Algorithm for checking if a string was built from a list of substrings

Question

You are given a string and an array of strings. How to quickly check, if this string can be built by concatenating some of the strings in the array?

This is a theoretica

Answer 1

If each substring must be used only once but not all of them must be used...

For each permutation of size N from the substrings that is equal in size to the original string check it, if none, do a permutation of N+1 items, end so forth, until you exhaust all the permutations.

Of course NP complete, slow as hell, but i think that no normal solutions exist.

To explain why the solutions where removing substrings from the original string won't ever work:

Have a string "1234123" and array "12","34","123". If you remove "123" from the start, you have a false negative. A similar example where removing from the end would be: "1234123" : "23,"41","123".

With backtracking with greedy: (m string length 7, n num elements 3) - take the longest: 123 - remove it from first occurence O(3) - try other two with the rest: no go + O((n-1)*(m-3)) - backtrack O(1) - remove from second: O(m-3) - try other two O((n-1)*m-3) = O(30)

Permutations of 1 + 2 + 3 = O(3) + O(4) + O(6) = O(13). So for small subset lenght permutations are actually faster than backtracking. This will change if you ask for a lot of substrings to find (in most cases but not all).

You can remove only the nonexisting substrings from the array to lower the number of permutations from n^n to n^(n-1) for each removed nonexisting substring.