[Paper Review] A Separation Between Run-Length SLPs and LZ77

Philip Bille, Travis Gagie+2 more|arXiv (Cornell University)|Nov 20, 2017

Algorithms and Data Compression6 references3 citations

TL;DR

This paper establishes a super-constant separation between run-length straight-line programs (RLSLPs) and LZ77 parsing by constructing an infinite family of strings where the smallest RLSLP is asymptotically larger by a factor of Ω(log n / log log n) than the LZ77 parse length. The result shows that RLSLPs cannot universally outperform LZ77, even with run-length rules, and highlights a fundamental limitation in grammar compression extensions relative to LZ77 compression.

ABSTRACT

In this paper we give an infinite family of strings for which the length of the Lempel-Ziv'77 parse is a factor $Ω(\log n/\log\log n)$ smaller than the smallest run-length grammar.

Motivation & Objective

• To investigate whether run-length SLPs (RLSLPs) can outperform LZ77 parsing in terms of compression size.
• To determine if RLSLPs, which extend SLPs with run-length rules, can achieve compression ratios matching or exceeding LZ77.
• To resolve the open question of whether RLSLPs are strictly more powerful than LZ77 compression.
• To construct an explicit infinite family of strings demonstrating a provable separation in compression efficiency between RLSLPs and LZ77.

Proposed method

• Construct an infinite family of strings based on the Diff-Thue-Morse sequence, which is squarefree and has known small LZ77 parse sizes.
• Use the squarefreeness of the Diff-Thue-Morse sequence to equate the smallest grammar size and smallest RLSLP size for strings derived from it.
• Leverage a known construction from Charikar et al. (2005) that produces strings with large minimal grammar sizes relative to LZ77 parse size.
• Prove that the minimal RLSLP size for these strings is Ω(log²k₁ / log log k₁), where k₁ is the largest exponent in the construction.
• Establish that the LZ77 parse size for the same strings is O(log k₁), using known bounds on LZ77 for the Thue-Morse sequence and its difference sequence.
• Derive the asymptotic ratio g*rl / z_no = Ω(log n / log log n) by relating k₁ to the total string length n.

Experimental results

Research questions

• RQ1 Can run-length SLPs (RLSLPs) achieve compression ratios that are asymptotically better than LZ77 parsing?
• RQ2 Is there an infinite family of strings for which the smallest RLSLP is significantly larger than the LZ77 parse size?
• RQ3 Does the addition of run-length rules to SLPs close the gap between grammar compression and LZ77 compression?
• RQ4 Can RLSLPs universally dominate LZ77 compression in terms of parse size?

Key findings

• An infinite family of strings exists for which the smallest RLSLP size is Ω(log²k₁ / log log k₁), where k₁ is the largest exponent in the string construction.
• The LZ77 parse size for the same strings is O(log k₁), based on known bounds for the Thue-Morse sequence and its difference sequence.
• The ratio between the smallest RLSLP size and the LZ77 parse size is Ω(log n / log log n), where n is the total string length.
• This ratio demonstrates that LZ77 compression with no overlaps is strictly better than RLSLP compression in some cases, disproving the possibility of RLSLPs universally dominating LZ77.
• The result implies that RLSLPs cannot capture the full power of LZ77 compression, even with run-length rules.