A system to place observers on a polyhedral terrain in polynomial time

摘要

The Art Gallery Problem deals with determining the number of observers necessary to cover an art gallery room such that every point is seen by at least one observer. This problem is well known and has a linear time solution for the 2D case, but little is known in the 3D case. In this paper we present a polynomial time solution for the 3D version of the Art Gallery Problem. Because the problem is NP-hard, the solution presented is an approximation, and we present the bounds to our solution. The solution uses techniques from: (i) computational geometry to first build a terrain hierarchy keeping the overall terrain's shape and to compute the visibility map for each observer; (ii) graph coloring to make a first placement of observers on the terrain; and (iii) set coverage to reduce the number of observers based on their visibility map. A complexity analysis is presented for each step and an analysis of the overall quality of the solution is given.