Fibonacci Numbers

Internal

• Mathematics

Overview

We define Fibonacci numbers by the following recurrence:

F₀ = 0

F₁ = 1

F_i = F_i-1 + F_i-2 for i ≥ 2.

TODO CLRS page 108.

Golden Ratio

φ=1.6180339...

Algorithms

Straightforward Recursive

A straightforward recursive algorithm looks like this:

public static long fib(long n) {
  if (n == 0) {
    return 0;
  }
  if (n == 1) {
    return 1;
  }
  return fib(n - 1) + fib(n - 2);
}

However, the running time of this method, computed with a recursion tree, is O(2ⁿ):

Fibonacci.png

This is a bad exponential time, and attempting to use the algorithm for anything larger than 50 gets problematic. The solution is to apply a dynamic programming method: