After a shift of the pattern, the naive algorithm has forgotten all information about previously matched symbols. So it is possible that it re-compares a text symbol with different pattern symbols again and again. This leads to its worst case complexity of Θ(nm) (n: length of the text, m: length of the pattern).
The algorithm of Knuth, Morris and Pratt [KMP 77] makes use of the information gained by previous symbol comparisons. It never re-compares a text symbol that has matched a pattern symbol. As a result, the complexity of the searching phase of the Knuth-Morris-Pratt algorithm is in O(n).
However, a preprocessing of the pattern is necessary in order to analyze its structure. The preprocessing phase has a complexity of O(m). Since m
n, the overall complexity of the Knuth-Morris-Pratt algorithm is in O(n).
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