Results 1  10
of
20
Random Mapping Statistics
 IN ADVANCES IN CRYPTOLOGY
, 1990
"... Random mappings from a finite set into itself are either a heuristic or an exact model for a variety of applications in random number generation, computational number theory, cryptography, and the analysis of algorithms at large. This paper introduces a general framework in which the analysis of ..."
Abstract

Cited by 87 (6 self)
 Add to MetaCart
Random mappings from a finite set into itself are either a heuristic or an exact model for a variety of applications in random number generation, computational number theory, cryptography, and the analysis of algorithms at large. This paper introduces a general framework in which the analysis of about twenty characteristic parameters of random mappings is carried out: These parameters are studied systematically through the use of generating functions and singularity analysis. In particular, an open problem of Knuth is solved, namely that of finding the expected diameter of a random mapping. The same approach is applicable to a larger class of discrete combinatorial models and possibilities of automated analysis using symbolic manipulation systems ("computer algebra") are also briefly discussed.
Boltzmann Samplers For The Random Generation Of Combinatorial Structures
 Combinatorics, Probability and Computing
, 2004
"... This article proposes a surprisingly simple framework for the random generation of combinatorial configurations based on what we call Boltzmann models. The idea is to perform random generation of possibly complex structured objects by placing an appropriate measure spread over the whole of a combina ..."
Abstract

Cited by 69 (2 self)
 Add to MetaCart
(Show Context)
This article proposes a surprisingly simple framework for the random generation of combinatorial configurations based on what we call Boltzmann models. The idea is to perform random generation of possibly complex structured objects by placing an appropriate measure spread over the whole of a combinatorial class  an object receives a probability essentially proportional to an exponential of its size. As demonstrated here, the resulting algorithms based on realarithmetic operations often operate in linear time. They can be implemented easily, be analysed mathematically with great precision, and, when suitably tuned, tend to be very efficient in practice.
On Random Walks For Pollard's Rho Method
 Mathematics of Computation
, 2000
"... . We consider Pollard's rho method for discrete logarithm computation. Usually, in the analysis of its running time the assumption is made that a random walk in the underlying group is simulated. We show that this assumption does not hold for the walk originally suggested by Pollard: its per ..."
Abstract

Cited by 33 (5 self)
 Add to MetaCart
(Show Context)
. We consider Pollard's rho method for discrete logarithm computation. Usually, in the analysis of its running time the assumption is made that a random walk in the underlying group is simulated. We show that this assumption does not hold for the walk originally suggested by Pollard: its performance is worse than in the random case. We study alternative walks that can be efficiently applied to compute discrete logarithms. We introduce a class of walks that lead to the same performance as expected in the random case. We show that this holds for arbitrarily large prime group orders, thus making Pollard's rho method for prime group orders about 20% faster than before. 1. Introduction Let G be a finite cyclic group, written multiplicatively, and generated by the group element g. We define the discrete logarithm problem (DLP) as follows: given a group element h, find the least nonnegative integer x such that h = g x . We write x = log g h and call it the discrete logarithm of h...
Random mappings, forests, and subsets associated with AbelCayleyHurwitz multinomial expansions
, 2001
"... Various random combinatorial objects, such as mappings, trees, forests, and subsets of a finite set, are constructed with probability distributions related to the binomial and multinomial expansions due to Abel, Cayley and Hurwitz. Relations between these combinatorial objects, such as Joyal&apo ..."
Abstract

Cited by 14 (9 self)
 Add to MetaCart
Various random combinatorial objects, such as mappings, trees, forests, and subsets of a finite set, are constructed with probability distributions related to the binomial and multinomial expansions due to Abel, Cayley and Hurwitz. Relations between these combinatorial objects, such as Joyal's bijection between mappings and marked rooted trees, have interesting probabilistic interpretations, and applications to the asymptotic structure of large random trees and mappings. 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. Research supported in part by N.S.F. Grants DMS 9703961 and DMS0071448 1 Contents 1
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 r ..."
Abstract

Cited by 13 (12 self)
 Add to MetaCart
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...
A Calculus for the Random Generation of Combinatorial Structures
, 1993
"... A systematic approach to the random generation of labelled combinatorial objects is presented. It applies to structures that are decomposable, i.e., formally specifiable by grammars involving set, sequence, and cycle constructions. A general strategy is developed for solving the random generation pr ..."
Abstract

Cited by 10 (2 self)
 Add to MetaCart
A systematic approach to the random generation of labelled combinatorial objects is presented. It applies to structures that are decomposable, i.e., formally specifiable by grammars involving set, sequence, and cycle constructions. A general strategy is developed for solving the random generation problem with two closely related types of methods: for structures of size n, the boustrophedonic algorithms exhibit a worstcase behaviour of the form O(n log n); the sequential algorithms haveworst case O(n²), while offering good potential for optimizations in the average case. (Both methods appeal to precomputed numerical tables of linear size.) A companion calculus permits to systematically compute the average case cost of the sequential generation algorithm associated to a given specification. Using optimizations dictated by the cost calculus, several random generation algorithms are developed, based on the sequential principle; most of them have expected complexity 1/2 n log n,thu...
Images and Preimages in Random Mappings
 SIAM Journal on Discrete Mathematics
, 1996
"... We present a general theorem which can be used to identify the limiting distribution for a class of combinatorial schemata. For example, many parameters in random mappings can be covered in this way. Especially we can derive the limiting distibution of those points with a given number of total prede ..."
Abstract

Cited by 9 (1 self)
 Add to MetaCart
We present a general theorem which can be used to identify the limiting distribution for a class of combinatorial schemata. For example, many parameters in random mappings can be covered in this way. Especially we can derive the limiting distibution of those points with a given number of total predecessors. 1 Introduction By a random mapping ' 2 Fn F = S n0 Fn we mean an arbitrary mapping ' : f1; : : : ; ng ! f1; : : : ; ng such that every mapping has equal probability n n . The main purpose of this paper is to obtain limit theorems, when n tends to innity, for special parameters in random mappings, e.g. for the number of image points. Since every random mapping ' 2 Fn has equal probability it suces to count the number of radom mappings ' 2 Fn satisfying a special property, e.g. that the number of image points equals k. By dividing this number by n n we get the probability of interest. In order to get the limit distribution for n !1 it is not necessary to know the exact v...
On the local time density of the reflecting Brownian bridge
 MR MR1768499 (2001h:60134
, 2000
"... Expressions for the multidimensional densities of Brownian bridge local time are derived by two different methods: A direct method based on Kac’s formula for Brownian functionals and an indirect one based on a limit theorem for strata of random mappings. ..."
Abstract

Cited by 4 (2 self)
 Add to MetaCart
(Show Context)
Expressions for the multidimensional densities of Brownian bridge local time are derived by two different methods: A direct method based on Kac’s formula for Brownian functionals and an indirect one based on a limit theorem for strata of random mappings.
Predecessors in Random Mappings
 Combin. Probab. Comput
, 1995
"... Let Fn be the set of random mappings ' : f1; : : : ; ng ! f1; : : : ; ng (such that every mapping is equally likely). For x 2 f1; : : : ; ng the elements S k0 ' k (fxg) are called the predecessors of x. Let Nr denote the random variable which counts the number of points x 2 f1; : : ..."
Abstract

Cited by 3 (1 self)
 Add to MetaCart
(Show Context)
Let Fn be the set of random mappings ' : f1; : : : ; ng ! f1; : : : ; ng (such that every mapping is equally likely). For x 2 f1; : : : ; ng the elements S k0 ' k (fxg) are called the predecessors of x. Let Nr denote the random variable which counts the number of points x 2 f1; : : : ; ng with exactly r predecessors. In this paper we identify the limiting distribution of Nr as n ! 1. If r = r(n) = o(n 2 3 ) then the limiting distribution is Gaussian, if r Cn 2 3 then it is Poisson, and in the remaining case rn 2 3 !1 it is degenerate. Furthermore it is shown that Nr is a Poisson approximation if r !1. 1 Introduction Let Fn be the set of mappings ' : f1; : : : ; ng ! f1; : : : ; ng. In what follows we will assume the uniform random model, i.e. each mapping ' 2 Fn appears with probability n n . We will call the elements of F = S n0 Fn random mappings. If X : F ! R is an arbitrary function then Xn = XjFn may considered as a sequence of random variables and it may as...