Compression in the Space of Permutations

TL;DR

This paper studies lossy compression of permutation data, analyzing rate-distortion trade-offs under various measures, and proposes simple, asymptotically optimal coding schemes, especially for the Mallows model.

Contribution

It provides a comprehensive analysis of permutation compression under multiple distortion measures and introduces explicit low-complexity codes that are asymptotically optimal.

Findings

Rate-distortion characteristics are derived for uniform permutation sources.

Explicit low-complexity codes achieve near-optimal compression.

Mallows model exhibits lower entropy and distortion at zero rate compared to uniform distribution.

Abstract

We investigate lossy compression (source coding) of data in the form of permutations. This problem has direct applications in the storage of ordinal data or rankings, and in the analysis of sorting algorithms. We analyze the rate-distortion characteristic for the permutation space under the uniform distribution, and the minimum achievable rate of compression that allows a bounded distortion after recovery. Our analysis is with respect to different practical and useful distortion measures, including Kendall-tau distance, Spearman's footrule, Chebyshev distance and inversion-$\ell_1$ distance. We establish equivalence of source code designs under certain distortions and show simple explicit code designs that incur low encoding/decoding complexities and are asymptotically optimal. Finally, we show that for the Mallows model, a popular nonuniform ranking model on the permutation space, both the entropy and the maximum distortion at zero rate are much lower than the uniform counterparts, which motivates the future design of efficient compression schemes for this model.