<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
class Solution:
    def maxSumSubmatrix(self, matrix: List[List[int]], k: int) -> int:
        R, C, N = len(matrix), len(matrix[0]), -1000000
        row_sum_dic = { (r,-1):0 for r in range(R) }
        for r in range(R):
            for c in range(C):
                row_sum_dic[ (r,c) ] = row_sum_dic[ (r,c-1) ] + matrix[r][c]
        for c1 in range(C):
            for c2 in range(c1, C):
                S = SortedList([0])
                s = 0
                for r in range(R):
                    s += row_sum_dic[(r,c2)] - row_sum_dic[(r,c1-1)]
                    r1 = S.bisect_left(s-k)
                    if r1!=len(S): N = max(N, s-S[r1]) 
                    S.add(s) 
        return N 
18600