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26
AverageCase Analysis of Algorithms and Data Structures
, 1990
"... This report is a contributed chapter to the Handbook of Theoretical Computer Science (NorthHolland, 1990). Its aim is to describe the main mathematical methods and applications in the averagecase analysis of algorithms and data structures. It comprises two parts: First, we present basic combinato ..."
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Cited by 96 (8 self)
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This report is a contributed chapter to the Handbook of Theoretical Computer Science (NorthHolland, 1990). Its aim is to describe the main mathematical methods and applications in the averagecase analysis of algorithms and data structures. It comprises two parts: First, we present basic combinatorial enumerations based on symbolic methods and asymptotic methods with emphasis on complex analysis techniques (such as singularity analysis, saddle point, Mellin transforms). Next, we show how to apply these general methods to the analysis of sorting, searching, tree data structures, hashing, and dynamic algorithms. The emphasis is on algorithms for which exact "analytic models" can be derived.
Random Mapping Statistics
 IN ADVANCES IN CRYPTOLOGY
, 1990
"... Random mappings from a finite set into itself are either a heuristic or an exact model for a variety of applications in random number generation, computational number theory, cryptography, and the analysis of algorithms at large. This paper introduces a general framework in which the analysis of ..."
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Cited by 78 (6 self)
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Random mappings from a finite set into itself are either a heuristic or an exact model for a variety of applications in random number generation, computational number theory, cryptography, and the analysis of algorithms at large. This paper introduces a general framework in which the analysis of about twenty characteristic parameters of random mappings is carried out: These parameters are studied systematically through the use of generating functions and singularity analysis. In particular, an open problem of Knuth is solved, namely that of finding the expected diameter of a random mapping. The same approach is applicable to a larger class of discrete combinatorial models and possibilities of automated analysis using symbolic manipulation systems ("computer algebra") are also briefly discussed.
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 59 (11 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.
Analytic Combinatorics of Noncrossing Configurations
, 1997
"... This paper describes a systematic approach to the enumeration of "noncrossing" geometric configurations built on vertices of a convex ngon in the plane. It relies on generating functions, symbolic methods, singularity analysis, and singularity perturbation. A consequence is exact and asymptotic c ..."
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Cited by 55 (8 self)
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This paper describes a systematic approach to the enumeration of "noncrossing" geometric configurations built on vertices of a convex ngon in the plane. It relies on generating functions, symbolic methods, singularity analysis, and singularity perturbation. A consequence is exact and asymptotic counting results for trees, forests, graphs, connected graphs, dissections, and partitions. Limit laws of the Gaussian type are also established in this framework; they concern a variety of parameters like number of leaves in trees, number of components or edges in graphs, etc.
Graphical Major Indices
 II, Sém. Lothar. Combin
, 1995
"... A generalization of the classical statistics "maj" and "inv" (the major index and number of inversions) on words is introduced, parameterized by arbitrary graphs on the underlying alphabet. The question of characterizing those graphs that lead to equidistributed "inv" and "maj" is posed and ans ..."
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Cited by 16 (1 self)
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A generalization of the classical statistics "maj" and "inv" (the major index and number of inversions) on words is introduced, parameterized by arbitrary graphs on the underlying alphabet. The question of characterizing those graphs that lead to equidistributed "inv" and "maj" is posed and answered.
Binomial Identities  Combinatorial and Algorithmic Aspects
, 1994
"... The problem of proving a particular binomial identity is taken as an opportunity to discuss various aspects of this field and to discuss various proof techniques in an examplary way. In particular, the unifying role of the hypergeometric nature of binomial identities is underlined. This aspect is al ..."
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Cited by 15 (3 self)
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The problem of proving a particular binomial identity is taken as an opportunity to discuss various aspects of this field and to discuss various proof techniques in an examplary way. In particular, the unifying role of the hypergeometric nature of binomial identities is underlined. This aspect is also basic for combinatorial models and techniques, developed during the last decade, and for the recent algorithmic proof procedures. Much of mathematics comes from looking at very simple examples from a more general perspective. Hypergeometric functions are a good example of this. R. Askey 1 Introduction In this article I want to highlight some aspects of "binomial identites" or "combinatorial sums" in an exemplary way. Writing such an article was motivated by a question that I was asked in spring 1992, and by my subsequent investigations on it: Can you show that the binomial identity n X k=0 ` n k ' 2 ` n + k k ' 2 = n X k=0 ` n k '` n + k k ' k X j=0 ` k j ' 3 (1)...
LambdaUpsilonOmega  The 1989 Cookbook
, 1989
"... LambdaUpsilonOmega ( \Upsilon\Omega ) is a research tool designed to assist the average case analysis of some well defined classes of algorithms and data structures. This cookbook consists of an informal introduction to the system together with eighteen examples of programmes that are automatica ..."
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Cited by 15 (6 self)
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LambdaUpsilonOmega ( \Upsilon\Omega ) is a research tool designed to assist the average case analysis of some well defined classes of algorithms and data structures. This cookbook consists of an informal introduction to the system together with eighteen examples of programmes that are automatically analyzed. Amongst the applications treated here, we find: addition chains, quantitative concurrency analysis of simple systems, symbolic manipulation algorithms such as formal differentiation, simplification and rewriting systems, as well as combinatorial models including various tree and permutation statistics and functional graphs with applications to integer factorisation.
Asymptotics of Poisson approximation to random discrete distributions: an analytic approach
 Advances in Applied Probability
, 1998
"... this paper, we shall describe the asymptotic behaviors of several distances of Poisson approximation to a wide class of discrete distributions covering many examples from number theory, combinatorics and arithmetic semigroups. Our aim is to show that whenever (analytic) generating functions of the r ..."
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Cited by 13 (9 self)
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this paper, we shall describe the asymptotic behaviors of several distances of Poisson approximation to a wide class of discrete distributions covering many examples from number theory, combinatorics and arithmetic semigroups. Our aim is to show that whenever (analytic) generating functions of the random variables in question are available, complexanalytic methods can be used to derive precise asymptotic results for the five distances above. Actually, we shall consider the following generalized distances: let ff ? 0 be a fixed positive number, (X; Y ) = FM (X; Y ) = (X; Y ) = sup K (X; Y ) = sup M (X; Y ) = jP(X = j) \Gamma P(Y = j) Note that d TV = d M . Besides the case ff = 1 (and ff = 1=2 for d M ), only the case d TV was previously studied by Franken [39] for Poisson approximation to the sum of independent but not identically distributed Bernoulli random variables. We take these quantities as our measures of degree of nearness of Poisson approximation, some of which may be interpreted as certain norms in suitable space as many authors did (cf. [12, 22, 23, 74, 96]). For a large class of discrete distributions, we shall derive an asymptotic main term together with an error estimate for each of these distances. Our results are thus "approximation theorems" rather than "limit theorems". The common form of the underlying structure of these distributions suggests the study of an analytic scheme as we did previously for normal approximation and large deviations (cf. [53, 54]). Many concrete examples from probabilistic number theory and combinatorial structures will justify the study of this scheme. Our treatment being completely general, many extensions can be further pursued with essentially the same line of methods. We shall di...
Weighted derangements and the linearization coefficients of orthogonal Sheffer polynomials
 Proc. London Math. Soc
, 1992
"... The present paper is devoted to a systematic study of the combinatorial interpretations of the moments and the linearization coefficients of the orthogonal Sheffer polynomials, i.e., Hermite, Charlier, Laguerre, Meixner and MeixnerPollaczek polynomials. In particular, we show that Viennot's combina ..."
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Cited by 11 (2 self)
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The present paper is devoted to a systematic study of the combinatorial interpretations of the moments and the linearization coefficients of the orthogonal Sheffer polynomials, i.e., Hermite, Charlier, Laguerre, Meixner and MeixnerPollaczek polynomials. In particular, we show that Viennot's combinatorial interpretations of the moments can be derived directly from their classical analytical expressions and that the linearization coefficients of MeixnerPollaczek polynomials have an interpretation in the model of derangements analogous to those of Laguerre and Meixner polynomials. 1.
Enumerative Applications of a Decomposition for Graphs and Digraphs
 Discrete Math
, 1993
"... A simple decomposition for graphs yields generating functions for counting graphs by edges and connected components. A change of variables gives a new interpretation to the Tutte polynomial of the complete graph involving inversions of trees. The relation between the Tutte polynomial of the compl ..."
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Cited by 9 (2 self)
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A simple decomposition for graphs yields generating functions for counting graphs by edges and connected components. A change of variables gives a new interpretation to the Tutte polynomial of the complete graph involving inversions of trees. The relation between the Tutte polynomial of the complete graph and the inversion enumerator for trees is generalized to the Tutte polynomial of an arbitrary graph. When applied to digraphs, the decomposition yields formulas for counting digraphs and acyclic digraphs by edges and initially connected components.