Communication-aware processor allocation for supercomputers: Finding point sets of small average distance

We give processor-allocation algorithms for grid architectures, where the objective is to select processors from a set of available processors to minimize the average number of communication hops. The associated clustering problem is as follows: Given n points in R ^d, find a size-k subset with minimum average pairwise L ₁ distance. We present a natural approximation algorithm and show that it is a 7/4-approximation for two-dimensional grids; in d dimensions, the approximation guarantee is 2-1/2d, which is tight. We also give a polynomial-time approximation scheme (PTAS) for constant dimension d, and we report on experimental results.