https://www.codeeval.com/open_challenges/105/

CodeEval: Largest Sub-matrix

// 0th O(n) on all candidate bottoms @B.
// 1th O(n) on candidate tops @T.
// 2th O(n) on finding the maximum @left/@right match.
int maxRect(vector<vector<int> >& mat) {
    int n               = mat.size();
    vector<vector<int> >& colSum = mat;

    for (int i = 1 ; i < n ; ++i) 
    for (int j = 0 ; j < n ; ++j)
        colSum[i][j] += colSum[i-1][j];

    int optrect = 0;
    for (int b = 0 ; b < n ; ++b) {
        for (int t = 0 ; t <= b ; ++t) {
            int minLeft = 0;
            int rowSum[n];
            for (int i = 0 ; i < n ; ++i) {
                int col = t == 0 ? colSum[b][i] : colSum[b][i] - colSum[t-1][i];
                rowSum[i] = i == 0? col : col + rowSum[i-1];
                optrect = max(optrect, rowSum[i] - minLeft); 
                minLeft = min(minLeft, rowSum[i]);
            }
        }
    }

    return optrect;
}