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Entropy Lower Bounds for Dictionary Compression

Author Michał Gańczorz

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Michał Gańczorz
  • Institute of Computer Science, University of Wrocław, Poland

Funding

Supported under National Science Centre, Poland project number 2017/26/E/ST6/00191.

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Michał Gańczorz. Entropy Lower Bounds for Dictionary Compression. In 30th Annual Symposium on Combinatorial Pattern Matching (CPM 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 128, pp. 11:1-11:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)
https://doi.org/10.4230/LIPIcs.CPM.2019.11

Abstract

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|.

Subject Classification

ACM Subject Classification
  • Theory of computation → Data compression
Keywords
  • compression
  • empirical entropy
  • parsing
  • lower bounds

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