Geodesic
Estimates of the mean size of the subset image under composition of random mappings
Diskretnaya Matematika, Tome 30 (2018) no. 2, pp. 27-36

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Let $\mathcal{X}_N$ be a set of $N$ elements and $F_1,F_2,\ldots$ be a sequence of random independent equiprobable mappings $\mathcal{X}_N\to\mathcal{X}_N$. For a subset $S_0\subset\mathcal{X}_N$, $|S_0|=m$, we consider a sequence of its images $S_t=F_t(\ldots F_2(F_1(S_0))\ldots)$, $t=1,2\ldots$ 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 $m,t,N\to\infty,\,mt=o(N)$ are derived. The results are of interest for the analysis of time-memory tradeoff algorithms.
Keywords: compositions of random mappings, time-memory tradeoff method. 
@article{DM_2018_30_2_a2,
     author = {A. M. Zubkov and A. A. Serov},
     title = {Estimates of the mean size of the subset image under composition of random mappings},
     journal = {Diskretnaya Matematika},
     pages = {27--36},
     publisher = {mathdoc},
     volume = {30},
     number = {2},
     year = {2018},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/DM_2018_30_2_a2/}
}
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A. M. Zubkov; A. A. Serov. Estimates of the mean size of the subset image under composition of random mappings. Diskretnaya Matematika, Tome 30 (2018) no. 2, pp. 27-36. http://geodesic.mathdoc.fr/item/DM_2018_30_2_a2/