Asymmetric Rényi Problem

Abstract

In 1960 Rényi, in his Michigan State University lectures, asked for the number of random queries necessary to recover a hidden bijective labelling ofndistinct 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 probabilityp> 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:Dnconverges in probability but not almost surely;HnandFnexhibit phase transitions with respect topin the second term. We prove that forp> 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 fromnindependent binary sequences generated by a biased(p) memoryless source.