Illumination with Orthogonal Floodlights? (Extended Abstract) College Station, Texas MS 3112, US.

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1 Illumination with Othogonal Floodlights (Extended Abstact) James Abello 1, Vladimi Estivill-Casto 2, Thomas Sheme 3, Joge Uutia 1 Depatment of Compute Science, Texas A & M Univesity, College Station, Texas MS 3112, US. abello@cs.tamu.edu. 2 Laboatoio Nacional de Infomatica Avanzada, Rebsamen 80, Xalapa, Veacuz 91000, Mexico. vlad@xalapa.lania.mx. 3 School of Compute Science, Simon Fase Univesity, Bunaby, B.C. V5A 1S6, Canada. sheme@cs.sfu.ca. Depatment of Compute Science, Univesity of Ottawa, Ottawa, Ontaio K1N 6N5, Canada. joge@csi.uottawa.ca. Abstact. We povide the st tight bound fo coveing a polygon with n vetices and h holes with vetex guads. In paticula, we povide tight bounds fo the numbe of oodlights, placed at vetices o on the bounday, sucient to illuminate the inteio o the exteio of an othogonal polygon with holes. Ou esults lead diectly to simple linea, and thus optimal, algoithms fo computing a coveing of an othogonal polygon. Resumen. En este tabajo popocionamos cotas exactas paa ilumina un polgono con n vetices y h hoyos mediante lampaas en los vetices. En paticula, popocionamos cotas excatas paa el numeo de eectoes de apetua =2 sucientes paa ilumina tanto el inteio como el exteio de un polgono otogonal con hoyos. Nuestos esultados conducen a algoitmos de complejidad lineal (y po lo tanto, de complejidad optima) que son a la vez sencillos de implanta. This wok was patially caied out unde gant CONACYT 3912-A902 in Mexico.

2 1 Intoduction The question of guading a polygonal at galley has aised many poblems anging fom polygon decomposition and poblem complexity to combinatoial stuctue of visibility gaphs [1, 11]. Moeove, the study of visibility in this type of geometic setting has not only been natually motivated by many applications, but it has also been fundamental in developing many theoetical and pactical esults [13]. Despite the many vaiants of the poblem, little egad has been placed to the assumption that guads can cove a complete 2 ange of oientations aound them. Rawlins [12] studied visibility along nitely oiented staicases and povided coesponding At Galley Theoems. Howeve, only ecently the question of studying visibility coves with oodlights (that is, static guads with esticted angle of vision) has been aised. The poblem of coveing a line with oodlights has been labeled the stage illumination poblem [3, 5] while the poblem of coveing the plane has also been discussed [3, 1]. Obseve that in O'Rouke's book [11] the st sentence of Chapte 5 says \One of the majo open poblems in the eld of at galley theoems is to establish a theoem fo polygons with holes". In this pape we extend the tight bound of b3(n 1)=8c [6] fo the numbe of othogonal oodlights sucient to cove an othogonal polygon with n vetices to the case of polygons with holes. In paticula, we show that b(3n + (h 1))=8c othogonal oodlights placed at vetices ae always sucient and sometimes necessay to cove an othogonal polygon with n vetices and h holes. Moeove, we show that the above bound is educed to b(n+2h)=c if the othogonal oodlights may be placed in the bounday of the polygon. Homan [8] poved the st tight bound pevious to these esults; namely, bn=c guads ae sometimes necessay and always sucient to cove an othogonal polygon independent of the numbe of holes. Howeve, Homann's esults equie that the guads be placed on points othe than the vetices o the bounday. Homan also claims an O(n 1:5 log 2 log log n) guad placement algoithm. Two goups independently [2] poved that b(n + h)=3c is the coesponding tight bound fo geneal polygons; but again, guads must be allowed in the inteio of the polygon. Othe eots to povide tight bounds fo geneal polygons with holes and othogonal polygons with holes have been successful fo only special cases [7, 13]. Thus, ou esults povide the st theoem with a tight bound fo vetex guading polygons with holes, fo all values of n and h. Moeove, ou poofs lead to linea algoithms that ae simple and pactical since they avoid complex polygon decomposition (we do not need to tiangulate o to nd a quadilatealization of the polygon). We obseve that, fo the case h = 0, the oiginal at galley theoem equies long poofs [10], howeve, ou poofs ae simple. Moeove, fom the point of view of total apetue used, the oiginal at galley theoems may equie an apetue of 3=2 fo each of the bn=c oodlights esulting in a total apetue of 3n=8 (in fact, Fig of O'Rouke's book [11] illustates a possible wost case whee all oodlights that the algoithm nds must be of apetue 3=2). This can be egaded as unnecessaily inecient, since fo example, fo polygons without holes ou esults demonstate that only b(3n )=8c oodlights, each of apetue =2, ae needed even if we ae equied to place them at vetices. The total apetue obtained with ou algoithms is half the naive use of the oiginal at galley theoem and coesponding algoithms. In Section 2 we demonstate that an apetue of =2 is necessay fo vetex oodlights to illuminate an othogonal polygon. We also show that b(3n + (h 1))=8c othogonal oodlights ae sometimes necessay. Section 3 poves that b(3n + (h 1))=8c othogonal oodlights placed at vetices ae always sucient and descibes the linea algoithm to nd the coveing. Section discusses the case whee the oodlights ae placed on the bounday. We show that b(n+2h)=c oodlights ae always sucient and sometimes necessay. Section 5 povides an At Galley Theoem and a coesponding linea algoithm fo illuminating the exteio of a simply-connected othogonal polygon with holes; namely that (n+)=2 othogonal oodlights ae always sucient and sometimes necessay. Section 6 povides some nal emaks. 2 Necessity Bounds Conside a polygonal at galley given by an othogonal polygon P in the plane with n vetices and h holes. Thee ae n 0 vetices on the oute bounday and n i vetices in the i-th hole, with P h n i=0 i = n. An -oodlight is a lamp that shines light in a wedge of apetue. We ae inteested in detemining a set of

3 -oodlights that illuminate (cove) the inteio o the exteio of P. If we equie that the oodlights be placed at vetices, we will call them vetex oodlights, o v-oodlights fo shot. Note that no moe than one oodlight is allowed at a point in P, othewise, we have one oodlight of lage apetue. The following esult demonstates that an apetue of =2 is the smallest apetue necessay fo othogonal polygons. Theoem 1. Fo all > 0, thee is an othogonal polygon that can not be illuminated with vetex oodlights in each vetex with apetue 2. Poof. Let be a line segment with endpoints a and b and middle point in the oigin O. Let the slope of be < and = 2. Let 0 be a segment othogonal to with the same length as and also with middle point in the oigin, but with endpoints c and d. We constuct an othogonal polygon with 12 vetices in the shape of an helix and whose eex vetices ae pecisely a, b, c and d; efe to Fig. 1. If the pongs of P ae u = 8 v = 7 B BB p c = 6 B 10 a = 9 b = 3 B OB 11 d = 12 (a) 5 BB B AA B AB E AB E hhhh H h A B E (b) 1 2 Fig. 1. This othogonal helix equies fou othogonal v-oodlights. lage enough, the line of slope + =2 though u intesects the side cv in a point p fa enough fom c; see Fig. 1 again. The eade can now veify that P can not be illuminated with oodlights of apetue 2 in each vetex. ut In what follows we concentate on oodlights with apetue =2; we will call them othogonal oodlights. The helix of Fig. 1 will povide the building block to pove a bound of the numbe of othogonal oodlights necessay to illuminate an othogonal polygon with n vetices and h holes. Fist, we will pove the bound fo the case h = 0, that is polygons without holes, and late the geneal case. (a) (b) (c) (d) Fig. 2. Othogonal polygons that equie one =2-oodlight fo each pong. Conside again the othogonal polygon in Fig. 1, whee the pongs ae long enough that a oodlight as shown in pat (b) can illuminate at most one pong. Clealy, at least othogonal oodlights ae equied to

4 cove this polygon. Now, conside the pogession illustated by Fig. 2. At each new stage, we mege a copy of the polygon in Fig. 1 by its left pong. Vetices 10 and 11 ae identied with the two ight-most vetices in the pevious gue. It is not had to see that at each stage, thee moe othogonal v-oodlights ae needed but only eight vetices ae added. It now follows that b(3n )=8c oodlights ae necessay. We now povide polygons that demonstate the necessity of b 3n+(h1) c othogonal oodlights. Conside 8 the polygon P 32 of Fig. 3 (a). It has 32 vetices and one hole. This polygon is constucted with fou copies of the helix of Fig. 1 joined to fom a polygon with one hole. Moeove, P 32 equies 12 othogonal oodlights, one fo each of the eight pongs and one fo each of the fou alleys. This polygon povides the basic building block. Fo lage values of h, we join the the ight-most edge of one copy of P 32 with the left-most edge of anothe. Fo example, Fig. 3 (b) is a polygon with 60 vetices and two holes that equies 23 othogonal oodlights. The pocess can be epeated to geneate polygons with (h 1) vetices and h holes that need (h 1) othogonal oodlights. Othe values of n can be obtained eliminating the exta pongs. (a) (b) Fig. 3. Polygons that equie one =2-oodlight fo each pong o alley. 3 Illuminating Vetex Floodlights We now pove that b(3n + (h 1))=8c othogonal v-oodlights ae always sucient to illuminate an othogonal at galley with n vetices and h holes. We use the following notation intoduced by Rawlins [12]. Given an othogonal polygon P, an edge e of P is said to be a Noth edge (N-edge fo shot), if the inteio of the polygon is immediately below e. East, West and South edges ae dened analogously. A vetex is said to be a Noth-East vetex (NE-vetex fo shot), if the polygonal edges that intesect at the vetex ae an N-edge and an E-edge. NW-vetices, SE-vetices and SW-vetices ae dened similaly. Since a vetex may also be convex o eex, thee ae eight possible types of vetices in an othogonal polygon. We dene the following oodlight placement ule. Denition 2. Noth-East ule (NE-ule): Fo each Noth edge e of the polygon, place a oodlight aligned with e at the East vetex of e. Fo each East edge e of the polygon, place a oodlight aligned with e at the Noth vetex of e. Fo a diagam of the NE-ule see Fig. (a). Lemma 3. Let P be an othogonal polygon with holes. The NE-ule poduces an assignment of oodlights that illuminates the inteio of P. Poof. Let p be a point in the inteio of P. Let x be the st point in the bode of P visible by a hoizontal ay fom p to the East. Clealy, x is in an E-edge e and p is visible fom x. Conside a point x 0 in e just above x and conside the ectangle R with exteme points at x 0 and p; see Fig. (b). Clealy, if x 0 is close

5 enough to x, the ectangle R is contained in P. Conside moving x 0 Noth until it cannot be moved futhe without R leaving P. This happens because x 0 has eached the Noth vetex of e, in which case, p is illuminated by a oodlight at this point; see Fig. (c), o the uppe side of the ectangle R has coincided with a Noth edge, in which case, p is illuminated by a oodlight at the East point of this Noth edge; see Fig. (d). In both cases, p is illuminated and the poof is complete. ut N-edge (a) E-edge R e x 0 p x (b) R x 0 e p x (c) R p (d) x 0 e x Fig.. Diagam illustating the placement of oodlights by the NE-ule Similaly, we can dene a NW-ule, a SE-ule and a SW-ule, each illuminating the polygon. We ae now eady to pove suciency. Lemma. Let P be an othogonal polygon with n vetices and h holes. A total of b(3n + (h 1))=8c othogonal oodlights ae sucient to illuminate P. Poof. Illuminate the polygon P by each of the fou ules poposed above. Let kxk denote the numbe of oodlights used by the X ule. Note that each edge of the polygon eceives at most two oodlights (fo example, a N-edge eceives a oodlight at its E-vetex in the NE-ule and at its W-vetex in the NW-ule) and the sets of oodlights of any pai of ules is disjoint. Moeove, in the NE-ule, a NE-convex vetex eceives only one oodlight. Thus, the numbe knek of oodlights used by the NE-ule is given by knek = ksek + knw k + knek c ; whee ksek is the numbe of SE-eex vetices, knw k is the numbe of NW-eex vetices, and knek c is the numbe of NE-convex vetices. Thus, the total numbe of othogonal oodlights used by the fou ules is given by knek + knw k + ksek + ksw k = 2 + c; whee is the numbe of eex vetices in the polygon P and c is the numbe of convex vetices. Since fo an othogonal polygon with no holes c = (n + )=2 [11] and = (n )=2 [11], and also, fo a polygon with holes, the convex vetices on a hole ae eex vetices fo the hole, while the eex vetices on a hole ae convex vetices fo the hole, we have that the coveing ule that uses the minimum numbe of oodlights uses 2 + c 2(0 + c 1 + : : : + c h ) + c : : : + h = oodlights, whee c i + i = n i and c i is the numbe of convex vetices in the i-th hole. Since n = P h n n=0 i, we have that b(2 + c)=c is h 6 2 n P h i=1 i n i+ + n0+ + P h 2 2 i=1 n i n + (h 1) = 8 : This completes the poof. ut We have poved the following esult.

6 Theoem 5. If P is an othogonal polygon with n vetices and h holes, then b(3n + (h 1))=8c othogonal oodlights ae always sucient and sometimes necessay to illuminate P. We claim that this esult is signicant despite the fact that it may suggest that moe guads ae equied than in the oiginal at galley theoem. We suppot this claim with thee obsevations: 1.- Using Theoem 5, the total apetue of 3n=8 poposed by the oiginal vesion of the othogonal at galley theoem, fo polygons without holes, has been educed by half. Moeove, fo polygons with holes, the total apetue using oodlights is always less than the n=2 poposed by Homann's esult. 2.- The placing ules lead diectly to a linea algoithm that is much simple than the algoithms fo guads that equie tapezoidization, quadilatealization o decomposition into L-shaped pieces [11]. The algoithm consists of a tavesal of the bounday of the polygon that counts the types of vetices. It computes the numbe of vetex oodlights equied by each of the fou ules, detemines which one uses the minimum and assigns oodlights in the coesponding vetices with a second tavesal of the bounday. 3.- The algoithm may place many fewe oodlights that b(3n + (h 1))=8c; fo example, in a staicase polygon, only one oodlight is used. Illuminating with Floodlights on the Bounday An othogonal at galley P with no holes and eex vetices can be patitioned into b=2c + 1 L-shaped pieces [11], and since each L-shaped piece can be illuminated with one oodlight, we have that bn=c othogonal oodlights ae sometimes necessay and always sucient to illuminate an othogonal polygon. This seems to contadict ou pevious esults; howeve, the missing detail is that using O'Rouke's algoithm to patition P into L-shaped pieces, some of the oodlights will be placed in the inteio of the polygon. This seems athe unsatisfactoy. In this section, we st show that we can illuminate a polygon P with no holes with bn=c othogonal oodlights placed at points in the bounday of P. We pove this by showing that b=2c + 1 othogonal oodlights at points in the bounday ae always sucient to illuminate P, whee is the numbe of eex vetices in P. Then, we demonstate that any othogonal polygon with n vetices and h holes can be illuminated with b(n + 2h)=c oodlights in its bounday. Fo a polygon with no holes, necessity of bn=c oodlights is established by the well-known \comb" example [11, Figue 2.18]. Suciency follows an inductive agument simila to O'Rouke's poof of the othogonal at galley theoem [11, Sections 2.5 and 2.6]. A hoizontal cut of an othogonal polygon P is an extension of the hoizontal edge incident to a eex vetex though the inteio of the polygon. A cut esolves a eex vetex in the sense that the vetex is no longe eex in eithe of the two pieces of the patition detemined by the cut. Clealy, a cut does not intoduce any eex vetices. A hoizontal cut is an odd-cut (also and H-odd-cut) if one of the halves contains an odd numbe of eex vetices. Lemma 6. Let P be an othogonal polygon and patition P into P 1 ; P 2 ; : : :; P t by dawing all H-odd-cuts and all H-cuts that ae visibility ays of two eex vetices. Then, each P i is in geneal hoizontal position and can be coveed with b i =2c + 1 oodlights in the bounday of P, whee i is the numbe of eex vetices in P i. In this extended abstact we omit the poof. Since the H-gaph can be constucted in linea time [11] and tapezoidization can be achieved in linea time [], it is not had to see that the above agument esults in a linea algoithm. Thus, we have obtained the following esult. Theoem 7. Let P be an othogonal polygon with n vetices, then bn=c othogonal oodlights placed in the bounday of P ae always sucient and sometimes necessay to illuminate the inteio of P. Moeove, such set of oodlights can be found in O(n) time. We ae now eady to discuss the case of othogonal polygons with holes.

7 Theoem 8. An othogonal polygon with n vetices and h holes can be illuminated with at most b n+2h c oodlights in its bounday. Moeove, this bound is tight, since thee ae othogonal polygons with n vetices and h holes that equie b n+2h c oodlights. Poof. To pove suciency, let P be a polygon with n vetices and h holes. Resolve all holes of P and constuct a polygon P 0 as follows. Let v 0 be a eex vetex that belongs to a hole P 0. Cut P along a hoizontal segment in the inteio of P that joins v 0 with the bounday of P. Include as two segments of the bounday. By Theoem 7, the polygon P 0 can be illuminated with b(n + 2h)=c oodlights in its bounday. Howeve, the cuts to constuct P 0 fom P wee always hoizontal, and the b(n+2h)=c oodlights povided by Theoem 7 ae placed on vetical edges of P 0. Thus, these oodlights ae on vetical edges of P. Moeove, it is not had to see that the oodlights illuminate P. To pove necessity, we conside the geneic polygon of Fig. 5. Fo each m > 0, this polygon can be congued to have m holes and 10m vetices and (m 1) s s s 10(m 1) (m 1) m Fig. 5. A polygon with m holes and 10m vetices that equies 3m =2-oodlights. equies 3m othogonal oodlights. ut 5 Illuminating the Exteio Two othe vaiants of the illumination poblem ae \The Fotess Poblem" and the \Pison Yad Poblem". The st asks fo the numbe of oodlights to illuminate the exteio of the polygon, while the second asks fo the numbe needed to see both the inteio and the exteio. In this section we demonstate that (n + )=2 othogonal oodlights ae sometimes necessay and always sucient to illuminate the exteio of an othogonal polygon P with n vetices and even with h holes. We pove suciency st. Let P be an othogonal polygon with n vetices and h holes. Again, let n 0 be the numbe of vetices in the oute bounday and n i the numbe of vetices in the i-th hole. Illuminating the exteio of P equies us to illuminate the inteio of the holes. Since P is simply-connected, the holes do not have holes of thei own, thus, by the esults of ealie sections, the i-th hole can be illuminated with fewe than n i =2 othogonal oodlights (ecall that n i must be even fo i = 0; : : :; h). If we pove that the exteio of the polygon P 0 dened by the n 0 vetices of the oute bounday of P can be illuminated with n 0 =2+2 oodlights, then we would have shown that 2+ P h n i=0 i=2 = 2+n=2 oodlights ae sucient. Recall that a set S in the plane is othogonally convex if any hoizontal o vetical line segment intesects S in a connected set. The othogonal convex hull C(S) of a set S is the smallest othogonally convex set containing S. Conside the othogonal convex hull of P 0. The bounday of C(P 0 ) is composed of at most fou staicases (and possibly fewe). Moeove, if this hull C(P 0 ) has k vetices, its exteio can be illuminated with k=2 + 2 vetices by taveling aound the bounday and placing one oodlight in evey othe vetex of each staicase and one on each of the Noth-most, South-most, East-most and West-most edges. Obseve also that, fo each new vetex intoduced to the bounday of C(P 0 ), a vetex is esolved fom P 0 and no oodlights ae assigned to a esolved vetex. It only emains to illuminate the bays of P 0 ; that is the egions exteio to P 0 but inteio to the hull C(P 0 ). We can not apply Theoem 5 diectly and conclude that a bay with b vetices can be illuminated with

8 b(3b )=8c othogonal oodlights since a bay may have a vetex not oiginally in P 0. Howeve, estimating the size of the second smallest oodlight class in Theoem 5 suces. Necessity is poved by an othogonally convex polygon. Theoem 9. If P is an othogonal polygon with n vetices and h holes, then n=2 + 2 othogonal oodlights (vetex o at the bounday) ae sometimes necessay and always sucient to illuminate the exteio of P. Obseve that the poof of this theoem leads again to a linea algoithm that does not equie polygon patitioning and thus is simple and pactical. Fo the Pison Yad Poblem, a ectangle illustates that thee is no solution fo vetex oodlights unless we allow two oodlights to be placed at one vetex. 6 Concluding Remaks We have established combinatoial tight bounds fo illuminating the inteio o exteio of othogonal polygons with holes by othogonal oodlights. Ou poofs ae simple and lead to linea algoithms on the numbe n of vetices of the polygon. The algoithms compute a coveing achieving the bounds, and since any algoithm must inspect all points of the input, the computational complexity of ou algoithms is optimal. We have shown that =2 is the smallest apetue necessay and in fact, it is not had to see that fo 2 [=2; 3=2) all ou esults hold fo -oodlights. It is tivial to note that fo 3=2, -oodlights ae geneal vetex guads in othogonal polygons. Howeve, seveal open poblems emain. What bounds can be found fo othe classes of polygons Is computing the minimum set of coveing -oodlights an NP-had o NP-complete poblem If the oodlights ae each allowed to have a dieent apetue k, what can be said about the poblem of nding a cove that optimizes the total angle powe given by P k i=1 k Refeences 1. J. Abello, O. Egecioglu, and Kuma K. Visibility gaphs of staicase polygons and the weak Buhat ode I: Fom polygons to maximal chains. Discete and Computational Geomety. in pess. 2. I. Bjoling-Sachs and D. L. Souvaine. A tight bound fo guading polygons with holes. Repot LCSR-TR-165, Lab. Comput. Sci. Res., Rutges Univ., New Bunswick, NJ, P. Bose, L. Guibas, A. Lubiw, M. Ovemas, D. Souvaine, and J. Uutia. The oodlight poblem. Int. J. On Computational Geomety. in pess.. B. Chazelle. Tiangulating a simple polygon in linea time. Discete Comput. Geom., 6:85{52, J. Czyzowicz, E. Rivea-Campo, and J. Uutia. Optimal oodlight illumination of stages. Poc. of the 5th CCCG, pages 393{398, Wateloo, Ontaio, August Univ. of Wateloo. 6. V. Estivill-Casto and J. Uutia. Optimal oodlight illumination of othogonal at galleies. In Poc. of the Sixth CCCG, pages 81{86, Saskatoon, Canada, August 199. Univesity of Saskatchewan. 7. Gyoi. E., F. Homan, K. Kiegel, and T. Sheme. Genealized guading and patitioning fo ectilinea polygons (extended abstact). In Poc. of the Sixth CCCG, pages 302{307, Saskatoon, Canada, August 199. U. of Saskatchewan. 8. F. Homann. On the ectilinea at galley poblem. In Poc. of the Int. Col. on Automata, Languages, and Pogamming 90, pages 717{728. Spinge-Velag Lectue Notes on Compute Science 3, F. Homann, M. Kaufmann, and K. Kiegel. The at galley theoem fo polygons with holes. In Poc. 32nd IEEE Sympos. Found. Comput. Sci., pages 39{8, J. Kahn, M. Klawe, and D. Kleitman. Taditional galleies equie fewe watchmen. SIAM J. Algebaic Discete Methods, :19{206, J. O'Rouke. At Galley Theoems and Algoithms. Oxfod Univesity Pess, New Yok, G. J. E. Rawlins. Exploations in esticted-oientation geomety. Ph.D. thesis, Univ. Wateloo, Wateloo, ON, T. Sheme. Recent esults in at galleies. Poceedings of the IEEE, 80:138{1399, W. Steige and I. Steuni. Positive and negative esults on the oodlight poblem. In Poc. of the Sixth CCCG, pages 87{92, Saskatoon, Canada, August 199. Univesity of Saskatchewan.

9 This aticle was pocessed using the LaT E X maco package with LLNCS style