Question

I would also like to know which algorithm has the worst case complexity of all for finding all occurrences of a string in another. Seems like Boyer–Moore’s algorithm has a linear time complexity.

Answer 1

The KMP algorithm has linear complexity for finding all occurrences of a pattern in a string, like the Boyer-Moore algorithm¹. If you try to find a pattern like “aaaaaa” in a string like “aaaaaaaaa”, once you have the first complete match,

aaaaaaaaa
aaaaaa
 aaaaaa
      ^

the border table contains the information that the next longest possible match (corresponding to the widest border of the pattern) of a prefix of the pattern is just one character short (a complete match is equivalent to a mismatch one past the end of the pattern in this respect). Thus the pattern is moved one place further, and since from the border table it is known that all characters of the pattern except possibly the last match, the next comparison is between the last pattern character and the aligned text character. In this particular case (find occurrences of a^m in aⁿ), which is the worst case for the naive matching algorithm, the KMP algorithm compares each text character exactly once.

In each step, at least one of

• the position of the text character compared
• the position of the first character of the pattern with respect to the text

increases, and neither ever decreases. The position of the text character compared can increase at most length(text)-1 times, the position of the first pattern character can increase at most length(text) - length(pattern) times, so the algorithm takes at most 2*length(text) - length(pattern) - 1 steps.

The preprocessing (construction of the border table) takes at most 2*length(pattern) steps, thus the overall complexity is O(m+n) and no more m + 2*n steps are executed if m is the length of the pattern and n the length of the text.

¹ Note that the Boyer-Moore algorithm as commonly presented has a worst-case complexity of O(m*n) for periodic patterns and texts like a^m and aⁿ if all matches are required, because after a complete match,

aaaaaaaaa
aaaaaa
 aaaaaa
      ^
  <- <-
 ^

the entire pattern would be re-compared. To avoid that, you need to remember how long a prefix of the pattern still matches after the shift following a complete match and only compare the new characters.