Which Regular Expression Patterns are Hard to Match?
Abstract
Regular expressions constitute a fundamental notion in formal language theory and are frequently used in computer science to define search patterns. A classic algorithm for these problems constructs and simulates a nondeterministic finite automaton corresponding to the expression, resulting in an $O(mn)$ running time (where $m$ is the length of the pattern and $n$ is the length of the text). This running time can be improved slightly (by a polylogarithmic factor), but no significantly faster solutions are known. At the same time, much faster algorithms exist for various special cases of regular expressions, including dictionary matching, wildcard matching, subset matching, word break problem etc. In this paper, we show that the complexity of regular expression matching can be characterized based on its {\em depth} (when interpreted as a formula). Our results hold for expressions involving concatenation, OR, Kleene star and Kleene plus. For regular expressions of depth two (involving any combination of the above operators), we show the following dichotomy: matching and membership testing can be solved in nearlinear time, except for "concatenations of stars", which cannot be solved in strongly subquadratic time assuming the Strong Exponential Time Hypothesis (SETH). For regular expressions of depth three the picture is more complex. Nevertheless, we show that all problems can either be solved in strongly subquadratic time, or cannot be solved in strongly subquadratic time assuming SETH. An intriguing special case of membership testing involves regular expressions of the form "a star of an OR of concatenations", e.g., $[aabbc]^*$. This corresponds to the socalled {\em word break} problem, for which a dynamic programming algorithm with a runtime of (roughly) $O(n\sqrt{m})$ is known. We show that the latter bound is not tight and improve the runtime to $O(nm^{0.44\ldots})$.
 Publication:

arXiv eprints
 Pub Date:
 November 2015
 arXiv:
 arXiv:1511.07070
 Bibcode:
 2015arXiv151107070B
 Keywords:

 Computer Science  Computational Complexity;
 Computer Science  Data Structures and Algorithms