Trishthedish / fibonacci

Fibonacci

Notes on Fibonacci Sequence

• The Fibonacci sequence is a famous mathematical algorithm where each number is the sum of the previous two. That is every number after the first two is the sum of the two preceding ones.
• So, we'll have to add the two previous numbers to calculate the next value.
  • 1 + 1 = 2 (this is our 3rd value)
  • 1 + 2 = 3 (4th)
  • 2 + 3 = 5 (5th)
  • 3 + 5 = 8 (6th)
  • 5 + 8 = 13 (7th)

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Fibonacci Run Time Chart

value of n	fibonacci's Value	Time Elapsed Recursive Solution	Time Elapsed Cache Solution	Time Elapsed Constructive	Time Elapsed Memoized
3	2	3.7e-05 seconds	3.9e-05 seconds	3.9e-05 seconds	4.0e-05 seconds
4	3	2.9e-05 seconds	4.5e-05 seconds	3.6e-05 seconds	4.2e-05 seconds
5	5	2.8e-05 seconds	3.2e-05 seconds	3.4e-05 seconds	3.9e-05 seconds
6	8	3.1e-05 seconds	3.7e-05 seconds	3.0e-05 seconds	3.1e-05 seconds
7	13	6.8e-05 seconds	4.1e-05 seconds	4.0e-05 seconds	4.1e-05 seconds
8	21	4.0e-05 seconds	4.3e-05 seconds	3.4e-05 seconds	4.0e-05 seconds
9	34	3.4e-05 seconds	4.3e-05 seconds	4.2e-05 seconds	4.0e-05 seconds
10	55	3.8e-05 seconds	3.1e-05 seconds	2.8e-05 seconds	3.0e-05 seconds
15	610	0.000104 seconds	3.5e-05 seconds	3.6e-05 seconds	3.2e-05 seconds
20	6765	0.000832 seconds	4.4e-05 seconds	4.2e-05 seconds	3.6e-05 seconds
30	832040	0.100245 seconds	3.7e-05 seconds	4.7e-05 seconds	4.4e-05 seconds
35	9227465	1.060006 seconds	3.9e-05 seconds	5.9e-05 seconds	4.8e-05 seconds
40	102334155	11.580249 seconds	4.0e-05 seconds	4.7e-05 seconds	4.4e-05 seconds
50				5.5e-05 seconds	4.9e-05 seconds
60				5.1e-05 seconds	4.3e-05 seconds
70				5.3e-05 seconds	5.7e-05 seconds
100					8.4e-05 seconds

Recursive Solution

demonstrates that as long as n is a relatively small number, we won’t really run into any problems. However, our fibonacci method has an exponential runtime. As the input n increases, it will take more and more time to run the method.

Problem each operation contains 2 sub-operations.

Cache Solution

Sacrifice some disk space as we hold on to data longer, but it will be able to evaluate inputs significantly faster.

Constructive Solution

Memoized

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Big "oh"	What it means
O(1)	Constant
O(n)	Linear
O(n^2)	Quadratic
O(n^3)	Cubic
O(2n)	Exponential
O(n!)	Factorial
O(log n)	Logarithmic