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Random Mapping Statistics
 IN ADVANCES IN CRYPTOLOGY
, 1990
"... Random mappings from a finite set into itself are either a heuristic or an exact model for a variety of applications in random number generation, computational number theory, cryptography, and the analysis of algorithms at large. This paper introduces a general framework in which the analysis of ..."
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Cited by 110 (6 self)
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Random mappings from a finite set into itself are either a heuristic or an exact model for a variety of applications in random number generation, computational number theory, cryptography, and the analysis of algorithms at large. This paper introduces a general framework in which the analysis of about twenty characteristic parameters of random mappings is carried out: These parameters are studied systematically through the use of generating functions and singularity analysis. In particular, an open problem of Knuth is solved, namely that of finding the expected diameter of a random mapping. The same approach is applicable to a larger class of discrete combinatorial models and possibilities of automated analysis using symbolic manipulation systems ("computer algebra") are also briefly discussed.
Limit Distributions and Random Trees Derived From the Birthday Problem With Unequal Probabilities
, 1998
"... Given an arbitrary distribution on a countable set S consider the number of independent samples required until the first repeated value is seen. Exact and asymptotic formulae are derived for the distribution of this time and of the times until subsequent repeats. Asymptotic properties of the repeat ..."
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Cited by 36 (12 self)
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Given an arbitrary distribution on a countable set S consider the number of independent samples required until the first repeated value is seen. Exact and asymptotic formulae are derived for the distribution of this time and of the times until subsequent repeats. Asymptotic properties of the repeat times are derived by embedding in a Poisson process. In particular, necessary and sufficient conditions for convergence are given and the possible limits explicitly described. Under the same conditions the finite dimensional distributions of the repeat times converge to the arrival times of suitably modified Poisson processes, and random trees derived from the sequence of independent Research supported in part by N.S.F. Grants DMS 9224857, 9404345, 9224868 and 9703691 trials converge in distribution to an inhomogeneous continuum random tree. 1 Introduction Recall the classical birthday problem: given that each day of the year is equally likely as a possible birthday, and that birth...
Random mappings, forests, and subsets associated with AbelCayleyHurwitz multinomial expansions
, 2001
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AbelCayleyHurwitz multinomial expansions associated with random mappings, forests, and subsets
, 1998
"... Extensions of binomial and multinomial formulae due to Abel, Cayley and Hurwitz are related to the probability distributions of various random subsets, trees, forests, and mappings. For instance, an extension of Hurwitz's binomial formula is associated with the probability distribution of the r ..."
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Cited by 11 (10 self)
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Extensions of binomial and multinomial formulae due to Abel, Cayley and Hurwitz are related to the probability distributions of various random subsets, trees, forests, and mappings. For instance, an extension of Hurwitz's binomial formula is associated with the probability distribution of the random set of vertices of a fringe subtree in a random forest whose distribution is defined by terms of a multinomial expansion over rooted labeled forests which generalizes Cayley's expansion over unrooted labeled trees. Contents 1 Introduction 2 Research supported in part by N.S.F. Grant DMS9703961 2 Probabilistic Interpretations 5 3 Cayley's multinomial expansion 11 4 Random Mappings 14 4.1 Mappings from S to S : : : : : : : : : : : : : : : : : : : : : : : : : : : : 15 4.2 The random set of cyclic points : : : : : : : : : : : : : : : : : : : : : : : 18 5 Random Forests 19 5.1 Distribution of the roots of a pforest : : : : : : : : : : : : : : : : : : : : 19 5.2 Conditioning on the set...
The Multinomial Distribution on Rooted Labeled Forests
, 1997
"... For a probability distribution (p s ; s 2 S) on a finite set S, call a random forest F of rooted trees labeled by S (with edges directed away from the roots) a pforest if given F has m edges the vector of outdegrees of vertices of F has a multinomial distribution with parameters m and (p s ; s 2 ..."
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Cited by 8 (8 self)
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For a probability distribution (p s ; s 2 S) on a finite set S, call a random forest F of rooted trees labeled by S (with edges directed away from the roots) a pforest if given F has m edges the vector of outdegrees of vertices of F has a multinomial distribution with parameters m and (p s ; s 2 S), and given also these outdegrees the distribution of F is uniform on all forests with the given outdegrees. The family of distributions of pforests is studied, and shown to be closed under various operations involving deletion of edges. Some related enumerations of rooted labeled forests are obtained as corollaries. 1 Introduction Let F(S) denote the set of all forests of rooted trees labeled by a finite set S of size jSj. Each f 2 F(S) is a directed graph labeled by S, that is a subset of S \Theta S, such that each Research supported in part by N.S.F. Grant DMS9703961 connected component of the graph is a tree with edges directed away from some root vertex. The notation v f ...
Forest volume decompositions and AbelCayleyHurwitz multinomial expansions
, 2001
"... This paper presents a systematic approach to the discovery, interpretation and verification of various extensions of Hurwitz's multinomial identities, involving polynomials defined by sums over all subsets of a finite set. The identities are interpreted as decompositions of forest volumes d ..."
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Cited by 3 (0 self)
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This paper presents a systematic approach to the discovery, interpretation and verification of various extensions of Hurwitz's multinomial identities, involving polynomials defined by sums over all subsets of a finite set. The identities are interpreted as decompositions of forest volumes defined by the enumerator polynomials of sets of rooted labeled forests. These decompositions involve the following basic forest volume formula, which is a refinement of Cayley's multinomial expansion: for R ` S the polynomial enumerating outdegrees of vertices of rooted forests labeled by S whose set of roots is R, with edges directed away from the roots, is ( P r2R x r )( P s2S x s ) jS j\GammajRj\Gamma1
Large Components of Bipartite Random Mappings
, 2000
"... A bipartite random mapping TK,L of a finite set V = V1 ∪ V2, V1  = K and V2  = L, into itself assigns independently to each i ∈ V1 its unique image j ∈ V2 with probability 1/L and to each i ∈ V2 its unique image j ∈ V1 with probability 1/K. We study the connected component structure of a rando ..."
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Cited by 2 (0 self)
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A bipartite random mapping TK,L of a finite set V = V1 ∪ V2, V1  = K and V2  = L, into itself assigns independently to each i ∈ V1 its unique image j ∈ V2 with probability 1/L and to each i ∈ V2 its unique image j ∈ V1 with probability 1/K. We study the connected component structure of a random digraph G(TK,L),representingTK,L, asK→ ∞ and L →∞. We show that, no matter how K and L tend to infinity relative to each other, the joint distribution of the normalized order statistics for the component sizes converges in distribution to the PoissonDirichlet distribution on the simplex ∇ = {{xi} : � xi ≤ 1,xi≥xi+1 ≥ 0 for every i ≥ 1}.
A new random mapping model
, 2006
"... In this paper we introduce a new random mapping model, T ˆ D n, which maps the set {1, 2,..., n} into itself. The random mapping T ˆ D n is constructed using a collection of exchangeable random variables ˆD1,...., ˆ Dn which satisfy � n i=1 ˆ Di = n. In the random digraph, G ˆ D n, which represents ..."
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Cited by 1 (1 self)
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In this paper we introduce a new random mapping model, T ˆ D n, which maps the set {1, 2,..., n} into itself. The random mapping T ˆ D n is constructed using a collection of exchangeable random variables ˆD1,...., ˆ Dn which satisfy � n i=1 ˆ Di = n. In the random digraph, G ˆ D n, which represents the mapping T ˆ D n, the indegree sequence for the vertices is given by the variables ˆ D1, ˆ D2,..., ˆ Dn, and, in some sense, G ˆ D n can be viewed as an analogue of the general independent degree models from random graph theory. We show that the distribution of the number of cyclic points, the number of components, and the size of a typical component can be expressed in terms of expectations of various functions of ˆ D1, ˆ D2,..., ˆ Dn. We also consider two special examples of T ˆ D n which correspond to random mappings with preferential and antipreferential attachment, respectively, and determine, for these examples, exact and asymptotic distributions for the statistics mentioned above. Results for the distribution of the number of successors and predecessors of a typical vertex in G ˆ D n in terms of expectations of various functions of ˆ D1, ˆ D2,..., ˆ Dn are obtained in a companion paper [23].
© Hindawi Publishing Corp. ON THE BIRTHDAY PROBLEM: SOME GENERALIZATIONS AND APPLICATIONS
, 2001
"... We study the birthday problem and some possible extensions. We discuss the unimodality of the corresponding exact probability distribution and express the moments and generating functions by means of confluent hypergeometric functions U(−;−;−) which are computable using the software Mathematica. The ..."
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We study the birthday problem and some possible extensions. We discuss the unimodality of the corresponding exact probability distribution and express the moments and generating functions by means of confluent hypergeometric functions U(−;−;−) which are computable using the software Mathematica. The distribution is generalized in two possible directions, one of them consists in considering a random graph with a single attracting center. Possible applications are also indicated. 2000 Mathematics Subject Classification: 60C05, 05A10, 05C05. 1. Introduction. In the present paper
Local properties of a random mapping model
, 2007
"... In this paper we investigate the ‘local ’ properties of a random mapping model, T D̂n, which maps the set {1, 2,..., n} into itself. The random mapping T D̂n was introduced in a companion paper [?] is constructed using a collection of exchangeable random variables D̂1,...., D̂n which satisfy ∑n i=1 ..."
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In this paper we investigate the ‘local ’ properties of a random mapping model, T D̂n, which maps the set {1, 2,..., n} into itself. The random mapping T D̂n was introduced in a companion paper [?] is constructed using a collection of exchangeable random variables D̂1,...., D̂n which satisfy ∑n i=1 D̂i = n. In the random digraph, G