Central and local limit theorems applied to asymptotic enumeration II: Multivariate generating functions

Bender, Edward A.; Richmond, L. Bruce

doi:10.1016/0097-3165(83)90062-6

Cited by 93 publications

(87 citation statements)

References 5 publications

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“…Next we state (and prove) a multivariate version of the so-called Quasi Power Theorem by H.-K. Hwang [11] (see also [9], similar theorems can be found in [1,2]). Note, too, that there exists a precise large deviation result for the univariate case [12].…”

Section: E Umentioning

confidence: 99%

Vertices of given degree in series‐parallel graphs

Drmota

Giménez

Noy

2009

Random Struct Algorithms

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Abstract. We show that the number of vertices of a given degree k in several kinds of series-parallel labelled graphs of size n satisfy a central limit theorem with mean and variance proportional to n, and quadratic exponential tail estimates. We further prove a corresponding theorem for the number of nodes of degree two in labelled planar graphs. The proof method is based on generating functions and singularity analysis. In particular we need systems of equations for multivariate generating functions and transfer results for singular representations of analytic functions. Statement of main resultsA graph is series-parallel if it does not contain the complete graph K 4 as a minor; equivalently, if it does not contain a subdivision of K 4 . Since both K 5 and K 3,3 contain a subdivision of K 4 , by Kuratowski's theorem a series-parallel graph is planar. An outerplanar graph is a planar graph that can be embedded in the plane so that all vertices are incident to the outer face. They are characterized as those graphs not containing a minor isomorphic to (or a subdivision of) either K 4 or K 2,3 . These are important subfamilies of planar graphs, as they are much simpler but often they already capture the essential structural properties of planar graphs. In particular, they are used as a natural first benchmark for many algorithmic problems and conjectures related to planar graphs The purpose of this paper is to study the number of vertices of given degree in certain classes of labelled planar graphs. In particular, we study labelled outerplanar graphs and series-parallel graphs; in what follows, all graphs are labelled.In order to state our results we introduce the notion of the degree distribution of a random outerplanar graph (the definition for series-parallel graphs is exactly the same). For every n we consider the class of all vertex labelled outerplanar graphs with n vertices. Let D n denote the degree of a randomly chosen vertex in this class of graphs 1 . Then we say that this class of graphs has a degree distribution if there exist non-negative numbers d k with k≥0 d k = 1 such that for all kIn a companion paper [6], we have established that the classes of 2-connected, connected or all outerplanar graphs, as well as the corresponding classes of seriesparallel graphs have a degree distribution. We describe briefly the degree distribution in the outerplanar case, which is the simplest one, and refer to [6] for the other 1 Alternatively we can define Dn as the degree of the vertex with label 1.

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Section: E Umentioning

confidence: 99%

Vertices of given degree in series‐parallel graphs

Drmota

Giménez

Noy

2009

Random Struct Algorithms

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“…Theorem 8 is a consequence of general theorems of Bender and Richmond [1,2] (see also Drmota [6]) in the case when r is bounded above and below by positive constants. Equivalenty, k ν ∈ [a, b] for constants 0 < a < b < A.…”

Section: Corollarymentioning

confidence: 99%

Numbers with fixed sum of digits in linear recurrent number systems

Wagner

2006

Ramanujan J

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We study the set of integers with a given sum of digits with respect to a linear recurrent digit system.An asymptotic formula for the number of integers ≤ N with given sum of digits is determined, and the distribution in residue classes is investigated, thus generalizing results due to Mauduit and Sárközy. It turns out that numbers with fixed sum of digits are uniformly distributed in residue classes under some very general conditions. Namely, the underlying linear recurring sequence must have the property that there is no prime factor P of the modulus such that all but finitely many members of the sequence leave the same residue modulo P . The key step in the proof is an estimate for exponential sums using known theorems from Diophantine approximation.

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“…It is natural to study these quantities jointly to gain information on the dependence structure beyond the pure marginal distributions of the parameters. For examples of multivariate limit laws in the field of combinatorial structures using various approaches see [1,29,18,30,31,26]. This paper is organized as follows: In section two we outline the type of divide-and-conquer structures for which the contraction method is extended to multivariate asymptotic analysis.…”

Section: Introductionmentioning

confidence: 99%

On a multivariate contraction method for random recursive structures with applications to Quicksort

Neininger

2001

Random Struct Algorithms

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The contraction method for recursive algorithms is extended to the multivariate analysis of vectors of parameters of recursive structures and algorithms. We prove a general multivariate limit law which also leads to an approach to asymptotic covariances and correlations of the parameters. As an application the asymptotic correlations and a bivariate limit law for the number of key comparisons and exchanges of median-of-(2t + 1) Quicksort is given. Moreover, for the Quicksort programs analyzed by Sedgewick the exact order of the standard deviation and a limit law follow, considering all the parameters counted by Sedgewick.

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Central and local limit theorems applied to asymptotic enumeration II: Multivariate generating functions

Cited by 93 publications

References 5 publications

Vertices of given degree in series‐parallel graphs

Vertices of given degree in series‐parallel graphs

Numbers with fixed sum of digits in linear recurrent number systems

On a multivariate contraction method for random recursive structures with applications to Quicksort

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