finding all cyclic substrings of a string

Question

I have been stuck on this problem for a while now; any help would be appreciated.

Given a string S, find the number of distinct substrings which are composed of 2 or more consecutive strings. For example, "abbccbbadad" has 3 because "bb" is composed of 2 "b"s, "cc" is composed of 3 "c"s, and "adad" is composed of 2 "ad"s.

My solution uses hashing and currently runs in n^2 time, which is fine because the length of the string is <= 5000. However, my program uses n^3 space. I am confident that a solution requiring n^2 space and n^2 time will pass. Is there a more efficient solution?

Answer 1

Any algorithm that runs in $O(n^2)$ time can access at most $O(n^2)$ different memory locations during the course of its execution. Thus, if you replace memory with a hashtable (instead of accessing address $a$, access the hashtable at key $a$), you obtain an algorithm with expected running time $O(n^2)$ and space $O(n^2)$. If you care about theoretical proofs, you can use a balanced binary search tree instead of a hashtable and obtain a worst-case time bound of $O(n^2 \log n)$ time and $O(n^2)$ space.