Finding minimum witness sets in orthogonal polygons

Abstract

A witness set W in a polygon P is a subset of P such that any set G⊂P that guards W is guaranteed to guard P. We study the problem of finding a minimum witness set for an orthogonal polygon under three models of orthogonal visibility.It is known that not all simple polygons admit a finite witness set under the traditional line-segment visibility and, if a polygon admits a finite minimal witness set, then the witnesses must lie on the boundary of the polygon [5]. In this paper, we prove that every orthogonal polygon with n vertices admits a finite witness set with O(n2) witnesses under rectangular, staircase, and k-periscope visibility. This upper bound is tight under staircase visibility. We also show an orthogonal polygon whose boundary is not a witness set for any of the three considered visibility models. We finally describe how to find a minimum witness set for a given orthogonal polygon in O(n4) time under rectangular and staircase visibility, and in O(n6) time under k-periscope visibility.