An O(n) time hierarchical tree algorithm for computing force field in n-body simulations

We consider the following force field computation problem: given a cluster of n particles in three-dimensional space, compute the force exerted on each particle by the other particles. Depending on different applications, the pairwise interaction could be either gravitational or Lennard-Jones. In both cases, the force between two particles vanishes as the distance between them approaches to infinity. Since there are n(n -1)/2 pairs, direct method requires Theta(n(2)) time for force-evaluation, which is very expensive for astronomical simulations. In 1985 and 1986, two famous O(n log n) time hierarchical tree algorithms were published by Appel (1985) and by Barnes and Hut (1986), respectively. In this paper, we show that Appel's algorithm can be implemented in O(n) time. (C) 1998-Elsevier Science B.V. All rights reserved.