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| About site: Programming/Languages/Regular Expressions/Perl - Perl Regular Expression Matching is NP-Complete |
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| About site: http://perl.plover.com/NPC/ |
Title: Programming/Languages/Regular Expressions/Perl - Perl Regular Expression Matching is NP-Complete A discussion of the complexity of regular expression matching algorithms in Perl.
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Perl Regular Expression Matching is NP-HardPerl Regular Expression Matching is NP-HardMatching ordinary regular expresions can be done in polynomialtime, proportional to MN, where M is the lengthof the regular expression and N is the length of the stringto be matched. The usual method for this is: Parse the regularexpression and construct an equivalent finite automaton (FA), whichwill have O(M) states; then simulate the actionof the FA on the input, which takes O(MN)time. My Regex.pm moduledemonstrates this.However, adding backreferences to regular expressions complicatesthem so that the usual FA construction does not work. Perl can handlethese extended regexes, but it sometimes takes a long time to find outif there is a match or not. For example, Perl takes exponential time(in the length of the string to be matched) to check whether or not astring matches the regex /^(?:(\d+)|::)*$/. One is tempted to wonder whether this difficulty is inherent, orwhether there might be a clever way of speeding up the matchingalgorithm. The answer is that there is probably no clever way tospeed up the algorithm, and that the exponential running time of thematching algorithm is probably unavoidable. We show below that regexmatching is NP-hard when regexes are allowed to havebackreferences.The usual method of showing that a problem P isNP-hard is by showing that some other NP-hard problem,Q, could be solved efficiently if P could be also. Ishowed that instances of the GRAPH 3-COLORABILITY problem can bereduced to instances of regex matching. Abigail showed that instancesof the 3CNF-SAT problem can be reduced to instances of regexmatching. Greg Bacon showed that instances of the CLIQUE problem canbe reduced to instances of regex matching. I thought I had a proof by reduction from the SUBSET-SUM problem, but I realized it had an error, so it's unavailable (except to Papadimitriou's cs270 students) until I can fix it, or maybe forever. Reduction from the 3CNF-SAT problem. (Due to Abigail.) Reduction from graph 3-colorability problem.Some Notes About NP-CompletenessThis page used to say "Regex Matching is NP-Complete". That maynot be correct. I've been meaning to fix it for years, but I nevergot around to it before. To show that regex matching is NP-complete,we need to show two things. First, we need to show that it isNP-hard; the proofs on this page do that. Second, we need to show thatregex matching is in NP. The proofs don't do that.Return to: Universe of Discourse main page |What's new page |Perl Paraphernaliamjd-perl-npc@plover.com |
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A | discussion | of | the | complexity | of | regular | expression | matching | algorithms | in | Perl. |
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Perl Regular Expression Matching is NP-Complete 2008 October
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A discussion of the complexity of regular expression matching algorithms in Perl.
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