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TreeValued Markov Chains Derived From GaltonWatson Processes.
 Ann. Inst. Henri Poincar'e
, 1997
"... Let G be a GaltonWatson tree, and for 0 u 1 let G u be the subtree of G obtained by retaining each edge with probability u. We study the treevalued Markov process (G u ; 0 u 1) and an analogous process (G u ; 0 u 1) in which G 1 is a critical or subcritical GaltonWatson tree conditio ..."
Abstract

Cited by 35 (9 self)
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Let G be a GaltonWatson tree, and for 0 u 1 let G u be the subtree of G obtained by retaining each edge with probability u. We study the treevalued Markov process (G u ; 0 u 1) and an analogous process (G u ; 0 u 1) in which G 1 is a critical or subcritical GaltonWatson tree conditioned to be infinite. Results simplify and are further developed in the special case of Poisson() offspring distribution. Running head. Treevalued Markov chains. Key words. Borel distribution, branching process, conditioning, GaltonWatson process, generalized Poisson distribution, htransform, pruning, random tree, sizebiasing, spinal decomposition, thinning. AMS Subject classifications 05C80, 60C05, 60J27, 60J80 Research supported in part by N.S.F. Grants DMS9404345 and 9622859 1 Contents 1 Introduction 2 1.1 Related topics : : : : : : : : : : : : : : : : : : : : : : : : : : : 4 2 Background and technical setup 5 2.1 Notation and terminology for trees : : : : : : : : : : : : : : :...
Limit Distributions and Random Trees Derived From the Birthday Problem With Unequal Probabilities
, 1998
"... Given an arbitrary distribution on a countable set S consider the number of independent samples required until the first repeated value is seen. Exact and asymptotic formulae are derived for the distribution of this time and of the times until subsequent repeats. Asymptotic properties of the repeat ..."
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Cited by 26 (14 self)
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Given an arbitrary distribution on a countable set S consider the number of independent samples required until the first repeated value is seen. Exact and asymptotic formulae are derived for the distribution of this time and of the times until subsequent repeats. Asymptotic properties of the repeat times are derived by embedding in a Poisson process. In particular, necessary and sufficient conditions for convergence are given and the possible limits explicitly described. Under the same conditions the finite dimensional distributions of the repeat times converge to the arrival times of suitably modified Poisson processes, and random trees derived from the sequence of independent Research supported in part by N.S.F. Grants DMS 9224857, 9404345, 9224868 and 9703691 trials converge in distribution to an inhomogeneous continuum random tree. 1 Introduction Recall the classical birthday problem: given that each day of the year is equally likely as a possible birthday, and that birth...
AbelCayleyHurwitz multinomial expansions associated with random mappings, forests, and subsets
, 1998
"... Extensions of binomial and multinomial formulae due to Abel, Cayley and Hurwitz are related to the probability distributions of various random subsets, trees, forests, and mappings. For instance, an extension of Hurwitz's binomial formula is associated with the probability distribution of the random ..."
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Cited by 13 (12 self)
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Extensions of binomial and multinomial formulae due to Abel, Cayley and Hurwitz are related to the probability distributions of various random subsets, trees, forests, and mappings. For instance, an extension of Hurwitz's binomial formula is associated with the probability distribution of the random set of vertices of a fringe subtree in a random forest whose distribution is defined by terms of a multinomial expansion over rooted labeled forests which generalizes Cayley's expansion over unrooted labeled trees. Contents 1 Introduction 2 Research supported in part by N.S.F. Grant DMS9703961 2 Probabilistic Interpretations 5 3 Cayley's multinomial expansion 11 4 Random Mappings 14 4.1 Mappings from S to S : : : : : : : : : : : : : : : : : : : : : : : : : : : : 15 4.2 The random set of cyclic points : : : : : : : : : : : : : : : : : : : : : : : 18 5 Random Forests 19 5.1 Distribution of the roots of a pforest : : : : : : : : : : : : : : : : : : : : 19 5.2 Conditioning on the set...
Computing A DiameterConstrained Minimum Spanning Tree
, 2001
"... In numerous practical applications, it is necessary to find the smallest possible tree with a bounded diameter. A diameterconstrained minimum spanning tree (DCMST) of a given undirected, edgeweighted graph, G, is the smallestweight spanning tree of all spanning trees of G which contain no path wi ..."
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Cited by 8 (0 self)
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In numerous practical applications, it is necessary to find the smallest possible tree with a bounded diameter. A diameterconstrained minimum spanning tree (DCMST) of a given undirected, edgeweighted graph, G, is the smallestweight spanning tree of all spanning trees of G which contain no path with more than k edges, where k is a given positive integer. The problem of finding a DCMST is NPcomplete for all values of k; 4 k (n  2), except when all edgeweights are identical. A DCMST is essential for the efficiency of various distributed mutual exclusion algorithms, where it can minimize the number of messages communicated among processors per critical section. It is also useful in linear lightwave networks, where it can minimize interference in the network by limiting the traffic in the network lines. Another practical application requiring a DCMST arises in data compression, where some algorithms compress a file utilizing a tree datastructure, and decompress a path in the tree to access a record. A DCMST helps such algorithms to be fast without sacrificing a lot of storage space. We present a survey of the literature on the DCMST problem, study the expected diameter of a random labeled tree, and present five new polynomialtime algorithms for an approximate DCMST. One of our new algorithms constructs an approximate DCMST in a modified greedy fashion, employing a heuristic for selecting an edge to be added to iii the tree in each stage of the construction. Three other new algorithms start with an unconstrained minimum spanning tree, and iteratively refine it into an approximate DCMST. We also present an algorithm designed for the special case when the diameter is required to be no more than 4. Such a diameter4 tree is also used for evaluating the quality of o...
Forest volume decompositions and AbelCayleyHurwitz multinomial expansions
, 2001
"... This paper presents a systematic approach to the discovery, interpretation and verification of various extensions of Hurwitz's multinomial identities, involving polynomials defined by sums over all subsets of a finite set. The identities are interpreted as decompositions of forest volumes define ..."
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Cited by 2 (0 self)
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This paper presents a systematic approach to the discovery, interpretation and verification of various extensions of Hurwitz's multinomial identities, involving polynomials defined by sums over all subsets of a finite set. The identities are interpreted as decompositions of forest volumes defined by the enumerator polynomials of sets of rooted labeled forests. These decompositions involve the following basic forest volume formula, which is a refinement of Cayley's multinomial expansion: for R ` S the polynomial enumerating outdegrees of vertices of rooted forests labeled by S whose set of roots is R, with edges directed away from the roots, is ( P r2R x r )( P s2S x s ) jS j\GammajRj\Gamma1
Colored looperased random walk on the complete graph
, 2006
"... Starting from a sequence regarded as a walk through some set of values, we consider the associated looperased walk as a sequence of directed edges, with an edge from i to j if the loop erased walk makes a step from i to j. We introduce a coloring of these edges by painting edges with a fixed color ..."
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Cited by 1 (0 self)
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Starting from a sequence regarded as a walk through some set of values, we consider the associated looperased walk as a sequence of directed edges, with an edge from i to j if the loop erased walk makes a step from i to j. We introduce a coloring of these edges by painting edges with a fixed color as long as the walk does not loop back on itself, then switching to a new color whenever a loop is erased, with each new color distinct from all previous colors. The pattern of colors along the edges of the looperased walk then displays stretches of consecutive steps of the walk left untouched by the looperasure process. Assuming that the underlying sequence generating the looperased walk is a sequence of independent random variables, each uniform on [N]: = {1, 2,..., N}, we condition the walk to start at N and stop the walk when it first reaches the subset [k], for some 1 ≤ k ≤ N − 1. We relate the distribution of the random length of this looperased walk to the distribution of the length of the first loop of the walk, via Cayley’s enumerations of trees, and via Wilson’s algorithm. For fixed N and k, and i = 1,2,..., let Bi denote the event that the looperased walk from N to [k] has i+1 or more edges, and the i th and (i+1) th of these edges are colored differently. We show that given that the looperased random walk has j edges for some 1 ≤ j ≤ N − k, the events Bi for 1 ≤ i ≤ j − 1 are independent, with the probability of Bi equal to 1/(k + i + 1). This determines the distribution of the sequence of random lengths of differently colored segments of the looperased walk, and yields asymptotic descriptions of these random lengths as N → ∞.
The Wiener Index of Random Digital Trees
, 2012
"... The Wiener index has been studied for simply generated random trees, nonplane unlabeled random trees and a huge subclass of random grid trees containing random binary search trees, random medianof(2k + 1) search trees, random mary search trees, random quadtrees, random simplex trees, etc. An impo ..."
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The Wiener index has been studied for simply generated random trees, nonplane unlabeled random trees and a huge subclass of random grid trees containing random binary search trees, random medianof(2k + 1) search trees, random mary search trees, random quadtrees, random simplex trees, etc. An important class of random trees for which the Wiener index was not studied so far are random digital trees. In this work, we close this gap. More precisely, we derive asymptotic expansions of moments of the Wiener index and show that a central limit law for the Wiener index holds. These results are obtained for digital search trees and bucket versions as well as tries and PATRICIA tries. Our findings answer in affirmative two questions posed by Neininger. 1