<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 dp(x1, x2, y1, y2):
            u, d, l, r = x2-1, x1, y2-1, y1

            for i in range(x1, x2):
                for j in range(y1, y2):
                    if grid[i][j]:
                        u = min(u, i)
                        d = max(d, i)
                        l = min(l, j)
                        r = max(r, j)
            
            return max(0, (d-u+1) * (r-l+1))
        
        r, c = len(grid), len(grid[0])
        res = r * c

        for x in range(1, r):
            for y in range(1, c):
                res = min(res, dp(0, r, 0, y) + dp(0, x, y, c) + dp(x, r, y, c))
                res = min(res, dp(0, x, 0, y) + dp(x, r, 0, y) + dp(0, r, y, c))
                res = min(res, dp(0, x, 0, c) + dp(x, r, 0, y) + dp(x, r, y, c))
                res = min(res, dp(0, x, 0, y) + dp(0, x, y, c) + dp(x, r, 0, c))
        
        for x1 in range(1, r-1):
            for x2 in range(x1+1, r):
                res = min(res, dp(0, x1, 0, c) + dp(x1, x2, 0, c) + dp(x2, r, 0, c))
        
        for y1 in range(1, c-1):
            for y2 in range(y1+1, c):
                res = min(res, dp(0, r, 0, y1) + dp(0, r, y1, y2) + dp(0, r, y2, c))
        
        return res
4363