An $O(n\log n)$-Time Algorithm for the k-Center Problem in Trees

TL;DR

This paper presents an optimal $O(n\,log n)$ time algorithm for the k-center problem in weighted trees, improving previous solutions and resolving a three-decade-old open problem.

Contribution

The paper introduces the first $O(n\,log n)$ time algorithm for the weighted k-center problem in trees, surpassing the prior $O(n\,log^2 n)$ solution.

Findings

Achieved $O(n\,log n)$ time complexity for the problem.

Resolved the open question of whether the problem can be solved faster than $O(n\,log^2 n)$.

Provides a practical and efficient algorithm for weighted k-center in trees.

Abstract

We consider a classical k-center problem in trees. Let T be a tree of n vertices and every vertex has a nonnegative weight. The problem is to find k centers on the edges of T such that the maximum weighted distance from all vertices to their closest centers is minimized. Megiddo and Tamir (SIAM J. Comput., 1983) gave an algorithm that can solve the problem in $O(n\log^2 n)$ time by using Cole's parametric search. Since then it has been open for over three decades whether the problem can be solved in $O(n\log n)$ time. In this paper, we present an $O(n\log n)$ time algorithm for the problem and thus settle the open problem affirmatively.