Hello Everyone,
The Fibonacci numbers are the numbers in the following integer sequence.
0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, ………
In mathematical terms, the sequence Fn of Fibonacci numbers is defined by the recurrence relation
Fn = Fn-1 + Fn-2
with seed values
F0 = 0 and F1 = 1.
Given a number n, print n-th Fibonacci Number.
Examples:
Input : n = 2 Output : 1 Input : n = 9 Output : 34
Write a function int fib(int n) that returns Fn. For example, if n = 0, then fib() should return 0. If n = 1, then it should return 1. For n > 1, it should return Fn-1 + Fn-2
For n = 9 Output:34
Following are different methods to get the nth Fibonacci number.
Method 1 (Use recursion)
A simple method that is a direct recursive implementation mathematical recurrence relation given above.
//Fibonacci Series using Recursion
#include<bits/stdc++.h>
using namespace std;
int fib( int n)
{
if (n <= 1)
return n;
return fib(n-1) + fib(n-2);
}
int main ()
{
int n = 9;
cout << fib(n);
getchar ();
return 0;
}
Output
34
Time Complexity: T(n) = T(n-1) + T(n-2) which is exponential.
We can observe that this implementation does a lot of repeated work (see the following recursion tree). So this is a bad implementation for nth Fibonacci number.
fib(5) / \ fib(4) fib(3) / \ / \ fib(3) fib(2) fib(2) fib(1) / \ / \ / \ fib(2) fib(1) fib(1) fib(0) fib(1) fib(0) / \ fib(1) fib(0)
Extra Space: O(n) if we consider the function call stack size, otherwise O(1).
Method 2 (Use Dynamic Programming)
We can avoid the repeated work done is method 1 by storing the Fibonacci numbers calculated so far.
// C++ program for Fibonacci Series
// using Dynamic Programming
#include<bits/stdc++.h>
using namespace std;
class GFG{
public :
int fib( int n)
{
// Declare an array to store
// Fibonacci numbers.
// 1 extra to handle
// case, n = 0
int f[n + 2];
int i;
// 0th and 1st number of the
// series are 0 and 1
f[0] = 0;
f[1] = 1;
for (i = 2; i <= n; i++)
{
//Add the previous 2 numbers
// in the series and store it
f[i] = f[i - 1] + f[i - 2];
}
return f[n];
}
};
// Driver code
int main ()
{
GFG g;
int n = 9;
cout << g.fib(n);
return 0;
}
Output
34
Method 3 (Space Optimized Method 2)
We can optimize the space used in method 2 by storing the previous two numbers only because that is all we need to get the next Fibonacci number in series.
// Fibonacci Series using Space Optimized Method
#include<bits/stdc++.h>
using namespace std;
int fib( int n)
{
int a = 0, b = 1, c, i;
if ( n == 0)
return a;
for (i = 2; i <= n; i++)
{
c = a + b;
a = b;
b = c;
}
return b;
}
// Driver code
int main()
{
int n = 9;
cout << fib(n);
return 0;
}
Output
34
Time Complexity: O(n)
Extra Space: O(1)