Count of possible paths from top left to bottom right of a M x N matrix by moving right, down or diagonally

Given 2 integers M and N, the task is to find the count of all the possible paths from top left to the bottom right of an M x N matrix with the constraints that from each cell you can either move only to right or down or diagonally

Examples:

Input:  M = 3, N = 3
Output: 13
Explanation: There are 13 paths as follows: VVHH, VHVH, HVVH, DVH, VDH, VHHV, HVHV, DHV, HHVV, HDV, VHD, HVD, and DD where V represents vertical, H represents horizontal, and D represents diagonal paths.

Input:  M = 2, N = 2
Output: 3

Approach: The idea is to use recursion to find the total number of paths. This approach is very similar to the one discussed in this article. Follow the steps below to solve the problem:

• If M or N equals 1, then return 1.
• Else create a recursive function numberOfPaths() call the same function for values {M-1, N}, {M, N-1} and {M-1, N-1} representing vertical, horizontal and diagonal movement respectively.

Below is the implementation of the above approach:

C++

// C++  program for the above approach
#include <iostream>
using namespace std;

// Returns count of possible paths to
// reach cell at row number M and column
// number N from the topmost leftmost
// cell (cell at 1, 1)
int numberOfPaths(int M, int N)
{

    // If either given row number or
    // given column number is first
    if (M == 1 || N == 1)
        return 1;

    // Horizontal Paths +
    // Vertical Paths +
    // Diagonal Paths
    return numberOfPaths(M - 1, N)
          + numberOfPaths(M, N - 1)
           + numberOfPaths(M - 1, N - 1);
}

// Driver Code
int main()
{
    cout << numberOfPaths(3, 3);
    return 0;
}

Java

// Java  program for the above approach
import java.util.*;
public class GFG
{

// Returns count of possible paths to
// reach cell at row number M and column
// number N from the topmost leftmost
// cell (cell at 1, 1)
static int numberOfPaths(int M, int N)
{

    // If either given row number or
    // given column number is first
    if (M == 1 || N == 1)
        return 1;

    // Horizontal Paths +
    // Vertical Paths +
    // Diagonal Paths
    return numberOfPaths(M - 1, N)
          + numberOfPaths(M, N - 1)
           + numberOfPaths(M - 1, N - 1);
}

// Driver Code
public static void main(String args[])
{
    System.out.print(numberOfPaths(3, 3));

}
}

// This code is contributed by Samim Hossain Mondal.

Python

# Python  program for the above approach

# Returns count of possible paths to
# reach cell at row number M and column
# number N from the topmost leftmost
# cell (cell at 1, 1)
def numberOfPaths(M, N):

    # If either given row number or
    # given column number is first
    if (M == 1 or N == 1):
        return 1

    # Horizontal Paths +
    # Vertical Paths +
    # Diagonal Paths
    return numberOfPaths(M - 1, N) \
        + numberOfPaths(M, N - 1) \
        + numberOfPaths(M - 1, N - 1)

# Driver Code
print(numberOfPaths(3, 3))

# This code is contributed by Samim Hossain Mondal.

C#

// C#  program for the above approach
using System;

public class GFG
{

// Returns count of possible paths to
// reach cell at row number M and column
// number N from the topmost leftmost
// cell (cell at 1, 1)
static int numberOfPaths(int M, int N)
{

    // If either given row number or
    // given column number is first
    if (M == 1 || N == 1)
        return 1;

    // Horizontal Paths +
    // Vertical Paths +
    // Diagonal Paths
    return numberOfPaths(M - 1, N)
          + numberOfPaths(M, N - 1)
           + numberOfPaths(M - 1, N - 1);
}

// Driver Code
public static void Main(String []args)
{
    Console.Write(numberOfPaths(3, 3));

}
}

// This code is contributed by 29AjayKumar

Javascript

<script>
       // JavaScript code for the above approach

       // Returns count of possible paths to
       // reach cell at row number M and column
       // number N from the topmost leftmost
       // cell (cell at 1, 1)
       function numberOfPaths(M, N) {

           // If either given row number or
           // given column number is first
           if (M == 1 || N == 1)
               return 1;

           // Horizontal Paths +
           // Vertical Paths +
           // Diagonal Paths
           return numberOfPaths(M - 1, N)
               + numberOfPaths(M, N - 1)
               + numberOfPaths(M - 1, N - 1);
       }

       // Driver Code
       document.write(numberOfPaths(3, 3));

 // This code is contributed by Potta Lokesh
   </script>

Output

13

Time Complexity: O(3^M*N)
Auxiliary Space: O(N), where N is recursion stack space.

Efficient Approach: Dp (using Memoization)

C++

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

static int dp[1001][1001];

// Returns count of possible paths to
// reach cell at row number M and column
// number N from the topmost leftmost
// cell (cell at 1, 1)
int numberOfPaths(int M, int N)
{

    // If either given row number or
    // given column number is first
    if (M == 1 || N == 1)
        return 1;

    // If a value already present
    // in t[][], return it
    if(dp[M][N] != -1) {
        return dp[M][N];
    }

    // Horizontal Paths +
    // Vertical Paths +
    // Diagonal Paths
    return dp[M][N] = numberOfPaths(M - 1, N)
                    + numberOfPaths(M, N - 1)
                    + numberOfPaths(M - 1, N - 1);
}

// Driver Code
int main()
{
    memset(dp,-1,sizeof(dp));
    cout << numberOfPaths(3, 3);
    return 0;
}

Java

// Java program for the above approach
import java.util.*;
public class GFG {
  static int[][] dp = new int[1001][1001];

  // Returns count of possible paths to
  // reach cell at row number M and column
  // number N from the topmost leftmost
  // cell (cell at 1, 1)
  static int numberOfPaths(int M, int N)
  {

    // If either given row number or
    // given column number is first
    if (M == 1 || N == 1)
      return 1;

    // If a value already present
    // in t[][], return it
    if (dp[M][N] != -1) {
      return dp[M][N];
    }

    // Horizontal Paths +
    // Vertical Paths +
    // Diagonal Paths
    dp[M][N] = numberOfPaths(M - 1, N)
      + numberOfPaths(M, N - 1)
      + numberOfPaths(M - 1, N - 1);
    return dp[M][N];
  }

  // Driver Code
  public static void main(String args[])
  {
    for (int i = 0; i < 1001; i++)
      for (int j = 0; j < 1001; j++)
        dp[i][j] = -1;
    System.out.println(numberOfPaths(3, 3));
  }
}

// This code is contributed by Samim Hossain Mondal.

Python

# Python program for the above approach

# Taking the matrix as globally
dp = [[-1 for i in range(1001)] for j in range(1001)]

# Returns count of possible paths to
# reach cell at row number M and column
# number N from the topmost leftmost
# cell (cell at 1, 1)
def numberOfPaths(M, N):

    # If either given row number or
    # given column number is first
    if (M == 1 or N == 1):
        return 1

    # If a value already present
    # in t[][], return it
    if(dp[M][N] != -1):
        return dp[M][N]

    # Horizontal Paths +
    # Vertical Paths +
    # Diagonal Paths
    dp[M][N] = numberOfPaths(M - 1, N) + numberOfPaths(M,
                     N - 1) + numberOfPaths(M - 1, N - 1)
    return dp[M][N]

# Driver Code
print(numberOfPaths(3, 3))

# This code is contributed by Samim Hossain Mondal.

C#

// C# program for the above approach
using System;
class GFG {
    static int[, ] dp = new int[1001, 1001];

    // Returns count of possible paths to
    // reach cell at row number M and column
    // number N from the topmost leftmost
    // cell (cell at 1, 1)
    static int numberOfPaths(int M, int N)
    {

        // If either given row number or
        // given column number is first
        if (M == 1 || N == 1)
            return 1;

        // If a value already present
        // in t[][], return it
        if (dp[M, N] != -1) {
            return dp[M, N];
        }

        // Horizontal Paths +
        // Vertical Paths +
        // Diagonal Paths
        dp[M, N] = numberOfPaths(M - 1, N)
                   + numberOfPaths(M, N - 1)
                   + numberOfPaths(M - 1, N - 1);
        return dp[M, N];
    }

    // Driver Code
    public static void Main()
    {
        for (int i = 0; i < 1001; i++)
            for (int j = 0; j < 1001; j++)
                dp[i, j] = -1;
        Console.Write(numberOfPaths(3, 3));
    }
}

// This code is contributed by ukasp.

Javascript

<script>
        // JavaScript Program of the above approach

  var dp = new Array(1001);
  // Loop to create 2D array using 1D array
    for (var i = 0; i < dp.length; i++) {
        dp[i] = new Array(1001);
    }

  // Returns count of possible paths to
  // reach cell at row number M and column
  // number N from the topmost leftmost
  // cell (cell at 1, 1)
  function numberOfPaths(M, N)
  {

    // If either given row number or
    // given column number is first
    if (M == 1 || N == 1)
      return 1;

    // If a value already present
    // in t[][], return it
    if (dp[M][N] != -1) {
      return dp[M][N];
    }

    // Horizontal Paths +
    // Vertical Paths +
    // Diagonal Paths
    dp[M][N] = numberOfPaths(M - 1, N)
      + numberOfPaths(M, N - 1)
      + numberOfPaths(M - 1, N - 1);
    return dp[M][N];
  }

        // Driver Code

      for (let i = 0; i < 1001; i++){
      for (let j = 0; j < 1001; j++){
        dp[i][j] = -1;
      }}
      document.write(numberOfPaths(3, 3));

// This code is contributed by avijitmondal1998
    </script>

Output

13

Time Complexity: O(M*N), The time complexity of this approach is O(M*N). Since there are MN subproblems, and each subproblem takes constant time to solve. This is because the solution to each subproblem is a sum of three previously computed subproblems. Therefore, the total time taken is proportional to the number of subproblems, which is M*N.

Auxiliary Space: O(M*N), The space complexity of this approach is also O(M*N). This is because we are using a two-dimensional array of size (M+1) * (N+1) to store the results of the subproblems. Each cell of this array requires constant space, and hence the total space required is proportional to the number of subproblems, which is M*N.

Efficient Approach: Dp (using Tables)

Follow below steps to solve the problem using bottom up approach:

1.  Initialize dp[1][1] = 1 and dp[1][1..N-1] = 1, dp[1..M-1][1] = 1
2.  For each i = 1 to M, do the following:
         a. For each j = 1 to N, do the following:
              i. Compute dp[i][j] as dp[i-1][j] + dp[i][j-1] + dp[i-1][j-1]
3.  Return dp[M][N]

Below is the implementation of the approach:

C++

// C++ program for the above approach

#include <bits/stdc++.h>
using namespace std;

// Returns count of possible paths to
// reach cell at row number M and column
// number N from the topmost leftmost
// cell (cell at 1, 1)
int numberOfPaths(int M, int N) {
    // Initialize a 2D array to store
    // intermediate results of subproblems
    int dp[M + 1][N + 1];

    // Set all elements of dp to 1
    // as there is only one path
    // to reach any cell in the first
    // row or column
    for (int i = 1; i <= M; i++) {
        for (int j = 1; j <= N; j++) {
            if (i == 1 || j == 1) {
                dp[i][j] = 1;
            }
        }
    }

    // Fill the dp array using bottom-up
    // approach
    for (int i = 2; i <= M; i++) {
        for (int j = 2; j <= N; j++) {
            dp[i][j] = dp[i - 1][j] + dp[i][j - 1]
                       + dp[i - 1][j - 1];
        }
    }

    // Return the value stored at dp[M][N]
    return dp[M][N];
}

// Driver code
int main() {
      // Function Call
    cout << numberOfPaths(3, 3);
    return 0;
}

// This code is contributed by Chandramani Kumar

Output

13

Time Complexity: O(M*N) as two nested loops are executing one from 1 to M and other from 1 to N where M and N are rows and columns respectively.

Space Complexity: O(M*N) as 2D array dp has been created of size M+1 and N+1 where M and N are rows and columns respectively.