<p>Alice and Bob take turns playing a game, with <strong>Alice starting first</strong>.</p>

<p>There are <code>n</code> stones arranged in a row. On each player&#39;s turn, they can <strong>remove</strong> either the leftmost stone or the rightmost stone from the row and receive points equal to the <strong>sum</strong> of the remaining stones&#39; values in the row. The winner is the one with the higher score when there are no stones left to remove.</p>

<p>Bob found that he will always lose this game (poor Bob, he always loses), so he decided to <strong>minimize the score&#39;s difference</strong>. Alice&#39;s goal is to <strong>maximize the difference</strong> in the score.</p>

<p>Given an array of integers <code>stones</code> where <code>stones[i]</code> represents the value of the <code>i<sup>th</sup></code> stone <strong>from the left</strong>, return <em>the <strong>difference</strong> in Alice and Bob&#39;s score if they both play <strong>optimally</strong>.</em></p>

<p>&nbsp;</p>
<p><strong class="example">Example 1:</strong></p>

<pre>
<strong>Input:</strong> stones = [5,3,1,4,2]
<strong>Output:</strong> 6
<strong>Explanation:</strong> 
- Alice removes 2 and gets 5 + 3 + 1 + 4 = 13 points. Alice = 13, Bob = 0, stones = [5,3,1,4].
- Bob removes 5 and gets 3 + 1 + 4 = 8 points. Alice = 13, Bob = 8, stones = [3,1,4].
- Alice removes 3 and gets 1 + 4 = 5 points. Alice = 18, Bob = 8, stones = [1,4].
- Bob removes 1 and gets 4 points. Alice = 18, Bob = 12, stones = [4].
- Alice removes 4 and gets 0 points. Alice = 18, Bob = 12, stones = [].
The score difference is 18 - 12 = 6.
</pre>

<p><strong class="example">Example 2:</strong></p>

<pre>
<strong>Input:</strong> stones = [7,90,5,1,100,10,10,2]
<strong>Output:</strong> 122</pre>

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

<ul>
	<li><code>n == stones.length</code></li>
	<li><code>2 &lt;= n &lt;= 1000</code></li>
	<li><code>1 &lt;= stones[i] &lt;= 1000</code></li>
</ul>

class Solution:
  def stoneGameVII(self, stones: List[int]) -> int:
    n = len(stones)

    # sum[i,j] = prefix_sums[j+1]-prefix_sums[i].
    prefix_sums = [0] * (n + 1)
    for i, num in enumerate(stones):
      prefix_sums[i + 1] = prefix_sums[i] + num

    def _get_sum(i, j):
      return prefix_sums[j + 1] - prefix_sums[i]

    # dp[(i, j, is_player1)]: for stones[i..j], and is_player1.
    dp = [[-1] * n for _ in range(0, n)]

    def _helper(i, j, is_player1):
      if i >= j:
        return 0, 0
      if dp[i][j] != -1:
        return dp[i][j]

      scl1, scl2 = _helper(i+1, j, not is_player1)
      scr1, scr2 = _helper(i, j-1, not is_player1)
      suml = _get_sum(i+1, j)
      sumr = _get_sum(i, j-1)

      if is_player1:
        if scl1 + suml - scl2 > scr1 + sumr - scr2:
          res = (scl1 + suml, scl2)
        else:
          res = (scr1 + sumr, scr2)
      else:
        if scl1 - (scl2 + suml) < scr1 - (scr2 + sumr):
          res = (scl1, scl2 + suml)
        else:
          res = (scr1, scr2 + sumr)
      dp[i][j] = res
      return dp[i][j]

    sc1, sc2 = _helper(0, n-1, True)
    return sc1 - sc2
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