Routing in a Polygonal Terrain with the Shortest Beacon Watchtower

In a paper by Biro et al. [2], a novel twist on guarding in art galleries, motivated by geographical greedy routing in sensor networks, is introduced. A beacon is a point that when activated induces a force of attraction, or repulsion that can move points within the environment. The effect of a beacon is similar to standard visibility with some additional properties. The effects of a beacon are asymmetric leading to separate algorithms to compute the “beacon kernel” and “inverse beacon kernel”. In Biro [1] O(n) algorithms are given to compute the beacon kernel and inverse beacon kernels in simple polygons. In this paper we revisit the problem of computing the shortest watchtower to guard a 2D terrain, using the properties of beacons, and we present an O(n log n) time algorithm that computes the shortest beacon watchtower. In doing this we introduce O(n log n) algorithms to compute the beacon kernel in a simple polygon and an inverse beacon kernel in a terrain polygon. Similar ideas are then used for an algorithm to compute the inverse beacon kernel in a monotone polygon in O(n log n) time.