Guard Placement in Rectilinear Polygons

Abstract

Guard placement problems have been extensively studied by mathematicians as well as computer scientists. The classical guard problem posed by Victor Klee is to determine the number of guards sufficient to see any art gallery given as an n -vertex simple polygon. Here we study traditional (or rectilinear) art-galleries, i.e. galleries in which all interior angles are either 270° or 90°. The Rectilinear Art Gallery Theorem originally proved by Kahn, Klawe and Kleitman states that any n-vertex rectilinear art gallery can always be guarded by at most [ n /4 J guards. Here we examine the problem from its computational point-of-view by providing an algorithmic proof of the Rectilinear Art Gallery Theorem. It is demonstrated that guard placement in monotone rectilinear polygons can be done in linear time, while the problem can be solved for arbitrary, nvertex, rectilinear polygons in 0 ( n loglog n ) time.