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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 derived ..."
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Cited by 24 (1 self)
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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...
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 22 (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.
An Iterated Eigenvalue Algorithm for Approximating Roots of Univariate Polynomials
 J. Symbolic Comput
, 2001
"... We present an iterative algorithm that approximates all roots of a univariate polynomial. The iteration uses floatingpoint eigenvalue computation of a generalized companion matrix. With some assumptions, we show that the algorithm approximates the roots within about log ae=ffl (P ) iterations, w ..."
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Cited by 21 (0 self)
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We present an iterative algorithm that approximates all roots of a univariate polynomial. The iteration uses floatingpoint eigenvalue computation of a generalized companion matrix. With some assumptions, we show that the algorithm approximates the roots within about log ae=ffl (P ) iterations, where ffl is the relative error of floatingpoint arithmetic, ae is the relative separation of the roots, and (P ) is the condition number of the polynomial. Each iteration requires an n\Thetan floatingpoint eigenvalue computation, n the polynomial degree, and evaluation of the polynomial to floatingpoint accuracy at up to n points. We describe a careful implementation of the algorithm, including many techniques that contribute to the practical efficiency of the algorithm. On some hard examples of illconditioned polynomials, e.g. highdegree Wilkinson polynomials, the implementation is an order of magnitude faster than the BiniFiorentino implementation mpsolve. 1
Polynomial Root Finding Using Iterated Eigenvalue Computation
 in « Proc. ISSAC », NewYork, ACM
, 2001
"... We present a novel iterative algorithm that approximates all roots of a univariate polynomial. The iteration uses floatingpoint eigenvalue computation of a generalized companion matrix. With some assumptions, we show that the algorithm approximates the roots to floatingpoint accuracy within about ..."
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Cited by 15 (1 self)
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We present a novel iterative algorithm that approximates all roots of a univariate polynomial. The iteration uses floatingpoint eigenvalue computation of a generalized companion matrix. With some assumptions, we show that the algorithm approximates the roots to floatingpoint accuracy within about log ae=ffl (P ) iterations, where ffl is the relative error of floatingpoint arithmetic, ae is the relative separation of the roots, and (P ) is the condition number of the polynomial. Each iteration requires an n\Thetan floatingpoint eigenvalue computation, n the polynomial degree, and evaluation of the polynomial to floatingpoint accuracy at n points. On some hard examples of illconditioned polynomials, e.g. highdegree Wilkinson polynomials, a careful implementation of the algorithm is an order of magnitude faster than the best alternative. 1 Introduction The algorithmic problem of approximating the roots of a univariate polynomial, presented by its coefficients, is classic in numeri...
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 like strings ..."
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Cited by 14 (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.
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" based o ..."
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Cited by 14 (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.
A Hybrid of Darboux’s Method and Singularity Analysis in Combinatorial Asymptotics, in "The Electronic
 n o 1, June 2006, R103, http://www.combinatorics.org/Volume_13/PDF/v13i1r103.pdf. Algo 11
"... Abstract. 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 su ..."
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Cited by 5 (1 self)
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Abstract. 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
Random Group Automata
, 2000
"... A group automaton is a complete deterministic automaton such that each letter of the alphabet acts on the set of states as a permutation [1, 5]. The aim is to describe an algorithm for the random generation of a minimal group automaton with n states. The treatment is largely based on properties of r ..."
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A group automaton is a complete deterministic automaton such that each letter of the alphabet acts on the set of states as a permutation [1, 5]. The aim is to describe an algorithm for the random generation of a minimal group automaton with n states. The treatment is largely based on properties of random permutations and random automata. 1. Properties A group automaton is a complete deterministic automaton such that each letter of the alphabet acts on the set of states as a permutation [1, 5]. We consider a group automaton A, with states 1, 2, : : : , n. The state 1 is the initial state; the set of final states is denoted by F , the alphabet by a, b, : : : , and the transitions by q 2 = ffi(q 1 ; a) or equivalently (q 1 ; a; q 2 ). 1 2 3 4 start a a b a b b a,b Figure 1. a group automaton Let us recall that two states q 1 and q 2 of an automaton are equivalent, notationally q 1 q 2 , if for every word u, the state ffi(q 1 ; u) belongs to F if and only if ffi(q 2 ; u) belongs to F . ...
A General Approach to Isolating Roots of a Bitstream Polynomial
"... Abstract. We describe a new approach to isolate the roots (either real or complex) of a squarefree polynomial F with real coefficients. It is assumed that each coefficient of F can be approximated to any specified error bound and refer to such coefficients as bitstream coefficients. The presented m ..."
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Abstract. We describe a new approach to isolate the roots (either real or complex) of a squarefree polynomial F with real coefficients. It is assumed that each coefficient of F can be approximated to any specified error bound and refer to such coefficients as bitstream coefficients. The presented method is exact, complete and deterministic. Compared to previous approaches [10, 12, 23] we improve in two aspects. Firstly, our approach can be combined with any existing subdivision method for isolating the roots of a polynomial with rational coefficients. Secondly, the approximation demand on the coefficients and the bit complexity of our approach is considerably smaller. In particular, we can replace the worstcase quantity σ(F) by the averagecase quantity ∏ n i=1 n √ σi, where σi denotes the minimal distance of the i−th root ξi of F to any other root of F, σ(F): = mini σi, and n = degF. For polynomials with integer coefficients, our method matches the best bounds known for existing practical algorithms that perform exact operations on the input coefficients.