Fibonacci Number

Problem Description

The Fibonacci numbers form a sequence where each number is the sum of the two preceding ones, starting from 0 and 1. Given an integer n, calculate F(n).

Key Insights

• The Fibonacci sequence starts with F(0) = 0 and F(1) = 1.
• The recursive relation F(n) = F(n - 1) + F(n - 2) can be used to compute Fibonacci numbers.
• There are multiple approaches to solve this problem: recursion, memoization, dynamic programming, and iterative techniques.

Space and Time Complexity

Time Complexity: O(n)
Space Complexity: O(1) for the iterative solution, O(n) for the recursive solution with memoization.

Solution

To solve the Fibonacci number problem, we can use various approaches:

1. Recursive Approach: This is a straightforward implementation of the Fibonacci formula, but it can be inefficient due to repeated calculations.
2. Memoization: This technique stores previously calculated Fibonacci numbers to avoid redundant computations, improving efficiency.
3. Dynamic Programming: We can build a table of Fibonacci numbers up to F(n) iteratively, which is space efficient.
4. Iterative Approach: This is the most efficient way to compute Fibonacci numbers in terms of both time and space.

We will primarily focus on the iterative approach for its simplicity and efficiency.

Code Solutions