Tree data structures for N-body simulation

In this paper, we study data structures for use in N-body simulation. We concentrate on the spatial decomposition tree used in particle-cluster force evaluation algorithms such as the Barnes-Hut algorithm. We prove that a k-d tree is asymptotically inferior to a spatially balanced tree. We show that the worst case complexity of the force evaluation algorithm using a k-d tree is Theta(n log(3) n log L) compared with Theta(n log L) for an oct-tree. (L is the separation ratio of the set of points.) We also investigate improving the constant factor of the algorithm and present several methods which improve over the standard oct-tree decomposition. Finally, we consider whether or not the bounding box of a point set should be "tight" and show that it is safe to use tight bounding boxes only for binary decompositions. The results are all directly applicable to practical implementations of N-body algorithms.