A characterization of polygonal regions searchable from the boundary

Abstract

We consider the problem of searching for a moving target with unbounded speed in a dark polygonal region by a searcher. The searcher continuously moves on the polygon boundary and can see only along the rays of the flashlights emanating from his position at a time. We present necessary and sufficient conditions for a polygon of n vertices to be searchable from the boundary. Our two main results are the following: 1. We present an O(n log n) time and O(n) space algorithm for testing the searchability of simple polygons. Moreover, a search schedule can be reported in time linear in its size I, if it exists. For the searcher having full 360° vision, I < 2n, and for the searcher having only one flashlight, I < 3n2. Our result improves upon the previous O(n2) time and space solution, given by LaValle et al [5]. Also, the linear bound for the searcher having full 360° vision solves an open problem posed by Suzuki et al [7]. 2. We show the equivalence of the abilities of the searcher having only one flashlight and the one having full 360° vision. Although the same result has been obtained by Suzuki et al [7], their proof is long and complicated, due to lack of the characterization of boundary search. © Springer-Verlag Berlin Heidelberg 2005.