Lossy compression of permutations

We investigate the lossy compression of permutations by analyzing the trade-off between the size of a source code and the distortion with respect to Kendall tau distance, Spearman's footrule, Chebyshev distance and ℓ1 distance of inversion vectors. We show that given two permutations, Kendall tau distance upper bounds the ℓ1 distance of inversion vectors and a scaled version of Kendall tau distance lower bounds the ℓ1 distance of inversion vectors with high probability, which indicates an equivalence of the source code designs under these two distortion measures. Similar equivalence is established for all the above distortion measures, every one of which has different operational significance and applications in ranking and sorting. These findings show that an optimal coding scheme for one distortion measure is effectively optimal for other distortion measures above.