Combinatorics or another approach?

Question

I've recently read this problem, first, I thought of a combinatorial approach, but it seems nobody - along the contestants - has submitted such a solution. Is a solution using combinatorics possible? If not, what's the solution? The problem briefly is: Given a dictionary of M words, where any two words can be joined together and possibly overlapping some letters with each other, find how many strings of length N can be formed from the dictionary. The combinatorial approach has a downer limit of M!, then for each two consecutive words, you should try intersecting them. That's what I thought of. I doubt it'd work. Please help?

You should have tried to summarize the relevant points here. This way we wouldn't be forced to follow a link and decipher a purposely misleading problem statement full of irrelevant and distracting info. (Also, the question title could be WAY more specific) — hugomg, Jul 02 '11 at 17:19
Sorry for any inconvenience, I'll summarize the problem. Thanks for the suggestion :D — Chris, Jul 02 '11 at 17:25
The direction I'd start off is to find out how exactly my words can overlap. — biziclop, Jul 02 '11 at 17:32

hugomg · Answer 1 · 2011-07-02T17:36:19.703

Don't confuse combinatorics with brute force. This is clearly a combinatorial counting problem but the time limit also rules out any brute force solution.

I think you can solve this with dinamic programming. For example, suppose our substring are "CAT and TCAT" and we need to cover a word of size 100

if we start with "CAT" we have 97 letters left if we start with "TCAT" we have 96 letters left.

So, if f(n) is the number of solutions for a matching of size n, we would have f(100) = f(97) + f(96).

Note, however, that this is clearly too simplified and not enough - you also need to cover the cases where the strings overlap and make sure that you don't count the same solution twice.

Oops, misread the problem this doesn't work. – hugomg Jul 02 '11 at 17:30 — hugomg, Jul 02 '11 at 17:30

score 0 · Answer 2 · answered Jul 02 '11 at 17:39

if you ignore the overlap, this is the subset sum problem. and considering the overlap, if you replace the overlapping combinations with their concatenations, you'll get a set of words that have no possiblity of overlap, and after that if you replace the strings with their lengths and then want to find sums that add up to N, that is exactly the subset sum problem.

Combinatorics or another approach?

2 Answers2