<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:
                n = len(grid)
                max_score = 0
                dp = defaultdict(int)
                dp[frozenset()] = 0  
                for row in grid:
                    new_dp = defaultdict(int, dp)  
                    pq = []

                    for value in row:
                        for value_set in dp.keys():
                            if value not in value_set:  
                                new_value_set = frozenset(value_set | {value})
                                new_score = dp[value_set] + value
                                if new_score > new_dp[new_value_set]:
                                    new_dp[new_value_set] = new_score
                                    heappush(pq, (-new_score, new_value_set))  # Use negative for max-heap
                    
                    #to avoid MLE
                    next_dp = defaultdict(int)
                    while pq and len(next_dp) < 2000:  
                        score, value_set = heappop(pq)
                        next_dp[value_set] = max(next_dp[value_set], -score)
                    
                    if not next_dp:  
                        next_dp = dp

                    dp = next_dp  

                return max(dp.values())


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