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.
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 the method 1 by storing the Fibonacci numbers calculated so far.
Time Complexity: O(n)
Extra Space: O(n)
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.
Please refer complete article on Program for Fibonacci numbers for more details!
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