Noisy (Binary) Searching: Simple, Fast and Correct

TL;DR

This work tackles noisy binary search and noisy graph search under independent noise $p\in[0,1/2)$, designing MWU-based adaptive strategies that achieve information-theoretic bounds on query complexity. By relating query answers to information gain $I(p)=1-H(p)$ and employing stopping-time analyses with Wald's identity, the authors obtain tight (up to lower-order terms) expected-length bounds and competitive worst-case bounds, both for uniform-prior targets and adversarial targets. The graph-search results extend the approach to medians and heavier-weight notions, yielding comparable bounds with additional $\mathcal{O}(\log\log n)$ and $\mathcal{O}(\sqrt{\log n\log\delta^{-1}})$ terms. Overall, the paper clarifies correctness issues in prior work, unifies noisy-search bounds across domains, and delivers simpler, tighter algorithms with broad applicability to graph-search problems.

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].

Theorems & Definitions (39)

• Theorem 1.1
• Corollary 1.2
• Theorem 1.3
• Theorem 1.4
• Theorem 1.5
• Proposition 2.1
• proof
• Proposition 2.2: Wald's Identity wald1945ams
• Lemma 2.4
• proof