Dynamic Compressed Hypertoctrees with Application to the N-Body Problem

Hyperoctree is a popular data structure for organizing multidimensional point data. The main drawback of this data structure is that its size and the run-time of operations supported by it are dependent upon the distribution of the points. Clarkson rectified the distributiondependency in the size of hyperoctrees by introducing compressed hyperoctrees. He presents an O(n log n) expected time randomized algorithm to construct a compressed hyperoctree. In this paper, we give three deterministic algorithms to construct a compressed hyperoctree in O(n log n) time, for any fixed dimension d. We present O(log n) algorithms for point and cubic region searches, point insertions and deletions. We propose a solution to the N-body problem in O(n) time, given the tree. Our algorithms also reduce the run-time dependency on the number of dimensions.