Guarding a Terrain by a Single k-modem Watchtower

In this paper we study the problem of guarding a 2-dimentional terrain by the shortest k-modem watchtower, for a given constant k. We refer to a 2-dimentional terrain as an x-monotone polygonal chain. A kmodem watchtower is a vertical segment whose lower endpoint lies on the terrain and the upper endpoint is k-visible to all points of the terrain. Two points are k-visible if and only if the segment connecting them crosses at most k edges. The watchtower problem has two versions according to whether the lower endpoint of the watchtower lies exactly on a vertex (discrete watchtower problem) or not (continuous watchtower problem). We present the very first algorithm to solve the shortest k-modem watchtower problem in both discrete and continuous versions. Our algorithm runs in O(n4 log n) time for the discrete version and O(n5 log n) time for the continuous version, where n is the number of vertices of the terrain. Given a simple polygon with n vertices and a k-modem placed inside this polygon, we also improve the time complexity of computing all regions that are k-visible to the k-modem to O(n log

کلیدواژه ها:

Computational Geometry ، Visibility ، Guarding a Terrain ، k-modem