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Asymmetric Rényi Problem

Part of: Theory of computing Algorithms - Computer Science

Published online by Cambridge University Press:  27 June 2018

M. DRMOTA ,
A. MAGNER  and
W. SZPANKOWSKI
M. DRMOTA
Affiliation:
Institute for Discrete Mathematics and Geometry, TU Wien, A-1040 Vienna, Austria (e-mail: michael.drmota@tuwien.ac.at)
A. MAGNER
Affiliation:
Coordinated Science Lab, UIUC, Champaign, IL 61820, USA (e-mail: anmagner@illinois.edu)
W. SZPANKOWSKI
Affiliation:
Department of Computer Science, Purdue University, IN 47907, USA (e-mail: szpan@purdue.edu)
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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

$$H_n=\log_{1/p} n +{\tfrac{1}{2}} \log_{p/(1-p)}\log n +o(\log \log n)$$
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
$$ H_n=\log_{2} n +\sqrt{2 \log_{2} n} +o(\sqrt{\log n})$$
 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.

MSC classification

Primary: 68Q87: Probability in computer science (algorithm analysis, random structures, phase transitions, etc.)
Secondary: 68W40: Analysis of algorithms
Type
Paper
Information
Combinatorics, Probability and Computing , Volume 28 , Special Issue 4: Analysis of Algorithms , July 2019 , pp. 542 - 573
Copyright
Copyright © Cambridge University Press 2018 

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Footnotes

†

Research partially supported by Austrian Science Foundation FWF grant F50-02.

‡

Research supported by NSF Center for Science of Information (CSoI) grant CCF-0939370.

§

Research partially supported by NSF Center for Science of Information (CSoI) grant CCF-0939370, and in addition by NSF grants CCF-1524312, and NIH grant 1U01CA198941-01.

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