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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.
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 i ..."
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Cited by 48 (7 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.
Bipartite Random Graphs and Cuckoo Hashing
 In Proceedings of the Fourth Colloquium on Mathematics and Computer Science
, 2006
"... The aim of this paper is to extend the analysis of Cuckoo Hashing of Devroye and Morin in 2003. In particular we make several asymptotic results much more precise. We show, that the probability that the construction of a hash table succeeds, is asymptotically 1 − c(ε)/m + O(1/m 2) for some explicit ..."
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Cited by 6 (1 self)
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The aim of this paper is to extend the analysis of Cuckoo Hashing of Devroye and Morin in 2003. In particular we make several asymptotic results much more precise. We show, that the probability that the construction of a hash table succeeds, is asymptotically 1 − c(ε)/m + O(1/m 2) for some explicit c(ε), where m denotes the size of each of the two tables, n = m(1 − ε) is the number of keys and ε ∈ (0, 1). The analysis rests on a generating function approach to the so called Cuckoo Graph, a random bipartite graph. We apply a double saddle point method to obtain asymptotic results covering tree sizes, the number of cycles and the probability that no complex component occurs.
NODES OF LARGE DEGREE IN RANDOM TREES AND FORESTS
"... Abstract. We study the asymptotic behaviour of the number Nk,n of nodes of given degree k in unlabeled random trees, when the tree size n and the node degree k both tend to infinity. It is shown that Nk,n is asymptotically normal if ENk,n → ∞ and asymptotically Poisson distributed if ENk,n → C> ..."
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Cited by 3 (3 self)
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Abstract. We study the asymptotic behaviour of the number Nk,n of nodes of given degree k in unlabeled random trees, when the tree size n and the node degree k both tend to infinity. It is shown that Nk,n is asymptotically normal if ENk,n → ∞ and asymptotically Poisson distributed if ENk,n → C> 0. If ENk,n → 0, then the distribution degenerates. The same holds for rooted, unlabeled trees and forests. 1.
A precise analysis of cuckoo hashing
, 2009
"... Cuckoo hashing was introduced by Pagh and Rodler in 2001. Its main feature is that it provides constant worst case search time. The aim of this paper is to present a precise average case analysis of Cuckoo hashing. In particular, we determine the probability that Cuckoo hashing produces no conflicts ..."
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Cited by 1 (0 self)
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Cuckoo hashing was introduced by Pagh and Rodler in 2001. Its main feature is that it provides constant worst case search time. The aim of this paper is to present a precise average case analysis of Cuckoo hashing. In particular, we determine the probability that Cuckoo hashing produces no conflicts and and give an upper bound for the construction time, that is linear in the size of the table. The analysis rests on a generating function approach to the so called Cuckoo Graph, a random bipartite graph and an application of a double saddle point method to obtain asymptotic expansions. Furthermore, we provide some results concerning the structure of these kind of random graphs. Our results extend an analysis of Devroye and Morin in 2003. Additionally, we provide numerical results confirming the mathematical analysis.
The Number Of Descendants In Simply Generated Random Trees
, 1999
"... We derive asymptotic results on the distribution of the number of descendants in simply generated trees. Our method is based on a generating function approach and complex contour integration. 1. ..."
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Cited by 1 (0 self)
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We derive asymptotic results on the distribution of the number of descendants in simply generated trees. Our method is based on a generating function approach and complex contour integration. 1.
ASYMPTOTIC EVOLUTION OF ACYCLIC RANDOM MAPPINGS
, 2007
"... Abstract. An acyclic mapping from an n element set into itself is a mapping ϕ such that if ϕ k (x) = x for some k and x, then ϕ(x) = x. Equivalently, ϕ ℓ = ϕ ℓ+1 =... for ℓ sufficiently large. We investigate the behavior as n → ∞ of a Markov chain on the collection of such mappings. At each step ..."
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Abstract. An acyclic mapping from an n element set into itself is a mapping ϕ such that if ϕ k (x) = x for some k and x, then ϕ(x) = x. Equivalently, ϕ ℓ = ϕ ℓ+1 =... for ℓ sufficiently large. We investigate the behavior as n → ∞ of a Markov chain on the collection of such mappings. At each step of the chain, a point in the n element set is chosen uniformly at random and the current mapping is modified by replacing the current image of that point by a new one chosen independently and uniformly at random, conditional on the resulting mapping being again acyclic. We can represent an acyclic mapping as a directed graph (such a graph will be a collection of rooted trees) and think of these directed graphs as metric spaces with some extra structure. Heuristic calculations indicate that the metric space valued process associated with the Markov chain should, after an appropriate time and “space ” rescaling, converge as n → ∞ to a real tree (Rtree) valued Markov process that is reversible with respect to a measure induced naturally by the standard reflected Brownian bridge. The limit process, which we construct using Dirichlet form methods, is a Hunt process with respect to a suitable GromovHausdorfflike metric. This process is similar to one that appears in earlier work by Evans and Winter as the limit of chains involving the subtree prune and regraft tree (SPR) rearrangements from phylogenetics. 1.
ON MOMENT SEQUENCES AND MIXED POISSON DISTRIBUTIONS
, 1403
"... ABSTRACT. In this article we survey properties of mixed Poisson distributions and probabilistic aspects of the Stirling transform: given a nonnegative random variable X with moment sequence (µs)s∈N we determine a discrete random variable Y, whose moment sequence is given by the Stirling transform o ..."
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ABSTRACT. In this article we survey properties of mixed Poisson distributions and probabilistic aspects of the Stirling transform: given a nonnegative random variable X with moment sequence (µs)s∈N we determine a discrete random variable Y, whose moment sequence is given by the Stirling transform of the sequence (µs)s∈N, and identify the distribution as a mixed Poisson distribution. We discuss properties of this family of distributions and present a simple limit theorem based on expansions of factorial moments. Moreover, we present several examples of mixed Poisson distributions in the analysis of random discrete structures, unifying and extending earlier results. We also add several entirely new results: we analyze triangular urn models, where the initial configuration or the dimension of the urn is not fixed, but may depend on the discrete time n. We discuss the branching structure of planeoriented recursive trees and its relation to table sizes in the Chinese restaurant process. Furthermore, we discuss root isolation procedures in Cayley trees, a parameter in parking functions, zero contacts in lattice paths consisting of bridges, and a parameter related to cyclic points and trees in graphs of random mappings, all leading to mixed PoissonRayleigh distributions. Finally, we indicate how mixed Poisson distributions naturally arise in the critical composition scheme of Analytic Combinatorics. 1.