Efficient Algorithms for One-Dimensional k-Center Problems

TL;DR

This paper introduces a simplified and efficient algorithm for the one-dimensional k-center problem on weighted points, achieving optimal or near-optimal time complexities and developing novel data structures for related geometric queries.

Contribution

It presents an easier O(n log n) algorithm for the weighted k-center problem on a line, avoiding complex methods like parametric search, and introduces new data structures for geometric intersection queries.

Findings

Achieves O(n log n) time complexity for the problem

In certain cases, solves the problem in O(n) time

Develops new data structures for half-plane intersection queries

Abstract

We consider the problem of finding k centers for n weighted points on a real line. This (weighted) k-center problem was solved in O(n log n) time previously by using Cole's parametric search and other complicated approaches. In this paper, we present an easier O(n log n) time algorithm that avoids the parametric search, and in certain special cases our algorithm solves the problem in O(n) time. In addition, our techniques involve developing interesting data structures for processing queries that find a lowest point in the common intersection of a certain subset of half-planes. This subproblem is interesting in its own right and our solution for it may find other applications as well.