Approximate String Matching in Sublinear Expected Time

The k differences approximate string matching problem specifies a text string of length n, a pattern string of length m, the number k of differences (insertions, deletions, substitutions) allowed in a match, and asks for every location in the text where a match occurs. Previous algorithms require at least O(nk) time. When k is as large as a fraction of m, no substantial progress has been made over O(nm) dynamic programming. We are interested in much faster algorithms for restricted cases of the problem, such as when the text string is random and errors are not too frequent. We have devised an algorithm that, for k < m/(logm + 0 ( 1 ) ) , runs in time O((n/m)k logm) on the average. In the worst case, our algorithm is O(nk) , but still an improvement in that it is very practical and uses only O(m) space compared to O(n) or O(m2). We define the approximate substring matching problem and give efficient algorithms based on our techniques. Special cases include several applications to genetics and molecular biology. For example, even allowing errors, we can find long common blocks of the text and pattern (local similarities), or select from among a set of text fragments ones that overlap one end of the pattern (sequence assembly). These are common tasks in DNA sequence analysis but are expensive to perform using previous techniques.