Optimally computing the shortest weakly visible Subedge of a simple polygon preliminary version (Preliminary version)

Abstract

Given an n-vertex simple polygon P, the problem of computing the shortest weakly visible subedge of P is that of finding a shortest line segment s on the boundary of P such that P is weakly visible from s (if s exists). In this paper, we present new geometric observations that are useful for solving this problem. Based on these geometric observations, we obtain optimal sequential and parallel algorithms for solving this problem. Our sequential algorithm runs in O(n) time, and our parallel algorithm runs in O(log n) time using O(n/log n) processors in the CREW PRAM computational model. Using the previously best known sequential algorithms to solve this problem would take O(n2) time. We also give geometric observations that lead to extremely simple and optimal algorithms for solving, both sequentially and in parallel, the case of this problem where the polygons are rectilinear.