<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>

from collections import defaultdict
import functools
class Solution:
    def maxScore(self, grid: List[List[int]]) -> int:
        n = len(grid)
        m = len(grid[0])
        used = set()
        mem = {}
        values = []
        for i in range(n):
            for j in range(m):
                values.append((grid[i][j], i, j))

        values.sort(reverse=True)
        l = len(values)
        def backtrack(step, mask):
            if step == l:
                # print(step, n, "return 0")
                return 0
            if (step, mask) in mem:
                return mem[(step, mask)]
            res = backtrack(step+1, mask)
            row = values[step][1]
            if not ((1 << row) & mask):
                j = step + 1
                # not allowed to use duplicates valeus
                while j < l and values[step][0] == values[j][0]:
                    j += 1
                res = max(values[step][0] + backtrack(j, mask | (1 << row)),
                       res); 

            mem[(step, mask)] = res
            return res

        return backtrack(0, 0)


        
128040