# Optimal Bounds for Noisy Sorting

## TL;DR

This paper establishes tight bounds on the number of noisy comparisons needed for sorting and binary search, improving previous bounds by precisely characterizing the impact of noise with information-theoretic measures.

## Contribution

It provides the first tight bounds for noisy sorting and binary search, connecting the number of comparisons to the capacity of the binary symmetric channel.

## Key findings

- Optimal bounds match necessary and sufficient comparisons for noisy sorting.
- Bounds improve previous results by incorporating channel capacity.
- Results extend to noisy binary search with error probability.

## Abstract

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 $\delta = 1/n^{o(1)}$.