# On the Approximation Ratio of Greedy Parsings

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 the optimal one when phrases can be copied only from the left. While z can be computed in linear time with a greedy algorithm, almost nothing has been known for decades about its approximation ratio 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 string 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 proceed by observing that Lempel-Ziv is just one particular case of greedy parse, and introduce a new parse where phrases can only be copied from lexicographically smaller text locations. We prove that the size v of the smallest parse of this kind has properties similar to z, including the same approximation ratio with respect to b. Interestingly, we also show that v = O(r), whereas r = o(z) holds on some particular classes of strings. On our way, we prove other relevant bounds between compressibility measures.

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