<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 getgrid(a, b, c, d):
    #         if b < a or d < c:
    #             return 1e9
            
    #         lx, rx, ty, by = 1e9, -1, 1e9, -1

    #         for i in range(a, b + 1):
    #             for j in range(c, d + 1):
    #                 if grid[i][j]:
    #                     lx = min(lx, j)
    #                     rx = max(rx, j)
    #                     ty = min(ty, i)
    #                     by = max(by, i)
    #         return (rx - lx + 1) * (by - ty + 1)
        def getgrid(a, b, c, d): # inclusive [a, b], [c, d]
            if b < a or d < c:
                return 1e9
            lx, rx, ty, dy = 1e9, -1, 1e9, -1
            for i in range(a, b+1):
                for j in range(c, d+1):
                    if grid[i][j]:
                        lx = min(lx, j)
                        rx = max(rx, j)
                        ty = min(ty, i)
                        dy = max(dy, i)
            return (rx-lx+1) * (dy - ty+1)
        
        m, n = len(grid), len(grid[0])
        area = 1e9
        for i in range(m):
            for j in range(n):
                area = min(area, getgrid(0, i, 0, n - 1) + getgrid(i + 1, m - 1, 0, j) + getgrid(i + 1, m - 1, j + 1, n - 1))

                area = min(area, getgrid(0, i, 0, j) + getgrid(0, i, j + 1, n - 1)+ getgrid(i + 1, m - 1, 0, n - 1))

                area = min(area, getgrid(0, m - 1, 0, j) + getgrid(0, i, j + 1, n - 1) +  getgrid(i + 1, m - 1, j + 1, n - 1))

                area = min(area, getgrid(0, i, 0, j) + getgrid(i + 1, m - 1, 0, j) + getgrid(0, m - 1, j + 1, n - 1))

        

        for i in range(m):
            for j in range(i, m):
                area = min(area, getgrid(0, i, 0, n - 1) + getgrid(i + 1, j, 0, n - 1) + getgrid(j + 1, m - 1, 0, n - 1))
        
        for i in range(n):
            for j in range(i, n):
                area = min(area, getgrid(0, m - 1, 0, i) + getgrid(0, m - 1, i + 1, j) + getgrid(0, m - 1, j + 1, n - 1))

        return area
5697