<p>Given a <code>m x n</code> matrix <code>mat</code> and an integer <code>threshold</code>, return <em>the maximum side-length of a square with a sum less than or equal to </em><code>threshold</code><em> or return </em><code>0</code><em> if there is no such square</em>.</p>

<p>&nbsp;</p>
<p><strong class="example">Example 1:</strong></p>
<img alt="" src="https://assets.leetcode.com/uploads/2019/12/05/e1.png" style="width: 335px; height: 186px;" />
<pre>
<strong>Input:</strong> mat = [[1,1,3,2,4,3,2],[1,1,3,2,4,3,2],[1,1,3,2,4,3,2]], threshold = 4
<strong>Output:</strong> 2
<strong>Explanation:</strong> The maximum side length of square with sum less than 4 is 2 as shown.
</pre>

<p><strong class="example">Example 2:</strong></p>

<pre>
<strong>Input:</strong> mat = [[2,2,2,2,2],[2,2,2,2,2],[2,2,2,2,2],[2,2,2,2,2],[2,2,2,2,2]], threshold = 1
<strong>Output:</strong> 0
</pre>

<p>&nbsp;</p>
<p><strong>Constraints:</strong></p>

<ul>
	<li><code>m == mat.length</code></li>
	<li><code>n == mat[i].length</code></li>
	<li><code>1 &lt;= m, n &lt;= 300</code></li>
	<li><code>0 &lt;= mat[i][j] &lt;= 10<sup>4</sup></code></li>
	<li><code>0 &lt;= threshold &lt;= 10<sup>5</sup></code></li>
</ul>

class Solution:
    def maxSideLength(self, mat: List[List[int]], threshold: int) -> int:
        rows = len(mat)
        cols = len(mat[0])

        prefix_mat = [[0] * (cols + 1) for _ in range(rows + 1)]

        for r in range(rows):
            for c in range(cols):
                prefix_mat[r+1][c+1] = prefix_mat[r][c+1] + prefix_mat[r+1][c] - prefix_mat[r][c] + mat[r][c]

        max_size = 0

        for r in range(rows):
            for c in range(cols):
                if min(r, c) >= max_size:
                    curr = prefix_mat[r + 1][c + 1]
                    top = prefix_mat[r - max_size][c + 1]
                    left = prefix_mat[r + 1][c - max_size]
                    top_left = prefix_mat[r - max_size][c - max_size]

                    sq_sum = curr - top - left + top_left
                    if sq_sum <= threshold:
                        max_size += 1

        return max_size
21600