<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:
        def area(dims):
            x1, y1, x2, y2 = dims
            r1, c1, r2, c2 = inf, inf, -inf, -inf
            for r in range(x1, x2 + 1):
                for c in range(y1, y2 + 1):
                    if grid[r][c] == 1:
                        r1 = min(r1, r)
                        c1 = min(c1, c)
                        r2 = max(r2, r)
                        c2 = max(c2, c)
            if r1 == inf: 
                return 0  
            return (r2 - r1 + 1) * (c2 - c1 + 1)
        
        def cut(dims, cuts):
            x1, y1, x2, y2 = dims
            if cuts == 2:
                return area(dims)

            def eval(r1, r2):
                min_area = area(r1) + cut(r2, cuts + 1)
                options = float('inf')
                if not cuts: 
                    option = area(r2) + cut(r1, cuts + 1)
                    options = min(min_area, option)
                return min(min_area, options)

            min_area = area(dims)
            for c in range(x1 + 1, x2 + 1):
                r1, r2 = (x1, y1, c - 1, y2), (c, y1, x2, y2)
                min_area = min(min_area, eval(r1, r2))
                 
            for c in range(y1 + 1, y2 + 1):
                r1, r2 = (x1, y1, x2, c - 1), (x1, c, x2, y2)
                min_area = min(min_area, eval(r1, r2))
            
            return min_area
        
        return cut((0, 0, len(grid) - 1, len(grid[0]) - 1), 0)
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