<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:
        @cache
        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, (r-l+1) * (d-u+1))

        rows, cols = len(grid), len(grid[0])
        res = rows * cols
        # case 1: horizontal * 3
        for a in range(1, rows - 1):
            for b in range(a + 1, rows):
                res = min(res, dp(0, a, 0, cols) + dp(a, b, 0, cols) + dp(b, rows, 0, cols))
        
        # case 2: vertical * 3
        for a in range(1, cols - 1):
            for b in range(a + 1, cols):
                res = min(res, dp(0, rows, 0, a) + dp(0, rows, a, b) + dp(0, rows, b, cols))
        
        for a in range(1, rows):
            for b in range(1, cols):
                # (horizontal) 1 + 2 
                res = min(res, dp(0, a, 0, cols) + dp(a, rows, 0, b) + dp(a, rows, b, cols))
                # (horizontal 2 + 1)
                res = min(res, dp(0, a, 0, b) + dp(0, a, b, cols) + dp(a, rows, 0, cols))
                # (vertical 1 + 2)
                res = min(res, dp(0, rows, 0, b) + dp(0, a, b, cols) + dp(a, rows, b, cols))
                # (vertical 2 + 1)
                res = min(res, dp(0, a, 0, b) + dp(a, rows, 0, b) + dp(0, rows, b, cols))

        print(dp(0, 1, 0, cols))
        print(dp(1, 2, 0, 2))
        if cols > 2:
            print(dp(1, 2, 2, cols))
        return res
1028