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23
Branching processes in the analysis of the heights of trees
 Acta Informatica
, 1987
"... Summary. It is shown how the theory of branching processes can be applied in the analysis of the expected height of random trees. In particular, we will study the height of random binary search trees, random kd trees, quadtrees and unionend trees under various models of randomization. For example, ..."
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Cited by 59 (19 self)
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Summary. It is shown how the theory of branching processes can be applied in the analysis of the expected height of random trees. In particular, we will study the height of random binary search trees, random kd trees, quadtrees and unionend trees under various models of randomization. For example, for the random binary search tree constructed from a random permutation of 1,..., n, it is shown that H„/(c log (n)) tends to 1 in probability and in the mean as n oo, where H „ is the height of the tree, and c =4.31107... is a solution of the equation c log (2e / = 1. In addition, we ~c ~ show that H „clog (n) = O (/log (n) loglog (n)) in probability.
An Optimal Algorithm for Generating Minimal Perfect Hash Functions
 Information Processing Letters
, 1992
"... A new algorithm for generating order preserving minimal perfect hash functions is presented. The algorithm is probabilistic, involving generation of random graphs. It uses expected linear time and requires a linear number words to represent the hash function, and thus is optimal up to constant facto ..."
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Cited by 42 (0 self)
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A new algorithm for generating order preserving minimal perfect hash functions is presented. The algorithm is probabilistic, involving generation of random graphs. It uses expected linear time and requires a linear number words to represent the hash function, and thus is optimal up to constant factors. It runs very fast in practice. Keywords: Data structures, probabilistic algorithms, analysis of algorithms, hashing, random graphs
Coalescent Random Forests
 J. COMBINATORIAL THEORY A
, 1998
"... Various enumerations of labeled trees and forests, including Cayley's formula n n\Gamma2 for the number of trees labeled by [n], and Cayley's multinomial expansion over trees, are derived from the following coalescent construction of a sequence of random forests (R n ; R n\Gamma1 ; : : : ; R 1 ..."
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Cited by 38 (18 self)
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Various enumerations of labeled trees and forests, including Cayley's formula n n\Gamma2 for the number of trees labeled by [n], and Cayley's multinomial expansion over trees, are derived from the following coalescent construction of a sequence of random forests (R n ; R n\Gamma1 ; : : : ; R 1 ) such that R k has uniform distribution over the set of all forests of k rooted trees labeled by [n]. Let R n be the trivial forest with n root vertices and no edges. For n k 2, given that R n ; : : : ; R k have been defined so that R k is a rooted forest of k trees, define R k\Gamma1 by addition to R k of a single edge picked uniformly at random from the set of n(k \Gamma 1) edges which when added to R k yield a rooted forest of k \Gamma 1 trees. This coalescent construction is related to a model for a physical process of clustering or coagulation, the additive coalescent in which a system of masses is subject to binary coalescent collisions, with each pair of masses of magnitude...
A recurrence related to trees
 Proceedings of the American Mathematical Society
, 1989
"... Abstract. The asymptotic behavior of the solutions to an interesting class of recurrence relations, which arise in the study of trees and random graphs, is derived by making uniform estimates on the elements of a basis of the solution space. We also investigate a family of polynomials with integer c ..."
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Cited by 34 (4 self)
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Abstract. The asymptotic behavior of the solutions to an interesting class of recurrence relations, which arise in the study of trees and random graphs, is derived by making uniform estimates on the elements of a basis of the solution space. We also investigate a family of polynomials with integer coefficients, which may be called the "tree polynomials." There are n" ~ (n 1)! sequences of edges between vertices (0.1) ux—vx.un_x—vn_x, \<uk<vk<n, that define a free tree on {1,...,«}, because there are n" ~ free trees on n labeled vertices and every such tree has n 1 edges. If we consider each of these n (n — 1)! sequences to be equally likely, the probability that unX and vn_x belong respectively to components of sizes k and n k based on the first « 2 edges is '^oer^r'■•<*< • ■ Knuth and Schönhage [9, §§912] considered treeconstruction algorithms whose analysis depended on the solution of the recurrence (°3) Xn=Cn+ E Pnk(xk+Xnk) 0<k<n for various sequences (cn). The purpose of the present note is to extend the results of [9] and to consider related sequences of functions whose exact and asymptotic values arise in a variety of algorithms. Much of the analysis below, as in [9], depends on properties of the formal power series tt\A \ TV \ \r^n"~[z " 2, 3 3, 8 4, 125 5, (0.4) T(z) = ^ — — = z + z +z +^z + —z +■■ ■, n>l Received by the editors March 18, 1988.
Singularity Analysis, Hadamard Products, and Tree Recurrences
, 2003
"... We present a toolbox for extracting asymptotic information on the coecients of combinatorial generating functions. This toolbox notably includes a treatment of the eect of Hadamard products on singularities in the context of the complex Tauberian technique known as singularity analysis. As a consequ ..."
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Cited by 28 (9 self)
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We present a toolbox for extracting asymptotic information on the coecients of combinatorial generating functions. This toolbox notably includes a treatment of the eect 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 divideandconquer algorithms, or equivalently random tree models, including several classical methods for sorting, searching, and dynamically managing equivalence relations.
A random tree model associated with random graphs
 RANDOM STRUCTURES AND ALGORITHMS
, 1990
"... Grow a tree on n vertices by starting with no edges and successively adding an edge chosen uniformly from the set of possible edges whose addition would not create a cycle. This process is closely related to the classical random graph process. We describe the asymptotic structure of the tree, as see ..."
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Cited by 22 (8 self)
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Grow a tree on n vertices by starting with no edges and successively adding an edge chosen uniformly from the set of possible edges whose addition would not create a cycle. This process is closely related to the classical random graph process. We describe the asymptotic structure of the tree, as seen locally from a given vertex. In particular, we give an explicit expression for the asymptotic degree distribution. Our results an be applied to study the random minimumweight spanning tree question, when the edgeweight distribution is allowed to vary almost arbitrarily with n.
On the Analysis of Linear Probing Hashing
, 1998
"... This paper presents moment analyses and characterizations of limit distributions for the construction cost of hash tables under the linear probing strategy. Two models are considered, that of full tables and that of sparse tables with a fixed filling ratio strictly smaller than one. For full tables, ..."
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Cited by 19 (8 self)
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This paper presents moment analyses and characterizations of limit distributions for the construction cost of hash tables under the linear probing strategy. Two models are considered, that of full tables and that of sparse tables with a fixed filling ratio strictly smaller than one. For full tables, the construction cost has expectation O(n3/2), the standard deviation is of the same order, and a limit law of the Airy type holds. (The Airy distribution is a semiclassical distribution that is defined in terms of the usual Airy functions or equivalently in terms of Bessel functions of indices − 1 2 3, 3.) For sparse tables, the construction cost has expectation O(n), standard deviation O ( √ n), and a limit law of the Gaussian type. Combinatorial relations with other problems leading to Airy phenomena (like graph connectivity, tree inversions, tree path length, or area under excursions) are also briefly discussed.
Limit laws for sums of functions of subtrees of random binary search trees
 SIAM Journal on Computing
, 2001
"... We consider sums of functions of subtrees of a random binary search tree, and obtain general laws of large numbers and central limit theorems. These sums correspond to L random recurrences of the quicksort type, Xn = XIn +X ′ n−1−In +Yn, n ≥ 1, where In is uniformly distributed on {0, 1,..., n − 1}, ..."
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Cited by 12 (1 self)
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We consider sums of functions of subtrees of a random binary search tree, and obtain general laws of large numbers and central limit theorems. These sums correspond to L random recurrences of the quicksort type, Xn = XIn +X ′ n−1−In +Yn, n ≥ 1, where In is uniformly distributed on {0, 1,..., n − 1}, Yn is a given random variable, Xk L = X ′ k for all k, and given In, XIn and X ′ n−1−In are independent. Conditions are derived such that (Xn −µn)/σ √ n L → N (0, 1), the normal distribution, for some finite constants µ and σ.