<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:
        m,n = len(grid),len(grid[0])

        @lru_cache(None)
        def min_area1(r0,c0,r1,c1):
            x0,y0,x1,y1 = m,n,0,0
            for i in range(r0,r1):
                for j in range(c0,c1):
                    if grid[i][j]:
                        x0,y0 = min(x0,i),min(y0,j)
                        x1,y1 = max(x1,i),max(y1,j)
            return (x1-x0+1)*(y1-y0+1)
        
        @lru_cache(None)
        def min_area2(r0,c0,r1,c1):
            ans = (r1-r0+1)*(c1-c0+1)
            for i in range(r0,r1):
                ans = min(ans, min_area1(r0,c0,i+1,c1)+min_area1(i+1,c0,r1,c1))
            for j in range(c0,c1):
                ans = min(ans, min_area1(r0,c0,r1,j+1)+min_area1(r0,j+1,r1,c1))
            return ans

        ans = m*n
        for i in range(m):
            ans = min(ans, min_area1(0,0,i+1,n)+min_area2(i+1,0,m,n))
            ans = min(ans, min_area1(m-i-1,0,m,n)+min_area2(0,0,m-i-1,n))
        for j in range(n):
            ans = min(ans, min_area1(0,0,m,j+1)+min_area2(0,j+1,m,n))
            ans = min(ans, min_area1(0,n-j-1,m,n)+min_area2(0,0,m,n-j-1))
        return ans
1599