The Two-Center Problem of Uncertain Points on Trees

TL;DR

This paper addresses the two-center problem for uncertain points on trees, proposing a more efficient algorithm that minimizes the maximum expected distance, improving upon previous solutions in terms of computational complexity.

Contribution

The paper introduces a simpler, faster algorithm for the weighted two-center problem of uncertain points on trees, reducing the time complexity from previous methods.

Findings

Achieved an $O(|T| + mn ext{log} mn)$ time algorithm.

Provided a more straightforward approach compared to existing solutions.

Enhanced efficiency in solving the two-center problem on trees with uncertain points.

Abstract

In this paper, we consider the (weighted) two-center problem of uncertain points on a tree. Given are a tree $T$ and a set $\calP$ of $n$ (weighted) uncertain points each of which has $m$ possible locations on $T$ associated with probabilities. The goal is to compute two points on $T$, i.e., two centers with respect to $\calP$, so that the maximum (weighted) expected distance of $n$ uncertain points to their own expected closest center is minimized. This problem can be solved in $O(|T|+ n^{2}\log n\log mn + mn\log^2 mn \log n)$ time by the algorithm for the general $k$-center problem. In this paper, we give a more efficient and simple algorithm that solves this problem in $O(|T| + mn\log mn)$ time.