Re-pair Achieves High-Order Entropy

Re-pair is a dictionary-based compression method invented in 1999 by J. Larsson and A. Moffat [Off-line dictionary-based compression. Proc. IEEE, 88(11):1722-1732, 2000], lacking up to now an efficiency analysis. We show that re-pair compresses a sequence T[1,n] over an alphabet of size sigma to at most 2nH_k + o(n log sigma) bits, for any k = o(log_sigma n), where H_k is either the classical information-theory or the empirical k-th order entropy (in the latter, the model is inferred from the sequence statistics).