13,112 research outputs found

    Limiting distributions for additive functionals on Catalan trees

    Full text link
    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 α0\alpha \downarrow 0]. 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

    Get PDF
    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

    Full text link
    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

    Full text link
    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

    Full text link
    Using recent results on singularity analysis for Hadamard products of generating functions, we obtain the limiting distributions for additive functionals on mm-ary search trees on nn keys with toll sequence (i) nαn^\alpha with α0\alpha \geq 0 (α=0\alpha=0 and α=1\alpha=1 correspond roughly to the space requirement and total path length, respectively); (ii) ln(nm1)\ln \binom{n}{m-1}, which corresponds to the so-called shape functional; and (iii) 1n=m1\mathbf{1}_{n=m-1}, 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

    Get PDF
    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

    Full text link
    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 (tn)(t_n) [roughly, tn=O(n1/2)t_n =O(n^{1 / 2})] we have asymptotic normality if m26m \leq 26 and typically periodic behavior if m27m \geq 27; (ii) for moderate toll sequences [roughly, tn=ω(n1/2)t_n = \omega(n^{1 / 2}) but tn=o(n)t_n = o(n)] we have convergence to non-normal distributions if mm0m \leq m_0 (where m026m_0 \geq 26) and typically periodic behavior if mm0+1m \geq m_0 + 1; and (iii) for large toll sequences [roughly, tn=ω(n)t_n = \omega(n)] 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.
    corecore