Approximate Regular Expression Matching with Multi-strings

Abstract

In this paper, we are interested in solving the approximate regular expression matching problem: we are given a regular expression R in advance and we wish to answer the following query: given a text T and a parameter k, find all the substrings of T which match the regular expression R with at most k errors (an error consist in deleting inserting, or substituting a character). There exists a well known solution for this problem in time O(mn) where m is the size of the regular expression (the number of operators and characters appearing in R) and n the length of the text. There also exists a solution for the case k = 0 (exact regular expression matching) which solves the problem in time O(dn), where d is the number of strings in the regular expression (a string is a sequence of characters connected with concatenation operator). In this paper, we show that both methods can be combined to solve the approximate regular approximate matching problem in time O(kdn) for arbitrary k. This bound can be much better than the bound O(mn/log _k + 2 n) achieved by the best actual regular expression matching algorithm in case $d < \frac{m} {k \log_{k+2} n}$ (that is k is not too large and R contains much less occurrences of ∪ and * than occurrences of (·)).

This work is supported by the french ANR-2010-COSI-004 project MAPPI.