<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>





int _ = [](){ std::ios_base::sync_with_stdio(false); std::cin.tie(nullptr); return 0; }();

class Solution {
public:
    int maxSumSubmatrix(vector<vector<int>>& grid, int max_area) {
        int const r = grid.size(), c = grid[0].size();
        int ans = INT_MIN;
    	std::vector<int> cols(c);
	    std::set<int> sigma;
        for(int i = 0; i < r; ++i) {
	        cols.clear(); cols.resize(c);
            for(int j = i; j < r; ++j) {
		        for(int k = 0; k < c; ++k) {
		            cols[k] += grid[j][k];
		        } 
                int curr = 0;
                sigma.clear();
                sigma.insert(curr);
                for(int const &sum: cols) {
                    curr += sum;
                    std::set<int>::const_iterator it = sigma.lower_bound(curr-max_area);
                    if(it != sigma.end()) {
                        ans = std::max(ans, curr - *it);
                        if(ans == max_area) { return ans; }
                    }
                    sigma.insert(curr);
                }
            }
        }
        return ans;
    }
};

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