Count zeros in a row wise and column wise sorted matrix

Given a N x N binary matrix (elements in matrix can be either 1 or 0) where each row and column of the matrix is sorted in ascending order, count number of 0s present in it.
Expected time complexity is O(N).

Examples:

Input: 
[0, 0, 0, 0, 1]
[0, 0, 0, 1, 1]
[0, 1, 1, 1, 1]
[1, 1, 1, 1, 1]
[1, 1, 1, 1, 1]

Output: 8


Input: 
[0, 0]
[0, 0]

Output: 4


Input: 
[1, 1, 1, 1]
[1, 1, 1, 1]
[1, 1, 1, 1]
[1, 1, 1, 1]

Output: 0

The idea is very simple. We start from the bottom-left corner of the matrix and repeat below steps until we find the top or right edge of the matrix.

1. Decrement row index until we find a 0. 
2. Add number of 0s in current column i.e. current row index + 1 to the result and move right to next column (Increment col index by 1).

The above logic will work since the matrix is row-wise and column-wise sorted. The logic will also work for any matrix containing non-negative integers.

Below is the implementation of above idea :

C++

// C++ program to count number of 0s in the given
// row-wise and column-wise sorted binary matrix.
#include <iostream>
using namespace std;
// define size of square matrix
#define N 5

// Function to count number of 0s in the given
// row-wise and column-wise sorted binary matrix.
int countZeroes(int mat[N][N])
{
    // start from bottom-left corner of the matrix
    int row = N - 1, col = 0;

    // stores number of zeroes in the matrix
    int count = 0;

    while (col < N)
    {
        // move up until you find a 0
        while (mat[row][col])

            // if zero is not found in current column,
            // we are done
            if (--row < 0)
                return count;

        // add 0s present in current column to result
        count += (row + 1);

        // move right to next column
        col++;
    }

    return count;
}

// Driver Program to test above functions
int main()
{
    int mat[N][N] =
    {
        { 0, 0, 0, 0, 1 },
        { 0, 0, 0, 1, 1 },
        { 0, 1, 1, 1, 1 },
        { 1, 1, 1, 1, 1 },
        { 1, 1, 1, 1, 1 }
    };

    cout << countZeroes(mat);

    return 0;
}

C

// C program to count number of 0s in the given
// row-wise and column-wise sorted binary matrix.
#include <stdio.h>

// define size of square matrix
#define N 5

// Function to count number of 0s in the given
// row-wise and column-wise sorted binary matrix.
int countZeroes(int mat[N][N])
{
    // start from bottom-left corner of the matrix
    int row = N - 1, col = 0;

    // stores number of zeroes in the matrix
    int count = 0;

    while (col < N)
    {
        // move up until you find a 0
        while (mat[row][col])

            // if zero is not found in current column,
            // we are done
            if (--row < 0)
                return count;

        // add 0s present in current column to result
        count += (row + 1);

        // move right to next column
        col++;
    }

    return count;
}

// Driver Program to test above functions
int main()
{
    int mat[N][N] =
    {
        { 0, 0, 0, 0, 1 },
        { 0, 0, 0, 1, 1 },
        { 0, 1, 1, 1, 1 },
        { 1, 1, 1, 1, 1 },
        { 1, 1, 1, 1, 1 }
    };

    printf("%d",countZeroes(mat));

    return 0;
}

// This code is contributed by kothavvsaakash.

Java

// Java program to count number of 0s in the given
// row-wise and column-wise sorted binary matrix
import java.io.*;

class GFG
{
    public static int N = 5;

    // Function to count number of 0s in the given
    // row-wise and column-wise sorted binary matrix.
    static int countZeroes(int mat[][])
    {
        // start from bottom-left corner of the matrix
        int row = N - 1, col = 0;

        // stores number of zeroes in the matrix
        int count = 0;

        while (col < N)
        {
            // move up until you find a 0
            while (mat[row][col] > 0)

                // if zero is not found in current column,
                // we are done
                if (--row < 0)
                    return count;

            // add 0s present in current column to result
            count += (row + 1);

            // move right to next column
            col++;
        }

        return count;
    }

    // Driver program
    public static void main (String[] args)
    {
        int mat[][] = { { 0, 0, 0, 0, 1 },
                        { 0, 0, 0, 1, 1 },
                        { 0, 1, 1, 1, 1 },
                        { 1, 1, 1, 1, 1 },
                        { 1, 1, 1, 1, 1 } };
        System.out.println(countZeroes(mat));
    }
}

// This code is contributed by Pramod Kumar

Python

# Python program to count number
# of 0s in the given row-wise
# and column-wise sorted
# binary matrix.

# Function to count number
# of 0s in the given
# row-wise and column-wise
# sorted binary matrix.
def countZeroes(mat):

    # start from bottom-left
    # corner of the matrix
    N = 5;
    row = N - 1;
    col = 0;

    # stores number of
    # zeroes in the matrix
    count = 0;

    while (col < N):

        # move up until
        # you find a 0
        while (mat[row][col]):

            # if zero is not found
            # in current column, we
            # are done
            if (row < 0):
                return count;
            row = row - 1;

        # add 0s present in
        # current column to result
        count = count + (row + 1);

        # move right to
        # next column
        col = col + 1;

    return count;

# Driver Code
mat = [[0, 0, 0, 0, 1],
       [0, 0, 0, 1, 1],
       [0, 1, 1, 1, 1],
       [1, 1, 1, 1, 1],
       [1, 1, 1, 1, 1]];

print( countZeroes(mat));

# This code is contributed
# by chandan_jnu

C#

// C# program to count number of
// 0s in the given row-wise and
// column-wise sorted binary matrix
using System;

class GFG
{
    public static int N = 5;

    // Function to count number of
    // 0s in the given row-wise and
    // column-wise sorted binary matrix.
    static int countZeroes(int [,] mat)
    {
        // start from bottom-left
        // corner of the matrix
        int row = N - 1, col = 0;

        // stores number of zeroes
        // in the matrix
        int count = 0;

        while (col < N)
        {
            // move up until you find a 0
            while (mat[row,col] > 0)

                // if zero is not found in
                // current column,
                // we are done
                if (--row < 0)
                    return count;

            // add 0s present in current
            // column to result
            count += (row + 1);

            // move right to next column
            col++;
        }

        return count;
    }

    // Driver Code
    public static void Main ()
    {
        int [,] mat = { { 0, 0, 0, 0, 1 },
                        { 0, 0, 0, 1, 1 },
                        { 0, 1, 1, 1, 1 },
                        { 1, 1, 1, 1, 1 },
                        { 1, 1, 1, 1, 1 } };
        Console.WriteLine(countZeroes(mat));
    }
}

// This code is contributed by KRV.

PHP

<?php
// PHP program to count number
// of 0s in the given row-wise
// and column-wise sorted
// binary matrix.

// Function to count number
// of 0s in the given
// row-wise and column-wise
// sorted binary matrix.
function countZeroes($mat)
{
    // start from bottom-left
    // corner of the matrix
    $N = 5;
    $row = $N - 1;
    $col = 0;

    // stores number of
    // zeroes in the matrix
    $count = 0;

    while ($col < $N)
    {
        // move up until
        // you find a 0
        while ($mat[$row][$col])

            // if zero is not found
            // in current column, we
            // are done
            if (--$row < 0)
                return $count;

        // add 0s present in
        // current column to result
        $count += ($row + 1);

        // move right to
        // next column
        $col++;
    }

    return $count;
}

// Driver Code
$mat = array(array(0, 0, 0, 0, 1),
             array(0, 0, 0, 1, 1),
             array(0, 1, 1, 1, 1),
             array(1, 1, 1, 1, 1),
             array(1, 1, 1, 1, 1));

echo countZeroes($mat);

// This code is contributed by Sam007
?>

Javascript

<script>

// JavaScript program to count number of 0s in the given
// row-wise and column-wise sorted binary matrix

    let N = 5;

    // Function to count number of 0s in the given
    // row-wise and column-wise sorted binary matrix.
    function countZeroes(mat)
    {
        // start from bottom-left corner of the matrix
        let row = N - 1, col = 0;

        // stores number of zeroes in the matrix
        let count = 0;

        while (col < N)
        {
            // move up until you find a 0
            while (mat[row][col] > 0)

                // if zero is not found in current column,
                // we are done
                if (--row < 0)
                    return count;

            // add 0s present in current column to result
            count += (row + 1);

            // move right to next column
            col++;
        }

        return count;
    }


    // Driver code

        let mat = [[ 0, 0, 0, 0, 1 ],
                        [ 0, 0, 0, 1, 1 ],
                        [ 0, 1, 1, 1, 1 ],
                        [ 1, 1, 1, 1, 1 ],
                        [ 1, 1, 1, 1, 1 ]];
        document.write(countZeroes(mat));

</script>

Output

8

Time complexity of above solution is O(n) since the solution follows single path from bottom-left corner to top or right edge of the matrix. 
Auxiliary space used by the program is O(1). since no extra space has been taken.

This article is contributed by Aditya Goel. If you like GeeksforGeeks and would like to contribute, you can also write an article using write.geeksforgeeks.org or mail your article to review-team@geeksforgeeks.org. See your article appearing on the GeeksforGeeks main page and help other Geeks.