Efficient Algorithms for One-Dimensional k-Center Problems

We present an O(n log n) time algorithm for the (weighted) k-center problem of n points on a real line. We show that the problem has an Ω(n log n) time lower bound, and thus our algorithm is optimal. We also show that the problem is solvable in O(n) time in certain special cases. Our techniques involve developing efficient 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.