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23
The connectivityprofile of random increasing ktrees
"... Random increasing ktrees represent an interesting, useful class of strongly dependent graphs for which analyticcombinatorial tools can be successfully applied. We study in this paper a notion called connectivityprofile and derive asymptotic estimates for it; some interesting consequences will als ..."
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Random increasing ktrees represent an interesting, useful class of strongly dependent graphs for which analyticcombinatorial tools can be successfully applied. We study in this paper a notion called connectivityprofile and derive asymptotic estimates for it; some interesting consequences will also be given. 1
Long and short paths in uniform random recursive dags
, 2009
"... Abstract. In a uniform random recursive kdag, there is a root, 0, and each node in turn, from 1 to n, chooses k uniform random parents from among the nodes of smaller index. If Sn is the shortest path distance from node n to the root, then we determine the constant σ such that Sn / logn → σ in prob ..."
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Abstract. In a uniform random recursive kdag, there is a root, 0, and each node in turn, from 1 to n, chooses k uniform random parents from among the nodes of smaller index. If Sn is the shortest path distance from node n to the root, then we determine the constant σ such that Sn / logn → σ in probability as n → ∞. We also show that max1≤i≤n Si / logn → σ in probability. Keywords and phrases. Uniform random recursive dag. Randomly generated circuit. Random web model. Longest paths. Probabilistic analysis of algorithms. Branching process.
On a functional contraction method
, 1202
"... Methods for proving functional limit laws are developed for sequences of stochastic processes which allow a recursive distributional decomposition either in time or space. Our approach is an extension of the socalled contraction method to the space C[0, 1] of continuous functions endowed with unifo ..."
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Methods for proving functional limit laws are developed for sequences of stochastic processes which allow a recursive distributional decomposition either in time or space. Our approach is an extension of the socalled contraction method to the space C[0, 1] of continuous functions endowed with uniform topology and the space D[0, 1] of càdlàg functions with the Skorokhod topology. The contraction method originated form the probabilistic analysis of algorithms and random trees where characteristics satisfy natural distributional recurrences. It is based on stochastic fixedpoint equations, where probability metrics can be used to obtain contraction properties and allow the application of Banach’s fixedpoint theorem. We develop the use of the Zolotarev metrics on the spaces C[0, 1] and D[0, 1] in this context. As an application a short proof of Donsker’s functional limit theorem is given. AMS 2010 subject classifications. Primary 60F17, 68Q25; secondary 60G18, 60C05. Key words. Functional limit theorem, contraction method, recursive distributional equation, Zolotarev
On the subtree size profile of binary search trees
 Combin. Probab. Comput
"... Abstract. For random trees T generated by the binary search tree algorithm from uniformly distributed input we consider the subtree size profile, which maps k ∈ N to the number of nodes in T that root a subtree of size k. Complementing earlier work by Devroye, by Feng, Mahmoud and Panholzer, and by ..."
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Abstract. For random trees T generated by the binary search tree algorithm from uniformly distributed input we consider the subtree size profile, which maps k ∈ N to the number of nodes in T that root a subtree of size k. Complementing earlier work by Devroye, by Feng, Mahmoud and Panholzer, and by Fuchs, we obtain results for the range of small kvalues and the range of kvalues proportional to the size n of T. In both cases emphasis is on the process view, i.e. the joint distributions for several kvalues. We also show that the dynamics of the tree sequence lead to a qualitative difference between the asymptotic behaviour of the lower and the upper end of the profile. 1.
Trickledown processes and their boundaries
, 2012
"... 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 onebyone at a distinguished source vertex, successive part ..."
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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 onebyone 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 twoparameter Chinese restaurant process, treegrowth models associated with Mallows ’ φ model of random permutations and with Schützenberger’s noncommutative qbinomial 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 DoobMartin compactifications, Poisson boundaries and tail σfields.
HIGHER MOMENTS OF BANACH SPACE VALUED RANDOM VARIABLES
"... Abstract. We define the k:th moment of a Banach space valued random variable as the expectation of its k:th tensor power; thus the moment (if it exists) is an element of a tensor power of the original Banach space. We study both the projective and injective tensor products, and their relation. Moreo ..."
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Abstract. We define the k:th moment of a Banach space valued random variable as the expectation of its k:th tensor power; thus the moment (if it exists) is an element of a tensor power of the original Banach space. We study both the projective and injective tensor products, and their relation. Moreover, in order to be general and flexible, we study three different types of expectations: Bochner integrals, Pettis integrals and Dunford integrals. One of the problems studied is whether two random variables with the same injective moments (of a given order) necessarily have the same projective moments; this is of interest in applications. We show that this holds if the Banach space has the approximation property, but not in general. Several sections are devoted to results in special Banach spaces, including Hilbert spaces, C pK q and Dr0, 1s. The latter space is nonseparable, which complicates the arguments, and we prove various preliminary results on e.g. measurability in Dr0, 1s that we need. One of the main motivations of this paper is the application to Zolotarev metrics and their use in the contraction method. This is sketched in an appendix. 1.
Towards More Realistic Probabilistic Models for Data Structures: The External Path Length in Tries under THE MARKOV MODEL
, 2012
"... Tries are among the most versatile and widely used data structures on words. They are pertinent to the (internal) structure of (stored) words and several splitting procedures used in diverse contexts ranging from document taxonomy to IP addresses lookup, from data compression (i.e., LempelZiv’77 sc ..."
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Cited by 1 (1 self)
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Tries are among the most versatile and widely used data structures on words. They are pertinent to the (internal) structure of (stored) words and several splitting procedures used in diverse contexts ranging from document taxonomy to IP addresses lookup, from data compression (i.e., LempelZiv’77 scheme) to dynamic hashing, from partialmatch queries to speech recognition, from leader election algorithms to distributed hashing tables and graph compression. While the performance of tries under a realistic probabilistic model is of significant importance, its analysis, even for simplest memoryless sources, has proved difficult. Rigorous findings about inherently complex parameters were rarely analyzed (with a few notable exceptions) under more realistic models of string generations. In this paper we meet these challenges: By a novel use of the contraction method combined with analytic techniques we prove a central limit theorem for the external path length of a trie under a general Markov source. In particular, our results apply to the LempelZiv’77 code. We envision that the methods described here will have further applications to other trie parameters and data structures.
Pólya urns via the contraction method
, 2013
"... We propose an approach to analyze the asymptotic behavior of Pólya urns based on the contraction method. For this a combinatorial discrete time embedding of the evolution of the composition of the urn into random rooted trees is used. A decomposition of the trees leads to a system of recursive distr ..."
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We propose an approach to analyze the asymptotic behavior of Pólya urns based on the contraction method. For this a combinatorial discrete time embedding of the evolution of the composition of the urn into random rooted trees is used. A decomposition of the trees leads to a system of recursive distributional equations which capture the distributions of the numbers of balls of each color. Ideas from the contraction method are used to study such systems of recursive distributional equations asymptotically. We apply our approach to a couple of concrete Pólya urns that lead to limit laws with normal limit distributions, with nonnormal limit distributions and with asymptotic periodic distributional behavior.