The naive method

Almost all discussions of exact matching begin with the naive method, and we follow this tradition. The naive method aligns the left end of P with the left end of T and then compares the characters of P and T left to right until either two unequal characters are found or until P is exhausted, in which case an occurrence of P is reported. In either case, P is then shifted one place to the right, and the comparisons are restarted from the left end of P. This process repeats until the right end of P shifts past the right end of T.

Using n to denote the length of P and m to denote the length of T, the worst-case number of comparisons made by this method is Θ(nm). In particular, if both P and T consist of the same repeated character, then there is an occurrence of P at each of the first m − n + 1 positions of T and the method performs exactly n(m − n + 1) comparisons. For example, if P = aaa and T = aaaaaaaaaa then n = 3, m = 10, and 24 comparisons are made.

The naive method is certainly simple to understand and program, but its worst-case running time of Θ(nm) may be unsatisfactory and can be improved. Even the practical running time of the naive method may be too slow for larger texts and patterns.