<p>You are given a binary matrix <code>matrix</code> of size <code>m x n</code>, and you are allowed to rearrange the <strong>columns</strong> of the <code>matrix</code> in any order.</p>

<p>Return <em>the area of the largest submatrix within </em><code>matrix</code><em> where <strong>every</strong> element of the submatrix is </em><code>1</code><em> after reordering the columns optimally.</em></p>

<p>&nbsp;</p>
<p><strong class="example">Example 1:</strong></p>
<img alt="" src="https://assets.leetcode.com/uploads/2020/12/29/screenshot-2020-12-30-at-40536-pm.png" style="width: 500px; height: 240px;" />
<pre>
<strong>Input:</strong> matrix = [[0,0,1],[1,1,1],[1,0,1]]
<strong>Output:</strong> 4
<strong>Explanation:</strong> You can rearrange the columns as shown above.
The largest submatrix of 1s, in bold, has an area of 4.
</pre>

<p><strong class="example">Example 2:</strong></p>
<img alt="" src="https://assets.leetcode.com/uploads/2020/12/29/screenshot-2020-12-30-at-40852-pm.png" style="width: 500px; height: 62px;" />
<pre>
<strong>Input:</strong> matrix = [[1,0,1,0,1]]
<strong>Output:</strong> 3
<strong>Explanation:</strong> You can rearrange the columns as shown above.
The largest submatrix of 1s, in bold, has an area of 3.
</pre>

<p><strong class="example">Example 3:</strong></p>

<pre>
<strong>Input:</strong> matrix = [[1,1,0],[1,0,1]]
<strong>Output:</strong> 2
<strong>Explanation:</strong> Notice that you must rearrange entire columns, and there is no way to make a submatrix of 1s larger than an area of 2.
</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;= 10<sup>5</sup></code></li>
	<li><code>matrix[i][j]</code> is either <code>0</code> or <code>1</code>.</li>
</ul>

class Solution {
public:
    int largestSubmatrix(vector<vector<int>>& matrix) {
        int n = matrix.size();
        int m = matrix[0].size();
        vector<vector<int>> prefix(n, vector<int>(m, 0));


        for (int i = 0; i < m; i++) {
            prefix[0][i] = matrix[0][i];
        }
        for (int i = 0; i < m; i++) {
            for (int j = 1; j < n; j++) {
                if (matrix[j][i] == 0) {
                    prefix[j][i] = 0;
                } else {
                    prefix[j][i] = prefix[j - 1][i] + 1;
                }
            }
        }

        int maxVal = 0;
        for (int i = 0; i < n; i++) {
            map<int, int, greater<int>> counts;
            for (int j = 0; j < m; j++) {
                if (counts.count(prefix[i][j]) == 0) {
                    counts.insert({prefix[i][j], 1});
                } else {
                    counts[prefix[i][j]]++;
                }
            }
            vector<int> countsPrefix(counts.size() + 1, 0);
            int index = 1;
            for (auto x = counts.begin(); x != counts.end(); ++x) {
                countsPrefix[index] = countsPrefix[index - 1] + x->second;
                index++;
            }
            index = 1;
            for (auto x = counts.begin(); x != counts.end(); ++x) {
                maxVal = max(maxVal, x->first * countsPrefix[index]);
                index++;
            }
        }
        return maxVal;
    }
};
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