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-ℓ1 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.
Bibliographical noteFunding Information:
This work was supported in part by the National Science Foundation under Grant CCF-1017772 and Grant CCF-1318093, and in part by the Air Force Office of Scientific Research under Grant FA9550-11-1-0183. The authors are grateful to an anonymous reviewer whose comment prompted an important correction to an earlier version of this paper.
- lossy compressions
- mallows model
- partial sorting
- permutation space