Tight bounds for the rectangular art gallery problem

Abstract

Consider a rectangular art gallery, subdivided into n rectangular rooms; any two adjacent rooms have a door connecting them. We show that ⌈n/2⌋ guards are always sufficient to protect all rooms in a rectangular art gallery; furthermore, their positioning can be determined in O(n) time. We show that the optimal positioning of the guards can be determined in linear time. We extend the result by proving that in an arbitrary orthogonal art gallery (not necessarily convex, possibly having holes) with n rectangular rooms and k walls, ⌈(n + k)/2⌋ guards are always sufficient and occasionally necessary to guard all the rooms in our gallery. A linear time algorithm to find the positioning of the guards is obtained.