On Romeo and Juliet Problems: Minimizing Distance-to-Sight

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1 On Romeo and Julie Problem: Minimizing Diance-o-Sigh Hee-Kap Ahn 1, Eunjin Oh 2, Lena Schlipf 3, Fabian Sehn 4, and Darren Srah 5 1 Deparmen of Compuer Science and Engineering, POSTECH, Souh Korea heekap@poech.ac.kr 2 Deparmen of Compuer Science and Engineering, POSTECH, Souh Korea jin9082@poech.ac.kr 3 Theoreiche Informaik, FernUniveriä in Hagen, Germany lena.chlipf@fernuni-hagen.de 4 Iniu für Informaik, Univeriä Bayreuh fabian.ehn@uni-bayreuh.de 5 Deparmen of Compuer Science, Colgae Univeriy, US. drah@c.colgae.edu Abrac We inroduce a varian of he wachman roue problem, which we call he quicke pair-viibiliy problem. Given wo peron anding a poin and in a imple polygon P wih no hole, we wan o minimize he diance hee peron ravel in order o ee each oher in P. We olve wo varian of hi problem, one minimizing he longer diance he wo peron ravel (min-max) and one minimizing he oal ravel diance (min-um), opimally in linear ime. 1 Inroducion In he wachman roue problem, a wachman ake a roue o guard a given region ha i, any poin in he region i viible from a lea one poin on he roue. I i deirable o make he roue a hor a poible o ha he enire area can be guarded a quickly a poible. The problem wa fir inroduced in 1986 by Chin and Nafo [4] and ha been exenively udied in compuaional geomery [3, 10]. Though he problem i NP-hard for polygon wih hole [4, 5, 7], an opimal roue can be compued in ime O(n 3 log n) for imple n-gon [6] when he our mu pa hrough a pecified poin, and O(n 4 log n) ime oherwie. In hi paper, we udy a varian we call he quicke pair-viibiliy problem, which can be aed a follow. Problem (quicke pair-viibiliy problem). Given wo poin and in a imple polygon P, compue he minimum diance ha and mu ravel in order o ee each oher in P. Thi problem may ound imilar o he hore pah problem beween and, in which he objecive i o compue he hore pah for o reach. However, hey differ even for a imple cae: for any wo poin lying in a convex polygon, he diance in he quicke pair-viibiliy problem i zero while in he hore pah problem i i heir Euclidean diance. The quicke pair-viibiliy problem occur in opimizaion ak. For example, mobile robo ha ue a line-of-igh communicaion model are required o move o muually-viible Thi work by Ahn and Oh wa uppored by he MSIT(Miniry of Science and ICT), Korea, under he SW Sarlab uppor program(iitp ) upervied by he IITP (Iniue for Informaion & communicaion Technology Promoion.). 34h European Workhop on Compuaional Geomery, Berlin, Germany, March 21 23, Thi i an exended abrac of a preenaion given a EuroCG 18. I ha been made public for he benefi of he communiy and hould be conidered a preprin raher han a formally reviewed paper. Thu, hi work i expeced o appear evenually in more final form a a conference wih formal proceeding and/or in a journal.

2 16:2 On Romeo and Julie Problem 1 P 2 1 v 2 v 4 = v 1 (a) (b) (c) v 3 v 5 v 6 = v i 1 v i v i+1 < 90 Figure 1 (a) The quicke pair-viibiliy problem find wo pah π(, 1) and π(, 1) uch ha 1 1 P and max{ π(, 1), π(, 1) } or π(, 1) + π(, 1) i minimized. The quicke viibiliy problem for query poin find a hore π(, 2) wih 2 P. (b) min-max: Every pair (, ), where i ome poin wihin he geodeic dik cenered in wih radiu π(, ), i an opimal oluion o he min-max problem. (c) min-um: Every pair (v i, v i+1) for 1 i < 6 i an opimal oluion o hi inance. poiion o eablih communicaion [8]. An opimizaion ak here i o find hore pah for he robo o mee he viibiliy requiremen for eablihing communicaion among hem. Wyner e al. [12] udied hi problem for wo agen acing in a polygonal domain in he preence of polygonal obacle and gave an O(nm)-ime algorihm for he min-um varian (where m i he number of edge of he viibiliy graph of all corner) and an O(n 3 log n)-ime algorihm for he min-max varian. A query verion of he quicke viibiliy problem ha alo been udied [1, 9, 11]. In he query problem, a polygon and a ource poin lying in he polygon are given, and he goal i o preproce hem and conruc a daa rucure ha allow, for a given query poin, o find he hore pah aken from he ource poin o ee he query poin efficienly. Khoravi and Ghodi [9] conidered he cae for a imple n-gon and preened an algorihm o conruc a daa rucure of O(n 2 ) pace o ha given a query, i find he hore viibiliy pah in O(log n) ime. Laer, Arkin e al. [1] improved he reul and preened an algorihm for he problem in a polygonal domain. Very recenly, Wang [11] preened an improved algorihm for hi problem for he cae ha he number of he hole in he polygon i relaively mall. Figure 1(a) illurae difference in hee problem for a imple polygon and wo poin, and, in he polygon. 1.1 Our reul In hi paper, we conider wo varian of he quicke pair-viibiliy problem for a imple polygon: eiher we wan o minimize he maximum lengh of a raveled pah (min-max varian) or we wan o minimize he um of he lengh of boh raveled pah (min-um varian). We give a weep-line-like approach ha roae he line-of-igh along verice on he hore pah beween he ar poiion, allowing u o evaluae a linear number of candidae oluion on hee line. Throughou he weep, we encouner oluion o boh varian of he problem. We furher how ha our echnique can be implemened in linear ime. 2 Preliminarie Le P be a imple polygon and P be i boundary. The verice of P are given in counerclockwie order along P. We denoe he hore pah wihin P beween wo poin p, q P by π(p, q) and i lengh by π(p, q). We ay a poin p P i viible from anoher poin q P (and q i viible from p) if and only if line egmen pq i compleely conained in P. For wo aring poin and, our ak i o compue a pair (, ) of poin uch ha

3 H. Ahn and E. Oh and L. Schlipf and F. Sehn and D. Srah 16:3 and are viible o each oher, where we wih o minimize he lengh of π(, ), and π(, ). In he min-max eing, we wih o minimize max{ π(, ), π(, ) }. For he min-um eing, we wih o minimize π(, ) + π(, ). Noe ha, for boh varian, he opimum i no necearily unique; ee Figure 1(b) and (c). For our dicuion, le (, ) be an opimal oluion for he inance a hand. Le V (p) denoe he viible region for a poin p in P, ha i, he porion of P ha i viible from p. Clearly, V (p) i a ar-haped polygon. Moreover, every boundary edge of V (p) i eiher (par of) an edge of P or a egmen vq ha i conained in P and parallel o pv, where v i a verex of P viible from p and q i a poin on he boundary of P. We call an edge of he laer ype a window edge of he viibiliy region. The rucure of V (p) may change a p move along a pah conained in P. I i known ha a change o he rucure of V (p) occur if and only if wo verice of P become collinear wih p [2]. Lemma 2.1. Unle and are viible o each oher, he egmen conain a verex v of he hore pah π(, ) from o. I i eay o ee by conradicion ha mu conain a verex v of he boundary of P ; uing hore pah properie, one can how ha v i a verex of π(, ). The full proof i omied due o pace conrain. 3 Compuing All Even for a Sweep-Line-Like Approach For each verex v on π(, ) we compue a finie collecion of line hrough v, each being a configuraion a which he combinaorial rucure of he hore pah π(, ) and/or π(, ) change. To be more precie, a hee line eiher he verice of π(, ) or π(, ) (excep for and ) change or he edge of P change ha i inereced by he exenion of. To explain how o compue hee line, we inroduce he concep of a line-of-igh. Definiion 3.1 (line-of-igh). We call a egmen l a line-of-igh if (i) l P, (ii) boh endpoin of l lie on P, and (iii) l i angen o π(, ) a a verex v π(, ). We ay a egmen g i angen o a pah π a a verex v if v g π and he local neighborhood of π a all inerecion g π i on he ame ide of g. The algorihm we preen i in many apec imilar o a weep-line raegy, excep ha we do no weep over he cene in a andard fahion bu roae a line-of-igh l in P around he verice of he hore pah π(, ) := ( = v 0 ), v 1,..., v k 1, ( = v k ). The proce will be iniialized wih a line-of-igh ha conain and v 1 and i hen roaed around v 1 (while remaining angen o v 1 ) unil i hi v 2, ee Figure 2(a). In general, he curren line-of-igh i roaed around v i in a way o ha i remain angen o v i (i i roaed in he inerior of P ) unil he line-of-igh conain v i and v i+1, hen he proce i ieraed wih v i+1 a he new roaion cener. The proce erminae a oon a he line-of-igh conain v k 1 and. While performing hee roaion around he hore pah verice, we encouner all combinaorially differen line-of-igh. A for a andard weep-line approach, we will compue and conider even a which he rucure of a oluion change: hi i eiher becaue he inerior verice of π(, ) or π(, ) change or becaue he line-of-igh ar or end a a differen edge of P. Thee even will be repreened by poin on P (acually, we inroduce he even a verice on P unle hey are already verice). Beween wo conecuive line-of-igh, we compue he local minima of he relevan diance for he varian a hand in conan ime and hence encouner all global minima evenually. There are hree even-ype o diinguih: E u r o C G 1 8

4 16:4 On Romeo and Julie Problem v 1 v 2 (a) (b) (c) Figure 2 Pah- and boundary-even. (a) The fir pah-even i he line-of-igh hrough v 1. The line-of-igh roae unil i hi he nex pah-even: he egmen hrough v 1v 2. (b) All pahand boundary-even: he even-queue i iniialized wih hee even. (c) A bend-even (marked wih a cro) occur beween he wo boundary-even. The hore pah from o hee egmen change a he bend-even. 1. Pah-Even are endpoin of line-of-igh ha conain wo conecuive verice of he hore pah π(, ). See Figure 2(a). 2. Boundary-Even are endpoin of line-of-igh ha are angen a a verex of π(, ) and conain a lea one verex of P \ π(, ) (poenially a an endpoin). See Figure 2(b). 3. Bend-Even are encounered when, he hore pah of (or ) o he line-of-igh gain or loe a verex while roaing he line-of-igh around a verex v. See Figure 2(c). Noe ha bend-even can coincide wih pah- or boundary-even. We will need o explicily know boh endpoin of he line-of-igh on P a each even and he correponding verex of π(, ) on which we roae. Lemma 3.2 (Compuing pah- and boundary-even). For a imple polygon P wih n verice and poin, P, he queue Q of all pah- and boundary-even of he roaional weep proce, ordered according o he equence in which he weeping line-of-igh encouner hem, can be iniialized in O(n) ime. Pah even coincide wih pecific verice of he hore pah map of (or of ) in P, wherea boundary even are endpoin of pecific edge of he hore pah ree of (or of ) in P. Thee rucure can be conruced and claified in linear ime, a full proof i omied due o pace conrain. Once we iniialized he even queue Q, we can now compue and proce bend-even a we proceed in our line-of-igh roaion. Lemma 3.3. All bend-even can be compued in O(n) ime, ored in he order a hey appear on he boundary of P. Due o pace limiaion, he proof of Lemma 3.3 i omied. 4 Algorihm Baed on a Sweep-Line-Like Approach In hi ecion, we preen a linear-ime algorihm for compuing he minimum diance ha wo poin and in a imple polygon P ravel in order o ee each order. We compue all even defined in Secion 3 in linear ime. The remaining ak i o handle he line-of-igh lying beween wo conecuive even. Lemma 4.1. For any wo conecuive even, he line-of-igh l lying beween hem ha minimize he um of he diance from and o l can be found in conan ime.

5 H. Ahn and E. Oh and L. Schlipf and F. Sehn and D. Srah 16:5 Proof. Le L be he e of all line-of-igh lying beween he wo conecuive even. Every line-of-igh in L conain a common verex v of π(, ). We aume ha L conain no verical line-of-igh. Oherwie, we conider he e conaining all line-of-igh of L wih poiive lope, and hen he e conaining all line-of-igh of L wih negaive lope. By conrucion, he econd o he la verex u of π(, l) (and π(, l)) for any l L remain he ame. We already obained v and u while compuing he even. We will give an algebraic funcion for he lengh of π(, l) for l L. An algebraic funcion for he lengh of π(, l) can be obained by changing he role of and. Since he opology of π(, l) for every l L remain he ame, we conider only he lengh of π(u, l). Oberve ha π(u, l) i a line egmen for any l L, and hu i lengh i he ame a he Euclidean diance beween u and l. The lengh i eiher he Euclidean diance beween u and he line conaining l, or he Euclidean diance beween u and he endpoin of l cloe o u. We how how o handle he fir cae only becaue he econd cae can be handled analogouly. To ue hi obervaion, we ue l(α) o denoe he line of lope α paing hrough v for any α > 0. There i an inerval I uch ha l(α) conain a line-of-igh in L if and only if α I. The Euclidean diance beween u and l(α) i he ame a he diance beween u and he line-of-igh conained in l(α). Thu, in he following, we conider he diance beween u and l(α) for every α I. Since l(α) pae hrough a common verex, he line l(α) can be repreened a he form of y = αx + f(α), where f(α) i a funcion linear in α. Then, he diance beween u and l(α) can be repreened a he form of c 1 α + c 2 / α 2 + 1, where c 1 and c 2 are conan depending only on v and u. Then our problem reduce o he problem of finding a minimum of he funcion of he form of ( c 1 α + c 2 + c 1α + c 2 )/ α for four conan c 1, c 2, c 1 and c 2, and for all α I. We can find a minimum in conan ime uing an elemenary analyi. Lemma 4.2. For any wo conecuive even, he line-of-igh l lying beween he hem ha minimize he maximum of he diance from and o l can be found in conan ime. Theorem 4.3. Given a imple n-gon P wih no hole and wo poin, P, a poin-pair (, ) uch ha i) P and ii) eiher π(, ) + π(, ) or max{ π(, ), π(, ) } i minimized can be compued in O(n) ime. Proof. Our algorihm fir compue all pah- and boundary-even a decribed in Lemma 3.2. The number of even inroduced during hi phae i bounded by he number of verice of he hore pah map, M and M, repecively, which are O(n). In he nex ep, i compue he bend-even on P a decribed in Lemma 3.3, which can be done in O(n) ime. Finally, our algorihm ravere he equence of even. Beween any wo conecuive even, i compue he repecive local opimum in conan ime by Lemma 4.1. I mainain he malle one among he local opima compued o far, and reurn i once all even are proceed. Therefore he running ime of he algorihm i O(n). For he correcne, conider he combinaorial rucure of a oluion and how i change. The pah-even enure ha all verice of π(, ) are conidered a being he verex lying on he egmen connecing he oluion (, ). While he line-of-igh roae around one fixed verex of π(, ), eiher he endpoin of line-of-igh weep over or become angen o a verex of P. Thee are exacly he boundary-even. Or he combinaorial rucure of π(, ) or π(, ) change a inerior verice of π(, ) or π(, ) appear or diappear. Thee happen exacly a bend even. Therefore, our algorihm reurn an opimal poin-pair. E u r o C G 1 8

6 16:6 On Romeo and Julie Problem Acknowledgmen Thi reearch wa iniiaed a he 19h Korean Workhop on Compuaional Geomery in Würzburg, Germany. Reference 1 E. M. Arkin, A. Efra, C. Knauer, J. S. B. Michell, V. Polihchuk, G. Roe, L. Schlipf, and T. Talviie. Shore pah o a egmen and quicke viibiliy querie. Journal of Compuaional Geomery, 7(2):77 100, B. Aronov, L. J. Guiba, M. Teichmann, and L. Zhang. Viibiliy querie and mainenance in imple polygon. Dicree Compu. Geom., 27(4): , S. Carlon, H. Jonon, and B. J. Nilon. Finding he hore wachman roue in a imple polygon. Dicree Compu. Geom., 22(3): , W. Chin and S. Nafo. Opimal wachman roue. In Proceeding of he 2nd ACM Sympoium on Compuaional Geomery, page 24 33, W. Chin and S. Nafo. Opimum wachman roue. Informaion Proceing Leer, 28(1):39 44, M. Dror, A. Efra, A. Lubiw, and J. S. B. Michell. Touring a equence of polygon. In Proc. 35h ACM Sympoium on Theory of Compuing, page , A. Dumirecu and C. D. Tóh. Wachman our for polygon wih hole. Compuaional Geomery, 45(7): , A. Ganguli, J. Core, and F. Bullo. Viibiliy-baed muli-agen deploymen in orhogonal environmen. In Proc. Am. Conrol Conf., page , R. Khoravi and M. Ghodi. The fae way o view a query poin in imple polygon. In Proc. European Workhop on Compuaional Geomery, page , J. S. B. Michell. Approximaing wachman roue. In Proceeding of he 24h Annual ACM-SIAM Sympoium on Dicree Algorihm, page , H. Wang. Quicke viibiliy querie in polygonal domain. In Proceeding of he 33rd Inernaional Sympoium on Compuaional Geomery, volume 77, page 61:1 61:16, E. L. Wyner and J. S. B. Michell. Shore pah for a wo-robo rendez-vou. In Proc. 5h Canadian Conference on Compuaional Geomery, page , 1993.