<p>There is a pizza with <code>3n</code> slices of varying size, you and your friends will take slices of pizza as follows:</p>

<ul>
	<li>You will pick <strong>any</strong> pizza slice.</li>
	<li>Your friend Alice will pick the next slice in the anti-clockwise direction of your pick.</li>
	<li>Your friend Bob will pick the next slice in the clockwise direction of your pick.</li>
	<li>Repeat until there are no more slices of pizzas.</li>
</ul>

<p>Given an integer array <code>slices</code> that represent the sizes of the pizza slices in a clockwise direction, return <em>the maximum possible sum of slice sizes that you can pick</em>.</p>

<p>&nbsp;</p>
<p><strong class="example">Example 1:</strong></p>
<img alt="" src="https://assets.leetcode.com/uploads/2020/02/18/sample_3_1723.png" style="width: 500px; height: 266px;" />
<pre>
<strong>Input:</strong> slices = [1,2,3,4,5,6]
<strong>Output:</strong> 10
<strong>Explanation:</strong> Pick pizza slice of size 4, Alice and Bob will pick slices with size 3 and 5 respectively. Then Pick slices with size 6, finally Alice and Bob will pick slice of size 2 and 1 respectively. Total = 4 + 6.
</pre>

<p><strong class="example">Example 2:</strong></p>
<img alt="" src="https://assets.leetcode.com/uploads/2020/02/18/sample_4_1723.png" style="width: 500px; height: 299px;" />
<pre>
<strong>Input:</strong> slices = [8,9,8,6,1,1]
<strong>Output:</strong> 16
<strong>Explanation:</strong> Pick pizza slice of size 8 in each turn. If you pick slice with size 9 your partners will pick slices of size 8.
</pre>

<p>&nbsp;</p>
<p><strong>Constraints:</strong></p>

<ul>
	<li><code>3 * n == slices.length</code></li>
	<li><code>1 &lt;= slices.length &lt;= 500</code></li>
	<li><code>1 &lt;= slices[i] &lt;= 1000</code></li>
</ul>

class Solution:
    """
        Idea: take the maximum sum subset of n non-adjacent slices.
        max_sum_subset_fst[slice_idx][slices_eaten] -> maximum sum subset of n non-adjacent slices up to slice slice_idx with taking first slice.
        max_sum_subset_nonfst[slice_idx][slices_eaten] -> maximum sum subset of n non-adjacent slices up to slice slice_idx without taking first slice.
    """
    def maxSizeSlices(self, slice_sizes: List[int]) -> int:
        assert(len(slice_sizes) % 3 == 0)
        slices_per_person: int = len(slice_sizes) // 3

        max_sum_subset_fst: List[List[int]] = [[-math.inf] * (slices_per_person + 1) for slice_idx in range(len(slice_sizes) + 2)]
        max_sum_subset_fst[1][1] = slice_sizes[0]

        max_sum_subset_nonfst: List[List[int]] = [[-math.inf] * (slices_per_person + 1) for slice_idx in range(len(slice_sizes) + 2)]
        for slice_idx in range(len(slice_sizes) + 2):
            max_sum_subset_nonfst[slice_idx][0] = 0

        for slice_idx, slice_size in enumerate(slice_sizes[1:], start = 2):
            for slices_eaten in range(1, slices_per_person + 1):
                max_sum_subset_fst[slice_idx][slices_eaten] = max(
                    max_sum_subset_fst[slice_idx - 2][slices_eaten - 1] + slice_sizes[slice_idx - 1],
                    max_sum_subset_fst[slice_idx - 1][slices_eaten],
                )
                max_sum_subset_nonfst[slice_idx][slices_eaten] = max(
                    max_sum_subset_nonfst[slice_idx - 2][slices_eaten - 1] + slice_sizes[slice_idx - 1],
                    max_sum_subset_nonfst[slice_idx - 1][slices_eaten],
                )
        print(max_sum_subset_nonfst)
        return max(
            max_sum_subset_fst[len(slice_sizes) - 1][slices_per_person],
            max_sum_subset_nonfst[len(slice_sizes)][slices_per_person]
        )

20400