<p>You are given a 2D <strong>binary</strong> array <code>grid</code>. You need to find 3 <strong>non-overlapping</strong> rectangles having <strong>non-zero</strong> areas with horizontal and vertical sides such that all the 1&#39;s in <code>grid</code> lie inside these rectangles.</p>

<p>Return the <strong>minimum</strong> possible sum of the area of these rectangles.</p>

<p><strong>Note</strong> that the rectangles are allowed to touch.</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 = [[1,0,1],[1,1,1]]</span></p>

<p><strong>Output:</strong> <span class="example-io">5</span></p>

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

<p><img alt="" src="https://assets.leetcode.com/uploads/2024/05/14/example0rect21.png" style="padding: 10px; background: rgb(255, 255, 255); border-radius: 0.5rem; width: 280px; height: 198px;" /></p>

<ul>
	<li>The 1&#39;s at <code>(0, 0)</code> and <code>(1, 0)</code> are covered by a rectangle of area 2.</li>
	<li>The 1&#39;s at <code>(0, 2)</code> and <code>(1, 2)</code> are covered by a rectangle of area 2.</li>
	<li>The 1 at <code>(1, 1)</code> is covered by a rectangle of area 1.</li>
</ul>
</div>

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

<div class="example-block">
<p><strong>Input:</strong> <span class="example-io">grid = [[1,0,1,0],[0,1,0,1]]</span></p>

<p><strong>Output:</strong> <span class="example-io">5</span></p>

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

<p><img alt="" src="https://assets.leetcode.com/uploads/2024/05/14/example1rect2.png" style="padding: 10px; background: rgb(255, 255, 255); border-radius: 0.5rem; width: 356px; height: 198px;" /></p>

<ul>
	<li>The 1&#39;s at <code>(0, 0)</code> and <code>(0, 2)</code> are covered by a rectangle of area 3.</li>
	<li>The 1 at <code>(1, 1)</code> is covered by a rectangle of area 1.</li>
	<li>The 1 at <code>(1, 3)</code> is covered by a rectangle of area 1.</li>
</ul>
</div>

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

<ul>
	<li><code>1 &lt;= grid.length, grid[i].length &lt;= 30</code></li>
	<li><code>grid[i][j]</code> is either 0 or 1.</li>
	<li>The input is generated such that there are at least three 1&#39;s in <code>grid</code>.</li>
</ul>

class Solution:
    def minimumSum(self, grid: List[List[int]]) -> int:

        @lru_cache(None)
        def area(r_beg: int, c_beg: int, r_end: int, c_end: int) -> int:

            maxRow = maxCol = -inf
            minRow = minCol = inf
            
            for r in range(r_beg, r_end + 1):
                for c in range(c_beg, c_end + 1):
                    if grid[r][c] == 1:
                        if r < minRow: minRow = r
                        if r > maxRow: maxRow = r
                        if c < minCol: minCol = c
                        if c > maxCol: maxCol = c
            if minRow == inf: return 0
            return (maxRow - minRow + 1) * (maxCol - minCol + 1)


        ans, m, n = inf, len(grid), len(grid[0])

        for r, c in product(range(m), range(n)):
            '''top, lft, bot, rght = (area(*(0, 0), *(r, n - 1))
                                   area(*(0, 0), *(r, n - 1)'''
                                   


            ans = min(ans, area(*(0, 0), *(r, n - 1)) + 
                           area(*(r + 1, 0), *(m - 1, c)) + 
                           area(*(r + 1, c + 1), *(m - 1, n - 1))  )

            ans = min(ans, area(*(r + 1, 0), *(m - 1, n - 1)) +
                           area(*(0, 0), *(r, c)) + 
                           area(*(0, c+1), *(r, n - 1)) )

            ans = min(ans,  area(*(0, 0), *(m - 1, c)) + 
                            area(*(0, c + 1), *(r, n - 1)) + 
                            area(*(r + 1, c + 1), *(m - 1, n - 1))   )

            ans = min(ans,  area(*(0, c + 1), *(m - 1, n - 1)) + 
                            area(*(0, 0), *(r, c)) + 
                            area(*(r + 1, 0), *(m-1, c)))


        for r1, r2 in combinations(range(m), 2):

            ans = min(ans,  area(*(0, 0)     , *(r1, n - 1)) + 
                            area(*(r1 + 1, 0), *(r2, n - 1)) + 
                            area(*(r2 + 1, 0), *(m-1, n - 1)))

        for c1, c2 in combinations(range(n), 2):

            ans = min(ans,  area(*(0, 0)     , *(m - 1, c1)) + 
                            area(*(0, c1 + 1), *(m - 1, c2)) + 
                            area(*(0, c2 + 1), *(m - 1, n - 1)))
 
        return ans
837