in a nework he roing raegy m be local. Informally, a roing raegy i local when he roing algorihm m decide which verex o forward a meage o baed olely on knowledge of he orce and deinaion verex, he crren verex and all verice direcly conneced o he crren verex. Roing algorihm are conidered geomeric when he graph ha i roed on i embedded in he plane, wih edge being raigh line egmen connecing pair of verice and weighed by he Eclidean diance beween heir endpoin. Geomeric roing algorihm are imporan in wirele enor nework (ee [11] and [12] for rvey of he area) ince hey offer roing Reearch ppored by NSERC, he Onario Miniry of Reearch and Innovaion, Carleon Univeriy Preiden 2010 Docoral Fellowhip, he Carleon-Field Podocoral Award, he Danih Concil for Independen Reearch, Naral Science, gran DFF , and JST ERATO Gran Nmber JPMJER1305, Japan. An exended abrac conaining ome of he rel in hi paper appeared in he 26h Inernaional Sympoim on Algorihm and Compaion (ISAAC 2015) [5]. School of Comper Science, Carleon Univeriy, Oawa, Canada, ji@c.carleon.ca, ander@cg.c.carleon.ca Deparmen of Mahemaic and Comper Science, Univeriy of Sohern Denmark, Odene, Denmark, rolf@imada.d.dk Naional Inie of Informaic, Tokyo, Japan, andre@nii.ac.jp JST, ERATO, Kawarabayahi Large Graph Projec

2 raegie ha e he coordinae of he verice o gide he earch, inead of he more radiional roing able. Mo of he reearch ha foced on he iaion where he nework i conrced by aking a bgraph of he complee Eclidean graph, i.e. he graph ha conain an edge beween every pair of verice and he lengh of hi edge i he Eclidean diance beween he wo verice. We dy hi problem in a more general eing wih he inrodcion of line egmen conrain. Specifically, le P be a e of verice in he plane and le S be a e of line egmen beween verice in P, wih no wo line egmen inerecing properly. The line egmen of S are called conrain. Two verice and v can ee each oher if and only if eiher he line egmen v doe no properly inerec any conrain or v i ielf a conrain. If wo verice and v can ee each oher, he line egmen v i a viibiliy edge. The viibiliy graph of P wih repec o a e of conrain S, denoed Vi(P, S), ha P a verex e and all viibiliy edge a edge e. In oher word, i i he complee graph on P min all non-conrain edge ha properly inerec one or more conrain in S. Thi naral exenion allow for more realiic nework modeling by exclding edge ha canno be ed, ch a one croing monain range or area of high inerference which wold cramble he meage if ed. A ch, hi eing ha been died exenively wihin he conex of moion planning amid obacle. Clarkon [9] wa one of he fir who died hi problem and howed how o conrc a (1 + ɛ)-panner of Vi(P, S) wih a linear nmber of edge. A bgraph H of G i called a -panner of G (for 1) if for each pair of verice and v, he hore pah in H beween and v ha lengh a mo ime he hore pah in G beween and v. The malle vale for which H i a -panner i he panning raio or rech facor of H. Following Clarkon rel, Da [10] howed how o conrc a panner of Vi(P, S) wih conan panning raio and conan degree. Boe and Keil [7] howed ha he Conrained Delanay Trianglaion (which conain an edge beween wo viible verice and v if and only if v i a conrain or here exi a circle wih and v on i bondary ha conain no verice viible o and v in i inerior) i a 2.42-panner of Vi(P, S). Recenly, he conrained half-θ 6 -graph (which i idenical o he conrained Delanay graph whoe empy viible region i an eqilaeral riangle, a formal definiion follow in Secion 2) wa hown o be a plane 2-panner of Vi(P, S) [4] and all conrained Θ-graph wih a lea 6 cone were hown o be panner a well [8]. However, hogh i i known ha hee graph conain hor pah, i i no known how o roe in a local fahion. In oher word, oher han by rnning ome global hore pah algorihm or flooding he nework wih meage, he verice are ill nable o commnicae wih each oher. To addre hi ie, we look a k-local roing algorihm in he conrained eing, i.e. roing algorihm ha m decide which verex o forward a meage o baed olely on knowledge of he orce and deinaion verex, he crren verex and all verice ha can be reached from he crren verex by following a mo k edge. Frhermore, we reqire or algorihm o be compeiive, i.e. he lengh of he rerned pah need o be relaed o he lengh of he hore pah in he graph. In he nconrained eing, here exi a 1-local 0-memory roing algorihm ha i 2-compeiive on he Θ 6 -graph and 5/ 3-compeiive on he half-θ 6 -graph (he Θ 6 -graph coni of he nion of wo half-θ 6 -graph) [6]. In he ame paper, he ahor alo how

3 ha hee raio are he be poible, i.e. here are maching lower bond. In hi paper, we how ha he iaion in he conrained eing i qie differen: no deerminiic 1-local roing algorihm i o( n)-compeiive on all pair of verice of he conrained Θ 6 -graph, regardle of he amon of memory (defined in Secion 2) i i allowed o e. Thi how ha roing in he conrained eing i coniderably harder han in he nconrained eing. Depie hi lower bond, we decribe a 1-local 0-memory roing algorihm beween any wo viible verice of he conrained Θ 6 -graph ha garanee ha he lengh of he pah raveled i a mo 2 ime he Eclidean diance beween he orce and deinaion. Addiionally, we provide a 1-local O(1)-memory 18-compeiive roing algorihm beween any wo viible verice in he conrained half-θ 6 -graph. To he be of or knowledge, hee are he fir local roing algorihm in he conrained eing wih garanee on he pah lengh. 2 Preliminarie We define a cone C o be he region in he plane beween wo ray originaing from a ingle verex. Thi verex i referred o a he apex of he cone. We le ix ray originae from each verex, wih angle o he poiive x-axi being mliple of π/3 (ee Figre 1). Each pair of conecive ray define a cone. We wrie Ci o indicae he i-h cone of a verex, or C i if he apex i clear from he conex. For eae of expoiion, we only conider poin e in general poiion: no wo verice define a line parallel o one of he ray ha define he cone and no hree verice are collinear. C 0,0 C 0,1 C 1,0 C 0 C 5 C 1 C 5,0 C 1,1 C 4,1 C 1,2 C 4 C 3 C 2 C 4,0 C 2,0 C 3,0 Figre 1: The cone having apex in he Θ 6 -graph. Figre 2: The bcone having apex in he conrained Θ 6 -graph. Conrain are hown a hick red line egmen. Le verex be an endpoin of a conrain and le he oher endpoin lie in cone Ci. The line hrogh all ch conrain pli C i ino everal bcone (ee Figre 2). We e Ci,j o denoe he j-h bcone, in clockwie order, of C i. When a conrain c = (, v) pli a cone of ino wo bcone, we define v o lie in boh of hee bcone. We conider

4 a cone ha i no pli o be a ingle bcone. The conrained Θ 6 -graph i conrced a follow: for each bcone C i,j of each verex, add an edge from o he cloe viible verex in ha bcone, where diance i meared along he biecor of he original cone, no he bcone (ee Figre 3). More formally, we add an edge beween wo verice and v if v can ee, v C i,j, and for all verice w C i,j ha can ee, v w, where v and w denoe he orhogonal projecion of v and w on he biecor of C i. Noe ha or general poiion ampion imply ha each verex add a mo one edge per bcone o he graph. v w Figre 3: Three verice are projeced ono he biecor of a cone of. Verex v i he cloe verex in he lef bcone and w i he cloe verex in he righ bcone. Nex, we define he conrained half-θ 6 -graph. Thi i a generalized verion of he half-θ 6 -graph a decribed by Bonichon e al. [2]. The conrained half-θ 6 -graph i imilar o he conrained Θ 6 -graph wih one major difference: edge are only added in every econd cone. More formally, i cone are caegorized a poiive and negaive. Le (C 0, C 2, C 1, C 0, C 2, C 1 ) be he eqence of cone in conerclockwie order aring from he poiive y-axi (ee Figre 4). The cone C 0, C 1, and C 2 are called poiive cone and C 0, C 1, and C 2 are called negaive cone. We add edge only in he poiive cone (and heir bcone). Noe ha by ing addiion and bracion modlo 3 on he indice, he poiive cone C i ha negaive cone C i+1 a clockwie nex cone and negaive cone C i 1 a conerclockwie nex cone. A imilar aemen hold for negaive cone. We e Ci and C i o denoe cone C i and C i wih apex. For any wo verice and v, we have v Ci if and only if C v i (ee Figre 4). Analogo o he bcone defined for he Θ 6 -graph, conrain can pli cone ino bcone. We call a bcone of a poiive cone a poiive bcone and a bcone of a negaive cone a negaive bcone (ee Figre 5). We look a he ndireced verion of hee graph, i.e. when an edge i added, boh verice are allowed o e i. Thi i conien wih previo work on Θ-graph. Given a verex w in a poiive cone Ci of verex, we define he canonical riangle T w o be he riangle defined by he border of Ci (no he border of he bcone of ha conain w) and he line hrogh w perpendiclar o he biecor of Ci (ee Fig. 6). Noe ha for each pair of verice here exi a niqe canonical riangle. Nex, we define or roing model. A deerminiic roing algorihm i k-local and e m-memory, if he verex o which a meage i forwarded from he crren verex i a fncion of,, N k (), and M, where and are he orce and deinaion verex, N k ()

5 C 0,0 C 0,1 C 1,0 C 0 C 2 C 1 C 2,0 C 1,1 C 1,1 C 1,2 C 1 C 0 C 2 C 1,0 C 2,0 C 0,0 Figre 4: The cone having apex in he half-θ 6 -graph. Figre 5: The bcone having apex in he conrained half-θ 6 -graph. Conrain are hown a hick red line egmen. a w b Figre 6: The canonical riangle T w. i he k-neighborhood of and M i a memory of ize m, ored wih he meage. The k-neighborhood of a verex i he e of verice in he graph ha can be reached from by following a mo k edge. For or prpoe, we conider a ni of memory o coni of log 2 n bi or a poin in R 2. Or model alo ame ha he only informaion ored a each verex of he graph i N k (). Since or graph are geomeric, we idenify each verex by i coordinae in he plane. Unle oherwie noed, all roing algorihm we conider in hi paper are deerminiic 0-memory algorihm. There are eenially wo noion of compeiivene of a roing algorihm on a bgraph of he viibiliy graph. One i o look a he Eclidean hore pah beween he wo verice, i.e. he hore pah in he viibiliy graph, and he oher i o compare he roing pah o he hore pah in he bgraph. A roing algorihm i c-compeiive wih repec o he Eclidean hore pah (rep. hore pah in he bgraph) provided ha he oal diance raveled by he meage i no more han c ime he Eclidean hore pah lengh (rep. hore pah lengh) beween orce and deinaion. The roing raio of an algorihm i he malle c for which i i c-compeiive. Since he hore pah in he bgraph beween wo verice i a lea a long a

6 he Eclidean hore pah beween hem, an algorihm ha i c-compeiive wih repec o he Eclidean hore pah i alo c-compeiive wih repec o he hore pah in he bgraph. We e compeiivene wih repec o he Eclidean hore pah when proving pper bond and wih repec o he hore pah in he bgraph when proving lower bond. Frhermore, we wan o be able o alk abo poin a inerecion of line, h we diingih beween verice and poin. A poin i any poin in R 2, while a verex i par of he inp. 3 Lower Bond on Local Roing We modify he proof by Boe e al. [3] (ha how ha no deerminiic roing algorihm i o( n)-compeiive for all rianglaion) o how he following lower bond. Theorem 1. No deerminiic 1-local roing algorihm i o( n)-compeiive wih repec o he hore pah on all pair of verice of he Θ 6 -graph of ize n, regardle of he amon of memory i i allowed o e. Proof. The following conrcion i illraed in Figre 7a-e. Conider a c c grid of verice for an ineger c and hif every econd row o he righ by half a ni. We rech he grid, ch ha each horizonal line egmen ha lengh 2c. Nex, we replace each horizonal line egmen by a conrain o preven verical viibiliy edge and we remove all oher line egmen. Afer ha, we add wo addiional verice, orce and deinaion, cenered horizonally a one ni below he boom row and one ni above he op row, repecively /2 c c (a) (b) (c) (d) (e) (f) Figre 7: Conrcing he lower bond: (a) he gird, (b) afer hifing, (c) afer reching (d) adding he conrain, (e) adding and, (f) conforming o general poiion. To conform o or general poiion ampion, we move all verice by a mo ome arbirarily mall amon ɛ, ch ha no wo verice define a line parallel o one of he ray ha define he cone and no hree verice are collinear (ee Figre 7f). A par of hi move,

7 we enre ha each verex on he boom row ha a i cloe verex in cone C 2 or C 4 (depending on wheher i lie o he righ or lef of ), and ha each verex on he op row ha a i cloe verex in cone C 1 or C 5 (again depending on wheher i lie o he lef or righ of ). Thi can be done e.g. by placing he boom row on he pper hll of an ellipe and placing he op row on he lower hll of an ellipe. On hi poin e and hee conrain, we bild he conrained Θ 6 -graph G (ee Figre 8). Noe ha verical edge only appear a he lef and righ grid bondarie. Figre 8: The conrained Θ 6 -graph aring from a grid, ing horizonal conrain o block verical edge, and he orange pah of he roing algorihm. Conider any deerminiic 1-local -memory roing algorihm and le π be he pah hi algorihm ake when roing from o. We noe ha by conrcion, π coni of a lea c + 1 ep. If π coni of more han c c non-verical ep, we rncae i afer he fir c c non-verical ep. Th, in he remainder of hi proof, we conider only pah having a mo k non-verical ep for k c c. The overall idea of he proof i o redce G o a Θ 6 -graph G of ize Θ(c + k) in a way which doe no change he pah π (p o i rncaion poin, if preen) aken by he algorihm, and hen o how ha π i no o( c + k)-compeiive wih repec o he hore pah in G. Thi prove ha no deerminiic 1-local -memory roing algorihm can be o( n)-compeiive wih repec o he hore pah on all Θ 6 -graph. To conrc G, we define he rronding of a verex v on π o be v ielf, he verice conneced o i by eiher an edge or a conrain in G, and he conrain in G beween hee verice. Th, for v in he inerior of G, i rronding are hexagonal in hape and conain even verice and for conrain (ee Figre 8). Informally, he nion of he rronding of verice of π can be een a weeping hi hexagonal hape along π. For v on he border of G, i rronding are lighly maller. For and, heir rronding conie he boom and op row, inclding he conrain in hee row. We le G be he Θ 6 -graph conrced on he nion of he rronding of all verice of π {} (he inclion of i only relevan if π wa rncaed). Thi conrcion i illraed in Figre 9. Clearly, he graph G ha O(c + k) verice and conrain. I i eay o check ha he 1-neighborhood of any verex v on π i he ame in G a in G, hence he roing algorihm m follow π alo in G. The boom row conain c verice. We now conider he 2 k horizonally mo cenral of hee, ha i, he fir k verice o he lef of and he fir k verice o he righ of. Seing c 16, he boom row doe conain a lea hee 2 k verice, by

8 Figre 9: The conrained Θ 6 -graph ha look he ame from he orange pah of he roing algorihm, b ha an moly verical dahed ble pah. k c c. Seing c a bi higher, we can ame ha i conain Ω(1) more verice a each end. Nex, conider a verical line hrogh each of hee 2 k verice. Le π be π min he verice and. We ay ha a verex of π oche ch a verical line if i rronding conain a poin on ha line. Hence, any verex along π oche O(1) verical line (ee Figre 8). Since he verical line are Ω(1) grid poiion away from he lef and righ ide of he grid, no verical ep of π can och any of hee line. Hence, he oal nmber of line oche by he verice along π i a mo O(k). Hence, on average, a line i oched O(k/ k) = O( k) ime. Thi implie ha here exi a verical line ha i oched O( k) ime. Le be verex on he boom row whoe verical line i oched he fewe nmber of ime. We now prove ha a moly verical pah from o he op row i conained in G, which will provide a pah G beween and mch horer han he pah π which he algorihm m follow. Ame fir ha he line of i oched zero ime. In he remainder of he proof, we e c o be odd, ch ha verice on he op and boom row align horizonally. Since he minimal horizonal diance beween verice in he grid i 2c, while he maximal verical diance i c, can ee exacly one verex in C 0, namely he verex i align horizonally wih in he op row. Th, here i a verical edge beween hee wo verice in G. If he line of i oched more han zero ime, each och cover a par of he line wih ome par of he hexagonal hape. The covering may overlap, and hey give rie o a naral decompoiion of he line ino maximal covered egmen wih non-covered egmen in beween. A core obervaion i ha a covered egmen verically exending h grid level can be ravered by h zig-zag edge in G, of oal lengh O(ch). Some example of hi are hown in Figre 10. Anoher core obervaion i ha for each ncovered egmen of he line, here will be a verical edge in G from he op verex of he covered egmen below o he boom verex of he covered egmen above (again de o he verex diance in he grid). Th, he verical edge from he cae of zero oche i broken p by zig-zag haped deor (one deor for each covered egmen). The reling pah ha lengh O(c k), ince he line hrogh i oched by a mo O( k) verice of π, each of which can cover only O(1) grid level of he line. Recalling ha he edge from o ha lengh a mo c k, we conclde ha G conain a pah from o of lengh O(c k): Follow he edge from o, follow he above pah from o he op row of G, and follow he edge o.

9 v v (a) (b) Figre 10: Two example of covered egmen and heir zig-zag deor: (a) when π ge cloe b doe no mee he verical line hrogh, (b) when π croe he verical line hrogh once. To complee he proof, we look a he nmber of non-verical edge of π, i.e. k. If k c, he roing pah follow a lea one verical edge along he bondary of G. I follow ha π ha lengh a lea Ω(c 2 ), a he lef and righ bondary of G are a diance Ω(c 2 ) from. Since he lengh of he moly verical pah i O(c k), π i no o(c/ k)-compeiive on a graph of ize Θ(c + k), which for k c implie ha π i no o( c)-compeiive on a graph of ize Θ(c). Hence, when we ake n = c, he heorem i proven for hi cae. If k > c, he lengh of π i dominaed by he non-verical edge of lengh c, leading o a pah lengh of Ω(ck). Since he lengh of he moly verical pah i O(c k), hi implie ha π i no o( k)-compeiive on a graph of ize Θ(k). Hence, when we ake n = k, he heorem i proven for hi cae. Th, ince G can be conrced for any deerminiic 1-local roing algorihm, we have hown ha no deerminiic 1-local roing algorihm i o( n)-compeiive on all pair of verice in a graph of ize O(n). 4 Roing on he Conrained Θ 6 -Graph In hi ecion, we provide a 1-local roing algorihm on he conrained Θ 6 -graph for any pair of viible verice. Since he conrained Θ 6 -graph i he nion of wo conrained half-θ 6 -graph, we decribe a roing algorihm for he conrained half-θ 6 -graph for he cae where he deinaion lie in a poiive bcone of he orce. Afer decribing hi algorihm and proving ha i i 2-compeiive, we decribe how o e i o roe 1-locally on he conrained Θ 6 -graph. Throgho hi ecion, we e he following axiliary lemma proven by Boe e al. [4]. We ay ha a region i empy if i doe no conain any verice of P. Lemma 1. Le, v, and w be hree arbirary poin in he plane ch ha w and vw are viibiliy edge and w i no he endpoin of a conrain inerecing he inerior of riangle vw. Then here exi a convex chain of viibiliy edge from o v in riangle vw, ch ha he polygon defined by w, wv and he convex chain i empy and doe no conain any conrain (ee Fig 11).

10 x y v w Figre 11: A convex chain from o v via x and y. Recall ha when working on pper bond, we e he noion of compeiivene wih repec o he Eclidean hore pah: A roing algorihm i c-compeiive wih repec o he Eclidean hore pah provided ha he oal diance raveled by he meage i no more han c ime he Eclidean hore pah lengh beween orce and deinaion. The roing raio of an algorihm wih repec o he Eclidean hore pah i he malle c for which i i c-compeiive wih repec o he Eclidean hore pah. 4.1 Poiive Roing on he Conrained Half-Θ 6 -Graph Before decribing how o roe on he conrained half-θ 6 -graph when lie in a poiive bcone of, we fir how ha here exi a pah in canonical riangle T. Lemma 2. Given wo verice and w ch ha and w ee each oher and w lie in a poiive bcone C i,j, here exi a pah beween and w in he riangle T w in he conrained half-θ 6 -graph. Proof. We ame wiho lo of generaliy ha w lie in C0,j. We prove he lemma by indcion on he area of he canonical riangle T w. Formally, we perform indcion on he rank of he riangle in he ordering, according o heir area, of he canonical riangle T xy of all pair of viible verice x and y. Bae cae: If T w i he malle canonical riangle, hen w i he cloe viible verex o in a poiive bcone of. Hence here i an edge beween and w and hi edge lie enirely inide T w. Indcion ep: We ame ha he indcion hypohei hold for all pair of verice ha can ee each oher and have a canonical riangle whoe area i maller han he area of T w. If w i an edge in he conrained half-θ 6 -graph, he indcion hypohei follow by he ame argmen a in he bae cae. If here i no edge beween and w, le v 0 be he verex cloe o in he poiive bcone ha conain w, and le a 0 and b 0 be he pper lef and righ corner of T v0 (ee Figre 12). We ame wiho lo of generaliy ha v 0 lie o he lef of w. Le x be he inerecion of w and a 0 b 0. By definiion x can ee and w. Since v 0 i he cloe viible verex o, v 0 can ee x a well. Oherwie Lemma 1 wold give a convex chain of verice connecing v 0 o x, all of which wold be cloer and able o ee, conradicing ha v 0 i he cloe viible verex o. By applying Lemma 1 o riangle

11 w v 2 v 1 a v 0 0 x b 0 Figre 12: An example of a convex chain from v 0 o w. v 0 xw, a convex chain v 0, v 1,..., v k = w of viibiliy edge connecing v 0 and w exi and he region bonded by x, v 0, v 1,..., v k = w i empy (ee Figre 12). Since every verex v i i viible o verex v i+1, we can apply indcion o each pair of conecive verice along he convex chain. Depending on wheher v i+1 C v i 0 or v i C v i+1 1, here exi a pah beween v i and v i+1 in T vi v i+1 or T vi+1 v i. Since each of hee riangle i conained in T w, hi give a pah beween and w ha lie inide T w. Poiive Roing Algorihm for he Conrained Half-Θ 6 -Graph Nex, we decribe how o roe from o, when can ee and lie in a poiive bcone Ci,j (ee Figre 13): When we are a, we follow he edge o he cloe verex in he bcone ha conain. When we are a any oher verex, we look a all edge in he bcone of Ci and all edge in he bcone of he adjacen negaive cone C ha i inereced by. An edge in a bcone of C i conidered only if i doe no cro. For example, in Figre 13, we do no conider he edge o v 1 ince i lie in C and croe. I follow ha we can cro only when we follow an edge in Ci. v 3 v 2 z v 1 Figre 13: An example of roing from o C0. The dahed line repreen he viibiliy line beween and. Le z be he inerecion of and he bondary of C ha i no a bondary of C i. We follow he edge v ha minimize he nigned angle zv. For example, in Figre 13, when we are a verex we follow he edge o v 2 ince, o of he wo remaining edge v 2 and v 3, zv 2 i maller han zv 3. We noe ha edge in C are added by he verice in

12 ha cone, ince lie in heir poiive cone C. We alo noe ha dring he roing proce, doe no necearily lie in Ci. Finally, ince he algorihm e only informaion abo he locaion of and and he neighbor of he crren verex, i i a 1-local roing algorihm. We proceed by proving ha he above roing algorihm can alway perform a ep, i.e. a every verex reached by he algorihm here exi an edge ha i conidered by he algorihm. Lemma 3. The roing algorihm can alway perform a ep in he conrained half-θ 6 -graph. Proof. Given wo verice and ch ha and can ee each oher, we ame wiho lo of generaliy ha C0. We mainain he following invarian (ee Figre 14): Invarian Le x be he la inerecion of an edge of he roing pah wih (iniially x i ), le v 0,..., v k denoe he endpoin of he edge following x a eleced by he algorihm, and le x be he inerecion of and he horizonal line hrogh v k. The imple polygon defined by x, v 0,..., v k, x i empy and doe no conain any conrain. v k x v k 1 v 0 x Figre 14: By he invarian, he gray region i empy and doe no conain any conrain. When he roing algorihm ar a, i look a he bcone ha conain. Since i viible from, hi bcone conain a lea one viible verex. Hence, i alo conain a cloe viible verex v 0 and by conrcion, ha an edge o v 0. Therefore, when he roing algorihm ar a, i can follow an edge. To ee ha he invarian i aified, we need o how ha riangle v 0 x i empy and doe no conain any conrain in i inerior. By conrcion canno be he endpoin of any conrain in he inerior of v 0 x, hence ince x and v 0 are viibiliy edge, any conrain ha a lea one endpoin in v 0 x. Th, i ffice o how ha v 0 x i empy. We prove hi by conradicion, o ame ha i i no empy. Since v 0 and x are viibiliy edge and by conrcion i no he endpoin of a conrain inerecing he inerior of v 0 x, Lemma 1 give a convex chain of viibiliy edge beween v 0 and x. Since he region bonded by v 0, x, and hi chain i empy and doe no conain any conrain, he verex along hi chain ha i cloe o i viible o. However ince every verex in v 0 x i cloer o han v 0, hi conradic he fac ha v 0 i he cloe viible verex o. Hence, riangle v 0 x m be empy and he invarian i aified.

13 When he roing algorihm i a verex ( ), we ame wiho lo of generaliy ha lie o he lef of. Le h be he halfplane below he horizonal line hrogh and le h be he halfplane o he lef of. We need o how ha ha a lea one edge in he nion of C 0 h and C 1 h h. We fir how ha here exi a verex ha i viible o in he nion of C 0 h and C 1 h h, by howing ha ch a verex exi in he nion of C 0 h h and C 1 h h. Since lie in hi region, we know ha i i no empy. Conider all verice in hi region and le v be he verex in hi region ha minimize x v. Noe ha we did no reqire here o be an edge beween and v. Since v minimize x v and no conrain can cro or x, v i viible from. We conider wo cae: v lie in a bcone of C 0 and v lie in a bcone of C 1. If v lie in C0 h h, i follow from Lemma 2 and he fac ha v i viible from ha here exi a pah beween and v ha lie inide T v. Since T v i conained in C0 h, here exi an edge in C 0 h and he roing algorihm can perform a ep. If v lie in C 1 h h, i follow from Lemma 2 and he fac ha v i viible from ha here exi a pah beween and v ha lie inide T v. Canonical riangle T v inerec hree cone of (ee Figre 15): C0, C 1, and C2. Since he roing algorihm follow edge in C0 or C 1, he roing pah reache by following edge v k 1 ha lie in eiher C 0 or C1. Thi implie ha T v C2 i conained in he region of he invarian and i herefore empy. Hence, he fir edge on he pah from o v lie in eiher C0 h or C 1 h h and he algorihm can perform a ep. v k 1 v x v 0 x Figre 15: By he invarian, he gray region i empy, o he pah beween and v lie inide T v (C 0 C 1). I remain o how ha afer he algorihm ake a ep, he invarian i aified a he new verex v. Le v be he edge ha he algorihm followed and le x be he inerecion of and he horizonal line hrogh v. We conider hree cae (ee Figre 16): (a) v lie in a bcone of C 1, (b) v lie in a bcone of C0 and v doe no cro, and (c) v lie in a bcone of C0 and v croe. Cae (a): If v lie in a bcone of C 1, we need o how ha he qadrilaeral vx x i empy and doe no conain any conrain (ee Figre 16a). We fir how ha canno be he endpoin of a conrain inerecing he inerior of vx x. We prove hi by conradicion, o ame i i and le y be he oher endpoin of he conrain. We fir

14 v x x q v v x x x v k 1 v k 1 v k 1 v 0 v 0 v 0 x x (a) (b) (c) x Figre 16: The hree ype of ep he algorihm can ake: (a) v lie in a bcone of C 1, (b) v lie in a bcone of C0 and v doe no cro, and (c) v lie in a bcone of C 0 and v croe. noe ha x y < x v. We look a C y 1,j, he bcone of Cy 1 ha lie below y, and le z be he lowe verex in hi bcone. If i he cloe viible verex in hi bcone, y wold be an edge, which conradic ha v minimize x v. Oherwie, ince z i he lowe verex in C y 1,j, he viible region of T z i empy and z i an edge. However, ince x z < x y < x v, we have a conradicion. Th canno be he endpoin of a conrain inerecing he inerior of vx x. Since i no he endpoin of a conrain inerecing he inerior of vx x, and v, x, and x x are viibiliy edge, any conrain inerecing he inerior of vx x ha a lea one endpoin in vx x. Th i ffice o how ha vx x i empy. We prove hi by conradicion, o ame ha vx x i no empy and le y be he lowe verex in vx x. Le C y 1,j be he bcone of Cy 1 ha conain. Verex i viible o y, ince any conrain croing y ha an endpoin in C 1 below y, conradicing ha y i he lowe verex, or in he region bonded by x, v 0,..., v k 1,, x which conradic he invarian. Hence y ha an edge in C y 1,j. Thi edge canno be o ince x y < x v. Since y i he lowe verex in vx x, i canno have an edge o a verex in vx x. Since by he invarian he region bonded by x, v 0,..., v k 1,, x i empy, he edge of y in C y 1,j m cro v. However, hi conradic he fac ha he conrained half-θ 6 -graph i plane. Th, vx x i empy of boh verice and conrain. Cae (b): If v lie in a bcone of C0 and v doe no cro, we again need o how ha he qadrilaeral vx x i empy and doe no conain any conrain (ee Figre 16b). We fir how ha vx x i empy. We prove hi by conradicion, o ame ha vx x i no empy and le y be he lowe verex in vx x. We conider wo cae: y lie in C 1 and y lie in C0. Since he cae where y lie in C 1 i analogo o he Cae (a), we foc on he cae where y lie in a bcone of C0. If y lie in a bcone of C 0 and y i viible o, y wold be an edge and x y < x v. So, ame ha y i no viible from. Thi mean ha here i a conrain ha croe y. Since he line and he edge of he region bonded by x, v 0,..., v k 1,, x are viibiliy edge, he lower endpoin of hi conrain m lie in x, v 0,..., v k 1,, v, x. By he

15 invarian, i canno lie in x, v 0,..., v k 1,, x, o i m lie in vx x and below y. However, hi conradic ha y i he lowe verex in vx x. Since we arrived a a conradicion in boh cae, we conclde ha qadrilaeral vx x i empy. Nex, we how ha vx x doe no conain any conrain. Since vx x i empy, a he only way a conrain can inerec i, i when i one of i endpoin. Hence, i remain o how ha canno be he endpoin of a conrain inerecing he inerior of vx x. We prove hi by conradicion, o ame i i and le y be he oher endpoin of he conrain. Since vx x i empy, y croe vx. Since i a viibiliy edge, y canno cro i. Verex y canno lie in C 1 h, ince hi wold imply ha eiher y i an edge or here exi a verex z in he bcone of y below y ha conain, which in combinaion wih Lemma 2 implie ha here exi a pah beween y and ha lie below y. Since boh alernaive conradic ha v minimize x v, y canno lie in C 1 h. Hence, i remain o conider he cae where y lie in a bcone of C0. Le C 0,j be he bcone of C 0 o he righ of y. If y lie below, C 0,j conain a cloe viible verex whoe angle wih x i le han x v, conradicing ha he roing algorihm roe o v. If y lie above, le z be he lowe verex in he nion of C0,j and C 1 h. Since hi region conain, i i no empy and ch a verex z exi. If z C0,j, i i he cloe verex in C0,j. If z C 1, i he cloe verex o z. We noe ha in boh cae z i viible o, ince any conrain blocking i wold have an endpoin below z. Hence, boh cae rel in an edge z. However, ince x z < x v, hi conradic ha he roing algorihm roed o v. Th, canno be he endpoin of a conrain inerecing he inerior of vx x. Cae (c): If v lie in a bcone of C0 and v croe, le q be he inerecion of v and. We need o how ha he riangle qx and qx v are empy and do no conain any conrain (ee Figre 16c). The proof ha qx i empy and doe no conain any conrain i analogo o he previo cae. We prove ha qx v i empy by conradicion, o ame ha qx v i no empy. Since qx and qv are viibiliy edge, we can apply Lemma 1 and we obain a verex y in qx v ha i viible from q. If y i viible from, v i no he cloe verex and edge v wold no exi. If y i no viible from, we noe ha q i viible and apply Lemma 1 on riangle yq. Thi give a verex z ha i viible o and cloer o han v, again conradicing he exience of edge v. Hence, riangle qx v i empy. Finally, we how ha qx v doe no conain any conrain. Since qx and qv are viibiliy edge and qx v i empy, any conrain inerecing he inerior of qx v m have q a an endpoin. However, ince q i no a verex, i canno be he endpoin of a conrain. Finally, we how ha he pah followed by he roing algorihm i 2-compeiive, wih repec o he Eclidean hore pah. Theorem 2. Given wo verice and in he half-θ 6 -graph ch ha and can ee each oher and lie in a poiive bcone of, here exi a 1-local roing algorihm ha roe from o and i 2-compeiive wih repec o he Eclidean diance.

16 Proof. We ame wiho lo of generaliy ha C0. The roing algorihm will h only ake ep in C v i 1, and C v i 2, where v i i an arbirary verex along he roing pah. 0, Cv i Le a and b be he pper lef and righ corner of T. To bond he lengh of he roing pah, we fir bond he lengh of each edge. We conider hree cae: (a) edge in bcone of C v i 1 or C v i 2, (b) edge in bcone of C v i ha cro. For eae of noaion we e v 0 and v k o denoe and. 0 ha do no cro, (c) edge in bcone of Cv i 0 a i v i+1 v i+1 a i v i+1 ai b i x v i v i (a) (b) (c) v i Figre 17: Bonding he edge lengh: (a) an edge in a bcone of C 1, (b) an edge in a bcone of C0 ha doe no cro, and (c) an edge in a bcone of C 0 ha croe. Cae (a): If edge v i v i+1 lie in a bcone of C v i 1, le a i be he pper corner of T vi+1 v i (ee Figre 17a). By he riangle ineqaliy, we have ha v i v i+1 v i a i + a i v i+1. The cae where v i v i+1 lie in C v i 2 i analogo. Cae (b): If edge v i v i+1 lie in a bcone of C v i 0 and doe no cro, le a i and b i be he pper lef and righ corner of T vi v i+1 (ee Figre 17b). If v i lie o he lef of, we e ha v i v i+1 v i a i + a i v i+1. If v i lie o he righ of, we e ha v i v i+1 v i b i + b i v i+1. Cae (c): If edge v i v i+1 lie in a bcone of C v i 0 and croe, we pli i ino wo par, one for each ide of (ee Figre 17c). Le x be he inerecion of and v i v i+1. If v i lie o he lef of, le a i be he pper lef corner of T vi x and le b i be he pper righ corner of T xvi+1. By he riangle ineqaliy, we have ha v i v i+1 v i a i + a i x + xb i + b i v i+1. If v i lie o he righ of, le a i be he pper lef corner of T xvi+1 and le b i be he pper righ corner of T vi x. By riangle ineqaliy, we have ha v i v i+1 v i b i + b i x + xa i + a i v i+1. To bond he lengh of he fll pah, le x and x be wo conecive poin where he roing pah croe and le v i v i+1 be he edge ha croe a x and le v i v i +1 be he edge ha croe a x. Le a x and b x be he pper lef and righ corner of T xx. If he pah beween x and x lie o he lef of, hi par of he pah i bonded by: i 1 xa i + a j v j+1 + j=i i j=i+1 v j a j + a i x. Since xa i and all v j a j are parallel o xa x and all a x v j+1 are horizonal, we have ha: xa i + i j=i+1 v j a j = xa x.

17 Similarly, ince a i x and all a j v j+1 are parallel and have dijoin projecion ono a x x, we have ha: i 1 a j v j+1 + a i x = a x x. j=i Th, he lengh of a pah o he lef of i a mo: xa x + a x x If he pah beween x and x lie o he righ of, hi par of he pah i bonded by (ee Figre 18a): i 1 i xb i + b j v j+1 + v j b j + b i x = xb x + b x x. j=i j=i+1 a b a b x b x a x x b x x x (a) (b) Figre 18: Bonding he oal lengh: (a) he bond (olid line) are nfolded (doed line) and (b) he nfolded bond (olid line) are flipped o he longer of he wo ide (doed line) and nfolded again (dahed line). Nex, we flip all nfolded bond o he longer of he wo ide a and b: if a b, we replace all bond of he form xb x + b x x by xa x + a x x and if a < b, we replace all bond of he form xa x + a x x by xb x + b x x (ee Figre 18b). Noe ha hi can only increae he lengh of he bond. Finally, we m hee bond and ge ha he m i eqal o max{ a + a, a + b }, which i a mo Roing on he Conrained Θ 6 -Graph To roe on he conrained Θ 6 -graph, we pli i ino wo conrained half-θ 6 -graph: he conrained half-θ 6 -graph oriened a in Figre 5 and he conrained half-θ 6 -graph where poiive and negaive cone are invered. When we wan o roe from o, we pick he conrained half-θ 6 -graph in which lie in a poiive bcone of, referred o a G + in he remainder of hi ecion, and apply he roing algorihm decribed in he previo

18 ecion. Since hi roing algorihm i 1-local and 2-compeiive, we obain a 1-local and 2-compeiive roing algorihm for he conrained Θ 6 -graph, provided ha we can deermine locally, while roing, wheher an edge i par of G +. When a a verex, we conider he edge in order of increaing angle wih he horizonal halfline hrogh ha inerec. Lemma 4. While execing he poiive roing algorihm for wo viible verice and, we can deermine locally a a verex for any edge v in he conrained Θ 6 -graph wheher i i par of G +. Proof. Sppoe we color he edge of he conrained Θ 6 -graph red and ble ch ha red edge form G + and ble edge form he conrained half-θ 6 -graph, where lie in a negaive bcone of. A a verex, we need o deermine locally wheher v i red. Since an edge can be par of boh conrained half-θ 6 -graph, i can be red and ble a he ame ime. Thi make i harder o deermine wheher an edge i red, ince deermining ha i i ble doe no imply ha i i no red. If v lie in a poiive bcone of, we need o deermine if i i he cloe verex in ha bcone. Since by conrcion of he conrained half-θ 6 -graph, i conneced o he cloe verex in hi bcone, i ffice o check wheher hi verex i v. Noe ha if v i a conrain, v lie in wo bcone of and hence we need o check if i i he cloe verex in a lea one of hee bcone. If v lie in a negaive bcone of, we know ha if i i no he cloe viible verex in ha bcone, v i red. Hence, i remain o deermine wheher he edge o he cloe verex i red: If i i he cloe viible verex, i i ble, b i may be red a well if i alo he cloe viible verex o v. Hence, we need o deermine wheher i he cloe verex in Ci,j v, a bcone of v ha conain. We conider wo cae: (a) v i a conrain, (b) v i no a conrain. Cae (a): Since v i a conrain, we know ha i canno cro. Since we are conidering v, we alo know ha all edge ha make a maller angle wih he horizonal halfline hrogh ha inerec are no red. Hence, v i eiher par of he bondary of he roing pah or he conrain i conained in he inerior of he region bonded by he roing pah and. However, by he invarian of Lemma 3, he region bonded by he roing pah and doe no conain any conrain in i inerior. Th, v i par of he bondary of he roing pah and v i red. Cae (b): If v i no a conrain, le region A and B be he inerecion of Ci v and he wo bcone of adjacen o C i and le C be he inerecion of Ci,j v and he negaive bcone of ha conain v (ee Figre 19). We fir noe ha ince v lie in a negaive bcone of, he invarian of Lemma 3 implie ha B i empy. Frhermore, ince v i he cloe viible verex o, C doe no conain any verice ha can ee or v. Since C doe no conain any verice ha can ee or v, any conrain in C i ha ha a an endpoin and lie above v, enre ha v canno ee A, i.e. i canno block viibiliy of hi region only parially. Hence, if ch a conrain exi, i he cloe viible verex o v in Ci,j v, ince neiher B nor C conain any verice viible o v. Therefore, v i red. If v can ee A, we how ha v i red if and only if he cloe viible verex in he

19 v A C B Figre 19: The hree region A, B, and C when deermining wheher an edge i par of he conrained half-θ 6 -graph. bcone of ha conain A doe no lie in A. We fir how ha if he cloe viible verex x in he bcone of ha conain A lie in A, hen v i no red. Since A i viible o v, i no he endpoin of a conrain in C i above v. Hence, we have wo viibiliy edge v and x and i no he endpoin of a conrain inerecing he inerior of riangle xv. Therefore, by Lemma 1, we have a convex chain of viibiliy verice beween x and v. Le y be he verex adjacen o v along hi chain. Since he polygon defined by x, v, and he convex chain i empy and doe no conain any conrain, y lie in Ci,j v. Th, i no he cloe viible verex in Ci,j v and v i no red. Nex, we how ha if he cloe viible verex x in he bcone of ha conain A doe no lie in A, hen v i red. We prove hi by conradicion, o ame ha v i no red. Thi implie ha here exi a verex y Ci,j v ha i viible o v and cloer han. Since B i empy and C doe no conain any verice ha can ee v, y lie in A. Since v and vy are viibiliy edge and v i no he endpoin of a conrain inerecing he inerior of riangle yv, by Lemma 1 here exi a convex chain of viibiliy edge beween and y. Frhermore, ince C doe no conain any verice ha can ee, he verex adjacen o along hi chain lie in A. Since any verex in A i cloer o han x, hi lead o a conradicion, compleing he proof. Roing Algorihm for he Conrained Θ 6 -Graph Hence, o roe on he conrained Θ 6 -graph, we apply he poiive roing algorihm on G +, while deermining which edge are par of hi conrained half-θ 6 -graph. The laer can be deermined a follow: If v lie in a poiive bcone, we need o check wheher i i he cloe verex in ha bcone. If v lie in a negaive bcone and i i no he cloe verex, i i par of he conrained half-θ 6 -graph. Finally, if v i he cloe verex in a negaive bcone, i i par of he conrained half-θ 6 -graph if i i a conrain or he inerecion of he cone of v ha conain and he bcone of Ci 1 adjacen o C i i empy. 4.3 Negaive Roing on he Conrained Half-Θ 6 -Graph We noe ha he roing algorihm provided in he previo ecion doe no ffice o alo roe on he conrained half-θ 6 -graph, ince i ame ha he deinaion lie in a poiive

20 bcone of he orce. Therefore, in hi ecion, we provide an O(1)-memory 1-local roing algorihm for he cae where he deinaion lie in a negaive bcone of he orce. For eae of expoiion, we ame ha lie in a bcone of C 0. The O(1)-memory 1-local roing algorihm find a pah from o of lengh a mo 2 and ravel a oal diance of a mo 18 o do o. We noe ha negaive roing i harder han poiive roing, ince here need no be an edge o a verex in he cone of ha conain. Thi phenomenon alo caed he eparaion beween panning raio and roing raio in he nconrained eing [6]. The remainder of hi ecion i rcred a follow: Fir, we idenify a e of condiion ha edge need o mee in order o be conidered by he roing algorihm. Unfornaely, one of hee condiion canno be checked 1-locally. Therefore, we replace i wih a e of condiion ha exclde edge ha are garaneed no o aify he original condiion and can be checked 1-locally. We proceed o decribe he edge conidered by he negaive roing algorihm. Given a verex v and all neighbor of v whoe projecion along he biecor of C0 i cloer o han he projecion of v, we nmber he neighbor 0,..., k of v in conerclockwie order, aring from he horizonal half-line o he lef of v (ee Figre 20). We creae k + 2 region arond v: We creae k rianglar region v i i+1 for 0 i k 1. We creae one nbonded region ing edge v 0 and he wo horizonal half-line aring a v and 0 direced oward he lef. We creae one nbonded region ing edge v k and he wo horizonal half-line aring a v and k direced oward he righ. v Figre 20: Triangle v 2 3 i he la region of v inereced by. The la region of v inereced by i defined a he la of hee region ha i enconered when following from o. In Figre 20, he region defined by v, 2, and 3 i he la region of v inereced by. We conider an edge v for or roing algorihm when i aifie he following hree condiion: 1. Verice and v lie inide or on he bondary of T.

21 2. Edge v i par of he la region of v ha i inereced by. 3. Edge v i he edge ha he poiive roing algorihm pick a when roing from o. Noe ha for hi condiion, we do no reqire ha i par of he poiive roing pah, b only ha hold he poiive roing pah reach, edge v i he edge i wold elec for i nex ep. Given and, he fir wo reqiremen can be checked ing only he locaion of and and 1-local informaion, i.e. he neighbor of he crren verex. The la reqiremen, on he oher hand, may need 2-local informaion a i involve he neighbor of he neighbor of v. Hence, inead of ing hi la reqiremen, we ignore he edge ha can never aify i and how ha we can roe compeiively and 1-locally on he graph G formed by he edge ha mee he fir wo reqiremen. Since lie in a bcone of C 0, he edge ha define he la inereced region of a verex v can lie in hree cone: C1 v, Cv 0, and C2 v. Since edge in Cv 1 and Cv 2 of he negaive roing algorihm correpond o edge in C 1 and C 2 of he poiive roing algorihm (applied from o ), he poiive roing algorihm never follow hee edge if hey inerec. Hence, hee edge need no be conidered by he negaive roing algorihm (ee Figre 21a). v v v (a) (b) (c) Figre 21: The edge ignored by he negaive roing algorihm: (a) edge 2 v i ignored ince i inerec, (b) edge v i ignored ince C v 2 i inereced by, (c) edge 1v i ignored ince i lie in a bcone ha i no inereced by and 1 v 2 i inereced by a conrain ha ha v a an endpoin. We alo do no need o conider edge in C1 v and Cv 2 when ha cone i inereced by (ee Figre 21b): Ame C1 v i inereced by. Since we are conidering edge v, i canno cro. Hence, inerec cone C1, b more imporanly inerec C 2. Hence, if he poiive roing algorihm reache, i conine by following an edge in C 2 or C0. Since Cv 1 correpond o C 1, no edge in hi cone i followed by he poiive roing algorihm, and we can ignore i. Finally, we ignore edge ha lie in a bcone ha i no inereced by when v i he endpoin of a conrain ha inerec he inerior of he la region of v ha i inereced by (ee Figre 21c): If v i he endpoin of a conrain ha inerec he inerior of he la region of v ha i inereced by, we do no conider he edge ha i no inereced by. We can ignore hi edge, ince by he invarian, he region beween he roing pah and doe no conain any conrain.

22 Since hee condiion can be checked ing only,, v, he neighbor of v, and he conrain inciden o v, we can deermine 1-locally wheher o conider an edge. Hence, he graph G on which we roe i he graph formed by all edge v ch ha: 1. Verice and v lie inide or on he bondary of T. 2. Edge v i par of he la region of v ha i inereced by. 3. Edge v doe no mee any of he following hree condiion: (a) Edge v lie in C1 v or Cv 2 and croe. (b) Edge v lie in C1 v or Cv 2 and hi cone i inereced by. (c) Edge v lie in a bcone ha i no inereced by and v i he endpoin of a conrain ha inerec he inerior of he la region of v ha i inereced by. Noe ha every edge v ha lie in C1 v or Cv 2 and croe, he cone ha conain v i inereced by. Hence, condiion 3a can be ignored a i i inclded in condiion 3b. In he remainder of hi ecion, for eae of expoiion, we conider each edge of G o be oriened pward: Le and v be he projecion of and v along he biecor of C 0. Edge v i oriened from o v if and only if v. Noe ha hi doe no imply ha lie in a negaive cone of v. We proceed o prove ha every verex wih wo incoming edge i par of he poiive roing pah when roing from o. Lemma 5. Every verex wih in-degree 2 in G ha i reached by he negaive roing algorihm i par of he poiive roing pah from o. Proof. Le v be a verex of in-degree 2 ha i reached by he negaive roing algorihm. Le and w be he oher endpoin of hee edge o v, ch ha he projecion of along he biecor of T i cloer o han he projecion of w (ee Figre 22). Since boh v and wv are par of he la inereced region of v, verice and w m lie on oppoie ide of. Thi implie ha he poiive roing algorihm reache a lea one of hem when roing from o, ince by he invarian he region beween he roing pah and i empy. Th i ffice o how ha from boh and w he poiive roing algorihm evenally reache v. If he poiive roing algorihm reache w, we how ha i wold follow he edge o v. Le x be he inerecion of v and he horizonal line hrogh w (ee Figre 22). Fir, we how ha riangle vwx i empy. If w lie in a bcone of C1 v or Cv 2, lie in a bcone of C v 0, ince oherwie one of he wo edge wold cro and be ignored. Since vw and vx are viibiliy edge and v i no he endpoin of a conrain inerecing he inerior of vwx, i follow from Lemma 1 ha if vwx i no empy, here exi a convex chain of viibiliy edge beween w and x and he region bonded by hi chain, vw, and vx i empy. Le y be he opmo verex along hi convex chain and noe ha y i viible o v. If y lie in he ame cone of v a w, i alo lie in he ame bcone of v a w, ince v i no he endpoin of a conrain inerecing he inerior of vwx. However, hi implie ha w i

23 v x w Figre 22: Verex v ha in-degree 2. no he cloe viible verex o v in hi bcone, conradicing ha vw i an edge. If y lie in C v 0, y ha an edge in i bcone ha conain v, ince v i a viible verex in ha bcone. Thi edge canno cro vw and v, ince he conrained half-θ 6 -graph i plane, and i canno be conneced o a verex in he region bonded by he convex chain, vw, and vx, ince i i empy. Finally, ince y i he opmo verex along he convex chain, he edge canno connec y o anoher verex of he convex chain. Hence, y wold have an edge o v, conradicing ha v and vw are conecive edge arond v. We conclde ha riangle vwx i empy. Uing an analogo argmen, i can be hown ha if lie in a bcone of C v 1 or C v 2, w lie in Cv 0 and he exience of a verex in vwx wold conradic ha v i an edge or ha and w are conecive edge arond v. If boh and w lie in a bcone of C v 0, he argmen redce o he cae where y lie in C v 0, again conradicing ha and w are conecive edge arond v. Hence, ince vwx i empy, he poiive roing algorihm roe o v when i reache w, ince i minimize angle xwv. Nex, we look a he cae where he poiive roing pah reache. If i follow edge v, we are done. If i doe no follow edge v, le z be he oher endpoin of he edge he poiive roing algorihm follow a. By conrcion of he poiive roing pah, we know ha he projecion of z on he biecor of C0 lie frher from han he projecion of. Since he conrained half-θ 6 -graph i plane, he pah from z o canno cro v or wv, and ince he poiive roing pah i monoone wih repec o he biecor of C0, i canno go down and arond or hrogh. Frhermore, ince he region encloed by he poiive roing pah and i empy, he pah alo canno go arond w wiho paing hrogh w. Finally, ince v and wv are conecive edge arond v, he pah from z o canno reach v by arriving from an edge beween v and wv. Hence, w m lie on he pah from z o. Th, ince we previoly howed ha when he poiive roing algorihm reache w, i roe o v, verex v i alo reached when he poiive roing pah reache.