<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)
        # dic = {}
        # for i in range(n):
        #     for j in grid[i]:
        #         if j not in dic:
        #             dic[j] = []
        #         dic[j].append(i)

        # valuelist = list(dic.items())

        # @cache
        # def dfs(mask, i):
        #     #return the max score you can get by starting from ith value with mask
        #     if i >= len(valuelist):
        #         return 0

        #     ans = dfs(mask, i+1)
        #     v = valuelist[i][0]
        #     for row in dic[v]:
        #         if not mask & (1<<row):
        #             newmask = mask | (1<<row)
        #             ans = max(ans, v+dfs(newmask, i+1))
        #     return ans
        
        # return dfs(0,0)





        #there are two constraints on this problem: the value needs to be unique and cannot be on the same row
        dic = {}#key: v value: row
        m = len(grid)
        n = len(grid[0])
        for i in range(m):
            for j in range(n):
                v = grid[i][j]
                if v not in dic:
                    dic[v] = []
                dic[v] += [i]
        
        ll = list(dic.keys())
        lval = len(ll)

        @cache
        def dfs(mask, vind):
            if vind >= len(ll):
                return 0

            v = ll[vind]
            ans = dfs(mask, vind+1)
            for r in dic[v]:
                if not mask & (1<<r):
                    ans = max(ans, v+dfs(mask | (1<<r), vind+1))
            return ans
        
        return dfs(0,0)

 















        
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