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Estimates of the mean size of the subset image under composition of random mappings. (English. Russian original) Zbl 1420.60038

Discrete Math. Appl. 28, No. 5, 331-338 (2018); translation from Diskretn. Mat. 30, No. 2, 27-36 (2018).
Summary: Let \(\mathcal{X}_N\) be a set of \(N\) elements and \(F_1,F_2,\dots\) be a sequence of random independent equiprobable mappings \(\mathcal{X}_N\rightarrow_N\). For a subset \(S_0 \subset \mathcal{X}_N\), \(|S_0|=m\), we consider a sequence of its images \(S_t=F_t(\dots F_2(F_1(S_0))\dots)\), \(t=1,2\dots\). An approach to the exact recurrent computation of distribution of \(|S_t|\) is described. Two-sided inequalities for \(\mathbf{M}\{|S_t|||S_0|=m\}\) such that the difference between the upper and lower bounds is \(o(m)\) for \(m,t,N\rightarrow\infty\), \(mt=o(N)\) are derived. The results are of interest for the analysis of time-memory tradeoff algorithms.

MSC:

60F05 Central limit and other weak theorems
60E05 Probability distributions: general theory
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