Kth diagonal from the top left of a given matrix

Given a squared matrix M[ ][ ] of N * N dimensions, the task is to find the K^th diagonal of the matrix, starting from the top-left towards the bottom-right, and print its contents.
Examples:

Input: N = 3, K = 2, M[ ][ ] = {{4, 7, 8}, {9, 2, 3}, {0, 4, 1} 
Output: 9 7 
Explanation:
5 possible diagonals starting from the top left to the bottom right for the given matrix are: 
1 -> {4} 
2 -> {9, 7} 
3 -> {0, 2, 8} 
4 -> {4, 3} 
5 -> {1} 
Since K = 2, the required diagonal is (9, 7).
Input: N = 2, K = 2, M[ ][ ] = {1, 5}, {9, 4}} 
Output: 1 
Explanation:
3 possible diagonals starting from the bottom left to the top right for the given matrix are: 
1 -> {1} 
2 -> {9, 5} 
3 -> {4} 
Since K = 2, the required diagonal is (9, 5). 

Naive Approach:
The simplest approach to solve this problem is to generate all diagonals starting from the top-left to the bottom right of the given matrix. After generating the K^th diagonal, print its contents. 
Time complexity: O(N²) 
Auxiliary Space: O(N)
Efficient Approach:
The above approach can be optimized by avoiding the traversal of the matrix. Instead, find the boundaries of the K^th diagonal i.e. bottom-left and top-right indices. Once these indices are obtained, print the required diagonal by just traversing the indices of the diagonal. 
The following observations are needed to find the position of the diagonal: 

If ((K – 1) < N): The diagonal is an upper diagonal 
Bottom-left boundary = M[K – 1][0]
Top-right boundary = M[0][K – 1]
If (K ? N): The diagonal is a lower diagonal. 
Bottom-left boundary = M[N-1][K-N]
Top-right boundary = M[K-N][N-1]

Follow the steps below to solve the problem: 

• If (K – 1) < N, set the starting row, startRow as K – 1 and column, startCol as 0.
• Otherwise, set startRow as N – 1 and startCol as K – N.
• Finally, print the elements of diagonal starting from M[startRow][startCol].

Below is the implementation of the above approach: 

C++

// C++ implementation of
// the above approach
#include <bits/stdc++.h>
using namespace std;

// Function returns required diagonal
void printDiagonal(int K, int N,
                vector<vector<int> >& M)
{
    int startrow, startcol;

    // Initialize values to
    // print upper diagonals
    if (K - 1 < N) {
        startrow = K - 1;
        startcol = 0;
    }

    // Initialize values to
    // print lower diagonals
    else {
        startrow = N - 1;
        startcol = K - N;
    }

    // Traverse the diagonal
    for (; startrow >= 0 && startcol < N;
        startrow--, startcol++) {

        // Print its contents
        cout << M[startrow][startcol]
            << " ";
    }
}

// Driver Code
int main()
{
    int N = 3, K = 4;
    vector<vector<int> > M = { { 4, 7, 8 },
                            { 9, 2, 3 },
                            { 0, 4, 1 } };

    printDiagonal(K, N, M);
    return 0;
}

Java

// Java implementation of
// the above approach
import java.util.*;

class GFG{

// Function returns required diagonal
static void printDiagonal(int K, int N,
                          int [][]M)
{
    int startrow, startcol;

    // Initialize values to
    // print upper diagonals
    if (K - 1 < N)
    {
        startrow = K - 1;
        startcol = 0;
    }

    // Initialize values to
    // print lower diagonals
    else
    {
        startrow = N - 1;
        startcol = K - N;
    }

    // Traverse the diagonal
    for(; startrow >= 0 && startcol < N;
          startrow--, startcol++)
    {

        // Print its contents
        System.out.print(M[startrow][startcol] + " ");
    }
}

// Driver Code
public static void main(String[] args)
{
    int N = 3, K = 4;
    int[][] M = { { 4, 7, 8 },
                  { 9, 2, 3 },
                  { 0, 4, 1 } };

    printDiagonal(K, N, M);
}
}

// This code is contributed by PrinciRaj1992

Python3

# Python3 implementation of the
# above approach
def printDiagonal(K, N, M):

    startrow, startcol = 0, 0

    # Initialize values to prupper
    # diagonals
    if K - 1 < N:
        startrow = K - 1
        startcol = 0
    else:
        startrow = N - 1
        startcol = K - N

    # Traverse the diagonals
    while startrow >= 0 and startcol < N:

        # Print its contents
        print(M[startrow][startcol], end = " ")
        startrow -= 1
        startcol += 1

# Driver code
if __name__ == '__main__':

    N, K = 3, 4
    M = [ [ 4, 7, 8 ],
          [ 9, 2, 3 ],
          [ 0, 4, 1 ] ]

    printDiagonal(K, N, M)

# This code is contributed by mohit kumar 29

C#

// C# implementation of
// the above approach
using System;
class GFG{

// Function returns required diagonal
static void printDiagonal(int K, int N,
                          int [,]M)
{
    int startrow, startcol;

    // Initialize values to
    // print upper diagonals
    if (K - 1 < N)
    {
        startrow = K - 1;
        startcol = 0;
    }

    // Initialize values to
    // print lower diagonals
    else
    {
        startrow = N - 1;
        startcol = K - N;
    }

    // Traverse the diagonal
    for(; startrow >= 0 && startcol < N;
          startrow--, startcol++)
    {

        // Print its contents
        Console.Write(M[startrow,startcol] + " ");
    }
}

// Driver Code
public static void Main(String[] args)
{
    int N = 3, K = 4;
    int[,] M = { { 4, 7, 8 },
                 { 9, 2, 3 },
                 { 0, 4, 1 } };

    printDiagonal(K, N, M);
}
}

// This code is contributed by PrinciRaj1992

Javascript

<script>

// JavaScript implementation of
// the above approach

// Function returns required diagonal
function printDiagonal(K,N,M)
{
    let startrow, startcol;

    // Initialize values to
    // print upper diagonals
    if (K - 1 < N)
    {
        startrow = K - 1;
        startcol = 0;
    }

    // Initialize values to
    // print lower diagonals
    else
    {
        startrow = N - 1;
        startcol = K - N;
    }

    // Traverse the diagonal
    for(; startrow >= 0 && startcol < N;
          startrow--, startcol++)
    {

        // Print its contents
        document.write(M[startrow][startcol] + " ");
    }
}

// Driver Code
let N = 3, K = 4;
let M = [[ 4, 7, 8 ],
    [ 9, 2, 3 ],
    [ 0, 4, 1 ]];

printDiagonal(K, N, M);


// This code is contributed by patel2127

</script>

Output: 

4 3 

Time complexity: O(N) 
Auxiliary Space: O(1)