A Dynamic programming problem

Question

Can anyone help me find an optimal Dynamic programming algorithm for this problem

On the way to dinner, the CCC competitors are lining up for their delicious curly f

Answer 1

I'd prefer not to solve an SPOJ problem in a practical manner for you, so take the following as an existence proof of a poly-time DP.

For K fixed, the set of strings that can dine is context-free. I'm going to use g and h instead of G and H. For example, for K = 3, one grammar looks like

S -> ε | g S g S g S G | h S h S h S H

G -> ε | g S G

H -> ε | h S H

The idea is that either there are no diners, or the first diner dines with at least K - 1 others, between any two of which (and the last and the end) there is a string that can dine.

Now use the weighted variant of CYK to find the minimum-weight parse, where nonempty S productions have weight 1, and all others have weight 0. For K fixed, the running time of CYK is O(N³).