Noisy Sorting Capacity

Sorting is the task of ordering $n$ elements using pairwise comparisons. It is well known that $m=\Theta(n\log n)$ comparisons are both necessary and sufficient when the outcomes of the comparisons are observed with no noise. In this paper, we study the sorting problem when each comparison is incorrect with some fixed yet unknown probability $p$. Unlike the common approach in the literature which aims to minimize the number of pairwise comparisons $m$ to achieve a given desired error probability, we consider randomized algorithms with expected number of queries $\textsf{E}[M]$ and aim at characterizing the maximal sorting rate $\frac{n\log n}{\textsf{E}[M]}$ such that the ordering of the elements can be estimated with a vanishing error probability asymptotically. The maximal rate is referred to as the noisy sorting capacity. In this work, we derive upper and lower bounds on the noisy sorting capacity. The two lower bounds -- one for fixed-length algorithms and one for variable-length algorithms -- are established by combining the insertion sort algorithm with the well-known Burnashev--Zigangirov algorithm for channel coding with feedback. Compared with existing methods, the proposed algorithms are universal in the sense that they do not require the knowledge of $p$, while maintaining a strictly positive sorting rate. Moreover, we derive a general upper bound on the noisy sorting capacity, along with an upper bound on the maximal rate that can be achieved by sorting algorithms that are based on insertion sort.