Fast Pattern Matching in Strings

An algorithm is presented which finds all occurrences of one. given string within another, in running time proportional to the sum of the lengths of the strings. The constant of proportionality is low enough to make this algorithm of practical use, and the procedure can also be extended to deal with some more general pattern-matching problems. A theoretical application of the algorithm shows that the set of concatenations of even palindromes, i.e., the language {can}*, can be recognized in linear time. Other algorithms which run even faster on the average are also considered. Text-editing programs are often required to search through a string of characters looking for instances of a given "pattern" string; we wish to find all positions, or perhaps only the leftmost position, in which the pattern occurs as a contiguous substring of the text. For example, c a e n a r y contains the pattern e n, but we do not regard c a n a r y as a substring. The obvious way to search for a matching pattern is to try searching at every starting position of the text, abandoning the search as soon as an incorrect character is found. But this approach can be very inefficient, for example when we are looking for an occurrence of aaaaaaab in aaaaaaaaaaaaaab. When the pattern is a"b and the text is a2"b, we will find ourselves making (n + 1) comparisons of characters. Furthermore, the traditional approach involves "backing up" the input text as we go through it, and this can add annoying complications when we consider the buffering operations that are frequently involved. In this paper we describe a pattern-matching algorithm which finds all occurrences of a pattern of length rn within a text of length n in O(rn + n) units of time, without "backing up" the input text. The algorithm needs only O(m) locations of internal memory if the text is read from an external file, and only O(log m) units of time elapse between consecutive single-character inputs. All of the constants of proportionality implied by these "O" formulas are independent of the alphabet size.