Covering uncertain points in a tree

Abstract

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 be 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 log2M) time algorithm, 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.