Question

I’m looking at the algorithm on http://en.wikipedia.org/wiki/Longest_common_substring_problem

They use dynamic programming, which gives them a time of O(nm). However, can’t the same time complexity be achieved with a brute force algorithm? I am doing a homework question, to find this algorithm in O(n*m) time, where n and m are the string lengths.

For a string A and string B, I check if A[i] equals any element in B. If it does equal some B[j], then check if A[i + 1] equals to B[j + 1], then if A[i + 2] = in B[i + 2] and so forth until no more match or end of string. If it is the no match case, then continue checking A[i] in B starting from the last element we checked in B. We repeat this process for every A element, while storing the start and end index for the max substring found so far. This algorithm appears to be O(n*m). If I’m not wrong about this, is there any reason why this approach is not used?

Thanks for any help.

Answer 1

If I’m reading your algorithm correctly, then I believe it is wrong.

Let A = "abac" and B = "ababac". Then, with i = 0, we see that the strings match at j = 0. So we start matching, and fail at j = 3 since b != c. So we start matching at j = 3, but fail immediately since b != a (note that we don’t start at j = 2 since we successfully matched a = a there). We would then conclude the longest substring is aba, which is incorrect.