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Lempel-Ziv-like Parsing in Small Space

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Daniel Valenzuela, Dmitry Kosolobov, Gonzalo Navarro, and Simon J. Puglisi

Lempel-Ziv (LZ77 or, briefly, LZ) is one of the most effective and widely-used
compressors for repetitive texts.
However, the existing efficient methods computing the exact LZ parsing have to
use linear or close to linear space to index the input text during the
construction of the parsing, which is prohibitive for long inputs.
An alternative is Relative Lempel-Ziv (RLZ), which indexes only a fixed reference
sequence, whose size can be controlled. Deriving the reference sequence by
sampling the text yields reasonable compression ratios for RLZ, but performance is
not always competitive with that of LZ and depends heavily on the similarity of
the reference to the text.
In this paper we introduce ReLZ, a technique that uses RLZ as a
preprocessor to approximate the LZ parsing using little memory. RLZ is first
used to produce a sequence of phrases, and these are regarded as metasymbols
that are input to LZ for a second-level parsing on a (most often) drastically
shorter sequence. This parsing is finally translated into one on the original
sequence.
We analyze the new scheme and prove that, like LZ, it achieves the *k*th order empirical
entropy compression *n H_k + o(n log s)* with
*k = o(log_s n)*, where *n* is the input length and *s* is the
alphabet size.
In fact, we prove this entropy bound not only for ReLZ but for a wide class of
LZ-like encodings.
Then, we establish a lower bound on ReLZ approximation ratio
showing that the number of phrases in it can be *Omega(log n)* times
larger than the number of phrases in LZ.
Our experiments show that ReLZ is orders of magnitude faster than
other alternatives to compute the (exact or approximate) LZ parsing, at the
reasonable price of an approximation factor below 2.0 in practice, and sometimes below
1.05, to the size of LZ.