Sorting is a fundamental problem in computer science. In the classical setting, it is well-known that $(1\pm o(1)) n\log_2 n$ comparisons are both necessary and sufficient to sort a list of $n$ elements. In this paper, we study the Noisy Sorting problem, where each comparison result is flipped independently with probability $p$ for some fixed $p\in (0, \frac 12)$. As our main result, we show that $$(1\pm o(1)) \left( \frac{1}{I(p)} + \frac{1}{(1-2p) \log_2 \left(\frac{1-p}p\right)} \right) n\log_2 n$$ noisy comparisons are both necessary and sufficient to sort $n$ elements with error probability $o(1)$ using noisy comparisons, where $I(p)=1 + p\log_2 p+(1-p)\log_2 (1-p)$ is capacity of BSC channel with crossover probability $p$. This simultaneously improves the previous best lower and upper bounds (Wang, Ghaddar and Wang, ISIT 2022) for this problem. For the related Noisy Binary Search problem, we show that $$ (1\pm o(1)) \left((1-\delta)\frac{\log_2(n)}{I(p)} + \frac{2 \log_2 \left(\frac 1\delta\right)}{(1-2p)\log_2\left(\frac {1-p}p\right)}\right) $$ noisy comparisons are both necessary and sufficient to find the predecessor of an element among $n$ sorted elements with error probability $\delta$. This extends the previous bounds of (Burnashev and Zigangirov, 1974), which are only tight for $\delta = 1/n^{o(1)}$.

翻译：排序是计算机科学中的基础问题。在经典设定下，已知对包含$n$个元素的列表进行排序所需的最优比较次数为$(1\pm o(1)) n\log_2 n$。本文研究含噪排序问题，其中每次比较结果以固定概率$p\in (0, \frac 12)$独立翻转。我们主要证明：当使用含噪比较时，以$o(1)$的误差概率对$n$个元素进行排序所需的最优含噪比较次数为$$(1\pm o(1)) \left( \frac{1}{I(p)} + \frac{1}{(1-2p) \log_2 \left(\frac{1-p}p\right)} \right) n\log_2 n$$ 其中$I(p)=1 + p\log_2 p+(1-p)\log_2 (1-p)$是交叉概率为$p$的二元对称信道容量。该结果同时改进了该问题此前的最优下界与上界（Wang, Ghaddar and Wang, ISIT 2022）。对于相关的含噪二分查找问题，我们证明：以误差概率$\delta$在$n$个已排序元素中查找某个元素的前驱所需的最优含噪比较次数为$$ (1\pm o(1)) \left((1-\delta)\frac{\log_2(n)}{I(p)} + \frac{2 \log_2 \left(\frac 1\delta\right)}{(1-2p)\log_2\left(\frac {1-p}p\right)}\right) $$ 该结果推广了(Burnashev and Zigangirov, 1974)的界，后者仅在$\delta = 1/n^{o(1)}$时紧确。