The Joint Distribution of Elastic Buckets in Multiway Search Trees

Abstract

Random search trees are studied when they grow under a general computer memory management scheme. In a general scheme, the space is released in buckets of certain predesignated sizes. For a search tree with branch factor m, the nodes may hold up to $m - 1$ keys. Suppose the buckets of the memory management scheme that can hold less than m keys have key capacities $c_1 , \ldots ,c_r $. The search tree must then be implemented with multitype nodes of these capacities. After n insertions, let $X_n^{(i)} $ be the number of buckets of type i (i.e., of capacity $c_i $,$1 \leqslant i \leqslant p$. The multivariate structure of the tree is investigated. For the vector ${\bf X}_n = ( \leqslant X_n^{(1)} , \ldots ,X_n^{(p)} )^T $, the asymptotic mean and covariance matrix are determined. Under practical memory management schemes, all variances and covariances experience a phase transition: For $3 \leqslant m \leqslant 26$, all variances and covariances are asymptotically linear in n; for higher branch factors the variances and covariances become a superlinear (but ubquadratic) function of n. The joint distribution of ${\bf X}_n $ is shown to be multivariate normal in a range of m. While the tree is growing, conversions between types are necessary. A multivariate problem concerning these conversions with an asymptotic multivariate normal distribution is also studied. The fixed bucket, exact fit, and buddy system allocation schemes will serve as illustrating examples.