Sorting by Swaps with Noisy Comparisons

TL;DR

This paper analyzes a noisy comparison-based sorting process using random swaps, providing theoretical and experimental insights into how comparison range affects convergence speed and solution quality.

Contribution

It introduces a model for sorting with noisy comparisons, analyzing the impact of comparison range on convergence and solution quality both theoretically and experimentally.

Findings

Faster convergence with larger comparison range

Better solution quality with smaller comparison range

Trade-off between convergence speed and solution quality

Abstract

We study sorting of permutations by random swaps if each comparison gives the wrong result with some fixed probability $p<1/2$. We use this process as prototype for the behaviour of randomized, comparison-based optimization heuristics in the presence of noisy comparisons. As quality measure, we compute the expected fitness of the stationary distribution. To measure the runtime, we compute the minimal number of steps after which the average fitness approximates the expected fitness of the stationary distribution. We study the process where in each round a random pair of elements at distance at most $r$ are compared. We give theoretical results for the extreme cases $r=1$ and $r=n$, and experimental results for the intermediate cases. We find a trade-off between faster convergence (for large $r$) and better quality of the solution after convergence (for small $r$).