Dice (I) lightoj 1145

** Things **:

There are several ways to arrange n k-faces in a column so that the numbers on the face are S?

** Question type **:

Dynamic Planning

** Solutions **:

DP [I] [J] There are several methods to save the sum of the first I partitions as J.
Transition equation: DP [I] [J] = DP [I-1] [J-1] + dp [I-1] [J-2] +... + dp [I-1] [J-K];

There are N * s of statuses, And the transfer complexity is K. The simple practice will definitely time out. How can we optimize it?

Imagine the DP array as a two-dimensional matrix of the plane, and then the transfer equation, we can find that the transfer equations of DP [I] [J] and DP [I] [J-1] overlap a lot, as shown below:

DP [I] [J] = DP [I-1] [J-1] + dp [I-1] [J-2] +... + dp [I-1] [J-K];

DP [I] [J-1] = DP [I-1] [J-2] +... + dp [I-1] [J-K] + dp [I-1] [j-K-1];

Therefore, we can obtain DP [I] [J] = DP [I] [J-1]-DP [I-1] [j-K-1] + dp [I-1] [J-1];

The transfer has been optimized to O (1) complexity, but the DP array is N * s = W complexity. Therefore, we need to optimize the array to O (s ).

Although this question is classified as DP, there are no special restrictions, so there can be more efficient methods, and the O (n) complexity can be achieved by the principle of rejection.

If f (I) = n operators, at least I have more than K operators, and the total number of workers is S;

Then the answer is F (0)-f (1) + F (2)-f (3)... ± F (n)

F (I) method: N operators select I with C (n, I) types (this I is guaranteed to exceed K, other super does not matter ), in each case, the s-I * K should be divided into N parts and each part should be at least 1. So f (I) = C (n, I) * C (s-I * K-1, N-1 );