Efficient parallel algorithms for geometric k-clustering problems

Abstract

We present efficient parallel algorithms for two geometric k-clustering problems in the CREW PRAM model of parallel computation. Given a point set P of n points in two dimensions, these problems are to find a k-point subset such that some measure for this subset is minimized. We consider the problems of finding a k-point subset with minimum L∞ perimeter and minimum L∞ diameter. For the L∞ perimeter case, our algorithm runs in O(log2n) time and O(n log2 n + nk2 log2k) work. For the L∞ diameter case, our algorithm runs in O(log2n + log2k loglog k log*k) time and O(n log2n) work. The work done (processor-time product) by our algorithms is close to the time complexity of best known sequential algorithms. Previously, no parallel algorithm was known for either of these problems.