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 Θ(nlog³nlogL) compared with Θ(nlogL) 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 only safe to use tight bounding boxes for binary decompositions. The results are all directly applicable to practical implementations of N-body algorithms