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.
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?

Solution: Use first row and column as auxiliary spaces instead of newly allocating ones.

class Solution {
public:
    void setZeroes(vector<vector<int> > &matrix) {
        if(matrix.empty()) return;
        int R = matrix.size();
        int C = matrix[0].size();
        bool firstRowZero = false;
        bool firstColZero = false;
        
        for(int j = 0; j < C; j++) {
            if(matrix[0][j] == 0) {
                firstRowZero = true;
                break;
            }
        }
        
        for(int i = 0; i < R; i++) {
            if(matrix[i][0] == 0) {
                firstColZero = true;
                break;
            }
        }
        
        for(int i = 1; i < R; i++) {
            for(int j = 1; j < C; j++) {
                if(matrix[i][j] == 0) {
                    matrix[i][0] = 0;
                    matrix[0][j] = 0;
                }
            }
        }
        
        for(int i = 1; i < R; i++)
            if(matrix[i][0] == 0)
                for(int j = 1; j < C; j++)
                    matrix[i][j] = 0;
        
        for(int j = 1; j < C; j++)
            if(matrix[0][j] == 0)
                for(int i = 1; i < R; i++)
                    matrix[i][j] = 0;
                    
        if(firstRowZero)
            for(int j = 0; j < C; j++)
                matrix[0][j] = 0;
                
        if(firstColZero)
            for(int i = 0; i < R; i++)
                matrix[i][0] = 0;
    }
};