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Quickselect and Dickman function
 Combinatorics, Probability and Computing
, 2000
"... We show that the limiting distribution of the number of comparisons used by Hoare's quickselect algorithm when given a random permutation of n elements for finding the mth smallest element, where m = o(n), is the Dickman function. The limiting distribution of the number of exchanges is also de ..."
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We show that the limiting distribution of the number of comparisons used by Hoare's quickselect algorithm when given a random permutation of n elements for finding the mth smallest element, where m = o(n), is the Dickman function. The limiting distribution of the number of exchanges is also derived. 1 Quickselect Quickselect is one of the simplest and e#cient algorithms in practice for finding specified order statistics in a given sequence. It was invented by Hoare [19] and uses the usual partitioning procedure of quicksort: choose first a partitioning key, say x; regroup the given sequence into two parts corresponding to elements whose values are less than and larger than x, respectively; then decide, according to the size of the smaller subgroup, which part to continue recursively or to stop if x is the desired order statistics; see Figure 1 for an illustration in terms of binary search trees. For more details, see Guibas [15] and Mahmoud [26]. This algorithm , although ine#cient in the worst case, has linear mean when given a sequence of n independent and identically distributed continuous random variables, or equivalently, when given a random permutation of n elements, where, here and throughout this paper, all n! permutations are equally likely. Let C n,m denote the number of comparisons used by quickselect for finding the mth smallest element in a random permutation, where the first partitioning stage uses n 1 comparisons. Knuth [23] was the first to show, by some di#erencing argument, that E(C n,m ) = 2 (n + 3 + (n + 1)H n (m + 2)Hm (n + 3 m)H n+1m ) , n, where Hm = 1#k#m k 1 . A more transparent asymptotic approximation is E(C n,m ) (#), (#) := 2 #), # Part of the work of this author was done while he was visiting School of C...
An iterated eigenvalue algorithm for approximating the roots of a univariate polynomial
 In Internat. Symposium on Symbolic and Algebraic Computation, ISSAC 2001
, 2001
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On Carlitz Compositions
 European J. Combin
, 1998
"... This paper deals with Carlitz compositions of natural numbers (adjacent parts have to be different). The following parameters are analyzed: Number of parts, number of equal adjacent parts in ordinary compositions, largest part, Carlitz compositions with zeros allowed (correcting an erroneous formula ..."
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Cited by 21 (6 self)
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This paper deals with Carlitz compositions of natural numbers (adjacent parts have to be different). The following parameters are analyzed: Number of parts, number of equal adjacent parts in ordinary compositions, largest part, Carlitz compositions with zeros allowed (correcting an erroneous formula from Carlitz). It is also briefly demonstrated that socalled 1compositions of a natural number can be treated in a similar style.  October 14, 1997  1.
The Average Case Analysis of Algorithms: Multivariate Asymptotics and Limit Distributions
, 1997
"... 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 develops a general approach to the distributional analysis of parameters of elementary combinatorial structures li ..."
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Cited by 16 (1 self)
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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 develops a general approach to the distributional analysis of parameters of elementary combinatorial structures like strings, trees, graphs, permutations, and so on. The methods are essentially analytic and relie on multivariate generating functions, singularity analysis, and continuity theorems. The limit laws that are derived mostly belong to the Gaussian, Poisson, or geometric type.
Polynomial root finding using iterated eigenvalue computation
 in « Proc. ISSAC », NewYork, ACM
, 2001
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The Complete Analysis of a Polynomial Factorization Algorithm Over Finite Fields
, 2001
"... This paper derives basic probabilistic properties of random polynomials over finite fields that are of interest in the study of polynomial factorization algorithms. We show that the main characteristics of random polynomial can be treated systematically by methods of "analytic combinatorics&quo ..."
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Cited by 16 (3 self)
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This paper derives basic probabilistic properties of random polynomials over finite fields that are of interest in the study of polynomial factorization algorithms. We show that the main characteristics of random polynomial can be treated systematically by methods of "analytic combinatorics" based on the combined use of generating functions and of singularity analysis. Our object of study is the classical factorization chain which is described in Fig. 1 and which, despite its simplicity, does not appear to have been totally analysed so far. In this paper, we provide a complete averagecase analysis.
Partial Fraction Decomposition in C(z) and Simultaneous Newton Iteration for Factorization in C[z]
, 1998
"... The subject of this paper is fast numerical algorithms for factoring univariate polynomials with complex coefficients and for computing partial fraction decompositions (PFDs) of rational functions in C(z). Numerically stable and computationally feasible versions of PFD are specified first for the sp ..."
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Cited by 5 (0 self)
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The subject of this paper is fast numerical algorithms for factoring univariate polynomials with complex coefficients and for computing partial fraction decompositions (PFDs) of rational functions in C(z). Numerically stable and computationally feasible versions of PFD are specified first for the special case of rational functions with all singularities in the unit disk (the ``bounded case'') and then for rational functions with arbitrarily distributed singularities. Two major algorithms for computing PFDs are presented: The first one is an extension of the ``splitting circle method' ' by A. Schonhage (``The Fundamental Theorem of Algebra in Terms of Computational Complexity,' ' Technical Report, Univ. Tubingen, 1982) for factoring polynomials in C[z] to an algorithm for PFD. The second algorithm is a Newton iteration for simultaneously improving the accuracy of all factors in an approximate factorization of a polynomial resp. all partial fractions of an approximate PFD of a rational function. Algorithmically useful starting value conditions for the Newton algorithm are provided. Three subalgorithms are of independent interest. They compute the product of a sequence of polynomials, the sum
A Hybrid of Darboux’s Method and Singularity Analysis in Combinatorial Asymptotics
 N O 1, JUNE 2006, R103, HTTP://WWW.COMBINATORICS.ORG/VOLUME_13/PDF/V13I1R103.PDF. ALGO 11
, 2006
"... A “hybrid method”, dedicated to asymptotic coefficient extraction in combinatorial generating functions, is presented, which combines Darboux’s method and singularity analysis theory. This hybrid method applies to functions that remain of moderate growth near the unit circle and satisfy suitable s ..."
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Cited by 5 (1 self)
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A “hybrid method”, dedicated to asymptotic coefficient extraction in combinatorial generating functions, is presented, which combines Darboux’s method and singularity analysis theory. This hybrid method applies to functions that remain of moderate growth near the unit circle and satisfy suitable smoothness assumptions—this, even in the case when the unit circle is a natural boundary. A prime application is to coefficients of several types of infinite product generating functions, for which full asymptotic expansions (involving periodic fluctuations at higher orders) can be derived. Examples relative to permutations, trees, and polynomials over finite fields are treated in this way.
Uniform asymptotics of some Abel sums arising in coding theory
"... We derive uniform asymptotic expressions of some Abel sums appearing in some problems in coding theory and indicate the usefulness of these sums in other fields, like empirical processes, machine maintenance, analysis of algorithms, probabilistic number theory, queuing models, etc. Key words: Abel s ..."
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We derive uniform asymptotic expressions of some Abel sums appearing in some problems in coding theory and indicate the usefulness of these sums in other fields, like empirical processes, machine maintenance, analysis of algorithms, probabilistic number theory, queuing models, etc. Key words: Abel sums, coding theory, Mellin transforms, Wfunction, uniform asymptotics. 1
Journal of Algorihms, to appear (accepted for publication January 2001).
"... c Academic Press, 2001. The complete analysis of a polynomial factorization algorithm over finite fields ..."
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c Academic Press, 2001. The complete analysis of a polynomial factorization algorithm over finite fields