# Covering Uncertain Points in a Tree

## TL;DR

This paper introduces a novel algorithm to optimally cover uncertain demand points in a tree with facilities, minimizing the number of centers needed to ensure expected coverage within a specified range.

## Contribution

It presents the first solution to a new coverage problem for uncertain points in trees, with an efficient algorithm and an application to the k-center problem.

## Key findings

- Algorithm runs in O(|T|+M log^2 M) time.
- Successfully solves the new coverage problem for uncertain points.
- Extends to solve the k-center problem on trees for uncertain points.

## Abstract

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 P_i\in P is uncertain but is known to appear 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 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 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=\sum_{P_i\in P}m_i. In addition, by using this algorithm, we solve a k-center problem on T for the uncertain points of P.