Routing in a polygonal terrain with the shortest beacon watchtower

In a paper by Biro et al. [3], a novel twist on guarding in art galleries, motivated by geographical greedy routing in sensor networks, is introduced. A beacon is a fixed point that when activated induces a force of attraction 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 [2]O(n2) time algorithms are given to compute the beacon kernel and the inverse beacon kernel 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(nlog⁡n) time algorithm that computes the shortest beacon watchtower. In doing this we introduce O(nlog⁡n) time algorithms to compute the beacon kernel and the inverse beacon kernel in a monotone polygon. We also prove that Ω(nlog⁡n) time is a lower bound for computing the beacon kernel of a monotone polygon.