13,112 research outputs found
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
On weighted depths in random binary search trees
Following the model introduced by Aguech, Lasmar and Mahmoud [Probab. Engrg.
Inform. Sci. 21 (2007) 133-141], the weighted depth of a node in a labelled
rooted tree is the sum of all labels on the path connecting the node to the
root. We analyze weighted depths of nodes with given labels, the last inserted
node, nodes ordered as visited by the depth first search process, the weighted
path length and the weighted Wiener index in a random binary search tree. We
establish three regimes of nodes depending on whether the second order
behaviour of their weighted depths follows from fluctuations of the keys on the
path, the depth of the nodes, or both. Finally, we investigate a random
distribution function on the unit interval arising as scaling limit for
weighted depths of nodes with at most one child
The mean, variance and limiting distribution of two statistics sensitive to phylogenetic tree balance
For two decades, the Colless index has been the most frequently used
statistic for assessing the balance of phylogenetic trees. In this article,
this statistic is studied under the Yule and uniform model of phylogenetic
trees. The main tool of analysis is a coupling argument with another well-known
index called the Sackin statistic. Asymptotics for the mean, variance and
covariance of these two statistics are obtained, as well as their limiting
joint distribution for large phylogenies. Under the Yule model, the limiting
distribution arises as a solution of a functional fixed point equation. Under
the uniform model, the limiting distribution is the Airy distribution. The
cornerstone of this study is the fact that the probabilistic models for
phylogenetic trees are strongly related to the random permutation and the
Catalan models for binary search trees.Comment: Published at http://dx.doi.org/10.1214/105051606000000547 in the
Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute
of Mathematical Statistics (http://www.imstat.org
Martingales and Profile of Binary Search Trees
We are interested in the asymptotic analysis of the binary search tree (BST)
under the random permutation model. Via an embedding in a continuous time
model, we get new results, in particular the asymptotic behavior of the
profile
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
Trickle-down processes and their boundaries
It is possible to represent each of a number of Markov chains as an evolving
sequence of connected subsets of a directed acyclic graph that grow in the
following way: initially, all vertices of the graph are unoccupied, particles
are fed in one-by-one at a distinguished source vertex, successive particles
proceed along directed edges according to an appropriate stochastic mechanism,
and each particle comes to rest once it encounters an unoccupied vertex.
Examples include the binary and digital search tree processes, the random
recursive tree process and generalizations of it arising from nested instances
of Pitman's two-parameter Chinese restaurant process, tree-growth models
associated with Mallows' phi model of random permutations and with
Schuetzenberger's non-commutative q-binomial theorem, and a construction due to
Luczak and Winkler that grows uniform random binary trees in a Markovian
manner. We introduce a framework that encompasses such Markov chains, and we
characterize their asymptotic behavior by analyzing in detail their Doob-Martin
compactifications, Poisson boundaries and tail sigma-fields.Comment: 62 pages, 8 figures, revised to address referee's comment
Transfer Theorems and Asymptotic Distributional Results for m-ary Search Trees
We derive asymptotics of moments and identify limiting distributions, under
the random permutation model on m-ary search trees, for functionals that
satisfy recurrence relations of a simple additive form. Many important
functionals including the space requirement, internal path length, and the
so-called shape functional fall under this framework. The approach is based on
establishing transfer theorems that link the order of growth of the input into
a particular (deterministic) recurrence to the order of growth of the output.
The transfer theorems are used in conjunction with the method of moments to
establish limit laws. It is shown that (i) for small toll sequences
[roughly, ] we have asymptotic normality if and
typically periodic behavior if ; (ii) for moderate toll sequences
[roughly, but ] we have convergence to
non-normal distributions if (where ) and typically
periodic behavior if ; and (iii) for large toll sequences
[roughly, ] we have convergence to non-normal distributions
for all values of m.Comment: 35 pages, 1 figure. Version 2 consists of expansion and rearragement
of the introductory material to aid exposition and the shortening of
Appendices A and B.
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