Split a string to a string of valid words using Dynamic Programming

Question

I need to find a dynamic programming algorithm to solve this problem. I tried but couldn\'t figure it out. Here is the problem:

You are given a string of n character

Answer 1

The string s[] can potentially be split into more than one ways. The method below finds the maximum number of words in which we can split s[]. Below is the sketch/pseudocode of the algorithm

bestScore[i] -> Stores the maximum number of words in which the first i characters can be split (it would be MINUS_INFINITY otherwise)

for (i = 1 to n){
     bestScore[i] = MINUS_INFINITY
     for (k = 1 to i-1){
        bestScore[i] = Max(bestSCore[i], bestScore[i-k]+ f(i,k))
     }
 }

Where f(i,k) is defined as:

f(i,k) = 1 : if s[i-k+1 to i] is in dictionary
       = MINUS_INFINITY : otherwise

bestScore[n] would store the maximum number of words in which s[] can be split (if the value is MINUS_INFINIY, s[] cannot be split)

Clearly the running time is O(n^2)

As this looks like a textbook exercise, I will not write the code to reconstruct the actual split positions.