<p>You are given a 2D matrix <code>grid</code> consisting of positive integers.</p>

<p>You have to select <em>one or more</em> cells from the matrix such that the following conditions are satisfied:</p>

<ul>
	<li>No two selected cells are in the <strong>same</strong> row of the matrix.</li>
	<li>The values in the set of selected cells are <strong>unique</strong>.</li>
</ul>

<p>Your score will be the <strong>sum</strong> of the values of the selected cells.</p>

<p>Return the <strong>maximum</strong> score you can achieve.</p>

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

<div class="example-block">
<p><strong>Input:</strong> <span class="example-io">grid = [[1,2,3],[4,3,2],[1,1,1]]</span></p>

<p><strong>Output:</strong> <span class="example-io">8</span></p>

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

<p><img alt="" src="https://assets.leetcode.com/uploads/2024/07/29/grid1drawio.png" /></p>

<p>We can select the cells with values 1, 3, and 4 that are colored above.</p>
</div>

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

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

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

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

<p><img alt="" src="https://assets.leetcode.com/uploads/2024/07/29/grid8_8drawio.png" style="width: 170px; height: 114px;" /></p>

<p>We can select the cells with values 7 and 8 that are colored above.</p>
</div>

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

<ul>
	<li><code>1 &lt;= grid.length, grid[i].length &lt;= 10</code></li>
	<li><code>1 &lt;= grid[i][j] &lt;= 100</code></li>
</ul>

class Solution:
    def maxScore(self, grid: List[List[int]]) -> int:
        m, n = len(grid), len(grid[0])

        values = []

        for r in range(m):
            for c in range(n):
                values.append((grid[r][c], r, c))
        
        values.sort(reverse=True)

        # @cache
        # def maxSum(l, selectedRows, prevVal):
        #     if l == len(values):
        #         return 0

        #     ans = maxSum(l+1, selectedRows, prevVal)

        #     val, r, c = values[l]
        #     if selectedRows & (1<<r) == 0 and prevVal != val:
        #         ans = max(ans, maxSum(l+1, selectedRows | (1<<r), val) + val)

        #     return ans


        # return maxSum(0, 0, -1)

        dp = {}
        dp[(0, -1)] = 0
        
        for val, r, c in values:
            next_dp = dp.copy()
            
            for (selectedRows, prevVal), score in dp.items():
                if selectedRows & (1 << r) == 0 and val != prevVal:
                    new_selectedRows = selectedRows | (1 << r)
                    new_score = score + val
                    
                    if (new_selectedRows, val) not in next_dp:
                        next_dp[(new_selectedRows, val)] = new_score
                    else:
                        next_dp[(new_selectedRows, val)] = max(next_dp[(new_selectedRows, val)], new_score)
            
            dp = next_dp
        
        return max(dp.values())
24440