The size of random bucket trees via urn models

Abstract.

We find the asymptotic average composition of a class of nonclassic Pólya urn models (not necessarily of fixed row sum) by embedding the discrete urn process into a renewal process with rewards. A subclass of the models considered has banded matrix urn schemes and serves as a natural modeling tool for the size of a class of random bucket trees. The class of urns considered extends known results for multicolor urns with constant row sums. The same asymptotic average results are shown to hold in the larger class. This provides an average-case analysis for the size of certain random bucketed multidimensional quad trees and k-d trees, which are all new results. Some bucket trees have urn schemes with constant row sum, a special case that helps detect phase changes in the limiting distribution of the (normed) size of the tree. For these special cases one can appeal to a more developed urn theory to find a limiting distribution of the normed size up to a threshold value of the capacity of a bucket. Once that cut-off point is surpassed, normality ceases to hold. This case appears in paged binary trees (threshold 116), m-ary search trees (threshold 26), and bucket recursive trees (threshold 26). The asymptotic normality results and the phase change after the threshold in these trees are already known and we only provide alternative proofs via a unified urn models approach.