Asymmetric R\'enyi Problem

In 1960 R\'enyi in his Michigan State University lectures asked for the number of random queries necessary to recover a hidden bijective labeling 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 labeling in its entirety ($H_n$), before at least one element is recovered ($F_n$), and to recover a randomly chosen element $(D_n)$. This problem exhibits several remarkable behaviors: $D_n$ converges in probability but not almost surely, $H_n$ and $F_n$ exhibit phase transitions with respect to $p$ in the second term. We prove that for $p>1/2$ with high probability (whp) we need $H_n=\log_{1/p} n +\frac 12 \log_{p/(1-p)}\log n +o(\log \log n)$ 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.