eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2019-06-06
11:1
11:18
10.4230/LIPIcs.CPM.2019.11
article
Entropy Lower Bounds for Dictionary Compression
Gańczorz, Michał
1
Institute of Computer Science, University of Wrocław, Poland
We show that a wide class of dictionary compression methods (including LZ77, LZ78, grammar compressors as well as parsing-based structures) require |S|H_k(S) + Omega (|S|k log sigma/log_sigma |S|) bits to encode their output. This matches known upper bounds and improves the information-theoretic lower bound of |S|H_k(S). To this end, we abstract the crucial properties of parsings created by those methods, construct a certain family of strings and analyze the parsings of those strings. We also show that for k = alpha log_sigma |S|, where 0 < alpha < 1 is a constant, the aforementioned methods produce an output of size at least 1/(1-alpha)|S|H_k(S) bits. Thus our results separate dictionary compressors from context-based one (such as PPM) and BWT-based ones, as the those include methods achieving |S|H_k(S) + O(sigma^k log sigma) bits, i.e. the redundancy depends on k and sigma but not on |S|.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol128-cpm2019/LIPIcs.CPM.2019.11/LIPIcs.CPM.2019.11.pdf
compression
empirical entropy
parsing
lower bounds