This post gives four solutions to Puzzle: Zero rows and columns in an array in C#. If you want to try the puzzle for yourself, see that post before you read this one.

This is a fairly long post, but it’s also fairly simple.

The obvious thing to try is to simply loop over the entries in the array and, when you find an entry that is zero, zero out the row an column containing that entry. Unfortunately that destroys information that you will need later. For example, suppose entry [3, 5] is 0. Then you need to zero out row 3 and column 5. Later when you consider entry [4, 5], that entry is 0 but you can’t tell whether it is 0 because it was originally zero or because you set it to 0 when you visited entry [3, 5].

The most obvious solution is to make a second array. That array can either indicate where the 0s are in the original array or where they belong in the result. The first solution available for download uses the following code to demonstrate the first approach approach.

private void ZeroRowsAndColumns(int[,] values) { // Make an array to hold values // indicating where there are zeros. bool[,] is_zero = new bool[ values.GetUpperBound(0) + 1, values.GetUpperBound(1) + 1]; // Set is_zero for values that are 0. for (int row = 0; row <= values.GetUpperBound(0); row++) { for (int col = 0; col <= values.GetUpperBound(1); col++) { is_zero[row, col] = (values[row, col] == 0); } } // Zero out the appropriate rows and columns. for (int row = 0; row <= values.GetUpperBound(0); row++) { for (int col = 0; col <= values.GetUpperBound(1); col++) { // See if this entry in the original array is zero. if (is_zero[row, col]) { // Zero out this entry's row and column. for (int r = 0; r <= values.GetUpperBound(0); r++) { values[r, col] = 0; } for (int c = 0; c <= values.GetUpperBound(1); c++) { values[row, c] = 0; } } } } }

The code makes an array of Booleans named `is_zero` that is big enough to hold an entry for each of the values in the original array. (Note that the code assumes that Boolean arrays are initialized to hold `false` in every entry. This is true in C#.)

The program then loops through the original array and sets the corresponding Boolean value to `true` for the original entries that are 0. The code then loops over the Boolean array and, when it finds an entry that is `true` (indicating that the original array had a 0 in that position), the code zeros out the corresponding row and column.

There is a slight improvement you can make to this algorithm. Suppose you are looping through the Boolean array and you find that `is_zero[3, 5]` is `true`. In that case you next zero out row 3 and column 5. As you do that, this version checks the `is_zero` entry for the items that it is zeroing. While zeroing out column 5, suppose you find that entry [1, 5] is `true`. In that case, the program already zeroed out column 5 when it considered `is_zero[1, 5]` so it doesn’t need to zero out that column again.

The second solution uses the following code to avoid re-zeroing a row or column if it has already done so. That saves some time if there are lots of zeros in the same row or column.

// Insert the algorithm here. private void ZeroRowsAndColumns(int[,] values) { // Make an array to hold values // indicating where there are zeros. bool[,] is_zero = new bool[ values.GetUpperBound(0) + 1, values.GetUpperBound(1) + 1]; // Set is_zero for values that are 0. for (int row = 0; row <= values.GetUpperBound(0); row++) { for (int col = 0; col <= values.GetUpperBound(1); col++) { is_zero[row, col] = (values[row, col] == 0); } } // Zero out the appropriate rows and columns. for (int row = 0; row <= values.GetUpperBound(0); row++) { for (int col = 0; col <= values.GetUpperBound(1); col++) { // See if this entry in the original array is zero. if (is_zero[row, col]) { // Zero out the column. // If this is the first row or this column has // not already been zeroed, zero it out. if ((row == 0) || (!is_zero[0, col])) { for (int r = 0; r <= values.GetUpperBound(0); r++) { values[r, col] = 0; } } else { Console.WriteLine("Skipping column " + col.ToString() + "."); } // Zero out the row. // If this is the first column or this row has // not already been zeroed, zero it out. if ((col == 0) || (!is_zero[row, 0])) { for (int c = 0; c <= values.GetUpperBound(1); c++) { values[row, c] = 0; } } else { Console.WriteLine("Skipping row " + row.ToString() + "."); } // Set is_zero to mark the row and column as zeroed. is_zero[row, 0] = true; is_zero[0, col] = true; } } } Console.WriteLine("Done"); }

This version uses the first row and column in `is_zero` to remember if the program has already zeroed out a row or column. After it zeros out row `r` and column `c`, it sets `is_zero[r, 0]` and `is_zero[0, c]` to `true`. Later, when the program must zero out column `C`, it checks `is_zero[0, C]`. If that entry is `true`, then this column has already been zeroed out so the program doesn’t need to do it again. Similarly when the program must zero out row `R`, it checks `is_zero[R, 0]` and, if that value is `true`, the code doesn’t need to zero out row `R` again.

The trickiest part here is that the code must always zero out a column if the program is considering row 0 and vice versa because at that point `is_zero` holds information about whether the original value is 0 and not information about whether the row/column has been zeroed.

This optimization saves some time if there are lots of 0s in the same row or column, but it still requires that you allocate and initialize the `is_zero` array. If the original array has M rows and N columns, then the new array has M * N entries. Unless the array is enormous, this extra space won’t be a big deal, but you can do better.

The book *Cracking the Coding Interview* by Gayle Laakmann McDowell points out that you can use two one-dimensional arrays to indicate which rows and columns should be zeroed. You pass through the values array once to initialize the new arrays. Then you pass through the new arrays zeroing out the appropriate rows and columns.

The third solution available for download uses the following code to demonstrate this approach.

private void ZeroRowsAndColumns(int[,] values) { // Make arrays to hold values // indicating where there are zeros. bool[] row_is_zero = new bool[values.GetUpperBound(0) + 1]; bool[] col_is_zero = new bool[values.GetUpperBound(1) + 1]; // Set is_zero for values that are 0. for (int row = 0; row <= values.GetUpperBound(0); row++) { for (int col = 0; col <= values.GetUpperBound(1); col++) { if (values[row, col] == 0) { row_is_zero[row] = true; col_is_zero[col] = true; } } } // Zero out the appropriate rows. for (int row = 0; row <= values.GetUpperBound(0); row++) { if (row_is_zero[row]) { for (int c = 0; c <= values.GetUpperBound(1); c++) { values[row, c] = 0; } } } // Zero out the appropriate columns. for (int col = 0; col <= values.GetUpperBound(1); col++) { if (col_is_zero[col]) { for (int r = 0; r <= values.GetUpperBound(0); r++) { values[r, col] = 0; } } } }

The code creates two arrays `row_is_zero` and `col_is_zero` to indicate whether a particular row or column should be zeroed. (Note that this code again assumes that Boolean arrays are initialized to hold `false` in every entry.) The code then loops through the `values` array, setting the `row_is_zero` and `col_is_zero` values to `true` for rows and columns that should be zeroed. Finally the code uses the two arrays to decide which rows and columns should be zeroed and zeros them.

If the values array has M rows and N columns, then this solution requires only M + N additional space. Like the previous solution, it also avoids zeroing the same row or column twice.

After a bit of thought, I realized you could do even better by using the first row and column of the `values` array to store the information in the `row_is_zero` and `col_is_zero arrays`. The only catch is what happens if there is a 0 in the first row or column. In that case, you would set `values[0, 0]` to 0. That would indicate that both the first row and column should be zero and that may not be the case.

For example, consider the following values:

5 7 9 0 8 4 7 3 9 1 2 3 2 5 1

The entry `values[0, 3]` so the code should set `values[0, 0]` to 0 to indicate that the first row should be zeroed, but that would also indicate that the first column should be zeroed and that’s incorrect.

To avoid losing information about the first row and column, the program must create and initialize two new Boolean variables. After that, it can use the first row and column to remember which columns and rows should be zeroed

The following code demonstrates this approach.

private void ZeroRowsAndColumns(int[,] values) { // See if the first row should be zero. bool row0_is_zero = false; for (int col = 0; col <= values.GetUpperBound(1); col++) { if (values[0, col] == 0) { row0_is_zero = true; break; } } // See if the first column should be zero. bool col0_is_zero = false; for (int row = 0; row <= values.GetUpperBound(0); row++) { if (values[row, 0] == 0) { col0_is_zero = true; break; } } // Copy zeros into the first row and column. for (int row = 1; row <= values.GetUpperBound(0); row++) { for (int col = 1; col <= values.GetUpperBound(1); col++) { if (values[row, col] == 0) { values[row, 0] = 0; values[0, col] = 0; } } } // Zero out the required rows. for (int row = 1; row <= values.GetUpperBound(0); row++) { if (values[row, 0] == 0) { // Zero out this row. for (int c = 1; c <= values.GetUpperBound(1); c++) { values[row, c] = 0; } } } // Zero out the required columns. for (int col = 1; col <= values.GetUpperBound(1); col++) { if (values[0, col] == 0) { // Zero out this column. for (int r = 1; r <= values.GetUpperBound(0); r++) { values[r, col] = 0; } } } // If the first row should be zero, zero it out. if (row0_is_zero) { for (int c = 0; c <= values.GetUpperBound(1); c++) { values[0, c] = 0; } } // If the first column should be zero, zero it out. if (col0_is_zero) { for (int r = 0; r <= values.GetUpperBound(0); r++) { values[r, 0] = 0; } } }

This version also uses only 2 extra Boolean variables instead of M + N or M * N extra pieces of memory so it uses the least extra space.

However, this code is longer than the previous solutions because it needs to handle the first row and column as special cases. The code is still fairly simple so it’s not too hard to read.

Here’s what I had come up with. It’s a simple approach but nothing fancy. Seems like it’s as efficient as your solutions?

I think if you plug that code in, you’ll find a problem. Suppose the values are as shown in the picture at the top of the post and the outer loops are considering row = 1 and col = 1. The value[1, 1] is 0 so the inner loops set all entries in row 1 and column 1 equal to 0.

Next the outer loops consider row = 1, col = 2. But the previous loop just set that entry equal to 0 so the code finds a zero (incorrectly this time). The inner loops set all entries in row 1 and column 2 equal to 0.

The code continues until it has filled all of the columns after column 1 with 0s. As it considers later rows, it also finds 0s so it sets each row to 0. In the end, only [0, 0] is not 0.

This is the solution described in the 3rd paragraph above that starts with “The obvious thing to try…”

It is a good try and you need to either think about it fairly hard or try it in a program to see why it fails.

D’oh! I missed that. Silly me.