Results 1  10
of
72
On the Distribution for the Duration of a Randomized Leader Election Algorithm
 Ann. Appl. Probab
, 1996
"... We investigate the duration of an elimination process for identifying a winner by coin tossing, or, equivalently, the height of a random incomplete trie. Applications of the process include the election of a leader in a computer network. Using direct probabilistic arguments we obtain exact expressio ..."
Abstract

Cited by 28 (10 self)
 Add to MetaCart
We investigate the duration of an elimination process for identifying a winner by coin tossing, or, equivalently, the height of a random incomplete trie. Applications of the process include the election of a leader in a computer network. Using direct probabilistic arguments we obtain exact expressions for the discrete distribution and the moments of the height. Elementary approximation techniques then yield asymptotics for the distribution. We show that no limiting distribution exists, as the asymptotic expressions exhibit periodic fluctuations. In many similar problems associated with digital trees, no such exact expressions can be derived. We therefore outline a powerful general approach, based on the analytic techniques of Mellin transforms, Poissonization, and dePoissonization, from which distributional asymptotics for the height can also be derived. In fact, it was this complex variables approach that led to our original discovery of the exact distribution. Complex analysis metho...
Hypergeometrics and the Cost Structure of Quadtrees
, 1995
"... Several characteristic parameters of randomly grown quadtrees of any dimension are analyzed. Additive parameters have expectations whose generating functions are expressible in terms of generalized hypergeometric functions. A complex asymptotic process based on singularity analysis and integral repr ..."
Abstract

Cited by 25 (2 self)
 Add to MetaCart
Several characteristic parameters of randomly grown quadtrees of any dimension are analyzed. Additive parameters have expectations whose generating functions are expressible in terms of generalized hypergeometric functions. A complex asymptotic process based on singularity analysis and integral representations akin to Mellin transforms leads to explicit values for various structure constants related to path length, retrieval costs, and storage occupation.
Profile of Tries
, 2006
"... Tries (from retrieval) are one of the most popular data structures on words. They are pertinent to (internal) structure of stored words and several splitting procedures used in diverse contexts. The profile of a trie is a parameter that represents the number of nodes (either internal or external) wi ..."
Abstract

Cited by 18 (8 self)
 Add to MetaCart
Tries (from retrieval) are one of the most popular data structures on words. They are pertinent to (internal) structure of stored words and several splitting procedures used in diverse contexts. The profile of a trie is a parameter that represents the number of nodes (either internal or external) with the same distance from the root. It is a function of the number of strings stored in a trie and the distance from the root. Several, if not all, trie parameters such as height, size, depth, shortest path, and fillup level can be uniformly analyzed through the (external and internal) profiles. Although profiles represent one of the most fundamental parameters of tries, they have been hardly studied in the past. The analysis of profiles is surprisingly arduous but once it is carried out it reveals unusually intriguing and interesting behavior. We present a detailed study of the distribution of the profiles in a trie built over random strings generated by a memoryless source. We first derive recurrences satisfied by the expected profiles and solve them asymptotically for all possible ranges of the distance from the root. It appears that profiles of tries exhibit several fascinating phenomena. When moving from the root to the leaves of a trie, the growth of the expected profiles vary. Near the root, the external profiles tend to zero in an exponentially rate, then the rate gradually rises to being logarithmic; the external profiles then abruptly tend to infinity, first logarithmically
On the Multiplicity of Parts in a Random Composition of a Large Integer
 SIAM J. Discrete Math
, 1999
"... In this paper we study the following question posed by H. S. Wilf: What is, asymptotically as n ! 1, the probablility that a randomly chosen part size in a random composition of an integer n has multiplicity m ? More specifically, given positive integers n and m, suppose that a composition of n is ..."
Abstract

Cited by 17 (4 self)
 Add to MetaCart
In this paper we study the following question posed by H. S. Wilf: What is, asymptotically as n ! 1, the probablility that a randomly chosen part size in a random composition of an integer n has multiplicity m ? More specifically, given positive integers n and m, suppose that a composition of n is selected uniformly at random and then, out of the set of part sizes in , a part size j is chosen uniformly at random. Let P(A (m) n ) be the probability that j has multiplicity m. We show that for fixed m, P(A (m) n ) goes to 0 at the rate 1= ln n. A more careful analysis uncovers an unexpected result: (ln n)P(A (m) n ) does not have a limit but instead oscillates around the value 1=m as n !1. This work is a counterpart of a recent paper of Corteel, Pittel, Savage and Wilf who studied the same problem in the case of partitions rather than compositions. 1 Introduction In this paper we consider the multiplicity of a randomly chosen part size in a random composition of an integer n. L...
On the nonholonomic character of logarithms, powers, and the nth prime function
, 2005
"... We establish that the sequences formed by logarithms and by “fractional” powers of integers, as well as the sequence of prime numbers, are nonholonomic, thereby answering three open problems of Gerhold [El. J. Comb. 11 (2004), R87]. Our proofs depend on basic complex analysis, namely a conjunction ..."
Abstract

Cited by 16 (6 self)
 Add to MetaCart
We establish that the sequences formed by logarithms and by “fractional” powers of integers, as well as the sequence of prime numbers, are nonholonomic, thereby answering three open problems of Gerhold [El. J. Comb. 11 (2004), R87]. Our proofs depend on basic complex analysis, namely a conjunction of the Structure Theorem for singularities of solutions to linear differential equations and of an Abelian theorem. A brief discussion is offered regarding the scope of singularitybased methods and several naturally occurring sequences are proved to be nonholonomic.
Regenerative partition structures
 Electron. J. Combin. 11 Research Paper
"... We consider Kingman’s partition structures which are regenerative with respect to a general operation of random deletion of some part. Prototypes of this class are the Ewens partition structures which Kingman characterised by regeneration after deletion of a part chosen by sizebiased sampling. We a ..."
Abstract

Cited by 14 (7 self)
 Add to MetaCart
We consider Kingman’s partition structures which are regenerative with respect to a general operation of random deletion of some part. Prototypes of this class are the Ewens partition structures which Kingman characterised by regeneration after deletion of a part chosen by sizebiased sampling. We associate each regenerative partition structure with a corresponding regenerative composition structure, which (as we showed in a previous paper) can be associated in turn with a regenerative random subset of the positive halfline, that is the closed range of a subordinator. A general regenerative partition structure is thus represented in terms of the Laplace exponent of an associated subordinator. We also analyse deletion properties characteristic of the twoparameter family of partition structures.
The Average Case Analysis of Algorithms: Mellin Transform Asymptotics
, 1996
"... This report is part of a series whose aim is to present in a synthetic way the major methods of "analytic combinatorics" needed in the averagecase analysis of algorithms. It reviews the use of MellinPerron formulae and of Mellin transforms in this context. Applications include: divideandconquer ..."
Abstract

Cited by 12 (0 self)
 Add to MetaCart
This report is part of a series whose aim is to present in a synthetic way the major methods of "analytic combinatorics" needed in the averagecase analysis of algorithms. It reviews the use of MellinPerron formulae and of Mellin transforms in this context. Applications include: divideandconquer recurrences, maxima finding, mergesort, digital trees and plane trees.
Continued Fractions, Comparison Algorithms, and Fine Structure Constants
, 2000
"... There are known algorithms based on continued fractions for comparing fractions and for determining the sign of 2x2 determinants. The analysis of such extremely simple algorithms leads to an incursion into a surprising variety of domains. We take the reader through a light tour of dynamical systems ..."
Abstract

Cited by 10 (2 self)
 Add to MetaCart
There are known algorithms based on continued fractions for comparing fractions and for determining the sign of 2x2 determinants. The analysis of such extremely simple algorithms leads to an incursion into a surprising variety of domains. We take the reader through a light tour of dynamical systems (symbolic dynamics), number theory (continued fractions), special functions (multiple zeta values), functional analysis (transfer operators), numerical analysis (series acceleration), and complex analysis (the Riemann hypothesis). These domains all eventually contribute to a detailed characterization of the complexity of comparison and sorting algorithms, either on average or in probability.
Combinatorics of Geometrically Distributed Random Variables: Value and Position of the rth LefttoRight Maximum
 Discrete Math
"... For words of length n, generated by independent geometric random variables, we consider the average value and the average position of the rth lefttoright maximum, for fixed r and n !1. 1. ..."
Abstract

Cited by 8 (5 self)
 Add to MetaCart
For words of length n, generated by independent geometric random variables, we consider the average value and the average position of the rth lefttoright maximum, for fixed r and n !1. 1.
The Number of Distinct Values in a Geometrically Distributed Sample
"... For words of length n, generated by independent geometric random variables, we consider the average and variance of the number of distinct values (= letters) that occur in the word. We then generalise this to the number of values which occur at least b times in the word. 1. ..."
Abstract

Cited by 8 (2 self)
 Add to MetaCart
For words of length n, generated by independent geometric random variables, we consider the average and variance of the number of distinct values (= letters) that occur in the word. We then generalise this to the number of values which occur at least b times in the word. 1.