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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 ..."
Abstract

Cited by 87 (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.
The number of components in a logarithmic combinatorial structure
 Ann. Appl. Probab
, 2000
"... Under very mild conditions, we prove that the number of components in a decomposable logarithmic combinatorial structure has a distribution which is close to Poisson in total variation. The conditions are satisfied for all assemblies, multisets and selections in the logarithmic class. The error in t ..."
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Cited by 1 (1 self)
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Under very mild conditions, we prove that the number of components in a decomposable logarithmic combinatorial structure has a distribution which is close to Poisson in total variation. The conditions are satisfied for all assemblies, multisets and selections in the logarithmic class. The error in the Poisson approximation is shown under marginally more restrictive conditions to be of exact order O1 / log n, by exhibiting the penultimate asymptotic approximation; similar results have previously been obtained by Hwang [20], under stronger assumptions. Our method is entirely probabilistic, and the conditions can readily be verified in practice.