53ms runtime using booleans and markers to store what gets zeroed.

The approach to this solution follows from the problem statement:

If one entry in a row must be zero, then all entries in that row must be zero.

Similarly, if one entry in a column must be zero, then all entries in that column must be zero.

The **obverse** of this is also true:

If we know that a row must be zeroed, then we can represent this by setting a single entry in that row to zero.

Similarly, if we know that a column must be zeroed, then we can represent this by setting a single entry in that column to zero.

Since every non-empty matrix has a first row and a first column:

(1) let's zero the first column of the first row when we find a zero in that row, and

(2) let's zero the first row of the first column when we find a zero in that column.

However, since we're using the first row and the first column to keep track of which things need to be zeroed, we can't zero them out until the very end.

Hence, we initialize two booleans to false, since we don't know yet if the first row or first column need to be zeroed.

Of course, if the incoming matrix has any zeros in the first row, that row needs to be zeroed, and the same goes for columns.

At that point, we'll set these booleans to true, and handle it just before return.

```
class Solution {
public:
void setZeroes(vector<vector<int>>& matrix) {
bool zeroFirstRow = false;
bool zeroFirstCol = false;
// Store frequent calls to meaningful variables
int m = matrix.size();
int n = matrix[0].size();
// Check for needing to zero initial row or column
if (matrix[0][0] == 0) {
zeroFirstRow = true;
zeroFirstCol = true;
}
else {
for (int i = 0; i < m; i++) {
if (matrix[i][0] == 0) {
zeroFirstCol = true;
break;
}
}
for (int j = 0; j < n; j++) {
if (matrix[0][j] == 0) {
zeroFirstRow = true;
break;
}
}
}
/**
* Scan the array for rows and columns
* to be zeroed.
*/
for (int i = 1; i < m; i++) {
for (int j = 1; j < n; j++) {
if (matrix[i][j] == 0) {
matrix[i][0] = 0;
matrix[0][j] = 0;
}
}
}
// Zero the marked rows
for (int i = 1; i < m; i++) {
if (matrix[i][0] == 0) {
for (int j = 1; j < n; j++) {
matrix[i][j] = 0;
}
}
}
// Zero the marked columns
for (int j = 1; j < n; j++) {
if (matrix[0][j] == 0) {
for (int i = 1; i < m; i++) {
matrix[i][j] = 0;
}
}
}
// Zero the first row, if needed
if (zeroFirstRow) {
for (int j = 0; j < n; j++) {
matrix[0][j] = 0;
}
}
// Zero the first column, if needed
if (zeroFirstCol) {
for (int i = 0; i < m; i++) {
matrix[i][0] = 0;
}
}
}
};
```