Optimal Binary Search with Two Unreliable Tests and Minimum Adaptiveness

• Authors: Ferdinando Cicalese;Daniele Mundici
• Venue: ESA '99 Proceedings of the 7th Annual European Symposium on Algorithms
• Year: 1999

What is the minimum number of yes-no questions needed to find an m bit number x in the set S = {0,1,...,2m -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 R茅nyi, 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 2q 驴 2m((q l)+(q l-1)+...+(q 2)+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 2m 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 驴 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.