Guarding rectangular art galleries

Abstract

Consider a rectangular art gallery divided into n rectangular rooms, such that any two rooms sharing a wall in common have a door connecting them. How many guards need to be stationed in the gallery so as to protect all of the rooms in our gallery? Notice that if a guard is stationed at a door, he will be able to guard two rooms. Our main aim in this paper is to show that Èn/2˘ guards are always sufficient to protect all rooms in a rectangular art gallery. Extensions of our result are obtained for non-rectangular galleries and for 3dimensional art galleries.

Guarding Orthogonal Galleries with Rectangular Rooms

“…Then we prove an upper bound on the number of guards for arbitrary orthogonal polygon with orthogonal holes. This result improves the previous bound by Czyzowicz et al [2] (even in the case of polygon without holes). …”

Coverage with k-Transmitters in the Presence of Obstacles

“…The largest color class has at least n/6 vertices and forms an independent set I. The set S − I has at most 5n/6 segments, so by the result of Czyzowicz et al [6], it has a set of 0-transmitters of cardinality at most ( 5n 6 + 1)/2 = (5n + 6)/12 that covers the entire plane. By Lemma 1, placing 1-transmitters at those points covers the entire plane with respect to S.…”

“…Czyzowicz et al [6] proved that (n + 1)/2 0-transmitters always suffice and are sometimes necessary to cover the plane in the presence of n disjoint orthogonal line segments. We generalize this to k-transmitters.…”

Matching Points with Squares

“…To address this issue, we proceed in a way similar to that used in [3]. Consider the three shaded squares and six points p 1 , .…”