AN OPTIMAL ALGORITHM FOR THE TWO-GUARD PROBLEM

Abstract

In this paper we give optimal solutions for several versions of the two-guard problem. Given a simple polygon P with vertices s and t, the straight walk problem asks whether we can move two points monotonically on P from s to t, one clockwise and one counter-clockwise, such that the points are always co-visible. In the counter walk problem, both points move clockwise, one from s to t and the other from t to s. We provide Θ(n)-time constructive algorithms for both problems. We also give a Θ(n)-time decision algorithm for a version called the general walk problem. We obtain our results by examining the structure of the restrictions placed on the motion of the two points, and by employing properties of shortest paths and shortest path trees.