<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>

#include <ext/pb_ds/assoc_container.hpp>
#include <ext/pb_ds/tree_policy.hpp>
using namespace __gnu_pbds;
#define ll int
template<class T> using pbds=tree<ll, null_type,less_equal<ll>, rb_tree_tag,tree_order_statistics_node_update>;
class Solution {
public:
    int maxSumSubmatrix(vector<vector<int>>& matrix, int k) {
        int n=matrix.size();
        int m=matrix[0].size();
        vector<vector<int>> presum(n+1,vector<int>(m+1,0));
        for(int i=1;i<=n;i++){
            for(int j=1;j<=m;j++){
                presum[i][j]=presum[i-1][j]+presum[i][j-1]+matrix[i-1][j-1]-presum[i-1][j-1];
            }
        }
        int ans=-1e9;
        for(int i=1;i<=n;i++){
            for(int h=1;h<=i;h++){
                pbds<int> s;
                // map<int,int> mp;
                // mp[0]=1;
                s.insert(0);
                for(int j=1;j<=m;j++){
                    int sum=presum[i][j]-presum[i-h][j];
                    // greater than or equal to required
                    int req=sum-k;
                    int small=s.order_of_key(req);
                    auto it =s.find_by_order(small);
                    // int diff=x-req;
                    if(it!=s.end()){
                        // cout<<*it<<endl;
                        // cout<<sum<<endl;
                        ans=max(ans,(sum-(*it)));
                    }
                    // ans+=s.size()-small;
                    s.insert(sum);
                }
            }
        }
        return ans;
        // for(int i=1;i<=n;i++){
        //     for(int j=1;j<=m;j++){
        //         cout<<presum[i][j]<<" ";
        //     }cout<<endl;
        // }

        // return 0;
    }
};
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