Efficient N-body Simulation : Fast Algorithms for Potential Field Evaluation and Trummer ’ s Problem ∗

In this paper, we describe an efficient approximation algorithm for the n-body problem. The algorithm is a non-trivial modification of the fast multipole method that works in both two and three dimensions. Due to the equivalence between the two-dimensional n-body problem and Trummer’s problem, our algorithm also gives the fastest known approximation algorithm for Trummer’s problem. Let A be the sum of the absolute values of the particle charges (or masses if the simulation is gravitational) in the n-body problem under consideration. To approximate the particle potentials with absolute error bound , we let p = dlog(A/ )e and give complexity bounds in terms of p. Note that, under reasonable assumptions on the particle charges, if we desire the output to be accurate to b bits, then p = Θ(b). In two dimensions, our algorithm runs in time O(n log p), which is a substantial improvement over the previous best algorithm which requires Θ(np log p) time. We also apply our new algorithm to the three dimensional problem and get a new algorithm that has time complexity O(np), an improvement over the best previouslyknown three-dimensional algorithm which requires Θ(np log p) time. Our algorithms do not make any assumptions about the input distribution, and are true worst-case bounds.