On the Approximation Ratio of Ordered 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 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 proceed by observing that Lempel-Ziv is just one particular case of greedy parses, meaning that the optimal value of $z$ is obtained by scanning the text and maximizing the phrase length at each step, and of ordered parses, meaning that there is an increasing order between phrases and their sources. As a new example of ordered greedy parses, we introduce {\em 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 $v = \Omega(b\log n)$.