Editorial for CCC '16 S4 - Combining Riceballs

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Author: FatalEagle

Suppose the riceballs are numbered from 11 to NN, with riceball ii having size A[i]A[i]. Define a boolean function f(i, j)f(i, j) to return 11 if the riceballs in the range [i, j][i, j] can all be combined into a single riceball, or 00 otherwise. The final answer to the problem is:

\displaystyle \max_{i=1}^N \max_{j=i}^N f(i, j) \cdot \sum_{k=i}^j A[k]\displaystyle \max_{i=1}^N \max_{j=i}^N f(i, j) \cdot \sum_{k=i}^j A[k]

Remember that no matter how you combine riceballs in an interval, after they are combined the size will always be the same.

We have the following recurrence:

\displaystyle f(i, j) = \begin{cases}
1 & \text{if } i \ge j \\
\max f(i, a) \cdot f(a+1, b-1) \cdot f(b, j) & \text{otherwise, where } \sum_{k=i}^a A[k] = \sum_{\ell=b}^j A[\ell]
\end{cases}\displaystyle f(i, j) = \begin{cases}
1 & \text{if } i \ge j \\
\max f(i, a) \cdot f(a+1, b-1) \cdot f(b, j) & \text{otherwise, where } \sum_{k=i}^a A[k] = \sum_{\ell=b}^j A[\ell]

To speed this up, notice that for every aa there is at most one bb that satisfies the size restriction, and when aa increases, bb decreases, so all valid pairs may be found with a two-pointers algorithm.

Complexity: \mathcal{O}(N^3)\mathcal{O}(N^3)


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