Counts Path in an Array

Given an array A consisting of positive integer, of size N. If the element in the array at index i is K then you can jump between index ranges (i + 1) to (i + K).
The task is to find the number of possible ways to reach the end with module 10⁹ + 7.
The starting position is considered as index 0.
Examples:

Input: A = {5, 3, 1, 4, 3}
Output: 6
Input: A = {2, 3, 1, 1, 2}
Output: 4

Naive Approach: We can form a recursive structure to solve the problem.
Let F[i] denotes the number of paths starting at index i, at every index i if the element A[i] is K then the total number of ways the jump can be performed is:

F(i) = F(i+1) + F(i+2) +...+ F(i+k), where i + k <= n, 
where F(n) = 1

By using this recursive formula we can solve the problem:

C++

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

const int mod = 1e9 + 7;

// Find the number of ways
// to reach the end
int ways(int i, int arr[], int n)
{
    // Base case
    if (i == n - 1)
        return 1;

    int sum = 0;

    // Recursive structure
    for (int j = 1;
         j + i < n && j <= arr[i];
         j++) {
        sum += (ways(i + j,
                     arr, n))
               % mod;
        sum %= mod;
    }

    return sum % mod;
}

// Driver code
int main()
{
    int arr[] = { 5, 3, 1, 4, 3 };

    int n = sizeof(arr) / sizeof(arr[0]);

    cout << ways(0, arr, n) << endl;

    return 0;
}

Java

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

class GFG
{
static int mod = 1000000000;

// Find the number of ways
// to reach the end
static int ways(int i, 
                int arr[], int n)
{
    // Base case
    if (i == n - 1)
        return 1;

    int sum = 0;

    // Recursive structure
    for (int j = 1; j + i < n && 
                    j <= arr[i]; j++)
    {
        sum += (ways(i + j,
                     arr, n)) % mod;
        sum %= mod;
    }
    return sum % mod;
}

// Driver code
public static void main (String[] args)
{
    int arr[] = { 5, 3, 1, 4, 3 };
    
    int n = arr.length;

    System.out.println (ways(0, arr, n));
}
}

// This code is contributed by ajit

Python3

# Python3 implementation of
# the above approach

mod = 1e9 + 7;

# Find the number of ways
# to reach the end
def ways(i, arr, n):
    
    # Base case
    if (i == n - 1):
        return 1;

    sum = 0;

    # Recursive structure
    for j in range(1, arr[i] + 1):
        if(i + j < n):
            sum += (ways(i + j, arr, n)) % mod;
            sum %= mod;

    return int(sum % mod);

# Driver code
if __name__ == '__main__':
    arr = [5, 3, 1, 4, 3];

    n = len(arr);

    print(ways(0, arr, n));

# This code is contributed by PrinciRaj1992

// C# implementation of
// the above approach
using System;
    
class GFG
{
static int mod = 1000000000;

// Find the number of ways
// to reach the end
static int ways(int i, 
                int []arr, int n)
{
    // Base case
    if (i == n - 1)
        return 1;

    int sum = 0;

    // Recursive structure
    for (int j = 1; j + i < n && 
                    j <= arr[i]; j++)
    {
        sum += (ways(i + j,
                     arr, n)) % mod;
        sum %= mod;
    }
    return sum % mod;
}

// Driver code
public static void Main (String[] args)
{
    int []arr = { 5, 3, 1, 4, 3 };
    
    int n = arr.Length;

    Console.WriteLine(ways(0, arr, n));
}
}

// This code is contributed by 29AjayKumar

JavaScript

<script>
    // Javascript implementation of the above approach
    
    let mod = 1000000000;
  
    // Find the number of ways
    // to reach the end
    function ways(i, arr, n)
    {
        // Base case
        if (i == n - 1)
            return 1;

        let sum = 0;

        // Recursive structure
        for (let j = 1; j + i < n && j <= arr[i]; j++)
        {
            sum += (ways(i + j, arr, n)) % mod;
            sum %= mod;
        }
        return sum % mod;
    }
    
    let arr = [ 5, 3, 1, 4, 3 ];
      
    let n = arr.length;
  
    document.write(ways(0, arr, n));
    
</script>

Output:

Efficient Approach: In the previous approach, there are some calculations that are being done more than once. It will be better to store these values in a dp array and dp[i] will store the number of paths starting at index i and ending at the end of the array.
Hence dp[0] will be the solution to the problem.
Below is the implementation of the approach:

C++

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

const int mod = 1e9 + 7;

// find the number of ways to reach the end
int ways(int arr[], int n)
{
    // dp to store value
    int dp[n + 1];

    // base case
    dp[n - 1] = 1;

    // Bottom up dp structure
    for (int i = n - 2; i >= 0; i--) {
        dp[i] = 0;

        // F[i] is dependent of
        // F[i+1] to F[i+k]
        for (int j = 1; ((j + i) < n
                         && j <= arr[i]);
             j++) {
            dp[i] += dp[i + j];
            dp[i] %= mod;
        }
    }

    // Return value of dp[0]
    return dp[0] % mod;
}

// Driver code
int main()
{
    int arr[] = { 5, 3, 1, 4, 3 };

    int n = sizeof(arr) / sizeof(arr[0]);

    cout << ways(arr, n) % mod << endl;
    return 0;
}

Java

// Java implementation of above approach
class GFG 
{
    static final int mod = (int)(1e9 + 7); 
    
    // find the number of ways to reach the end 
    static int ways(int arr[], int n) 
    { 
        // dp to store value 
        int dp[] = new int[n + 1]; 
    
        // base case 
        dp[n - 1] = 1; 
    
        // Bottom up dp structure 
        for (int i = n - 2; i >= 0; i--) 
        { 
            dp[i] = 0; 
    
            // F[i] is dependent of 
            // F[i+1] to F[i+k] 
            for (int j = 1; ((j + i) < n && 
                              j <= arr[i]); j++)
            { 
                dp[i] += dp[i + j]; 
                dp[i] %= mod; 
            } 
        } 
    
        // Return value of dp[0] 
        return dp[0] % mod; 
    } 
    
    // Driver code 
    public static void main (String[] args)
    { 
        int arr[] = { 5, 3, 1, 4, 3 }; 
    
        int n = arr.length; 
    
        System.out.println(ways(arr, n) % mod); 
    } 
}

// This code is contributed by AnkitRai01

Python3

# Python3 implementation of above approach
mod = 10**9 + 7

# find the number of ways to reach the end 
def ways(arr, n):
    
    # dp to store value 
    dp = [0] * (n + 1)
    
    # base case 
    dp[n - 1] = 1
    
    # Bottom up dp structure 
    for i in range(n - 2, -1, -1):
        dp[i] = 0
        
        # F[i] is dependent of 
        # F[i + 1] to F[i + k] 
        j = 1
        while((j + i) < n and j <= arr[i]):
            dp[i] += dp[i + j] 
            dp[i] %= mod 
            j += 1
    
    # Return value of dp[0] 
    return dp[0] % mod 

# Driver code 
arr = [5, 3, 1, 4, 3 ]
n = len(arr) 
print(ways(arr, n) % mod)

# This code is contributed by SHUBHAMSINGH10

// C# implementation of above approach
using System;
    
class GFG 
{
    static readonly int mod = (int)(1e9 + 7); 
    
    // find the number of ways to reach the end 
    static int ways(int []arr, int n) 
    { 
        // dp to store value 
        int []dp = new int[n + 1]; 
    
        // base case 
        dp[n - 1] = 1; 
    
        // Bottom up dp structure 
        for (int i = n - 2; i >= 0; i--) 
        { 
            dp[i] = 0; 
    
            // F[i] is dependent of 
            // F[i+1] to F[i+k] 
            for (int j = 1; ((j + i) < n && 
                              j <= arr[i]); j++)
            { 
                dp[i] += dp[i + j]; 
                dp[i] %= mod; 
            } 
        } 
    
        // Return value of dp[0] 
        return dp[0] % mod; 
    } 
    
    // Driver code 
    public static void Main (String[] args)
    { 
        int []arr = { 5, 3, 1, 4, 3 }; 
    
        int n = arr.Length; 
    
        Console.WriteLine(ways(arr, n) % mod); 
    } 
}

// This code is contributed by Rajput-Ji

JavaScript

<script>

    // Javascript implementation 
    // of above approach
    
    let mod = (1e9 + 7); 
      
    // find the number of ways 
    // to reach the end 
    function ways(arr, n) 
    { 
        // dp to store value 
        let dp = new Array(n + 1); 
        dp.fill(0);
      
        // base case 
        dp[n - 1] = 1; 
      
        // Bottom up dp structure 
        for (let i = n - 2; i >= 0; i--) 
        { 
            dp[i] = 0; 
      
            // F[i] is dependent of 
            // F[i+1] to F[i+k] 
            for (let j = 1; ((j + i)
            < n && j <= arr[i]); j++)
            { 
                dp[i] += dp[i + j]; 
                dp[i] %= mod; 
            } 
        } 
      
        // Return value of dp[0] 
        return dp[0] % mod; 
    } 
    
    let arr = [ 5, 3, 1, 4, 3 ]; 
      
    let n = arr.length; 

    document.write(ways(arr, n) % mod); 
    
</script>

Output:

Time Complexity: O(K)

Auxiliary Space: O(n)

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