Distribution-independent hierarchical N-body methods

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. The problem has applications in several domains including astrophysics, molecular dynamics, fluid dynamics, radiosity methods in computer graphics and numerical complex analysis. Research efforts have focused on reducing the O(iV^) time per iteration required by the naive algorithm of computing each pairwise interaction. Widely respected among these are the BarnesHut and Greengard methods. Greengard claims his algorithm reduces the complexity to 0{N) time per iteration. Throughout this thesis, we concentrate on rigorous, distribution-independent, worst-case analysis of the N-body methods. We show that Greengard's algorithm is not 0{N), as claimed. Both Barnes-Hut and Greengard's methods depend on the same data structure, which we show is distribution-dependent. For the distribution that results in the smallest running time, we show that Greengard's algorithm is D,{Nlog'^N) in two dimensions and Q,{Nlog'^N) in three dimensions. Both algorithms are unbounded for arbitrary distributions. We have designed a hierarchical data structure whose size depends entirely upon the number of particles and is independent of the distribution of the particles. We show that both Greengard's and Barnes-Hut algorithms can be used in conjunction