28 research outputs found
A repertoire for additive functionals of uniformly distributed m-ary search trees
Using recent results on singularity analysis for Hadamard products of
generating functions, we obtain the limiting distributions for additive
functionals on -ary search trees on keys with toll sequence (i)
with ( and correspond roughly
to the space requirement and total path length, respectively); (ii) , which corresponds to the so-called shape functional; and (iii)
, which corresponds to the number of leaves.Comment: 26 pages; v2 expands on the introduction by comparing the results
with other probability model
Destruction of very simple trees
We consider the total cost of cutting down a random rooted tree chosen from a
family of so-called very simple trees (which include ordered trees, -ary
trees, and Cayley trees); these form a subfamily of simply generated trees. At
each stage of the process an edge is chose at random from the tree and cut,
separating the tree into two components. In the one-sided variant of the
process the component not containing the root is discarded, whereas in the
two-sided variant both components are kept. The process ends when no edges
remain for cutting. The cost of cutting an edge from a tree of size is
assumed to be . Using singularity analysis and the method of moments,
we derive the limiting distribution of the total cost accrued in both variants
of this process. A salient feature of the limiting distributions obtained
(after normalizing in a family-specific manner) is that they only depend on
.Comment: 20 pages; Version 2 corrects some minor error and fixes a few typo
Limiting distributions for additive functionals on Catalan trees
Additive tree functionals represent the cost of many divide-and-conquer
algorithms. We derive the limiting distribution of the additive functionals
induced by toll functions of the form (a) n^\alpha when \alpha > 0 and (b) log
n (the so-called shape functional) on uniformly distributed binary search
trees, sometimes called Catalan trees. The Gaussian law obtained in the latter
case complements the central limit theorem for the shape functional under the
random permutation model. Our results give rise to an apparently new family of
distributions containing the Airy distribution (\alpha = 1) and the normal
distribution [case (b), and case (a) as ]. The main
theoretical tools employed are recent results relating asymptotics of the
generating functions of sequences to those of their Hadamard product, and the
method of moments.Comment: 30 pages, 4 figures. Version 2 adds background information on
singularity analysis and streamlines the presentatio
Singularity analysis, Hadamard products, and tree recurrences
We present a toolbox for extracting asymptotic information on the
coefficients of combinatorial generating functions. This toolbox notably
includes a treatment of the effect of Hadamard products on singularities in the
context of the complex Tauberian technique known as singularity analysis. As a
consequence, it becomes possible to unify the analysis of a number of
divide-and-conquer algorithms, or equivalently random tree models, including
several classical methods for sorting, searching, and dynamically managing
equivalence relationsComment: 47 pages. Submitted for publicatio