Noisy Sorting Capacity
Ziao Wang, Nadim Ghaddar, Banghua Zhu, Lele Wang

TL;DR
This paper investigates the maximum rate at which elements can be reliably sorted when each comparison is noisy, deriving bounds on this capacity and proposing universal algorithms that do not require knowing the noise level.
Contribution
It introduces the concept of noisy sorting capacity, providing bounds and universal algorithms that do not depend on the noise probability, extending classical sorting theory to noisy comparison settings.
Findings
Derived upper and lower bounds on noisy sorting capacity.
Proposed universal algorithms that do not require knowledge of noise probability.
Established that the sorting rate can be positive despite noisy comparisons.
Abstract
Sorting is the task of ordering elements using pairwise comparisons. It is well known that 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 . Unlike the common approach in the literature which aims to minimize the number of pairwise comparisons to achieve a given desired error probability, we consider randomized algorithms with expected number of queries and aim at characterizing the maximal sorting rate 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…
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