Efficient algorithms for the one-dimensional k-center problem

Danny Ziyi Chen, Jian Li, Haitao Wang

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(nlog⁡n)<math><mi is="true">O</mi><mo stretchy="false" is="true">(</mo><mi is="true">n</mi><mi mathvariant="normal" is="true">log</mi><mo is="true">⁡</mo><mi is="true">n</mi><mo stretchy="false" is="true">)</mo></math> time previously by using Cole's parametric search and other complicated approaches. In this paper, we present an easier O(nlog⁡n)<math><mi is="true">O</mi><mo stretchy="false" is="true">(</mo><mi is="true">n</mi><mi mathvariant="normal" is="true">log</mi><mo is="true">⁡</mo><mi is="true">n</mi><mo stretchy="false" is="true">)</mo></math> time algorithm that avoids the parametric search, and in certain special cases our algorithm solves the problem in O(n)<math><mi is="true">O</mi><mo stretchy="false" is="true">(</mo><mi is="true">n</mi><mo stretchy="false" is="true">)</mo></math> 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.