<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:
        R = len(grid)
        C = len(grid[0])
        best = math.inf
        if len(grid) >= 3:
            for er1 in range(1, len(grid) - 1):
                for er2 in range(er1+1, len(grid)):
                    best = min(best, self.minimumAreaH(grid, 0, er1, 0, C) + self.minimumAreaH(grid, er1, er2, 0, C) + self.minimumAreaH(grid, er2, R, 0, C))
        if len(grid[0]) >= 3:
            for ec1 in range(1, C - 1):
                for ec2 in range(ec1+1, C):
                    best = min(best, self.minimumAreaH(grid, 0, R, 0, ec1) + self.minimumAreaH(grid, 0, R, ec1, ec2) + self.minimumAreaH(grid, 0, R, ec2, C))
        if len(grid) >= 2 and len(grid[0]) >= 2:
            for er1 in range(1, len(grid)):
                for ec1 in range(1, C):
                    best = min(best, self.minimumAreaH(grid, 0, er1, 0, C) + self.minimumAreaH(grid, er1, R, 0, ec1) + self.minimumAreaH(grid, er1, R, ec1, C))
                    best = min(best, self.minimumAreaH(grid, er1,R, 0, C) +  self.minimumAreaH(grid, 0, er1, 0, ec1) + self.minimumAreaH(grid, 0, er1, ec1, C))
        if len(grid) >= 2 and len(grid[0]) >= 2:
            for ec1 in range(1, C):
                for er1 in range(1, R):
                    best = min(best, self.minimumAreaH(grid, 0, R, 0, ec1) + self.minimumAreaH(grid, 0, er1, ec1, C) + self.minimumAreaH(grid, er1, R, ec1, C))
                    best = min(best, self.minimumAreaH(grid, 0, R, ec1, C) + self.minimumAreaH(grid, 0, er1, 0, ec1) + self.minimumAreaH(grid, er1, R, 0, ec1))
        return best


    def minimumAreaH(self, grid: List[List[int]], sr, er, sc, ec) -> int:
        top = math.inf
        bottom = -math.inf
        left = math.inf
        right = -math.inf
        for r in range(sr, er):
            for c in range(sc, ec):
                if grid[r][c] == 1:
                    top = min(top, r)
                    bottom = max(bottom, r)
                    left = min(left, c)
                    right = max(right, c)
        width = abs(top - (bottom + 1))
        height = abs(left - (right + 1))
        return width * height
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