<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 = len(grid)
    n = len(grid[0])
    ans = m * n

    for i in range(m):
      top = self._minimumArea(grid, 0, i, 0, n - 1)
      for j in range(n):
        ans = min(ans, top +
                  self._minimumArea(grid, i + 1, m - 1, 0, j) +
                  self._minimumArea(grid, i + 1, m - 1, j + 1, n - 1))

    for i in range(m):
      bottom = self._minimumArea(grid, i, m - 1, 0, n - 1)
      for j in range(n):
        ans = min(ans, bottom +
                  self._minimumArea(grid, 0, i - 1, 0, j) +
                  self._minimumArea(grid, 0, i - 1, j + 1, n - 1))

    for j in range(n):
      left = self._minimumArea(grid, 0, m - 1, 0, j)
      for i in range(m):
        ans = min(ans, left +
                  self._minimumArea(grid, 0, i, j + 1, n - 1) +
                  self._minimumArea(grid, i + 1, m - 1, j + 1, n - 1))

    for j in range(n):
      right = self._minimumArea(grid, 0, m - 1, j, n - 1)
      for i in range(m):
        ans = min(ans, right +
                  self._minimumArea(grid, 0, i, 0, j - 1) +
                  self._minimumArea(grid, i + 1, m - 1, 0, j - 1))

    for i1 in range(m):
      for i2 in range(i1 + 1, m):
        ans = min(ans, self._minimumArea(grid, 0, i1, 0, n - 1) +
                  self._minimumArea(grid, i1 + 1, i2, 0, n - 1) +
                  self._minimumArea(grid, i2 + 1, m - 1, 0, n - 1))

    for j1 in range(n):
      for j2 in range(j1 + 1, n):
        ans = min(ans, self._minimumArea(grid, 0, m - 1, 0, j1) +
                  self._minimumArea(grid, 0, m - 1, j1 + 1, j2) +
                  self._minimumArea(grid, 0, m - 1, j2 + 1, n - 1))

    return ans

  def _minimumArea(
      self,
      grid: list[list[int]],
      si: int,
      ei: int,
      sj: int,
      ej: int,
  ) -> int:
    x1 = math.inf
    y1 = math.inf
    x2 = 0
    y2 = 0
    for i in range(si, ei + 1):
      for j in range(sj, ej + 1):
        if grid[i][j] == 1:
          x1 = min(x1, i)
          y1 = min(y1, j)
          x2 = max(x2, i)
          y2 = max(y2, j)
    return 0 if x1 == math.inf else (x2 - x1 + 1) * (y2 - y1 + 1)
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