Set Matrix Zeroes

Given a m x n matrix, if an element is 0, set its entire row and column to 0. Do it in place.

class Solution {//passed without editing. worst case O(mn) space
    void zeroPropagate(vector<vector<int> > &matrix, int row, int col){
        for(int ii = 0; ii < matrix.size(); ++ii){
            matrix[ii][col] = 0;
        }

        for(int ii = 0; ii < matrix[0].size(); ++ii){
            matrix[row][ii] = 0;
        }
    }
public:
    void setZeroes(vector<vector<int> > &matrix) {
        // Start typing your C/C++ solution below
        // DO NOT write int main() function
        if (matrix.empty() || matrix[0].empty()) return;
        vector<pair<int,int> > record;
        for(int ii = 0; ii < matrix.size(); ++ii){
            for(int jj = 0; jj < matrix[0].size(); ++jj){
                if (matrix[ii][jj] == 0) record.push_back(pair<int,int>(ii,jj));
            }
        }
        for(int ii = 0; ii < record.size(); ++ii){
            zeroPropagate(matrix, record[ii].first, record[ii].second);
        }
    }
};

Follow up:
Did you use extra space?
A straight forward solution using O(mn) space is probably a bad idea.
A simple improvement uses O(m + n) space, but still not the best solution.
Could you devise a constant space solution?

class Solution {//follow up: constant extra space
public:
    void setZeroes(vector<vector<int> > &matrix) {
        //mark zeroes at the edge of matrix[][]
        if (matrix.empty() || matrix[0].empty()) return;
        bool init_col_zero = false;
        bool init_row_zero = false;
        for(int ii = 0; ii < matrix.size(); ++ii){
            if (matrix[ii][0] == 0){
                init_col_zero = true;
                break;
            }
        }
        
        for(int jj = 0; jj < matrix[0].size(); ++jj){
            if (matrix[0][jj] == 0){
                init_row_zero = true;
                break;
            }
        }
        
        for(int ii = 1; ii < matrix.size();++ii){
            for(int jj = 1; jj < matrix[0].size(); ++jj){
                if (matrix[ii][jj] == 0){
                    matrix[0][jj] = 0;
                    matrix[ii][0] = 0;
                }
            }//for jj
        }//for ii

        for(int ii = 1; ii < matrix.size(); ++ii){//paint rows
            if (matrix[ii][0] == 0) matrix[ii].assign(matrix[0].size(), 0);
        }
        for(int jj = 1; jj < matrix[0].size(); ++jj){//paint columns
            if (matrix[0][jj] == 0){
                for(int ii = 1; ii < matrix.size(); ++ii){
                    matrix[ii][jj] = 0;
                }
            }
        }
        if (init_row_zero) matrix[0].assign(matrix[0].size(), 0);
        if (init_col_zero){
            for(int ii = 0; ii < matrix.size(); ++ii){
                matrix[ii][0] = 0;
        }

        }
    }
};