Distribution-Independent Hierarchical Algorithms for the N-body Problem

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The N-body problem is to simulate the motion of N particles under the influence of mutual force fields based on an inverse square law. Greengards algorithm claims to compute the cumulative force on each particle in O(N) time for a fixed precision irrespective of the distribution of the particles. In this paper, we show that Greengards algorithm is distribution dependent and has a lower bound of ­(N log 2 N) in two dimensions and ­(N log 4 N) in three dimensions. We analyze the Greengard and Barnes-Hut algorithms and show that they are unbounded for arbitrary distributions. We also present a truly distribution independent algorithm for the N-body problem that runs in O(N log N) time for any fixed dimension.

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