An Improved Upper-Bound for Rivest et al.’s Half-Lie Problem

Abstract

Ulam proposed the problem of determining an optimum strategy for finding an integer x ∈ {1, 2, ..., n} using binary queries (i.e., queries with yes/no answer) in which the responses to up to k queries (for a fixed k) can be incorrect. This problem has been extensively studied for the past fifty years. The paper by Rivest et al. [9] that made a major advance in Ulam’s problem introduced a restricted type of error in responses known as half-lies. Rivest et al. presented a lower-bound on the minimax complexity of the half-lie version of Ulam’s search problem. Here we present a new algorithm that improves the previous upper-bound for the half-lie problem (in the case of k = 1) for all sufficiently large values of n. Specifically, we show that the number of queries of the form ’Is x > s?’ sufficient (in the worst-case) to find an unknown integer x ∈ {1, 2, ..., n}, when the responder’s ’yes’ answers are always true, but at most one of the ’no’ answers may be false, is at most ⌈log₂((n + 4.5) ln (n + 4.5) − 4.5 ln (4.5))⌉. We also present an improvement to Rivest et al.’s lower-bound for the special case of n = 10⁶.