Empirical Analysis of Space-Filling Curves for Scientific Computing Applications

Space-Filling Curves are frequently used in parallel processing applications to order and distribute inputs while preserving proximity. Several different metrics have been proposed for analyzing and comparing the efficiency of different space-filling curves, particularly in database settings. In this paper, we introduce a general new metric, called Average Communicated Distance, that models the average pair wise communication cost expected to be incurred by an algorithm that makes use of an arbitrary space-filling curve. For the purpose of empirical evaluation of this metric, we modeled the communications structure of the Fast Multipole Method for n-body problems. Using this model, we empirically address a number of interesting questions pertaining to the effectiveness of space-filling curves in reducing communication, under different combinations of network topology and input distribution settings. We consider these problems from the perspective of ordering the input data, as well as using space-filling curves to assign ranks to the processors. Our results for these varied scenarios point towards a list of recommendations based on specific knowledge about the input data. In addition, we present some new empirical results, relating to proximity preservation under the average nearest neighbor stretch metric, that are application independent.