Noisy (Binary) Searching: Simple, Fast and Correct

Abstract

This work considers the problem of the noisy binary search in a sorted array. The noise is modeled by a parameter $p$ that dictates that a comparison can be incorrect with probability $p$, independently of other queries. We state two types of upper bounds on the number of queries: the worst-case and expected query complexity scenarios. The bounds improve the ones known to date, i.e., our algorithms require fewer queries. Additionally, they have simpler statements, and work for the full range of parameters. All query complexities for the expected query scenarios are tight up to lower order terms. For the problem where the target prior is uniform over all possible inputs, we provide an algorithm with expected complexity upperbounded by $(\log_2 n + \log_2 δ^{-1} + 3)/I(p)$, where $n$ is the domain size, $0\le p < 1/2$ is the noise ratio, and $δ>0$ is the failure probability, and $I(p)$ is the information gain function. As a side-effect, we close some correctness issues regarding previous work. Also, en route, we obtain new and improved query complexities for the search generalized to arbitrary graphs. This paper continues and improves the lines of research of Burnashev--Zigangirov [Prob. Per. Informatsii, 1974], Ben-Or and Hassidim [FOCS 2008], Gu and Xu [STOC 2023], and Emamjomeh-Zadeh et al. [STOC 2016], Dereniowski et al. [SOSA@SODA 2019].