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38
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 "to ..."
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Cited by 48 (19 self)
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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.
Distinctness of compositions of an integer: A probabilistic analysis
 RANDOM STRUCTURES AND ALGORITHMS
, 2001
"... Compositions of integers are used as theoretical models for many applications. The degree of distinctness of a composition is a natural and important parameter. In this paper, we use as measure of distinctness the number of distinct parts (or components). We investigate, from a probabilistic point o ..."
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Cited by 32 (13 self)
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Compositions of integers are used as theoretical models for many applications. The degree of distinctness of a composition is a natural and important parameter. In this paper, we use as measure of distinctness the number of distinct parts (or components). We investigate, from a probabilistic point of view, the first empty part, the maximum part size and the distribution of the number of distinct part sizes. We obtain asymptotically, for the classical composition of an integer, the moments and an expression for a continuous distribution F, the (discrete) distribution of the number of distinct part sizes being computable from F. We next analyze another composition: the Carlitz one, where two successive parts are dierent. We use tools such as analytical depoissonization, Mellin transforms, Markov chain potential theory, limiting hitting times, singularity analysis and perturbation analysis.
Analysis of an Asymmetric Leader Election Algorithm
 Electronic J. Combin
, 1996
"... We consider a leader election algorithm in which a set of distributed objects (people, computers, etc.) try to identify one object as their leader. The election process is randomized, that is, at every stage of the algorithm those objects that survived so far flip a biased coin, and those who rec ..."
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Cited by 32 (9 self)
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We consider a leader election algorithm in which a set of distributed objects (people, computers, etc.) try to identify one object as their leader. The election process is randomized, that is, at every stage of the algorithm those objects that survived so far flip a biased coin, and those who received, say a tail, survive for the next round. The process continues until only one objects remains. Our interest is in evaluating the limiting distribution and the first two moments of the number of rounds needed to select a leader. We establish precise asymptotics for the first two moments, and show that the asymptotic expression for the duration of the algorithm exhibits some periodic fluctuations and consequently no limiting distribution exists. These results are proved by analytical techniques of the precise analysis of algorithms such as: analytical poissonization and depoissonization, Mellin transform, and complex analysis.
2003b). On the contraction method with degenerate limit equation
"... A class of random recursive sequences (Yn) with slowly varying variances as arising for parameters of random trees or recursive algorithms leads after normalizations to degenerate limit equations of the form X L = X. For nondegenerate limit equations the contraction method is a main tool to establis ..."
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Cited by 24 (12 self)
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A class of random recursive sequences (Yn) with slowly varying variances as arising for parameters of random trees or recursive algorithms leads after normalizations to degenerate limit equations of the form X L = X. For nondegenerate limit equations the contraction method is a main tool to establish convergence of the scaled sequence to the “unique ” solution of the limit equation. In this paper we develop an extension of the contraction method which allows us to derive limit theorems for parameters of algorithms and data structures with degenerate limit equation. In particular, we establish some new tools and a general convergence scheme, which transfers information on mean and variance into a central limit law (with normal limit). We also obtain a convergence rate result. For the proof we use selfdecomposability properties of the limit normal distribution which allow us to mimic the recursive sequence by an accompanying sequence in normal variables.
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 ..."
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Cited by 17 (7 self)
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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
Analysis Of A Splitting Process Arising In Probabilistic Counting And Other Related Algorithms
, 1996
"... We present an analytical method of analyzing a class of "splitting algorithms" that include probabilistic counting, selecting the leader, estimating the number of questions necessary to identify distinct objects, searching algorithms based on digital tries, approximate counting, and so for ..."
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Cited by 11 (8 self)
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We present an analytical method of analyzing a class of "splitting algorithms" that include probabilistic counting, selecting the leader, estimating the number of questions necessary to identify distinct objects, searching algorithms based on digital tries, approximate counting, and so forth. In our discussion we concentrate on the analysis of a generalized probabilistic counting algorithm. Our technique belongs to the toolkit of the analytical analysis of algorithms, and it involves solutions of functional equations, analytical poissonization and depoissonization as well as Mellin transform. In particular, we deal with an instance of the functional equation g(z) = fia(z)g(z=2) + b(z) where a(z) and b(z) are given functions, and fi ! 1 is a constant. With respect to our generalized probabilistic counting algorithm, we obtain asymptotic expansions of the first two moments of an estimate of the cardinality of a set that is computed by the algorithm. We also derive the asymptotic distrib...
The oscillatory distribution of distances in random tries
 ANNALS OF APPLIED PROBABILITY
, 2005
"... We investigate ∆n, the distance between randomly selected pairs of nodes among n keys in a random trie, which is a kind of digital tree. Analytical techniques, such as the Mellin transform and an excursion between poissonization and depoissonization, capture small fluctuations in the mean and varian ..."
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Cited by 10 (3 self)
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We investigate ∆n, the distance between randomly selected pairs of nodes among n keys in a random trie, which is a kind of digital tree. Analytical techniques, such as the Mellin transform and an excursion between poissonization and depoissonization, capture small fluctuations in the mean and variance of these random distances. The mean increases logarithmically in the number of keys, but curiously enough the variance remains O(1), as n → ∞. It is demonstrated that the centered random variable ∆ ∗ n = ∆n − ⌊2log 2 n ⌋ does not have a limit distribution, but rather oscillates between two distributions.
Analysis in Distribution of Two Randomized Algorithms for Finding the Maximum in a Broadcast Communication Model
, 2002
"... The limit laws of three cost measures are derived of two algorithms for finding the maximum in a singlechannel broadcast communication model. Both algorithms use coin flips and comparisons. Besides the ubiquitous normal limit law, the Dickman distribution also appears in a natural way. The method o ..."
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Cited by 6 (3 self)
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The limit laws of three cost measures are derived of two algorithms for finding the maximum in a singlechannel broadcast communication model. Both algorithms use coin flips and comparisons. Besides the ubiquitous normal limit law, the Dickman distribution also appears in a natural way. The method of proof proceeds along the line via the method of moments and the "asymptotic transfers", which roughly bridges the asymptotics of the "conquering cost of the subproblems" and that of the total cost. Such a general approach has proved very fruitful for a number of problems in the analysis of recursive algorithms. 1