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(n
³
) 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 c
³
in latency (number of messages) and c
²
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.