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?

Code:

class Solution {
public:
    void setZeroes(vector<vector<int> > &matrix) {
        int m=matrix.size();
        int n=matrix[0].size();
        int firstrow=0;
        int firstcol=0;
        for(int i=0;i<n;i++)
            if(matrix[0][i]==0){
                firstrow=1;
                break;
            }
        for(int j=0;j<m;j++)
            if(matrix[j][0]==0){
                firstcol=1;
                break;
            }
        for(int j=0;j<m;j++)
            for(int i=0;i<n;i++){
                if(matrix[j][i]==0){
                    matrix[j][0]=0;
                    matrix[0][i]=0;
                }
            }
        for(int i=1;i<n;i++)
            if(matrix[0][i]==0)
                for(int j=1;j<m;j++)
                    matrix[j][i]=0;
        for(int j=1;j<m;j++)
            if(matrix[j][0]==0)
                for(int i=1;i<n;i++)
                    matrix[j][i]=0;
        if(firstrow==1)
            for(int i=0;i<n;i++)
                    matrix[0][i]=0;
        if(firstcol==1)
            for(int j=1;j<m;j++)
                    matrix[j][0]=0;
    }
};