A fast bit-vector algorithm for approximate string matching based on dynamic programming

The approximate string matching problem is to find all locations at which a query of length m matches a substring of a text of length n with k -or-fewer differences. Simple and practical bit-vector algorithms have been designed for this problem, most notably the one used in agrep . These algorithms compute a bit representation of the current state-set of the k -difference automaton for the query, and asymptotically run in either O ( nm/w ) or O ( nm log σ/ w ) time where w is the word size of the machine (e.g., 32 or 64 in practice), and σ is the size of the pattern alphabet. Here we present an algorithm of comparable simplicity that requires only O ( nm/w) time by virtue of computing a bit representation of the relocatable dynamic programming matrix for the problem. Thus, the algorithm's performance is independent of k , and it is found to be more efficient than the previous results for many choices of k and small m . Moreover, because the algorithm is not dependent on k , it can be used to rapidly compute blocks of the dynamic programming matrix as in the 4-Russians algorithm of Wu et al.(1996). This gives rise to an O(kn/w) expected-time algorithm for the case where m may be arbitrarily large. In practice this new algorithm, that computes a region of the dynamic progr amming (d.p.) matrx w entries at a time using the basic algorithm as a subroutine is significantly faster than our previous 4-Russians algorithm, that computes the same region 4 or 5 entries at a time using table lookup. This performance improvement yields a code that is either superior or competitive with all existing algorithms except for some filtration algorithms that are superior when k/m is sufficiently small.