Results 1  10
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85
Mellin transforms and asymptotics: Finite differences and Rice's integrals
, 1995
"... High order differences of simple number sequences may be analysed asymptotically by means of integral representations, residue calculus, and contour integration. This technique, akin to Mellin transform asymptotics, is put in perspective and illustrated by means of several examples related to combin ..."
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Cited by 81 (8 self)
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High order differences of simple number sequences may be analysed asymptotically by means of integral representations, residue calculus, and contour integration. This technique, akin to Mellin transform asymptotics, is put in perspective and illustrated by means of several examples related to combinatorics and the analysis of algorithms like digital tries, digital search trees, quadtrees, and distributed leader election.
Generating Functions for Generating Trees
 PROCEEDINGS OF 11TH FORMAL POWER SERIES AND ALGEBRAIC COMBINATORICS
, 1999
"... Certain families of combinatorial objects admit recursive descriptions in terms of generating trees: each node of the tree corresponds to an object, and the branch leading to the node encodes the choices made in the construction of the object. Generating trees lead to a fast computation of enumerati ..."
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Cited by 68 (18 self)
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Certain families of combinatorial objects admit recursive descriptions in terms of generating trees: each node of the tree corresponds to an object, and the branch leading to the node encodes the choices made in the construction of the object. Generating trees lead to a fast computation of enumeration sequences (sometimes, to explicit formulae as well) and provide efficient random generation algorithms. We investigate the links between the structural properties of the rewriting rules defining such trees and the rationality, algebraicity, or transcendence of the corresponding generating function.
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 ..."
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Cited by 67 (2 self)
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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.
Basic Analytic Combinatorics of Directed Lattice Paths
 Theoretical Computer Science
, 2001
"... This paper develops a unified enumerative and asymptotic theory of directed 2dimensional lattice paths in halfplanes and quarterplanes. The lattice paths are speci ed by a finite set of rules that are both time and space homogeneous, and have a privileged direction of increase. (They are then ess ..."
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Cited by 58 (10 self)
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This paper develops a unified enumerative and asymptotic theory of directed 2dimensional lattice paths in halfplanes and quarterplanes. The lattice paths are speci ed by a finite set of rules that are both time and space homogeneous, and have a privileged direction of increase. (They are then essentially 1dimensional objects.) The theory relies on a specific "kernel method" that provides an important decomposition of the algebraic generating functions involved, as well as on a generic study of singularities of an associated algebraic curve. Consequences are precise computable estimates for the number of lattice paths of a given length under various constraints (bridges, excursions, meanders) as well as a characterization of the limit laws associated to several basic parameters of paths.
Random maps, coalescing saddles, singularity analysis, and Airy phenomena
 Random Structures & Algorithms
, 2001
"... A considerable number of asymptotic distributions arising in random combinatorics and analysis of algorithms are of the exponentialquadratic type, that is, Gaussian. We exhibit a class of "universal" phenomena that are of the exponentialcubic type, corresponding to distributions that involve the ..."
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Cited by 45 (6 self)
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A considerable number of asymptotic distributions arising in random combinatorics and analysis of algorithms are of the exponentialquadratic type, that is, Gaussian. We exhibit a class of "universal" phenomena that are of the exponentialcubic type, corresponding to distributions that involve the Airy function. In this paper, such Airy phenomena are related to the coalescence of saddle points and the confluence of singularities of generating functions. For about a dozen types of random planar maps, a common Airy distribution (equivalently, a stable law of exponent 3/2) describes the sizes of cores and of largest (multi)connected components. Consequences include the analysis and fine optimization of random generation algorithms for multiply connected planar graphs. Based on an extension of the singularity analysis framework suggested by the Airy case, the paper also presents a general classification of compositional schemas in analytic combinatorics.
Mathematics by Experiment: Plausible Reasoning in the 21st Century, extended second edition, A K
 2008. EXPERIMENTATION AND COMPUTATION 19
, 2008
"... If mathematics describes an objective world just like physics, there is no reason why inductive methods should not be applied in mathematics just the same as in physics. (Kurt Gödel, 1951) Paper Revised 09–09–04 This paper is an extended version of a presentation made at ICME10, related work is elab ..."
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Cited by 38 (15 self)
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If mathematics describes an objective world just like physics, there is no reason why inductive methods should not be applied in mathematics just the same as in physics. (Kurt Gödel, 1951) Paper Revised 09–09–04 This paper is an extended version of a presentation made at ICME10, related work is elaborated in references [1–7]. 1 I shall generally explore experimental and heuristic mathematics and give (mostly) accessible, primarily visual and symbolic, examples. The emergence of powerful mathematical computing environments like Maple and Matlab, the growing
Faster subtree isomorphism
 Journal of Algorithms
, 1999
"... We study the subtree isomorphism problem: Given trees H and G, find a subtree of G which is isomorphic to H or decide that there is no such subtree. We give an O((k 1.5 / log k)n)time algorithm for this problem, where k and n are the number of vertices in H and G, respectively. This improves over t ..."
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Cited by 32 (2 self)
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We study the subtree isomorphism problem: Given trees H and G, find a subtree of G which is isomorphic to H or decide that there is no such subtree. We give an O((k 1.5 / log k)n)time algorithm for this problem, where k and n are the number of vertices in H and G, respectively. This improves over the O(k 1.5 n) algorithms of Chung and Matula. We also give a randomized (Las Vegas) O(k 1.376 n)time algorithm for the decision problem. 1
Twenty combinatorial examples of asymptotics derived from multivariate generating functions
"... Abstract. Let {ar: r ∈ Nd} be a ddimensional array of numbers for which the generating function F (z): = ∑ r arzr is meromorphic in a neighborhood of the origin. For example, F may be a rational multivariate generating function. We discuss recent results that allow the effective computation of asym ..."
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Cited by 32 (14 self)
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Abstract. Let {ar: r ∈ Nd} be a ddimensional array of numbers for which the generating function F (z): = ∑ r arzr is meromorphic in a neighborhood of the origin. For example, F may be a rational multivariate generating function. We discuss recent results that allow the effective computation of asymptotic expansions for the coefficients of F. Our purpose is to illustrate the use of these techniques on a variety of problems of combinatorial interest. The survey begins by summarizing previous work on the asymptotics of univariate and multivariate generating functions. Next we describe the Morsetheoretic underpinnings of some new asymptotic techniques. We then quote and summarize these results in such a way that only elementary analyses are needed to check hypotheses and carry out computations. The remainder of the survey focuses on combinatorial applications, such as enumeration of words with forbidden substrings, edges and cycles in graphs, polyominoes, and descents in permutations. After the individual examples, we discuss three broad classes of examples, namely, functions derived via the transfer matrix method, those derived via the kernel method, and those derived via the method of Lagrange inversion. These methods have
Singularity Analysis, Hadamard Products, and Tree Recurrences
, 2003
"... We present a toolbox for extracting asymptotic information on the coecients of combinatorial generating functions. This toolbox notably includes a treatment of the eect of Hadamard products on singularities in the context of the complex Tauberian technique known as singularity analysis. As a consequ ..."
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Cited by 28 (9 self)
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We present a toolbox for extracting asymptotic information on the coecients of combinatorial generating functions. This toolbox notably includes a treatment of the eect of Hadamard products on singularities in the context of the complex Tauberian technique known as singularity analysis. As a consequence, it becomes possible to unify the analysis of a number of divideandconquer algorithms, or equivalently random tree models, including several classical methods for sorting, searching, and dynamically managing equivalence relations.
Some Probabilistic Aspects Of Set Partitions
 American Mathematical Monthly
, 1996
"... this paper, section (1.2) offers an elementary combinatorial proof of Dobinski's formula which seems simpler than other proofs in the literature (Rota [35], Berge [5], p. 44, Comtet [9], p. 211). This argument involves identities whose probabilistic interpretations are brought out later in the paper ..."
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Cited by 22 (2 self)
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this paper, section (1.2) offers an elementary combinatorial proof of Dobinski's formula which seems simpler than other proofs in the literature (Rota [35], Berge [5], p. 44, Comtet [9], p. 211). This argument involves identities whose probabilistic interpretations are brought out later in the paper. 1.1 Notation