<p>Given an <code>m x n</code> matrix <code>matrix</code> and an integer <code>k</code>, return <em>the max sum of a rectangle in the matrix such that its sum is no larger than</em> <code>k</code>.</p>

<p>It is <strong>guaranteed</strong> that there will be a rectangle with a sum no larger than <code>k</code>.</p>

<p>&nbsp;</p>
<p><strong class="example">Example 1:</strong></p>
<img alt="" src="https://assets.leetcode.com/uploads/2021/03/18/sum-grid.jpg" style="width: 255px; height: 176px;" />
<pre>
<strong>Input:</strong> matrix = [[1,0,1],[0,-2,3]], k = 2
<strong>Output:</strong> 2
<strong>Explanation:</strong> Because the sum of the blue rectangle [[0, 1], [-2, 3]] is 2, and 2 is the max number no larger than k (k = 2).
</pre>

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

<pre>
<strong>Input:</strong> matrix = [[2,2,-1]], k = 3
<strong>Output:</strong> 3
</pre>

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

<ul>
	<li><code>m == matrix.length</code></li>
	<li><code>n == matrix[i].length</code></li>
	<li><code>1 &lt;= m, n &lt;= 100</code></li>
	<li><code>-100 &lt;= matrix[i][j] &lt;= 100</code></li>
	<li><code>-10<sup>5</sup> &lt;= k &lt;= 10<sup>5</sup></code></li>
</ul>

<p>&nbsp;</p>
<p><strong>Follow up:</strong> What if the number of rows is much larger than the number of columns?</p>

from sortedcontainers import SortedList
import sys

class Solution:
    def maxSumSubmatrix(self, matrix: List[List[int]], k: int) -> int:
        def find_max_sum_no_larger_than_k(arr, k):
            sorted_list = SortedList([0])
            prefix_sum = 0
            max_sum = -sys.maxsize

            for num in arr:
                prefix_sum += num
                idx = sorted_list.bisect_left(prefix_sum - k)
                if idx < len(sorted_list):
                    max_sum = max(max_sum, prefix_sum - sorted_list[idx])
                sorted_list.add(prefix_sum)

            return max_sum
        
        m, n = len(matrix), len(matrix[0])
        result = -sys.maxsize

        for left in range(n):
            temp = [0] * m
            for right in range(left, n):
                for i in range(m):
                    temp[i] += matrix[i][right]
                
                result = max(result, find_max_sum_no_larger_than_k(temp, k))
                if result == k:
                    return k

        return result

    
17800