Fibonacci Numbers: Difference between revisions
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[[File:Fibonacci_Recursion_Tree.png|900px]] | [[File:Fibonacci_Recursion_Tree.png|900px]] | ||
Running time < 2<sup>0</sup> + 2<sup>1</sup> + ... + 2<sup>n-1</sup>. | |||
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|dynamic programming]] method: | 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|dynamic programming]] method: | ||
==Dynamic Programming== | ==Dynamic Programming== | ||
Revision as of 21:39, 11 November 2021
Internal
Overview
We define Fibonacci numbers by the following recurrence:
F0 = 0
F1 = 1
Fi = Fi-1 + Fi-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):
Running time < 20 + 21 + ... + 2n-1.
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: