<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 solve1(i1, i2, j1, j2):
            imin = jmin = inf
            imax = jmax = -1
            for i in range(i1, i2):
                for j in range(j1, j2):
                    if grid[i][j]:
                        imin = min(imin, i)
                        imax = max(imax, i)
                        jmin = min(jmin, j)
                        jmax = max(jmax, j)
            if imax == -1:
                return 0
            return (imax - imin + 1) * (jmax - jmin + 1)
        def solve2(i1, i2, j1, j2):
            ans = inf
            for i in range(i1+1, i2):
                ans = min(ans, solve1(i1, i, j1, j2) + solve1(i, i2, j1, j2))
            for j in range(j1+1, j2):
                ans = min(ans, solve1(i1, i2, j1, j) + solve1(i1, i2, j, j2))
            return ans
        def solve3(i1, i2, j1, j2):
            ans = inf
            for i in range(i1+1, i2):
                ans = min(ans, solve2(i1, i, j1, j2) + solve1(i, i2, j1, j2))
                ans = min(ans, solve1(i1, i, j1, j2) + solve2(i, i2, j1, j2))
            for j in range(j1+1, j2):
                ans = min(ans, solve2(i1, i2, j1, j) + solve1(i1, i2, j, j2))
                ans = min(ans, solve1(i1, i2, j1, j) + solve2(i1, i2, j, j2))
            return ans
        return solve3(0, len(grid), 0, len(grid[0]))
1409