On the internal structure of random recursive circuits
Journal of Computational and Applied Mathematics - Special issue: Probabilistic methods in combinatorics and combinatorial optimization
A multivariate view of random bucket digital search trees
Journal of Algorithms - Analysis of algorithms
Journal of Algorithms - Analysis of algorithms
m-ary search trees when m 27: a strong asymptotics for the space requirements
Random Structures & Algorithms
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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 $c_1, \ldots,c_p$. 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 $c_i$, $1{\leq}i{\leq}p$). The multivariate structure of the tree is investigated. For the vector {\bf X}$_n = (X_n^{(1)}, \ldots, X_n^{(p)})^T$, the asymptotic mean and covariance matrix are determined. Under practical memory management schemes, all variances and covariances experience a phase transition: For $3 \leq m \leq 26$, all variances and covariances are asymptotically linear in $n$; for higher branch factors the variances and covariances become a superlinear (but subquadratic) 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.