Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Which Regular Expression Patterns are Hard to Match?   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
non-deterministic 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 near-linear time, except for "concatenations of
stars", which cannot be solved in strongly sub-quadratic 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 sub-quadratic time, or cannot be solved in
strongly sub-quadratic 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., $[a|ab|bc]^*$. This
corresponds to the so-called {\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
$O(nm^{0.44\ldots})$.

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The categories of this paper are: bayes-an
15
The categories of this paper are: nucl-ex
13
The categories of this paper are: quant-ph
8
The categories of this paper are: hep-ex
20
The categories of this paper are: acc-phys
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The categories of this paper are: eess
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The categories of this paper are: q-fin
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The categories of this paper are: physics
9
The categories of this paper are: nlin
6
The categories of this paper are: astro-ph
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The categories of this paper are: stat
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The categories of this paper are: comp-gas
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The categories of this paper are: mtrl-th
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The categories of this paper are: atom-ph
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The categories of this paper are: q-alg
0
The categories of this paper are: hep-ph