KMP模式匹配算法的时间复杂度可以是O（m * n）？

Question

I have seen few answers here, but still need some clarification. KMP has O(n+m) worst case time complexity. But what I understood, in worst case scenario it should be O(n*m) . Lets take an example:

string(n):  aaaaaaaaa
pattern(m): aaab

First 3 characters match. Then there is a mismatch. Proper prefix suffix table for "aaa" would return 2. So only (3-2)=1 character move can be made according to KMP. So we are comparing n*(m-1) times before we can conclude that there is no match in this case. Effectively time complexity is O(n*m) .

Can somebody please explain how it is O(m+n) in this case. Do I have to consider anything else besides proper prefix suffix table and treat it as a special case. Is it mentioned in KMP? A detailed explanation for this particular scenario would be really helpful.

Answer 1

So only (3-2)=1 character move can be made according to KMP.

Yes.

So we are comparing n*(m-1) times before we can conclude that there is no match in this case.

No. The next thing we compare is a (4th of text) against a (3rd of pattern), which gives 2+1=3 as the length of the current match.

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Let us work the T= aaaaaaaaa text and P= aaab pattern example.

• On position 1, we have T[1]=P[1], so the length of the match is 1.
• On position 2, we have T[2]=P[2], so the length of the match is 2.
• On position 3, we have T[3]=P[3], so the length of the match is 3.
• On position 4, we have T[4]≠P[4]. The length of the match reduces to pref(3)=2 ( negative step ).
• On position 4 again, we have T[4]=P[3], so the length of the match is 3.
• On position 5, we have T[5]≠P[4]. The length of the match reduces to pref(3)=2 ( negative step ).
• On position 5 again, we have T[5]=P[3], so the length of the match is 3.
• And so on.

As you can see, starting from position 4, for each position, we make two steps instead of one. But we are not comparing substrings at any moment, only individual characters. The number of positions is |T|, the number of negative steps is at most |T|, so the total number of steps is linear with respect to |T|.