Optimal Algorithms for Two-Guard Walkability of Simple Polygons

A polygon P admits a walk from a boundary point s to another boundary point t if two guards can simultaneously walk along the two boundary chains of P from s to t such that they are always visible to each other. A walk is called a straight walk if no backtracking is required during the walk. A straight walk is discrete if only one guard is allowed to move at a time, while the other guard waits at a vertex. We present simple, optimal O(n) time algorithms to determine all pairs of points of P which admit walks, straight walks and discrete straight walks. The chief merits of the algorithms are that these require simple data structures and do not assume a triangulation of P. Furthermore, the previous algorithms for the straight walk and the discrete straight walk versions ran in O(n log n) time even after assuming a triangulation.