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 target prior is uniform over all possible inputs, we provide algorithm with expected complexity upperbounded by $(\log_2 n + \log_2 \delta^{-1} + 3)/I(p)$, where $n$ is the domain size, $0\le p < 1/2$ is the noise ratio, and $\delta>0$ 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 upon 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].

翻译：本文研究有序数组中的噪声二分搜索问题。噪声由参数 $p$ 建模，该参数表示每次比较有概率 $p$ 独立于其他查询而错误。我们给出了两类查询次数上界：最坏情形和期望查询复杂度场景。这些上界改进了现有已知结果，即我们的算法所需查询次数更少。此外，它们具有更简洁的表述，且适用于完整参数范围。所有期望查询场景的查询复杂度在低阶项意义下都是紧的。对于目标先验均匀分布于所有可能输入的问题，我们提供了一种算法，其期望复杂度上界为 $(\log_2 n + \log_2 \delta^{-1} + 3)/I(p)$，其中 $n$ 为域规模，$0\le p < 1/2$ 为噪声率，$\delta>0$ 为失败概率，$I(p)$ 为信息增益函数。作为副产品，我们纠正了以往工作中的一些正确性问题。同时，在此过程中，我们为推广到任意图的搜索问题获得了新的且改进的查询复杂度。本文延续并改进了 Burnashev-Zigangirov [Prob. Per. Informatsii, 1974]、Ben-Or 和 Hassidim [FOCS 2008]、Gu 和 Xu [STOC 2023]、Emamjomeh-Zadeh 等人 [STOC 2016] 以及 Dereniowski 等人 [SOSA@SODA 2019] 的研究路线。