Optimal Binary Search with Two Unreliable Tests and Minimum Adaptiveness

Abstract

What is the minimum number of yes-no questions needed to find an m bit number x in the set S = {0,1,...,2^m — 1} if up to ℓ answers may be erroneous/false ? In case when the (t+1)th question is adaptively asked after receiving the answer to the tth question, the problem, posed by Ulam and Rényi, is a chapter of Berlekamp’s theory of error-correcting communication with feedback. It is known that, with finitely many exceptions, one can find x asking Berlekamp’s minimum number qℓ(m) of questions, i.e., the smallest integer q such that 2^q ≥ 2^m(( ^l _q + ( ^l-1 _q ) + ··· + ² _q +q+1): At the opposite, nonadaptive extreme, when all questions are asked in a unique batch before receiving any answer, a search strategy with qℓ(m) questions is the same as an ℓ-error correcting code of length qℓ(m) having 2^m codewords. Such codes in general do not exist for ℓ > 1. Focusing attention on the case ℓ = 2; we shall show that, with the exception of m = 2 and m = 4, one can always find an unknown m bit number x ∈ S by asking q2(m) questions in two nonadaptive batches. Thus the results of our paper provide shortest strategies with as little adaptiveness/interaction as possible.

Partially supported by ENEA Phd Grant

Partially supported by COST ACTION 15 on Many-valued logics for computer science applications, and by the Italian MURST Project on Logic.