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arXiv 2302.12440cs.DScs.ITmath.IT
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噪声排序的最优界

Optimal Bounds for Noisy Sorting

Yuzhou Gu, Yinzhan Xu

备注：
To appear at STOC'23; fixed issues in the previous version
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摘要：

排序是计算机科学中的一个基本问题。在经典设置中，众所周知，$(1\pm o(1)) n\log_2 n$比较对于对$n$元素列表进行排序是必要且充分的。本文研究了带噪排序问题，其中每个比较结果都以概率$p$独立翻转，且$p\in (0, \frac 12)$是固定的$p\in (0, \frac 12)$。作为我们的主要结果，我们表明，$$(1\pm o(1)) \left( \frac{1}{I(p)} + \frac{1}{(1-2p) \log_2 \left(\frac{1-p}p\right)} \right) n\log_2 n$$噪声比较是必要的和充分的排序$n$元素与错误概率$o(1)$使用噪声比较，其中$I(p)=1 + p\log_2 p+(1-p)\log_2 (1-p)$是BSC信道的容量与交叉概率$p$。这同时改进了这个问题的以前的最佳下限和上限（Wang，Ghaddar和Wang，ISIT 2022）。对于相关的噪声二进制搜索问题，我们表明，$$ (1\pm o(1)) \left((1-δ)\frac{\log_2(n)}{I(p)} + \frac{2 \log_2 \left(\frac 1δ\right)}{(1-2p)\log_2\left(\frac {1-p}p\right)}\right) $$嘈杂的比较是必要的和足够的，以找到一个元素的前身之间的$n$排序元素的错误概率$δ$。这扩展了（Burnashev和Zigangirov，1974）的先前边界，其仅对$δ= 1/n^{o(1)}$是紧的。

英文摘要：

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 $$ (1\pm o(1)) \left((1-δ)\frac{\log_2(n)}{I(p)} + \frac{2 \log_2 \left(\frac 1δ\right)}{(1-2p)\log_2\left(\frac {1-p}p\right)}\right) $$ noisy comparisons are both necessary and sufficient to find the predecessor of an element among $n$ sorted elements with error probability $δ$. This extends the previous bounds of (Burnashev and Zigangirov, 1974), which are only tight for $δ= 1/n^{o(1)}$.

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