Complexity of Sequential Pattern Matching Algorithms
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This paper considers the exact number of character comparisons needed to find all occurrences of a pattern of length $m$ in a text of length $n$ using on-line and general algorithms. For on-line algorithms, a lower bound of about $(1+\frac{9}{4(m+1)})\cdot n$ character comparisons is obtained. For general algorithms, a lower bound of about $(1+\frac{2}{m+3})\cdot n$ character comparisons is obtained. These lower bounds complement an on-line upper bound of about $(1+\frac{8}{3(m+1)})\cdot n$ comparisons obtained recently by Cole and Hariharan. The lower bounds are obtained by finding patterns with interesting combinatorial properties. It is also shown that for some patterns off-line algorithms can be more efficient than on-line algorithms.