A Computation- and Communication-Optimal Parallel Direct 3-Body Algorithm

Traditional particle simulation methods are used to calculate pair wise potentials, but some problems require 3-body potentials that calculate over triplets of particles. A direct calculation of 3-body interactions involves O(n3) interactions, but has significant redundant computations that occur in a nested loop formulation. In this paper we explore algorithms for 3-body computations that simultaneously optimize three criteria: computation minimization through symmetries, communication optimality, and load balancing. We present a new 3-body algorithm that is both communication and computation optimal. Its optional replication factor, c, saves c3 in latency (number of messages) and c2 in bandwidth (volume), with bounded load imbalance. We also consider the k-body case and discuss an algorithm that is optimal if there is a cut off distance of less than 1/3 of the domain. The 3-body algorithm demonstrates 99% efficiency on tens of thousands of cores, showing strong scaling properties with order of magnitude speedups over the nïve algorithm.