Results 1  10
of
48
Mathematical results on scalefree random graphs
 Handbook of Graphs and Networks
, 2003
"... Recently there has been much interest in studying largescale realworld networks and attempting to model their properties using random graphs. Although the study of realworld networks as graphs goes back some time, recent activity perhaps started with the paper of Watts and Strogatz [55] about the ..."
Abstract

Cited by 102 (2 self)
 Add to MetaCart
Recently there has been much interest in studying largescale realworld networks and attempting to model their properties using random graphs. Although the study of realworld networks as graphs goes back some time, recent activity perhaps started with the paper of Watts and Strogatz [55] about the ‘smallworld
On Convergence Rates in the Central Limit Theorems for Combinatorial Structures
, 1998
"... Flajolet and Soria established several central limit theorems for the parameter "number of components" in a wide class of combinatorial structures. In this paper, we shall prove a simple theorem which applies to characterize the convergence rates in their central limit theorems. This theorem is a ..."
Abstract

Cited by 67 (8 self)
 Add to MetaCart
Flajolet and Soria established several central limit theorems for the parameter "number of components" in a wide class of combinatorial structures. In this paper, we shall prove a simple theorem which applies to characterize the convergence rates in their central limit theorems. This theorem is also applicable to arithmetical functions. Moreover, asymptotic expressions are derived for moments of integral order. Many examples from different applications are discussed.
Phase Change of Limit Laws in the Quicksort Recurrence Under Varying Toll Functions
, 2001
"... We characterize all limit laws of the quicksort type random variables defined recursively by Xn = X In + X # n1In + Tn when the "toll function" Tn varies and satisfies general conditions, where (Xn ), (X # n ), (I n , Tn ) are independent, Xn . . . , n 1}. When the "toll function" Tn ..."
Abstract

Cited by 44 (18 self)
 Add to MetaCart
We characterize all limit laws of the quicksort type random variables defined recursively by Xn = X In + X # n1In + Tn when the "toll function" Tn varies and satisfies general conditions, where (Xn ), (X # n ), (I n , Tn ) are independent, Xn . . . , n 1}. When the "toll function" Tn (cost needed to partition the original problem into smaller subproblems) is small (roughly lim sup n## log E(Tn )/ log n 1/2), Xn is asymptotically normally distributed; nonnormal limit laws emerge when Tn becomes larger. We give many new examples ranging from the number of exchanges in quicksort to sorting on broadcast communication model, from an insitu permutation algorithm to tree traversal algorithms, etc.
Large deviations of combinatorial distributions II: Local limit theorems
, 1997
"... This paper is a sequel to our paper [17] where we derived a general central limit theorem for probabilities of large deviations applicable to many classes of combinatorial structures and arithmetic functions; we consider corresponding local limit theorems in this paper. More precisely, given a seq ..."
Abstract

Cited by 32 (5 self)
 Add to MetaCart
This paper is a sequel to our paper [17] where we derived a general central limit theorem for probabilities of large deviations applicable to many classes of combinatorial structures and arithmetic functions; we consider corresponding local limit theorems in this paper. More precisely, given a sequence of integral random variables n#1 each of maximal span 1 (see below for definition), we are interested in the asymptotic behavior of the probabilities n = m} (m N, m = n x n # n , n := n , # n := n ), ##, where x n can tend to with n at a rate that is restricted to O(# n ). Our interest here is not to derive asymptotic expression for n = m} valid for the widest possible range of m, but to show that for m lying in the interval n O(# n ), very precise asymptotic formulae can be obtained. These formulae are in close connection with our results in [17]. Although local limit theorems receive a constant research interest [2, 3, 7, 14, 13, 24], our approach and results, especially Theorem 1, seem rarely discussed in a systematic manner. Recall that a lattice random variable X is said to be of maximal span h if X takes only values of the form b + hk, k Z, for some constants b and h > 0; and there does not exist b # and h # > h such that X takes only values of the form b # + h # k
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 ..."
Abstract

Cited by 28 (9 self)
 Add to MetaCart
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.
Search costs in quadtrees and singularity perturbation asymptotics
 Discrete Comput. Geom
, 1994
"... Abstract. Quadtrees constitute a classical data structure for storing and accessing collections of points in multidimensional space. It is proved that, in any dimension, the cost of a random search in a randomly grown quadtree has logarithmic mean and variance and is asymptotically distributed as a ..."
Abstract

Cited by 22 (5 self)
 Add to MetaCart
Abstract. Quadtrees constitute a classical data structure for storing and accessing collections of points in multidimensional space. It is proved that, in any dimension, the cost of a random search in a randomly grown quadtree has logarithmic mean and variance and is asymptotically distributed as a normal variable. The limit distribution property extends to quadtrees of all dimensions a result only known so far to hold for binary search trees. The analysis is based on a technique of singularity perturbation that appears to be of some generality. For quadtrees, this technique is applied to linear differential equations satisfied by intervening bivariate generating functions 1.
An asymptotic theory for CauchyEuler differential equations with applications to the analysis of algorithms
, 2002
"... CauchyEuler differential equations surfaced naturally in a number of sorting and searching problems, notably in quicksort and binary search trees and their variations. Asymptotics of coefficients of functions satisfying such equations has been studied for several special cases in the literature. We ..."
Abstract

Cited by 22 (10 self)
 Add to MetaCart
CauchyEuler differential equations surfaced naturally in a number of sorting and searching problems, notably in quicksort and binary search trees and their variations. Asymptotics of coefficients of functions satisfying such equations has been studied for several special cases in the literature. We study in this paper the most general framework for CauchyEuler equations and propose an asymptotic theory that covers almost all applications where CauchyEuler equations appear. Our approach is very general and requires almost no background on differential equations. Indeed the whole theory can be stated in terms of recurrences instead of functions. Old and new applications of the theory are given. New phase changes of limit laws of new variations of quicksort are systematically derived. We apply our theory to about a dozen of diverse examples in quicksort, binary search trees, urn models, increasing trees, etc.
Analytic Urns
 March
, 2003
"... This article describes a purely analytic approach to urn models of the generalized or extended PólyaEggenberger type, in the case of two types of balls and constant "balance", i.e., constant row sum. (Under such models, an urn may contain balls of either of two colours and a fixed 2 × 2matri ..."
Abstract

Cited by 20 (1 self)
 Add to MetaCart
This article describes a purely analytic approach to urn models of the generalized or extended PólyaEggenberger type, in the case of two types of balls and constant "balance", i.e., constant row sum. (Under such models, an urn may contain balls of either of two colours and a fixed 2 × 2matrix determines the replacement policy when a ball is drawn and its colour is observed.) The treatment starts from a quasilinear firstorder partial differential equation associated with a combinatorial renormalization of the model and bases itself on elementary conformal mapping arguments coupled with singularity analysis techniques. Probabilistic consequences are new representations for the probability distribution of the urn's composition at any time n, structural information on the shape of moments of all orders, estimates of the speed of convergence to the Gaussian limits, and an explicit determination of the associated large deviation function. In the general case, analytic solutions involve Abelian integrals over the Fermat curve x = 1. Several urn models, including a classical one associated with balanced trees (23 trees and fringebalanced search trees) and related to a previous study of Panholzer and Prodinger as well as all urns of balance 1 or 2, are shown to admit of explicit representations in terms of Weierstraß elliptic functions. Other consequences include a unification of earlier studies of these models and the detection of stable laws in certain classes of urns with an offdiagonal entry equal to zero.
Profiles of random trees: Limit theorems for random recursive trees and binary search trees
, 2005
"... We prove convergence in distribution for the profile (the number of nodes at each level), normalized by its mean, of random recursive trees when the limit ratio ˛ of the level and the logarithm of tree size lies in Œ0; e/. Convergence of all moments is shown to hold only for ˛ 2 Œ0; 1 (with only con ..."
Abstract

Cited by 17 (11 self)
 Add to MetaCart
We prove convergence in distribution for the profile (the number of nodes at each level), normalized by its mean, of random recursive trees when the limit ratio ˛ of the level and the logarithm of tree size lies in Œ0; e/. Convergence of all moments is shown to hold only for ˛ 2 Œ0; 1 (with only convergence of finite moments when ˛ 2.1; e/). When the limit ratio is 0 or 1 for which the limit laws are both constant, we prove asymptotic normality for ˛ D 0 and a “quicksort type ” limit law for ˛ D 1, the latter case having additionally a small range where there is no fixed limit law. Our tools are based on contraction method and method of moments. Similar phenomena also hold for other classes of trees; we apply our tools to binary search trees and give a complete characterization of the profile. The profiles of these random trees represent concrete examples for which the range of convergence in distribution differs from that of convergence of all moments.
Profiles of random trees: planeoriented recursive trees
, 2005
"... We derive several limit results for the profile of random planeoriented recursive trees. These include the limit distribution of the normalized profile, asymptotic bimodality of the variance, asymptotic approximation to the expected width and the correlation coefficients of two level sizes. Most of ..."
Abstract

Cited by 17 (5 self)
 Add to MetaCart
We derive several limit results for the profile of random planeoriented recursive trees. These include the limit distribution of the normalized profile, asymptotic bimodality of the variance, asymptotic approximation to the expected width and the correlation coefficients of two level sizes. Most of our proofs are based on a method of moments. We also discover an unexpected connection between the profile of planeoriented recursive trees (with logarithmic height) and that of random binary trees (with height proportional to the square root of tree size).