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 $\mathcal {P}$ of n (weighted) demand points, and the location of each demand point $P_i\in \mathcal {P}$ is uncertain but is known to be in one of $m_i$ points on T each associated with a probability. Given a covering range $\lambda$, 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 $P_i\in \mathcal {P}$, the expected distance from $P_i$ to at least one center is no more than $\lambda$. The problem has not been studied before. We present an $O(|T|+M\log ^2 M)$ time algorithm, where |T| is the number of vertices of T and M is the total number of locations of all uncertain points of $\mathcal {P}$, i.e., $M=\sum _{P_i\in \mathcal {P}}m_i$.

This research was supported in part by NSF under Grant CCF-1317143.