Efficient algorithms for the one-dimensional k-center problem

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)O(nlog⁡n) time previously by using Cole's parametric search and other complicated approaches. In this paper, we present an easier O(nlog⁡n)O(nlog⁡n) time algorithm that avoids the parametric search, and in certain special cases our algorithm solves the problem in O(n)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.