A Variation on the Boyer-Moore Algorithm

String-matching consists in finding all the occurrences of a word w in a text t. Several algorithms have been found for solving this problem. They are presented by Aho in a recent book [l]. Among these algorithms, the Boyer-Moore approach [S, 1 l] seems to lead to the fastest algorithms for the search phase. Even if the original version of the Bayer-Moore algorithm has a quadratic worst case, its behavior in practice seems to be sublinear. Furthermore, other authors [9,2] have improved this worst-case time complexity for the search phase so that it becomes linear in the length of the text. The best bound for the number of letter comparisons is due to Apostolico and Giancarlo [2] and is 2n-m+ 1, where n is the length of the text and m the length of the word. Another particularity of the Boyer-Moore algorithm is that the study of its complexity is not obvious; see [lo, 73. Basically, the Boyer-Moore algorithm tries to find for a given position in the text the longest suffix of the word which ends at that position. A new approach can possess the ability for a given position in the text to compute the length of the longest prefix of the word which ends at that position. When we know this length, we are able to compute a better shift than the Boyer-Moore approach. In the first version we make a new attempt at matching, forgetting all the previous prefixes matched. This leads to a very simple algorithm but it has a quadratic worst-case running time. In an improved version we memorize the position where the previous longest prefix found ends and we make a new attempt at matching only the number of characters corresponding to the complement of this prefix. We are then able to compute a shift without reading again backwards more than half the characters of the prefix found in the previous attempt. This leads to a linear-time algorithm which scans the text characters at most three times each.