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 ( n 2 ) 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 ( n 4 ) time under rectangular and staircase visibility, and in O ( n 6 ) time under k-periscope visibility.