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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 52 (14 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.
Some exactly solvable models of urn process theory
, 2006
"... We establish a fundamental isomorphism between discretetime balanced urn processes and certain ordinary differential systems, which are nonlinear, autonomous, and of a simple monomial form. As a consequence, all balanced urn processes with balls of two colours are proved to be analytically solvable ..."
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Cited by 48 (1 self)
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We establish a fundamental isomorphism between discretetime balanced urn processes and certain ordinary differential systems, which are nonlinear, autonomous, and of a simple monomial form. As a consequence, all balanced urn processes with balls of two colours are proved to be analytically solvable in finite terms. The corresponding generating functions are expressed in terms of certain Abelian integrals over curves of the Fermat type (which are also hypergeometric functions), together with their inverses. A consequence is the unification of the analyses of many classical models, including those related to the coupon collector’s problem, particle transfer (the Ehrenfest model), Friedman’s “adverse campaign ” and Pólya’s contagion model, as well as the OK Corral model (a basic case of Lanchester’s theory of conflicts). In each case, it is possible to quantify very precisely the probable composition of the urn at any discrete instant. We study here in detail “semisacrificial ” urns, for which the following are obtained: a Gaussian limiting distribution with speed of convergence estimates as well as a characterization of the large and extreme large deviation regimes. We also work out explicitly the case of 2dimensional triangular models, where local limit laws of the stable type are obtained. A few models of dimension three or greater, e.g., “autistic ” (generalized Pólya), cyclic chambers
Twenty combinatorial examples of asymptotics derived from multivariate generating functions
, 2008
"... 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 e ..."
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Cited by 44 (16 self)
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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
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 43 (5 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
COMBINATORICS OF GEOMETRICALLY DISTRIBUTED RANDOM VARIABLES: VALUE AND POSITION OF LARGE LEFT–TO–RIGHT MAXIMA
, 2008
"... For words of length n, generated by independent geometric random variables, we consider the average value and the average position of the rth left–to–right maximum counted from the right, for fixed r and n → ∞. This complements previous research [5] where the analogous questions were considered fo ..."
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Cited by 42 (13 self)
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For words of length n, generated by independent geometric random variables, we consider the average value and the average position of the rth left–to–right maximum counted from the right, for fixed r and n → ∞. This complements previous research [5] where the analogous questions were considered for the rth left–to–right maximum counted from the left.
A Survey of Alternating Permutations
, 2009
"... Abstract. A permutation a1a2 · · · an of 1, 2,..., n is alternating if a1> a2 < a3> a4 < · · ·. We survey some aspects of the theory of alternating permutations, beginning with the famous result of André that if En is the number of alternating permutations of 1, 2,..., n, then P xn n ..."
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Cited by 29 (1 self)
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Abstract. A permutation a1a2 · · · an of 1, 2,..., n is alternating if a1> a2 < a3> a4 < · · ·. We survey some aspects of the theory of alternating permutations, beginning with the famous result of André that if En is the number of alternating permutations of 1, 2,..., n, then P xn n≥0 En = sec x + tan x. n! Topics include refinements and qanalogues of En, various occurrences of En in mathematics, longest alternating subsequences of permutations, umbral enumeration of special classes of alternating permutations, and the connection between alternating permutations and the cdindex of the symmetric group. Dedicated to Reza Khosrovshahi on the occasion of his 70th birthday 1. Basic enumerative properties. Let Sn denote the symmetric group of all permutations of [n]: = {1, 2,..., n}. A permutation w = a1a2 · · · an ∈ Sn is called alternating if a1> a2 < a3> a4 < · · ·. In other words, ai < ai+1 for i even, and ai> ai+1 for i odd. Similarly w is reverse alternating if a1 < a2> a3 < a4> · · ·. (Some authors reverse these definitions.) Let En denote the number of alternating permutations in Sn. (Set
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 28 (12 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.
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 27 (9 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.
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 (11 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.