Covering Uncertain Points in a Tree

In this paper, we consider a coverage problem for uncertain points in a tree. Let T be a tree containing a set P of n (weighted) demand points, and the location of each demand point Pi ∈ P is uncertain but is known to appear in one of mi points on T each associated with a probability. Given a covering range λ, the problem is to find a minimum number of points (called centers) on T to build facilities for serving (or covering) these demand points in the sense that for each uncertain point Pi ∈ P , the expected distance from Pi to at least one center is no more than λ. The problem has not been studied before. We present an O(|T |+M log M) time algorithm for the problem, where |T | is the number of vertices of T and M is the total number of locations of all uncertain points of P , i.e., M = ∑ Pi∈P mi. In addition, by using this algorithm, we solve a k-center problem on T for the uncertain points of P .