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Publications in Scientific Journals:

M. Drmota, A. Magner, W. Szpankowski:
"Asymmetric Renyi Problem";
Combinatorics, Probability and Computing, 28 (2019), 4, Special Issue (Analysis and Algorithms); 542 - 573.



English abstract:
In 1960 Rényi, in his Michigan State University lectures, asked for the number of random queries necessary to recover a hidden bijective labelling of n distinct objects. In each query one selects a random subset of labels and asks, which objects have these labels? We consider here an asymmetric version of the problem in which in every query an object is chosen with probability p > 1/2 and we ignore `inconclusive´ queries. We study the number of queries needed to recover the labelling in its entirety (Hn), before at least one element is recovered (Fn), and to recover a randomly chosen element (Dn). This problem exhibits several remarkable behaviours: Dn converges in probability but not almost surely; Hn and Fn exhibit phase transitions with respect to p in the second term. We prove that for p > 1/2 with high probability we need


queries to recover the entire bijection. This should be compared to its symmetric (p = 1/2) counterpart established by Pittel and Rubin, who proved that in this case one requires

queries. As a bonus, our analysis implies novel results for random PATRICIA tries, as the problem is probabilistically equivalent to that of the height, fillup level, and typical depth of a PATRICIA trie built from n independent binary sequences generated by a biased(p) memoryless source.


"Official" electronic version of the publication (accessed through its Digital Object Identifier - DOI)
http://dx.doi.org/10.1017/S0963548318000329


Created from the Publication Database of the Vienna University of Technology.