Results 1  10
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21
The Wiener Index Of Simply Generated Random Trees
 Random Struct. Alg
, 2003
"... Asymptotics are obtained for the mean, variance and higher moments as well as for the distribution of the Wiener index of a random tree from a simply generated family (or, equivalently, a critical Galton Watson tree). We also establish a joint asymptotic distribution of the Wiener index and the in ..."
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Cited by 32 (12 self)
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Asymptotics are obtained for the mean, variance and higher moments as well as for the distribution of the Wiener index of a random tree from a simply generated family (or, equivalently, a critical Galton Watson tree). We also establish a joint asymptotic distribution of the Wiener index and the internal path length, as well as asymptotics for the covariance and other mixed moments. The limit laws are described using functionals of a Brownian excursion. The methods include both Aldous' theory of the continuum random tree and analysis of generating functions. 1.
An asymptotic theory for CauchyEuler differential equations with applications to the analysis of algorithms
, 2002
"... CauchyEuler differential equations surfaced naturally in a number of sorting and searching problems, notably in quicksort and binary search trees and their variations. Asymptotics of coefficients of functions satisfying such equations has been studied for several special cases in the literature. We ..."
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Cited by 24 (10 self)
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CauchyEuler differential equations surfaced naturally in a number of sorting and searching problems, notably in quicksort and binary search trees and their variations. Asymptotics of coefficients of functions satisfying such equations has been studied for several special cases in the literature. We study in this paper the most general framework for CauchyEuler equations and propose an asymptotic theory that covers almost all applications where CauchyEuler equations appear. Our approach is very general and requires almost no background on differential equations. Indeed the whole theory can be stated in terms of recurrences instead of functions. Old and new applications of the theory are given. New phase changes of limit laws of new variations of quicksort are systematically derived. We apply our theory to about a dozen of diverse examples in quicksort, binary search trees, urn models, increasing trees, etc.
Analytic Variations on the Airy Distribution
, 2001
"... The Airy distribution (of the “area ” type) occurs as a limit distribution of cumulative parameters in a number of combinatorial structures, like path length in trees, area below walks, displacement in parking sequences, and it is also related to basic graph and polyomino enumeration. We obtain cur ..."
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Cited by 21 (4 self)
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The Airy distribution (of the “area ” type) occurs as a limit distribution of cumulative parameters in a number of combinatorial structures, like path length in trees, area below walks, displacement in parking sequences, and it is also related to basic graph and polyomino enumeration. We obtain curious explicit evaluations for certain moments of the Airy distribution, including moments of orders −1, −3, −5, etc., as well as + 1 5 11 7 13 19 3, − 3, − 3, etc. and − 3, − 3, − 3, etc. Our proofs are based on integral transforms of the Laplace and Mellin type and they rely essentially on “nonprobabilistic ” arguments like analytic continuation. A byproduct of this approach is the existence of relations between moments of the Airy distribution, the asymptotic expansion of the Airy function Ai(z) at +∞, and power symmetric functions of the zeros −αk of Ai(z).
Brownian excursion area, Wright’s constants in graph enumeration, and other Brownian areas
 PROBABILITY SURVEYS
, 2007
"... This survey is a collection of various results and formulas by different authors on the areas (integrals) of five related processes, viz. Brownian motion, bridge, excursion, meander and double meander; for the Brownian motion and bridge, which take both positive and negative values, we consider bot ..."
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Cited by 11 (4 self)
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This survey is a collection of various results and formulas by different authors on the areas (integrals) of five related processes, viz. Brownian motion, bridge, excursion, meander and double meander; for the Brownian motion and bridge, which take both positive and negative values, we consider both the integral of the absolute value and the integral of the positive (or negative) part. This gives us seven related positive random variables, for which we study, in particular, formulas for moments and Laplace transforms; we also give (in many cases) series representations and asymptotics for density functions and distribution functions. We further study Wright’s constants arising in the asymptotic enumeration of connected graphs; these are known to be closely connected to the moments of the Brownian excursion area. The main purpose is to compare the results for these seven Brownian areas by stating the results in parallel forms; thus emphasizing both the similarities and the differences. A recurring theme is the Airy function which
Planar Maps and Airy Phenomena
, 2000
"... A considerable number of asymptotic distributions arising in random combinatorics and analysis of algorithms are of the exponentialquadratic type (e x 2 ), that is, Gaussian. We exhibit here a new class of \universal" phenomena that are of the exponentialcubic type (e ix 3 ), correspondi ..."
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Cited by 11 (3 self)
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A considerable number of asymptotic distributions arising in random combinatorics and analysis of algorithms are of the exponentialquadratic type (e x 2 ), that is, Gaussian. We exhibit here a new class of \universal" phenomena that are of the exponentialcubic type (e ix 3 ), corresponding to nonstandard distributions that involve the Airy function. Such Airy phenomena are expected to be found in a number of applications, when conuences of critical points and singularities occur. About a dozen classes of planar maps are treated in this way, leading to the occurrence of a common Airy distribution that describes the sizes of cores and of largest (multi)connected components. Consequences include the analysis and ne optimization of random generation algorithms for multiply connected planar graphs.
Asymptotic distributions for the cost of linear probing hashing
 RANDOM STRUCTURES AND ALGORITHMS
, 2001
"... We study moments and asymptotic distributions of the construction cost, measured as the total displacement, for hash tables using linear probing. Four different methods are employed for different ranges of the parameters; together they yield a complete description. This extends earlier results by F ..."
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Cited by 11 (3 self)
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We study moments and asymptotic distributions of the construction cost, measured as the total displacement, for hash tables using linear probing. Four different methods are employed for different ranges of the parameters; together they yield a complete description. This extends earlier results by Flajolet, Poblete and Viola. The average cost of unsuccessful searches is considered too.
Moment Convergence In Conditional Limit Theorems
, 2000
"... . Consider a sum P N 1 Y i of random variables conditioned on a given value of the sum P N 1 X i of some other variables, where X i and Y i are dependent but the pairs (X i ; Y i ) form an i.i.d. sequence. We prove, for a triangular array (X ni ; Y ni ) of such pairs satisfying certain condi ..."
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Cited by 5 (4 self)
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. Consider a sum P N 1 Y i of random variables conditioned on a given value of the sum P N 1 X i of some other variables, where X i and Y i are dependent but the pairs (X i ; Y i ) form an i.i.d. sequence. We prove, for a triangular array (X ni ; Y ni ) of such pairs satisfying certain conditions, both convergence of the distribution of the conditioned sum (after suitable normalization) to a normal distribution, and convergence of its moments. The results are motivated by an application to hashing with linear probing; we give also some other applications to occupancy problems, random forests, and branching processes. 1. Introduction Many random variables arising in different areas of probability theory, combinatorics and statistics turn out to have the same distribution as a sum of independent random variables conditioned on a specific value of another such sum. More precisely, we are concerned with variables with the distribution of P N 1 Y i conditioned on P N 1 X...
On the profile of random forests
 in Mathematics and Computer Science
, 2002
"... Abstract. An approach via generating functions is used to derive multivariate asymptotic distributions for the number of nodes in strata of random forests. For a certain range for the strata numbers we obtain a weak limit theorem to Brownian motion as well. Moreover, a moment convergence theorem for ..."
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Cited by 5 (5 self)
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Abstract. An approach via generating functions is used to derive multivariate asymptotic distributions for the number of nodes in strata of random forests. For a certain range for the strata numbers we obtain a weak limit theorem to Brownian motion as well. Moreover, a moment convergence theorem for the width of random forests is derived. 1.
Individual displacements for linear probing hashing with different insertion policies
 ACM Transactions on Algorithms
, 2005
"... Abstract. We study the distribution of the individual displacements in hashing with linear probing for three different versions: First Come, Last Come and Robin Hood. Asymptotic distributions and their moments are found when the the size of the hash table tends to infinity with the proportion of occ ..."
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Cited by 4 (1 self)
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Abstract. We study the distribution of the individual displacements in hashing with linear probing for three different versions: First Come, Last Come and Robin Hood. Asymptotic distributions and their moments are found when the the size of the hash table tends to infinity with the proportion of occupied cells converging to some α, 0 < α < 1. (In the case of Last Come, the results are more complicated and less complete than in the other cases.) We also show, using the diagonal Poisson transform studied by Poblete, Viola and Munro, that exact expressions for finite m and n can be obtained from the limits as m, n → ∞. We end with some results, conjectures and questions about the shape of the limit distributions. These have some relevance for computer applications. 1.