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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 37 (17 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
Information Propagation Speed in Mobile and Delay Tolerant Networks
, 2009
"... The goal of this paper is to increase our understanding of the fundamental performance limits of mobile and Delay Tolerant Networks (DTNs), where endtoend multihop paths may not exist and communication routes may only be available through time and mobility. We use analytical tools to derive gene ..."
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Cited by 31 (10 self)
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The goal of this paper is to increase our understanding of the fundamental performance limits of mobile and Delay Tolerant Networks (DTNs), where endtoend multihop paths may not exist and communication routes may only be available through time and mobility. We use analytical tools to derive generic theoretical upper bounds for the information propagation speed in large scale mobile and intermittently connected networks. In other words, we upperbound the optimal performance, in terms of delay, that can be achieved using any routing algorithm. We then show how our analysis can be applied to specific mobility and graph models to obtain specific analytical estimates. In particular, when nodes move at speed v and their density ν is small (the network is sparse and surely disconnected), we prove that the information propagation speed is upper bounded by (1 + O(ν 2))v in the random waypoint model, while it is upper bounded by O ( √ νvv) for other mobility models (random walk, Brownian motion). We also present simulations that confirm the validity of the bounds in these scenarios.
A functional limit theorem for the profile of search trees
 Annals of Applied Probability
, 2008
"... We study the profile Xn,k of random search trees including binary search trees and mary search trees. Our main result is a functional limit theorem of the normalized profile Xn,k/EXn,k for k =⌊α log n ⌋ in a certain range of α. A central feature of the proof is the use of the contraction method to ..."
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Cited by 26 (14 self)
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We study the profile Xn,k of random search trees including binary search trees and mary search trees. Our main result is a functional limit theorem of the normalized profile Xn,k/EXn,k for k =⌊α log n ⌋ in a certain range of α. A central feature of the proof is the use of the contraction method to prove convergence in distribution of certain random analytic functions in a complex domain. This is based on a general theorem concerning the contraction method for random variables in an infinitedimensional Hilbert space. As part of the proof, we show that the Zolotarev metric is complete for a Hilbert space. 1. Introduction. Search
Tessellations of random maps of arbitrary genus
, 2009
"... We investigate Voronoilike tessellations of bipartite quadrangulations on surfaces of arbitrary genus, by using a natural generalization of a bijection of Marcus and Schaeffer allowing to encode such structures into labeled maps with a fixed number of faces. We investigate the scaling limits of the ..."
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Cited by 25 (3 self)
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We investigate Voronoilike tessellations of bipartite quadrangulations on surfaces of arbitrary genus, by using a natural generalization of a bijection of Marcus and Schaeffer allowing to encode such structures into labeled maps with a fixed number of faces. We investigate the scaling limits of the latter. Applications include asymptotic enumeration results for quadrangulations, and typical metric properties of randomly sampled quadrangulations. In particular, we show that scaling limits of these random quadrangulations are such that almost
AUTOMATIC CLASSIFICATION OF RESTRICTED LATTICE WALKS
"... Abstract. We propose an experimental mathematics approach leading to the computerdriven discovery of various structural properties of general counting functions coming from enumeration of walks. 1. ..."
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Cited by 24 (6 self)
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Abstract. We propose an experimental mathematics approach leading to the computerdriven discovery of various structural properties of general counting functions coming from enumeration of walks. 1.
Euclidean algorithms are Gaussian
, 2003
"... Abstract. We prove a Central Limit Theorem for a general class of costparameters associated to the three standard Euclidean algorithms, with optimal speed of convergence, and error terms for the mean and variance. For the most basic parameter of the algorithms, the number of steps, we go further an ..."
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Cited by 23 (10 self)
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Abstract. We prove a Central Limit Theorem for a general class of costparameters associated to the three standard Euclidean algorithms, with optimal speed of convergence, and error terms for the mean and variance. For the most basic parameter of the algorithms, the number of steps, we go further and prove a Local Limit Theorem (LLT), with speed of convergence O((log N) −1/4+ǫ). This extends and improves the LLT obtained by Hensley [27] in the case of the standard Euclidean algorithm. We use a “dynamical analysis ” methodology, viewing an algorithm as a dynamical system (restricted to rational inputs), and combining tools imported from dynamics, such as the crucial transfer operators, with various other techniques: Dirichlet series, Perron’s formula, quasipowers theorems, the saddle point method. Dynamical analysis had previously been used to perform averagecase analysis of algorithms. For the present (dynamical) analysis in distribution, we require precise estimates on the transfer operators, when a parameter varies along vertical lines in the complex plane. Such estimates build on results obtained only recently by Dolgopyat in the context of continuoustime dynamics [20]. 1.
Graph classes with given 3connected components: asymptotic enumeration and random graphs
, 2009
"... Consider a family T of 3connected graphs of moderate growth, and let G be the class of graphs whose 3connected components are graphs in T. We present a general framework for analyzing such graphs classes based on singularity analysis of generating functions, which generalizes previously studied c ..."
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Cited by 20 (7 self)
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Consider a family T of 3connected graphs of moderate growth, and let G be the class of graphs whose 3connected components are graphs in T. We present a general framework for analyzing such graphs classes based on singularity analysis of generating functions, which generalizes previously studied cases such as planar graphs and seriesparallel graphs. We provide a general result for the asymptotic number of graphs in G, based on the singularities of the exponential generating function associated to T. We derive limit laws, which are either normal or Poisson, for several basic parameters, including the number of edges, number of blocks and number of components. For the size of the largest block we find a fundamental dichotomy: classes similar to planar graphs have almost surely a unique block of linear size, while classes similar to seriesparallel graphs have only sublinear blocks. This dichotomy also applies to the size of the largest 3connected component. For some classes under study both regimes occur, because of a critical phenomenon as the edge density in the class varies.
The concrete Tetrahedron
, 2011
"... We give an overview over computer algebra algorithms for dealing with symbolic sums, recurrence equations, generating functions, and asymptotic estimates, and we will illustrate how to apply these algorithms to problems arising in discrete mathematics. ..."
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Cited by 19 (4 self)
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We give an overview over computer algebra algorithms for dealing with symbolic sums, recurrence equations, generating functions, and asymptotic estimates, and we will illustrate how to apply these algorithms to problems arising in discrete mathematics.
Boltzmann oracle for combinatorial systems
 In Algorithms, Trees, Combinatorics and Probabilities
, 2008
"... Boltzmann random generation applies to welldefined systems of recursive combinatorial equations. It relies on oracles giving values of the enumeration generating series inside their disk of convergence. We show that the combinatorial systems translate into numerical iteration schemes that provide s ..."
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Cited by 16 (6 self)
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Boltzmann random generation applies to welldefined systems of recursive combinatorial equations. It relies on oracles giving values of the enumeration generating series inside their disk of convergence. We show that the combinatorial systems translate into numerical iteration schemes that provide such oracles. In particular, we give a fast oracle based on Newton iteration.