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Images of subset of finite set under iterations of random mappings

Images of subset of finite set under iterations of random mappings Abstract Let N be a set of N elements and F 1 , F 2 ,... be a sequence of random independent equiprobable mappings N → N. For a subset S 0 ⊂ N, |S 0 | = n, we consider a sequence of its images S k = F k (. . . F 2 (F 1 (S 0 )) . . .), k = 1, 2... , and a sequence of their unions Ψ k = S 1 ⋃ ... ⋃ S k , k = 1, 2 ... An approach to the exact computation of distribution of |S k | and |Ψ k | for moderate values of N is described. Two-sided inequalities for M|S k | and M|Ψ k | such that upper bound are asymptotically equivalent to lower ones for N, n, k → ∞, nk = o(N) are derived. The results are of interest for the analysis of time-memory tradeoff algorithms. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Discrete Mathematics and Applications de Gruyter

Images of subset of finite set under iterations of random mappings

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References (14)

Publisher
de Gruyter
Copyright
Copyright © 2015 by the
ISSN
0924-9265
eISSN
1569-3929
DOI
10.1515/dma-2015-0017
Publisher site
See Article on Publisher Site

Abstract

Abstract Let N be a set of N elements and F 1 , F 2 ,... be a sequence of random independent equiprobable mappings N → N. For a subset S 0 ⊂ N, |S 0 | = n, we consider a sequence of its images S k = F k (. . . F 2 (F 1 (S 0 )) . . .), k = 1, 2... , and a sequence of their unions Ψ k = S 1 ⋃ ... ⋃ S k , k = 1, 2 ... An approach to the exact computation of distribution of |S k | and |Ψ k | for moderate values of N is described. Two-sided inequalities for M|S k | and M|Ψ k | such that upper bound are asymptotically equivalent to lower ones for N, n, k → ∞, nk = o(N) are derived. The results are of interest for the analysis of time-memory tradeoff algorithms.

Journal

Discrete Mathematics and Applicationsde Gruyter

Published: Jun 1, 2015

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