Guarding Art Galleries: The Extra Cost for Sculptures Is Linear

Art gallery problems are, broadly speaking, the study of the relation between the shapes of regions in the plane and the number of points needed to guard them. These problems have been extensively studied over the last decade and have found different type of applications in practical situation. Normally the number of sides of a polygon or the general shape of the polygon is used as a measure of the complexity of the problem. The aim of this paper is to present and explore another measure of complexity, namely, the number of guards required to guard the boundary, or the walls, of the gallery. We prove that if n guards are necessary to guard the walls of an art gallery, then an additional team of at most 4n − 6 will guard the whole gallery. This result improves a previously known quadratic bound, and is a step towards a possibly optimal value of n − 2 additional guards. The proof is algorithmic, uses ideas from graph theory (visibility graph induced on the already placed guards), and is mainly based on the definition of a new reduction operator which recursively eliminates the simple parts of the polygon. We also use the fact that every gallery with c right-turn angles can be guarded by at most 2c − 4 guards. This latter result is optimal. 1 Painting Galleries Introduction. Art gallery problems are, broadly speaking, the study of the relation between the shapes of regions in the plane and the number of points needed to guard them. The problem of determining how many guards are sufficient to see every point in the interior of an n-wall art gallery room was first posed by Victor Klee [Hon76]. Conceptually, the room is a simple polygon P with n vertices, and the guards are stationary points in P that can see any point of P connected to them by a straight line segment that lies entirely within P . The first ”art gallery theorem” was proved by Chvátal [Chv75], who demonstrated that given any simple polygon with n sides, the interior of the polygon can be guarded with at most ⌊n3 ⌋ guards and that this number of guards is sometimes necessary. Fisk [Fis78] later found a simpler proof which lends itself to an O(n logn) algorithm developed by Avis and Toussaint [AT81] for locating these ⌊n3 ⌋ stationary guards. With some restriction on the shape of the polygon, for example if the polygon is rectilinear, that is, the edges of the polygon are either horizontal or vertical, Kahn et al. [KKK83] have shown that [n4 ] guards are sufficient and sometimes necessary. Sack [Sac82] and Edelsbrunner, et al. [EOW84] have, based on the results of [KKK83] and [O’R83], respectively, devised an O(n logn) algorithm for locating these ⌊n4 ⌋ guards. These classical results in the theory of art galleries have spawned a plethora of research (see the monograph by O’Rourke [O’R87], and the surveys [Urr00,Sza97,She92] for overviews of previous work). In particular, since then the art gallery problems have emerged as a research area that stress complexity and algorithmic aspects of visibility and illumination in configurations comprising obstacles and guards. In fact by creating rather idealised situations the theory succeeds in abstracting the algorithmic essence ⋆ This project has been supported by the European project IST FET AEOLUS. 2 Louigi Addario-Berry, Omid Amini, Jean-Sébastien Sereni Stéphan Thomassé Fig. 1. Three guards are enough the guard the paintings (on the walls), but not the sculpture in the shaded area. Dashed lines are lines of sight of the guards. of many visibility problems (like in partitioning theorems, mobile guard configurations, visibility graphs, . . . ) thus significantly facilitating the study of their computational complexity, see [She92] and [KP94]. In most of the reseach papers in the field, the number of sides of a polygon or restriction on the shape of the polygon is used as a very natural measure of the ”complexity” of the polygon. The aim of this paper is to present and explore another measure of complexity, namely, the number of guards required to guard the boundary, or the walls, of the gallery. As we will see in the next sections, this new complexity measure can be regarded as a mixture of the two named ones: the shape and the number of sides, but remains different and has its own characteristics. As shown in Figure 1, a team of guards inside a gallery can see the walls (where paintings are hung), without necessarily guarding the whole gallery (where sculptures are displayed), showing that these two notions of complexity are in general quite different. More precisely, the question we investigate in this paper is the following: given that the interior walls of a polygon can be guarded with at most n guards, how many additional guards may be needed to guard the whole interior? This question has been first explored by Dujmović and Bose [DB07], who proved that an additional number of at most 3n/2 guards can guard the whole gallery. Main Results. The main result of this paper provides the following linear bound: Theorem 1. Let M be a polygonal gallery. If the walls of M can be guarded with at most n guards, an additional set of 4n− 6 guards is sufficient to guard the interior of M . Observe that when n = 1, the unique guard sees all the walls, hence sees the whole gallery. The previous bound is then sharp for n equal to one, but is most likely not for larger values of n. We offer the following conjecture. Conjecture 1 Let n be an integer greater than one. If the interior walls of a gallery can be guarded by n guards, then n− 2 additional guards are sufficient to guard the whole gallery. As n − 2 is also the number of triangles in a triangulation of an n-sided polygon, it is tempting to try to prove the above theorem by finding a relationship between the original polygon P and some auxiliary polygon whose vertices are the set of guards that guard the interior boundary of P . We were not able to prove the conjecture in this way. If Conjecture 1 is true then the given value would be optimal, as is shown by following example. In Figure 2, n− 2 “small rooms” are attached by narrow entrances to a main room. Guarding the walls requires at most n guards (as shown): one guard in each of the ”small rooms” off the main room, and one guard each of the two far corners of the main room. These latter two guards each have a line of sight along one wall of each small room. However, with such a set of guards the parts of the gallery’s interior shaded in dark grey are left unguarded. To guard the whole gallery requires two guards in each of the “small rooms”, and an additional two guards in the main room. Guarding Art Galleries: The Extra Cost for Sculptures is Linear 3