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Pattern Matching with Variables: Fast Algorithms and New Hardness Results

Authors Henning Fernau, Florin Manea, Robert Mercas, Markus L. Schmid

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Henning Fernau
Florin Manea
Robert Mercas
Markus L. Schmid

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Henning Fernau, Florin Manea, Robert Mercas, and Markus L. Schmid. Pattern Matching with Variables: Fast Algorithms and New Hardness Results. In 32nd International Symposium on Theoretical Aspects of Computer Science (STACS 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 30, pp. 302-315, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)
https://doi.org/10.4230/LIPIcs.STACS.2015.302

Abstract

A pattern (i. e., a string of variables and terminals) maps to a word, if this is obtained by uniformly replacing the variables by terminal words; deciding this is NP-complete. We present efficient algorithms\footnote{The computational model we use is the standard unit-cost RAM with logarithmic word size. Also, all logarithms appearing in our time complexity evaluations are in base 2.} that solve this problem for restricted classes of patterns. Furthermore, we show that it is NP-complete to decide, for a given number k and a word w, whether w can be factorised into k distinct factors; this shows that the injective version (i.e., different variables are replaced by different words) of the above matching problem is NP-complete even for very restricted cases.
Keywords
  • combinatorial pattern matching
  • combinatorics on words
  • patterns with variables
  • ${cal NP}$-complete string problems

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