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arXiv · 1705.09538

On Two LZ78-style Grammars: Compression Bounds and Compressed-Space Computation

Golnaz Badkobeh·Travis Gagie·Shunsuke Inenaga·Tomasz Kociumaka·Dmitry Kosolobov·Simon J. Puglisi
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Abstract

We investigate two closely related LZ78-based compression schemes: LZMW (an old scheme by Miller and Wegman) and LZD (a recent variant by Goto et al.). Both LZD and LZMW naturally produce a grammar for a string of length $n$; we show that the size of this grammar can be larger than the size of the smallest grammar by a factor $\Omega(n^{\frac{1}3})$ but is always within a factor $O((\frac{n}{\log n})^{\frac{2}{3}})$. In addition, we show that the standard algorithms using $\Theta(z)$ working space to construct the LZD and LZMW parsings, where $z$ is the size of the parsing, work in $\Omega(n^{\frac{5}4})$ time in the worst case. We then describe a new Las Vegas LZD/LZMW parsing algorithm that uses $O (z \log n)$ space and $O(n + z \log^2 n)$ time w.h.p..

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Golnaz Badkobeh, Travis Gagie, Shunsuke Inenaga, Tomasz Kociumaka, Dmitry Kosolobov, Simon J. Puglisi. 2017-05-26. On Two LZ78-style Grammars: Compression Bounds and Compressed-Space Computation. https://arxiv.org/abs/1705.09538

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