Count number of ways to cover a distance | Set 2

Given a distance N. The task is to count the total number of ways to cover the distance with 1, 2 and 3 steps.
Examples:

Input: N = 3 
Output: 4 
All the required ways are (1 + 1 + 1), (1 + 2), (2 + 1) and (3).
Input: N = 4 
Output: 7 

Approach: In previous article, a recursive and dynamic programming based approach has been discussed. Here we will reduce the space complexity. It can be observed that to calculate the number of steps to cover the distance i, only the last three states are required (i – 1, i – 2, i – 3). So, the result can be calculated using the last three states.
Below is the implementation of the above approach: 

C++

// C++ implementation of the approach
#include <iostream>
using namespace std;

// Function to return the count of the
// total number of ways to cover the
// distance with 1, 2 and 3 steps
int countWays(int n)
{
    // Base conditions
    if (n == 0)
        return 1;
    if (n <= 2)
        return n;

    // To store the last three stages
    int f0 = 1, f1 = 1, f2 = 2, ans;

    // Find the numbers of steps required
    // to reach the distance i
    for (int i = 3; i <= n; i++) {
        ans = f0 + f1 + f2;
        f0 = f1;
        f1 = f2;
        f2 = ans;
    }

    // Return the required answer
    return ans;
}

// Driver code
int main()
{
    int n = 4;

    cout << countWays(n);

    return 0;
}

Java

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

class GFG
{

// Function to return the count of the
// total number of ways to cover the
// distance with 1, 2 and 3 steps
static int countWays(int n)
{
    // Base conditions
    if (n == 0)
        return 1;
    if (n <= 2)
        return n;

    // To store the last three stages
    int f0 = 1, f1 = 1, f2 = 2;
    int ans=0;

    // Find the numbers of steps required
    // to reach the distance i
    for (int i = 3; i <= n; i++)
    {
        ans = f0 + f1 + f2;
        f0 = f1;
        f1 = f2;
        f2 = ans;
    }

    // Return the required answer
    return ans;
}

// Driver code
public static void main (String[] args)
{

    int n = 4;
    System.out.println (countWays(n));
}
}

// This code is contributed by jit_t

Python

# Python3 implementation of the approach

# Function to return the count of the
# total number of ways to cover the
# distance with 1, 2 and 3 steps
def countWays(n):

    # Base conditions
    if (n == 0):
        return 1
    if (n <= 2):
        return n

    # To store the last three stages
    f0 = 1
    f1 = 1
    f2 = 2
    ans = 0

    # Find the numbers of steps required
    # to reach the distance i
    for i in range(3, n + 1):
        ans = f0 + f1 + f2
        f0 = f1
        f1 = f2
        f2 = ans

    # Return the required answer
    return ans

# Driver code
n = 4

print(countWays(n))

# This code is contributed by mohit kumar 29

C#

// C# implementation of the approach
using System;

class GFG
{

// Function to return the count of the
// total number of ways to cover the
// distance with 1, 2 and 3 steps
static int countWays(int n)
{
    // Base conditions
    if (n == 0)
        return 1;
    if (n <= 2)
        return n;

    // To store the last three stages
    int f0 = 1, f1 = 1, f2 = 2;
    int ans = 0;

    // Find the numbers of steps required
    // to reach the distance i
    for (int i = 3; i <= n; i++)
    {
        ans = f0 + f1 + f2;
        f0 = f1;
        f1 = f2;
        f2 = ans;
    }

    // Return the required answer
    return ans;
}

// Driver code
public static void Main(String[] args)
{
    int n = 4;
    Console.WriteLine (countWays(n));
}
}

// This code is contributed by PrinciRaj1992

Javascript

<script>
    // Javascript implementation of the approach

    // Function to return the count of the
    // total number of ways to cover the
    // distance with 1, 2 and 3 steps
    function countWays(n)
    {
        // Base conditions
        if (n == 0)
            return 1;
        if (n <= 2)
            return n;

        // To store the last three stages
        let f0 = 1, f1 = 1, f2 = 2;
        let ans = 0;

        // Find the numbers of steps required
        // to reach the distance i
        for (let i = 3; i <= n; i++)
        {
            ans = f0 + f1 + f2;
            f0 = f1;
            f1 = f2;
            f2 = ans;
        }

        // Return the required answer
        return ans;
    }

    let n = 4;
    document.write(countWays(n));

// This code is contributed by suresh07.
</script>

Output:

7

Time Complexity: O(N) 
Space Complexity O(1)