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On the Approximation Ratio of Ordered Parsings

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Gonzalo Navarro, Carlos Ochoa, and Nicola Prezza

Shannon's entropy is a clear lower bound for statistical compression. The
situation is not so well understood for dictionary-based compression. A
plausible lower bound is *b*, the least number of phrases of a general
bidirectional parse of a text, where phrases can be copied from anywhere
else in the text. Since computing *b* is NP-complete, a popular gold standard
is *z*, the number of phrases in the Lempel-Ziv parse of the text, which is
computed in linear time and yields the least number of phrases when those
can be copied only from the left. Almost nothing has been known for decades
about the approximation ratio of *z* with respect to *b*. In this paper we
prove that *z=O(b log(n/b))*, where *n* is the text length. We also show that
the bound is tight as a function of *n*, by exhibiting a text family where
*z = Omega(b log n)*. Our upper bound is obtained by building a run-length
context-free grammar based on a locally consistent parsing of the text. Our
lower bound is obtained by relating *b* with *r*, the number of equal-letter
runs in the Burrows-Wheeler transform of the text.
We continue by observing that Lempel-Ziv is just one particular case of
*greedy* parses--meaning that it obtains the smallest parse by
scanning the text and maximizing the phrase length at each step--, and of
*ordered* parses--meaning that phrases are larger than their sources
under some order. As a new example of ordered greedy parses, we introduce
*lexicographical* parses, where phrases can
only be copied from lexicographically smaller text locations. We prove
that the size *v* of the optimal lexicographical parse is also obtained greedily
in *O(n)* time, that *v=O(b log(n/b))*, and that there exists a text family
where *v = Omega(b log n)*. Interestingly, we also show that *v =
O(r)*
because *r* also induces a lexicographical parse,
whereas *z = Omega(r log n)* holds on some text families.
We obtain some results on parsing complexity and size that hold on some general
classes of greedy ordered parses.
In our way, we also prove other relevant bounds between compressibility measures,
especially with those related to smallest grammars of various types generating (only) the text.