text: abstract: in 1960 rényi 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, what is the set of objects that 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 (the height), to recover at least one single element (the fillup level), and to recover a randomly
  chosen element (the typical depth). this problem exhibits several remarkable behaviors: the depth dn converges
  in probability but not almost surely and while it satisfies the central limit theorem its local limit theorem doesn’t
  hold; the height hn and the fillup level fn exhibit phase transitions with respect to p in the second term. to obtain
  these results, we take a unified approach via the analysis of the external profile defined at level k as the number of
  elements recovered by the kth query. we first establish new precise asymptotic results for the average and variance,
  and a central limit law, for the external profile in the regime where it grows polynomially with n. we then extend the
  external profile results to the boundaries of the central region, leading to the solution of our problem for the height
  and fillup level. as a bonus, our analysis implies novel results for random patricia tries, as it turns out that the
  problem is probabilistically equivalent to the analysis of the height, fillup level, typical depth, and external profile of
  a patricia trie built from n independent binary sequences generated by a biased(p) memoryless source.
  keywords: rényi problem, patricia trie, profile, height, fillup level, analytic combinatorics, mellin transform,
  depoissonization