<p>You are given a <code>m x n</code> 2D array <code>board</code> representing a chessboard, where <code>board[i][j]</code> represents the <strong>value</strong> of the cell <code>(i, j)</code>.</p>

<p>Rooks in the <strong>same</strong> row or column <strong>attack</strong> each other. You need to place <em>three</em> rooks on the chessboard such that the rooks <strong>do not</strong> <strong>attack</strong> each other.</p>

<p>Return the <strong>maximum</strong> sum of the cell <strong>values</strong> on which the rooks are placed.</p>

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

<div class="example-block">
<p><strong>Input:</strong> <span class="example-io">board = </span>[[-3,1,1,1],[-3,1,-3,1],[-3,2,1,1]]</p>

<p><strong>Output:</strong> 4</p>

<p><strong>Explanation:</strong></p>

<p><img alt="" src="https://assets.leetcode.com/uploads/2024/08/08/rooks2.png" style="width: 294px; height: 450px;" /></p>

<p>We can place the rooks in the cells <code>(0, 2)</code>, <code>(1, 3)</code>, and <code>(2, 1)</code> for a sum of <code>1 + 1 + 2 = 4</code>.</p>
</div>

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

<div class="example-block">
<p><strong>Input:</strong> <span class="example-io">board = [[1,2,3],[4,5,6],[7,8,9]]</span></p>

<p><strong>Output:</strong> <span class="example-io">15</span></p>

<p><strong>Explanation:</strong></p>

<p>We can place the rooks in the cells <code>(0, 0)</code>, <code>(1, 1)</code>, and <code>(2, 2)</code> for a sum of <code>1 + 5 + 9 = 15</code>.</p>
</div>

<p><strong class="example">Example 3:</strong></p>

<div class="example-block">
<p><strong>Input:</strong> <span class="example-io">board = [[1,1,1],[1,1,1],[1,1,1]]</span></p>

<p><strong>Output:</strong> <span class="example-io">3</span></p>

<p><strong>Explanation:</strong></p>

<p>We can place the rooks in the cells <code>(0, 2)</code>, <code>(1, 1)</code>, and <code>(2, 0)</code> for a sum of <code>1 + 1 + 1 = 3</code>.</p>
</div>

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

<ul>
	<li><code>3 &lt;= m == board.length &lt;= 100</code></li>
	<li><code>3 &lt;= n == board[i].length &lt;= 100</code></li>
	<li><code>-10<sup>9</sup> &lt;= board[i][j] &lt;= 10<sup>9</sup></code></li>
</ul>

def top3_elements(arr):
    top1, top2, top3 = float('-inf'), float('-inf'), float('-inf')
    idx1, idx2, idx3 = -1, -1, -1

    for idx, val in enumerate(arr):
        if val > top1:
            top3, top2, top1 = top2, top1, val
            idx3, idx2, idx1 = idx2, idx1, idx
        elif val > top2:
            top3, top2 = top2, val
            idx3, idx2 = idx2, idx
        elif val > top3:
            top3 = val
            idx3 = idx

    return [(top1, idx1), (top2, idx2), (top3, idx3)]

class Solution:
    def maximumValueSum(self, board: List[List[int]]) -> int:
        n_row, n_col = len(board), len(board[0])

        matrix = []

        for row in range(n_row):
            temp = top3_elements(board[row])
            matrix.append(temp)
        
        dp = {}
        
        def solve(row, selected_col):
            if len(selected_col) == 3:
                return 0
            
            if row == n_row:
                return float("-inf")
            
            key = (row, tuple(sorted(selected_col)))
            if key in dp:
                return dp[key]

            ans = float("-inf")
            
            for val, col in matrix[row]:
                if col not in selected_col:
                    selected_col.add(col)
                    ans = max(ans, val + solve(row+1, selected_col))
                    selected_col.remove(col)
            
            ans = max(ans, solve(row+1, selected_col))
            dp[key] = ans
            return ans
        
        return solve(0, set())
73718