Guarding curvilinear art galleries with edge or mobile guards

Transcription

1 Guaring curvilinear ar galleries wih ege or mobile guars Menelaos I. Karavelas Deparmen of Applie Mahemaics, Universiy of Cree, GR-1 09 Heraklion, Greece, an Insiue of Applie an Compuaional Mahemaics, FO.R.T.H., P.O. Box 18, GR Heraklion, Greece Absrac In his paper we consier he problem of monioring an ar gallery moele as a polygon, he eges of which are arcs of curves. We consier wo ypes of guars: ege guars (hese are eges of he polygon) an mobile guars (hese are eiher eges or sraigh-line iagonals of he polygon). Our focus is on piecewise-convex polygons, i.e., polygons ha are locally convex, excep possibly a he verices, an heir eges are convex arcs. We reuce he problem of monioring a piecewise-convex polygon o he problem of 2-ominaing a consraine riangulaion graph wih eges or iagonals, where 2-ominance requires ha every riangle in he riangulaion graph has a leas wo of is verices in he 2-ominaing se. We show ha, given a riangulaion graph T P of a polygon P wih n verices: (1) n+1 iagonal guars are always sufficien an someimes necessary, an (2) 2n+1 ege guars are al- ways sufficien an 2n ege guars are someimes necessary, in orer o 2-ominae T P. We also show ha a iagonal (resp., ege) 2-ominaing se of size n+1 (resp., n ) can be compue in O(n) ime an space, whereas an ege 2-ominaing se of size 2n+1 can be compue in O(n 2 ) ime an O(n) space. Base on hese resuls we prove ha, in orer o monior a piecewiseconvex polygon P wih n 2 verices: (1) n+1 mobile guars or 2n+1 ege guars are always sufficien, an (2) n mobile or ege guars are someimes necessary. A mobile (resp., ege) guar se for P of size n+1 (resp., 2n+1 or n ) can be compue in O(n log n + T(n)) ime an O(n) space, where T(n) enoes he ime for compuing a iagonal (resp., ege) 2-ominaing se of size n+1 (resp., 2n+1 or n ) for a riangulaion graph wih n verices. CR Caegories: F.2.2 [Theory of Compuaion]: Analysis of Algorihms an Problem Complexiy Nonnumerical Algorihms an Problems; I.. [Compuing Mehoologies]: Compuer Graphics Compuaional Geomery an Objec Moeling 1 Inroucion In recen years compuaional geomery has mae a shif owars curvilinear objecs. Recen works have aresse boh combinaorial properies an algorihmic aspecs, as well as he necessary algebraic echniques for resolving preicaes involving curvilinear objecs. The perinen lieraure is quie exensive; he inerese reaer may consul he recen book eie by Boissonna an Teillau [Boissonna an Teillau 200] for a collecion of recen resuls for various classical compuaional geomery problems involving curvilinear objecs. Despie he apparen shif owars he curvilinear worl, an espie he vas range of applicaion areas for ar gallery problems, incluing roboics [Kuc an Siegel 198; Xie e al. 1986], moion planning [Lozano-Pérez an Wesley 199; Michell 1989], compuer vision [Sensrom an Connolly 1986; Yachia 1986; Avis an ElGiny 198; Toussain 1980], graphics [McKenna 198; Chazelle an Incerpi 198], CAD/CAM [Bronsvoor 1988; Eo an Kyung 1989] an wireless neworks [Eppsein e al. 200], here are very few works aressing he well-known ar gallery an illuminaion class of problems when he objecs involve are curvilinear [Urruia an Zaks 1989; Coullar e al. 1989; Czyzowicz e al. 199; Czyzowicz e al. 199; Karavelas an Tsigarias 2008]. The original ar gallery problem was pose by Klee o Chváal: given a simple polygon P wih n verices, how many verex guars are require in orer o monior he inerior of P? Chváal [Chváal 19] prove ha n verex guars are always sufficien an someimes necessary, while Fisk [Fisk 198], a few years laer, gave exacly he same resul using a much simpler proof echnique base on -coloring a riangulaion of he polygon. In he conex of curvilinear polygons, i.e., polygons he eges of which may be line segmens or arcs of curves, Karavelas an Tsigarias [Karavelas an Tsigarias 2008] have shown ha is always possible o monior piecewise-convex polygons (i.e., polygons ha are Keywors: ar gallery, curvilinear polygons, riangulaion graphs, 2-ominance, ege guars, iagonal guars, mobile guars, piecewise-convex polygons mkaravel@em.uoc.gr

2 locally convex, excep possibly a he verices, an heir eges are convex arcs) wih 2n verex guars, whereas here exis classes of such polygons ha require a minimum of n 1 verex or n 2 poin guars. They also show ha 2n poin guars are always sufficien an someimes necessary in orer o monior piecewiseconcave polygons, i.e., polygons ha are locally concave, excep possibly a he verices, an heir eges are convex arcs. Soon afer he firs resuls on monioring polygons wih verex or poin guars, oher ypes of guaring moels where consiere. Toussain inrouce in 1981 he noion of ege guars. A poin p in he inerior of he polygon is consiere o be moniore if i is visible from a leas one poin of an ege in he guar se. Ege guars were inrouce as a guaring moel in which guars were allowe o move along he eges of he polygon. Anoher variaion, aing back o 198, is ue o O Rourke: guars are allowe o move along he eges or iagonals of he polygon. This ype of guars is calle mobile guars. Toussain conjecure ha, excep for a few polygons, n ege guars are always sufficien. There are only wo known counerexamples o his conjecure, wih n =, 11, ue o Paige an Shermer [Shermer 1992], requiring n+1 ege guars, whereas here exiss a family of polygons ha require n ege guars. The firs sep owars Toussain s conjecure was mae by O Rourke [O Rourke 198; O Rourke 198] who prove ha n mobile guars are always sufficien an occasionally necessary in orer o monior any polygon wih n verices. The echnique of O Rourke, for proving he upper boun, amouns o reucing he problem of monioring a simple polygon o ha of ominaing a riangulaion graph of he polygon. A riangulaion graph is a maximal ouerplanar graph, all inernal faces of which are riangles. Dominance means ha a leas one of he verices of each riangle in he riangulaion graph is incien o a mobile guar. The reucion o he problem of riangulaion graph ominance is applicable o he case of ege guars. Shermer [Shermer 1992] sele he problem of ominaing riangulaion graphs wih ege guars by showing ha n ege guars are always sufficien an someimes necessary, excep for n =, 6 or 1, in which case one exra 10 ege guar may be necessary; his, also, consiues he bes known upper boun on he number of ege guars ha are sufficien in orer o monior an n-verex polygon. In his paper we consier he problem of monioring piecewiseconvex polygons wih ege or mobile guars; in our conex, a mobile guar is eiher an ege or a sraigh-line iagonal of he polygon. Our proof echnique capializes on he echnique use by O Rourke o prove igh bouns on he number of mobile guars ha are sufficien for monioring sraigh-line polygons [O Rourke 198]. Unlike O Rourke s paraigm, where he soluion for he ominance problem is rivially a soluion for he geomeric guaring problem, in our paraigm we firs reuce he geomeric problem o a combinaorial problem, an hen map of he soluion for he combinaorial problem o a soluion for he geomeric problem. More precisely, in orer o monior piecewise-convex polygons wih mobile or ege guars, we firs reuce he problem of monioring a piecewise-convex polygon P o he problem of 2-ominaing a consraine riangulaion graph. Given a riangulaion graph T P of a polygon P, a se of eges an/or iagonals of T P is a 2-ominaing se of T P if every riangle in T P has a leas wo of is verices incien o an ege or iagonal in he 2- ominaing se. We prove ha n+1 iagonal guars (i.e., eges or iagonals of T P ) are always sufficien an someimes necessary in orer o 2-ominae T P, whereas 2n+1 ege guars are always sufficien an 2n ege guars are someimes necessary in orer o 2-ominae T P. The proofs of sufficiency are inucive on he number of verices of P. In he case of iagonal 2-ominance, our proof yiels a linear ime an space algorihm. In he case of ege 2-ominance, he inucive sep incorporaes ege conracion operaions, hus yieling an O(n 2 ) ime an O(n) space algorihm, where n is he size of P. A linear ime an space algorihm can be aaine by slighly relaxing he size of he ege 2-ominaing se. More precisely, we have shown ha we can 2-ominae T P wih n eges; he proof oes no make use of ege conracions an is analogous, hough more complicae, o he proof, presene in his paper, for he case of iagonal 2-ominance. Focusing back o he geomeric guaring problem, he riangulaion graph T P of he piecewise-convex polygon P is a consraine riangulaion graph: we require ha cerain iagonals of T P are presen. The remaining non-riangular subpolygons of T P are sraigh-line polygons an may be riangulae arbirarily. For he ege guaring problem, any ege 2-ominaing se compue for T P is also an ege guar se for P. A iagonal 2-ominaing se D of T P, however, may conain iagonals of T P ha are no embeable as sraigh-line iagonals of P. To prouce a mobile guar se for P, we keep all eges an sraigh-line iagonals of P in D an map non-sraigh-line iagonals in D o eges of P. In summary, we can compue: (1) a mobile guar se for P of size a mos n+1 in O(nlog n) ime an O(n) space; (2) an ege guar se for P of size a mos 2n+1 in O(n 2 ) ime an O(n) space; () an ege guar se for P of size a mos n in O(n log n) ime an O(n) space. Finally, we show ha n ege or mobile guars are someimes necessary in orer o monior P. The res of he paper is srucure as follows. Secion 2 is evoe o 2-ominance of riangulaion graphs using iagonal or ege guars. In Secion we iscuss he problem of monioring piecewise-convex polygons wih mobile or ege guars. Finally, in Secion we conclue wih a iscussion of our resuls an open problems. 2 2-ominance of riangulaion graphs Given a polygon P wih n verices, is riangulaion graph T P is a maximal ouerplanar graph, i.e., a Hamilonian planar graph consising of n verices an 2n eges, all inernal faces of which are riangles (cycles of size ). The riangulaion graph of a sraighline polygon, i.e., a polygon he eges of which are line segmens, is he planar graph we ge when he polygon has been riangulae. A ominaing se D of a riangulaion graph T P is a se of verices, eges or iagonals of T P such ha a leas one of he verices of each riangle in T P belongs o D. An ege (resp., iagonal) ominaing se of T P is a ominaing se of T P consising of only eges (resp., eges or iagonals) of P. A 2-ominaing se D of T P is a ominaing se of T P ha has he propery ha every riangle in T P has a leas wo of is verices in D. In a similar manner, an ege (resp., iagonal) 2-ominaing se of T P is a 2-ominaing se of T P consising only of eges (resp., eges or iagonals) of T P. Before proceeing wih he main resuls of his secion, we sae he following lemma, which is a irec generalizaion of Lemmas 1.1 an.6 in [O Rourke 198]. Lemma 1 Consier an ineger λ 2. Le P be a polygon of n 2λ verices, an T P a riangulaion graph of P. There exiss a iagonal in T P ha pariions T P ino wo pieces, one of which conains k arcs corresponing o eges of P, where λ k 2(λ 1). Proof. Choose o be a iagonal of T P ha separaes off a minimum number of polygon eges ha is a leas λ. Le k λ be his minimum number, an label he verices of P wih he labels 0,1,..., n 1, such ha is (0, k). The iagonal suppors a riangle whose apex is a verex, 0 k. Since k is minimal λ 1 an k λ 1. Thus, λ k 2(λ 1).

3 Diagonal guars Using Lemma 1 for λ =, yiels he following heorem concerning he wors-case number of iagonals ha are sufficien an necessary in orer o 2-ominae a riangulaion graph. The inucive proof ha follows is no he simples possible. The inerese reaer may fin a much simpler alernaive proof in [Karavelas 2008]. The simpler proof, however, makes use of ege conracions, which make i unsuiable as a basis for a linear ime an space algorihm. On he oher han, he proof presene below can be implemene in linear ime an space, as will be iscusse below. The proof ha follows is a eaile, raher echnical, case-by-case analysis; we presen i, however, unconense, so as o illusrae he eails ha perain o our linear ime an space algorihm. Theorem 2 Every riangulaion graph T P of a polygon P wih n verices can be 2-ominae by n+1 iagonal guars. This boun is igh in he wors-case. Proof. The proof for n is sraighforwar an is omie. Le us now assume ha n 8 an ha he heorem hols for all n such ha n < n. By means of Lemma 1 wih λ =, here exiss a iagonal ha pariions T P ino wo riangulaion graphs T 1 an T 2, where T 1 conains k bounary eges of T P wih k 6. Le v i, 0 i k, be he k + 1 verices of T 1, as we encouner hem while raversing P counerclockwise, an le v k be he common ege of T 1 an T 2. For each value of k we are going o efine a iagonal 2-ominaing se D for T P of size n+1. In wha follows ij enoes he iagonal v iv j, whereas e i enoes he ege v i i+1. Consier each value of k separaely. k =. In his case T 2 conains n verices. By our inucion hypohesis we can 2-ominae T 2 wih f(n ) = n+1 1 iagonal guars. Le D 2 be he iagonal 2-ominaing se for T 2. A leas one of an v is in D 2. The cases are symmeric, so we can assume wihou loss of generaliy ha D 2. Consier he following cases (see Fig. 1 2 ): 1 T 1. Se D = D 2 { 1}. 2 T 1. Se D = D 2 { 2}. 02, 0 T 1. Se D = D 2 {e 2}. k =. The presence of iagonals 0 an 1 woul violae he minimaliy of k. Le be he riangle suppore by in T 1. The apex v of his riangle can eiher be or v. The wo cases are symmeric, so we assume, wihou loss of generaliy ha he apex of is. Consier he riangulaion graph T = T 2 {}. I has n verices, hence, by our inucion hypohesis, i can be 2-ominae wih f(n ) = n+1 1 iagonal guars. Le D be he 2-ominaing se for T. Consier he following cases (see Fig. 2): 02 D 2. Se D = D {e }. 02 D 2. If 2 D, se D = (D \ { 2}) { 02, e }. Oherwise, canno belong o D (boh eges of T incien o o no belong o D ). However, he riangle is 2- ominae in T, which implies ha boh an v belong o D. Hence, se D = D {e 2}. k = 6. The presence of iagonals 0, 0, 16 an 26 woul violae he minimaliy of k. Le be he riangle suppore by in T 1. The apex v of his riangle mus be v. Le be he secon riangle in T 1 beyon suppore by he iagonal 0, an le v be is verex opposie o 0. Symmerically, le be he secon riangle in T 1 beyon suppore by he iagonal 6, an le v be is verex opposie o 6. Consier he riangulaion graph T = T 2 {, }. I has n verices, hence, by our inucion hypohesis, i can be 2-ominae wih 1 Inices are consiere o be evaluae moulo n. 2 In all figures, eges/iagonals in a ominaing/guar se are shown as hick soli/ashe lines, while verices in a ominaing/guar se are ransparen. v v v v v Figure 1: The case k =. Lef: 1 T 1. Mile: 2 T 1. Righ: 02, 0 T 1. v v v v v v v v v Figure 2: The case k =. Lef: 02 D. Mile: 02 D an 2 D. Righ: 02, 2 D. f(n ) = n+1 1 iagonal guars. Le D be he 2- ominaing se for T. Le us firs consier he case v. Le be he unique iagonal of he quarilaeral v v v v 6. Consier he following cases (see Fig. ): 02 D. Se D = D { }. 02 D. We furher isinguish beween he following wo cases: 6 D. If D, simply se D = (D \ { 6}) {e 2, e }. If D, he iagonal 0 canno belong o D. Therefore, in orer for he riangle o be 2- ominae by D, we mus have ha e 2 in D. Thus, se D = (D \ { 6}) {e 0, e }. 6 D. In orer for o be 2-ominae by D we mus have ha eiher 0 D or e 2 D. If 0 D, se D = (D \ { 0}) { 02, }; oherwise, se D = (D \ {e 2}) { 02, }. The siuaion is enirely symmeric if v v. Hence, he only remaining case is he case where v an v v. Consier he following cases (see Fig. ): 1 D. Se D = D {e }. 1 D. We furher isinguish beween he following wo cases: 0 D. Se D = (D \ { 0}) {e 0, }. v v v v v v v v v v v v v 6 v 6 v 6 v 6 Figure : The case k = 6 wih v. Lef: 02 D ; i is assume in his subfigure ha. Mile lef: 02 D an 6 D an D. Mile righ: 02 D an 6 D an D. Righ: 02, 6 D ; i is assume in his subfigure ha 6. v v v v v v 1 v v 6 v 6 Figure : The case k = 6 wih v an v v. Lef: 1 D ; also 1, 0, e 0 D. Righ: 1 D an 0 D ; also 1, 0 D an e 0 D. v

4 T 1 T 2 v v v T v m+1 v m v m 6 vm 1 v m 2 v m v m v m Figure : Three riangulaion graphs T i, i = 1,2,, wih n = m + i 1 verices, respecively. All hree riangulaion graphs require a leas n+1 iagonal guars in orer o be 2-ominae. 0 D. If e 0 D, se D = D { }. Oherwise, i.e., if e 0 D, canno be in D. Since he riangle is 2-ominae in D, boh an v have o belong o D. Since he iagonal 0 oes no belong o D, he iagonal 6 has o belong o D in orer for v o be in D. Thus, se D = (D \ { 6}) { 1, e }. Le us now urn our aenion o esablishing he lower boun. Consier he riangulaion graphs T i, i = 1,2,, wih n = m + i 1 verices, shown in Fig., an le D i be he iagonal 2-ominaing se of T i. The cenral par of T i is riangulae arbirarily. Noice ha each subgraph of T i, shown in eiher ligh or ark gray, requires a leas one among is eges or iagonals o be in D i in orer o be 2-ominae. This observaion immeiaely esablishes a lower boun of n. Le us now assume ha D = n. Uner his assumpion, each shae subgraph in T mus have exacly one among is eges/iagonals in D. Moreover, none of he iagonals in he cenral par of T (no shown in Fig. (boom)) can belong o D, since hen we woul have D > n. Consier he riangulae hexagon H := v m v m 2v m 1v mv m+1. In orer for H o be 2-ominae wih exacly one of is eges/iagonals, boh an v m have o be in D ue o eges/iagonals in he neighboring shae subgraphs, while he unique ege or iagonal of H in D mus be he iagonal m 2,m. Since we require ha v m mus belong o D via an ege/iagonal of he quarilaeral v m 6v m v m v em, an a he same ime we require ha exacly one of he eges/iagonals of v m 6v m v m v em o be in D, e m mus belong o D an v m 6 mus be in D ue o an ege/iagonal in he quarilaeral v m 9v m 8v m v em 6. Cascaing his argumen, we conclue ha, since v mus belong o D ue o an ege/iagonal of he quarilaeral v, an a he same ime exacly one of he eges/iagonals of v mus be in D, e 2 mus belong o D an mus belong o D ue o an ege/iagonal in H. This yiels a conraicion, since he unique ege/iagonal of H in D is m 2,m, which is, obviously, no incien o. The proof of Theorem 2 can almos immeiaely be ransforme ino a linear ime an space algorihm. The riangulaion graph T P of P is assume o be represene via a half-ege represenaion. Half-eges an verices in our represenaion are assume o have aiional flags for inicaing wheher a half-ege is a bounary Figure 6: The four possible configuraions for he ual rees 1 for k 6, shown as hick soli lines. The iagonal separaes T 1 from T 2. ege of he polygon, or wheher a half-ege or a verex of T P is marke as being in he iagonal 2-ominaing se of T P. Uner hese assumpions, aing or removing a half-ege or a verex from he sough-for 2-ominaing se, querying a half-ege or a verex for membership in he 2-ominaing se, as well as forming he riangulaion graph for he recursive calls, all ake O(1) ime. Consier a iagonal ha separaes T P ino wo riangulaion graphs T 1 an T 2, where T 1 conains k =, or 6 eges of P ; recall from he proof of Lemma 1 (for λ = ) ha he value of k is minimal. Le be he ual ree of T P, 1 (resp., 2) he ual ree of T 1 (resp., T 2) an 1 = 1 { }, where is he ual ege of in. 1 consiss of a subree of wih 2, or eges of, connece wih he res of via a egree-2 or a egree- noe (see Fig. 6). Moreover, for n 1, he subrees 1 corresponing o ifferen iagonals of T P mus be ege isjoin (oherwise he number of verices of P woul be less han 1). Having mae hese observaions we can now escribe he algorihm for compuing he iagonal 2-ominaing se D for T P. We firs escribe he iniializaion seps: (1) iniialize D o be empy; (2) creae a queue Q, an iniialize i o be empy. Q will consis of iagonals of T P ; () for each iagonal of T P eermine wheher i separaes off, or 6 eges of P in T P an is size is minimal. In oher wors, eermine if he ual ege of in is ajacen o subrees of he form shown in Fig. 6. If so, pu in Q. The recursive par of he algorihm is as follows: 1. If he number of verices of T P is less han 1, fin a iagonal 2-ominaing se D an reurn. 2. If Q is no empy: (a) Pop a iagonal ou of Q. (b) If T 2 has less han 1 verices, empy he queue Q an fin a 2-ominaing se D 2 for T 2. Base on D 2, an accoring o he cases in he proof of Theorem 2, compue D an reurn. (c) Deermine he case in he proof of Theorem 2, o which correspons. Le ˆT be he riangulaion graph for which we are suppose o fin he 2-ominaing se recursively, an le ˆ be he ual ree of ˆT. Le V be he se of verices in ( ˆ 1). For any v V eermine if v is a leaf-noe o a subree of ˆ like he subrees in Fig. 6. If so, a he corresponing iagonal o Q. () Recursively, fin a iagonal 2-ominaing ˆD for ˆT, using Q as he queue. (e) Consruc from ˆD a iagonal 2-ominaing se D for T P an reurn. I is sraighforwar o verify ha he ime T(n) spen for he recursive par of our algorihm saisfies he recursion T(n) = T(n ) + O(1), which gives T(n) = O(n). Since iniializaion akes linear ime, an our space requiremens are obviously linear in he size of P (we o no uplicae pars of T P for he recursive calls, bu raher se appropriaely he bounary flags for some halfeges), we arrive a he following heorem. Theorem Given he riangulaion graph T P of a polygon P wih

5 n verices, we can compue a iagonal 2-ominaing se for T P of size a mos n+1 in O(n) ime an space. Ege guars Applying Lemma 1 for λ = we can prove ha 2n+1 ege guars are sufficien in orer o 2-ominae he riangulaion of an n-verex piecewise-convex polygon. The proof is similar o he proof of Theorem 2; however, exacly like he simple (omie) proof of Theorem 2, i makes use of ege conracions, yieling an O(n 2 ) ime an O(n) space algorihm. A linear ime an space algorihm is feasible by relaxing he requiremen on he size of he ege 2-ominaing se. More precisely, applying Lemma 1 for λ = 6, we have shown ha we can 2-ominae he riangulaion graph of a piecewise-convex polygon wih n ege guars. Alhough his resul is weaker, i oes no use ege conracions. We can, hus, evise a linear ime an space algorihm for compuing an ege 2-ominaing se of size a mos n, in exacly he same manner as in he case of iagonal 2-ominance. The following heorem summarizes our resuls, incluing our wors-case lower boun on he number of ege guars require o 2-ominae he riangulaion graph of a piecewise-convex polygon. Theorem ([Karavelas 2008]) Given he riangulaion graph T P of a polygon P wih n verices, we can eiher compue: (1) an ege 2-ominaing se for T P of size a mos 2n+1 (excep for n =, where one aiional ege is require) in O(n 2 ) ime an O(n) space, or (2) an ege 2-ominaing se for T P of size a mos n (excep for n =, where one aiional ege is require) in O(n) ime an space. Finally, here exiss a family of riangulaion graphs wih n verices ha require 2n ege guars in orer o be 2-ominae. Piecewise-convex polygons Le,..., v n, n 2, be a sequence of poins an a 1,..., a n a se of curvilinear arcs, such ha a i has as enpoins he poins v i an v i+1. We will assume ha he arcs a i an a j, i j, o no inersec, excep when j = i 1 or j = i + 1, in which case hey inersec only a he poins v i an v i+1, respecively. We efine a curvilinear polygon P o be he close region of he plane elimie by he arcs a i. The poins v i are calle he verices of P. An arc a i is a convex arc if every line on he plane inersecs a i a a mos wo poins or along a line segmen. A polygon P is calle a locally convex polygon if P is locally convex excep possibly a is verices (see Fig. (lef)). A polygon P is calle a piecewiseconvex polygon, if i is locally convex an is eges are convex arcs (see Fig. (righ)). Le a i be an ege of a piecewise-convex polygon P wih enpoins v i an v i+1. We call he convex region r i elimie by a i an v iv i+1 a room, where xy enoes he line segmen from x o y. A room is calle egenerae if he arc a i is a line segmen. For p, q a i, pq is calle a chor of a i; he chor of r i is v iv i+1. An empy room is a non-egenerae room ha oes no conain any verex of P in he inerior of r i or in he inerior of v iv i+1. A non-empy room is a non-egenerae room ha conains a leas one Figure : Lef: A locally convex polygon. Righ: A piecewiseconvex polygon. Figure 8: Lef: A piecewise-convex polygon P. Righ: The riangulaion graph T P of P. The bounary eges of T P are shown as hick soli lines. The crescens of P are shown in ligh gray, whereas he sars of P are shown in ark gray. verex of P in he inerior of r i or in he inerior of v iv i+1. We say ha a poin p in he inerior of a piecewise-convex polygon P is visible from a poin q if pq lies in he closure of P. We say ha P is moniore by a guar se G if every poin in P is visible from a leas one poin belonging o some guar in G. An ege (resp., mobile) guar is an ege (resp., ege or iagonal) of P belonging o a guar se G of P. An ege (resp., mobile) guar se is a guar se ha consiss of only ege (resp., mobile) guars. Le P be a piecewise-convex polygon wih n verices. Consier a convex arc a i of P, wih enpoins v i an v i+1, an le r i be he corresponing room. If r i is a non-empy room, le X i be he se of verices of P ha lie in he inerior of v iv i+1, an le R i be he se of verices of P in he inerior of r i or in X i. If R i X i, le C i be he se of verices in he convex hull of he verex se (R i \ X i) {v i, v i+1}; if R i = X i, le C i = X i {v i, v i+1}. Finally, le Ci = C i \ {v i, v i+1}. If r i is an empy room, le C i = {v i, v i+1} an Ci =. Le T P be he sough-for riangulaion graph of P. The verex se of T P is he se of verices of P. The eges an iagonals of T P, as well as heir embeing, are efine as follows (see also Fig. 8): If a i is a line segmen or r i is an empy room, he ege (v i, v i+1) is an ege in T P, an is embee as v iv i+1. If r i is a non-empy room, he following eges or iagonals belong o T P : 1. (v i, v i+1), 2. (c i,j, c i,j+1), for 1 j K i 1, where K i = C i, c i,1 v i an c i,ki v i+1. The remaining c i,j s are he verices of P in C i as we encouner hem when walking insie r i an on he convex hull of he poin se C i from v i o v i+1, an. (v i, c i,j), for j K i 1, provie ha K i. We call hese iagonals weak iagonals. The iagonals (c i,j, c i,j+1), 1 j K i 1 are embee as c i,j, c i,j+1, whereas he iagonals (v i, c i,j), j K i 1, are embee as curvilinear segmens. Finally, he eges (v i, v i+1) are embee as curvilinear segmens, namely, he arcs a i. The eges (v i, v i+1), along wih he iagonals (c i,j, c i,j+1), 1 j K i 1, pariion P ino subpolygons of wo ypes: (1) subpolygons ha lie enirely insie a non-empy room, calle crescens, an (2) subpolygons elimie by eges of he polygon P, as well as iagonals of he ype (c i,j, c i,j+1), calle sars. In general, a piecewise-convex polygon may only have crescens, or only sars, or boh. The crescens are riangulae by means of he iagonals (v i, c i,j), j K i 1. To finish he efiniion of he riangulaion graph T P, we simply nee o riangulae all sars insie P. Since sars are sraigh-line polygons, any polygon riangulaion algorihm may be use o riangulae hem. In irec analogy o he ypes of subpolygons we can have insie P, we have wo possible ypes of riangles in T P : (1) riangles insie sars, calle sar riangles, an (2) riangles insie a crescen, calle crescen riangles. Crescen riangles have a leas one ege ha is

6 a weak iagonal, excep when he number of verices of P in he inerior of he corresponing room r is exacly one, in which case none of he hree eges of he unique crescen riangle in r is a weak iagonal. A crescen riangle ha has a leas one weak iagonal among is eges is calle a weak riangle. Mobile guars Le G TP be a iagonal 2-ominaing se of T P. Base on G TP we efine a se G of eges or sraigh-line iagonals of P as follows (see also Fig. 9): (1) a o G every non-weak iagonal of G TP, an (2) for every weak iagonal in G TP, a o G he ege of P elimiing he crescen ha conains he weak iagonal. Clearly, G G TP. Lemma Le P be a piecewise-convex polygon wih n verices, T P is consraine riangulaion graph, an G TP a iagonal 2-ominaing se of T P. The se G of mobile guars, efine by mapping every non-weak iagonal of G TP o iself, an every weak iagonal of G TP o he corresponing convex arc of P elimiing he crescen ha conains, is a mobile guar se for P. Proof. Le q be a poin in he inerior of P. q is eiher insie: (1) an empy room r i of P, (2) a sar riangle s of T P, () a non-weak crescen riangle nw of T P, or () a weak crescen riangle w of T P. In any of he four cases, q is visible from a leas wo verices u 1 an u 2 of T P ha are connece via an ege or a iagonal in T P. In he firs case, q is visible from he wo enpoins v i an v i+1 of a i. In he secon case, q is visible from all hree verices of s. The hir case arises when q is insie a non-empy room r j wih C j = 1 ( nw is he unique crescen riangle in r j), in which case q is visible from a leas wo of he hree verices v j, v j+1 an c j,1. Finally, in he fourh case, q has o lie insie he crescen of a non-empy room r j wih C j 2, an is visible from a leas wo consecuive verices c j,k an c j,k+1 of C j. Since G is a iagonal 2-ominaing se for T P, an (u 1, u 2) T P, a leas one of u 1 an u 2 belongs o G TP. Wihou loss of generaliy, le us assume ha u 1 G TP. If u 1 G, q is moniore by u 1. If u 1 G, u 1 has o be an enpoin of a weak iagonal w in G TP. Le r l be he room, insie he crescen of which lies w. Since w G TP, we have ha a l G. If q lies insie he crescen of he room r l (his can only happen in case () above), q is visible from a l, an hus moniore by a l. Oherwise, u 1 canno be an enpoin of a l (a l G, whereas u 1 G), which implies ha u 1 Cl, i.e., u 1 c l,m, wih 2 m K l 1. Bu hen q lies insie he cone wih apex c l,m, elimie by he rays c l,m c l,m 1 an c l,m c l,m+1, an conaining a leas one of v l an v l+1 in is inerior. Since, q is visible from he inersecion poin of he line qu 1 wih a l, q is moniore by a l. Our approach for compuing he mobile guar se G of P consiss of hree major seps: (1) Consruc he consraine riangulaion T P of P ; (2) Compue a iagonal 2-ominaing se G TP for he riangulaion graph T P ; () Map G TP o G. The ses C i, neee in orer o consruc he consraine riangulaion T P of P can be compue in O(nlog n) ime an O(n) space (cf. [Karavelas an Tsigarias 2008]). Once we have he ses C i, he consraine riangulaion T P of P can be consruce in linear ime an space. By Theorem, compuing G TP akes linear ime; fur- hermore G TP n+1 n+1, which implies ha G. Finally, he consrucion of G from G TP akes O(n) ime an space: for every iagonal in G TP we nee o eermine if i is a weak iagonal, in which case we nee o a he ege of P elimiing he crescen in which lies o G; by appropriae bookkeeping a he ime of consrucion of T P hese operaions can ake O(1) per iagonal. Summarizing, by Theorem 2, Lemma an our analysis above, we arrive a he following heorem. The case n = 2 can be rivially esablishe. Theorem 6 Le P be a piecewise-convex polygon wih n 2 verices. We can compue a mobile guar se for P of size a mos in O(nlog n) ime an O(n) space. n+1 Ege guars Le G TP be an ege 2-ominaing se of T P (see Fig. 10). The se G of ege guars, efine by mapping every ege in G TP o he corresponing convex arc of P, is an ege guar se for P (cf. [Karavelas 2008]). By Theorem, we can eiher compue an ege 2-ominaing se G TP of size 2n+1 in O(n 2 ) ime an O(n) space, or an ege 2-ominaing se G TP of size n (excep for n = where one aiional ege is neee) in linear ime an space. Since T P can be compue in O(n log n) ime an O(n) space, an G = G TP, we arrive a he following heorem. The case n = 2 is rivial, since in his case any of he wo eges of P is an ege guar se for P. Theorem Le P be a piecewise-convex polygon wih n 2 verices. We can eiher: (1) compue an ege guar se for P of size 2n+1 (excep for n =, where one aiional ege guar is require) in O(n 2 ) ime an O(n) space, or (2) compue an ege guar se for P of size n (excep for n = 2,, where one aiional ege guar is require) in O(n log n) ime an O(n) space. Lower boun consrucion Consier he piecewise-convex polygon P of Fig. 11. Each spike consiss of hree eges, namely, wo line segmens an a convex arc. In orer for poins in he nonempy room of he convex arc o be moniore, eiher one of he hree eges of he spike, or a iagonal a leas one enpoin of which is an enpoin of he convex arc, has o be in any guar se of P : he chosen ege or iagonal in a spike canno monior he non-empy room insie anoher spike of P. Since P consiss of k spikes, yieling n = k verices, we nee a leas k ege or mobile guars o Figure 10: Lef: an ege 2-ominaing se for he riangulaion graph T P from Fig. 8. Righ: he corresponing ege guar se. Figure 9: Lef: a iagonal 2-ominaing se for he riangulaion graph T P from Fig. 8. Righ: he corresponing mobile guar se. Figure 11: The lower boun consrucion: he polygon shown conains n = k verices, an requires k = n ege or mobile guars in orer o be moniore.

7 monior P. We, hus, conclue ha P requires a leas n ege or mobile guars in orer o be moniore. Discussion an open problems As far as he problem of 2-ominance of riangulaion graphs is concerne, we have no ye foun a way o compue an ege 2- ominaing se of size a mos 2n+1 in o(n 2 ) ime, whereas we have shown ha is i is possible o compue an ege 2-ominaing se of size a mos n in linear ime an space. I, hus, remains an open problem how o compue an ege 2-ominaing se of size a mos 2n+1 in o(n 2 ) ime an linear space. Moreover, we conjecure ha here exis riangulaion graphs ha require a minimum of 2n+1 ege guars; hus far we have foun such riangulaion graphs for n =, 12, 1, 22. Once a 2-ominaing se D has been foun for he consraine riangulaion graph of a piecewise-convex polygon P, we eiher prove ha D is also a guar se for P (his is he case for ege guars) or we map D o a mobile guar se for P. In he case of ege guars, he piecewise-convex polygon is acually moniore by he enpoins of he eges in he guar se. In he case of mobile guars, inerior poins of he eges may also be neee in orer o monior he inerior of he polygon. The laer observaion shoul be conrase agains he corresponing resuls for he class of sraighline polygons, where, for boh ege an mobile guars, he polygon is essenially moniore by he enpoins of hese guars (cf. [O Rourke 198]). Anoher imporan observaion, ue o he lower boun in Theorem, is ha he proof echnique of his paper canno possibly yiel beer resuls for he ege guaring problem. If we are o close he gap beween he upper an lower bouns, a funamenally ifferen echnique will have o use. Thus far we have limie our aenion o he class of piecewiseconvex polygons. I woul be ineresing o aain similar resuls for locally concave polygons (i.e., curvilinear polygons ha are locally concave excep possibly a he verices), for piecewise-concave polygons (i.e., locally concave polygons he eges of which are convex arcs), or for curvilinear polygons wih holes. Acknowlegemens The auhor was parially suppore by he IST Programme of he EU (FET Open) Projec uner Conrac No IST-0061 (ACS - Algorihms for Complex Shapes wih Cerifie Numerics an Topology). References AVIS, D., AND ELGINDY, H A combinaorial approach o polygon similariy. IEEE Trans. Inform. Theory IT-2, BOISSONNAT, J.-D., AND TEILLAUD, M., Es Effecive Compuaional Geomery for Curves an Surfaces. Mahemaics an Visualizaion. Springer. BRONSVOORT, W Bounary evaluaion an irec isplay of CSG moels. Compuer-Aie Design 20, CHAZELLE, B., AND INCERPI, J Triangulaion an shapecomplexiy. ACM Trans. Graph., 2, CHVÁTAL, V. 19. A combinaorial heorem in plane geomery. J. Combin. Theory Ser. B 18, 9 1. COULLARD, C., GAMBLE, B., LENHART, W., PULLEYBLANK, W., AND TOUSSAINT, G On illuminaing a se of isks. Manuscrip. CZYZOWICZ, J., RIVERA-CAMPO, E., URRUTIA, J., AND ZAKS, J Proecing convex ses. Graphs an Combinaorics 19, CZYZOWICZ, J., GAUJAL, B., RIVERA-CAMPO, E., URRUTIA, J., AND ZAKS, J Illuminaing higher-imensionall convex ses. Geom. Deicaa 6, EO, K., AND KYUNG, C Hybri shaow esing scheme for ray racing. Compuer-Aie Design 21, 8 8. EPPSTEIN, D., GOODRICH, M. T., AND SITCHINAVA, N Guar placemen for efficien poin-in-polygon proofs. In Proc. 2r Annu. ACM Sympos. Compu. Geom., 2 6. FISK, S A shor proof of Chváal s wachman heorem. J. Combin. Theory Ser. B 2,. KARAVELAS, M. I., AND TSIGARIDAS, E. P., Guaring curvilinear ar galleries wih verex or poin guars. arxiv: v1 [cs.cg]. KARAVELAS, M. I., Guaring curvilinear ar galleries wih ege or mobile guars via 2-ominance of riangulaion graphs. arxiv: v1 [cs.cg]. KUC, R., AND SIEGEL, M Efficien represenaion of reflecing srucures for a sonar navigaion moel. In Proc. 198 IEEE In. Conf. Roboics an Auomaion, LOZANO-PÉREZ, T., AND WESLEY, M. A An algorihm for planning collision-free pahs among polyheral obsacles. Commun. ACM 22, 10, MCKENNA, M Wors-case opimal hien-surface removal. ACM Trans. Graph. 6, MITCHELL, J. S. B An algorihmic approach o some problems in errain navigaion. In Geomeric Reasoning, D. Kapur an J. Muny, Es. MIT Press, Cambrige, MA. O ROURKE, J Galleries nee fewer mobile guars: a variaion on Chváal s heorem. Geom. Deicaa 1, O ROURKE, J Ar Gallery Theorems an Algorihms. The Inernaional Series of Monographs on Compuer Science. Oxfor Universiy Press, New York, NY. SHERMER, T. C Recen resuls in ar galleries. Proc. IEEE 80, 9 (Sep.), STENSTROM, J., AND CONNOLLY, C Builing wire frames for muliple range views. In Proc IEEE Conf. Roboics an Auomaion, TOUSSAINT, G. T Paern recogniion an geomerical complexiy. In Proc. h IEEE Inerna. Conf. Paern Recogn., URRUTIA, J., AND ZAKS, J Illuminaing convex ses. Technical Repor TR-89-1, Dep. Compu. Sci., Univ. Oawa, Oawa, ON. XIE, S., CALVERT, T., AND BHATTACHARYA, B Planning views for he incremenal consrucion of boy moels. In Proc. In. Conf. Paern Recongiion, 1 1. YACHIDA, M D aa acquisiion by muliple views. In Roboics Research: Thir In. Symp.,