Understanding the Fibonacci Sequence: Algorithms and Applications

1. Introduction:Fibonacci Numbers http://iti.wtf Algorithms and Data Structures Artem A. Golubnichiy artem@golubnichij.ru Department of Software of Computer Facilities and Automated Systems KatanovKhakass State University

2. STRUCTURE OF THE CLASS • Problem Overview; • Naive Algorithm; • Efficient Algorithm.

3. DEFINITION In mathematics, the Fibonacci numbers are the numbers in the following integer sequence, called the Fibonacci sequence, and characterized by the fact that every number after the first two is the sum of the two preceding ones.

4. DEFINITION In mathematics, the Fibonacci numbers are the numbers in the following integer sequence, called the Fibonacci sequence, and characterized by the fact that every number after the first two is the sum of the two preceding ones.

5. DEFINITION In mathematics, the Fibonacci numbers are the numbers in the following integer sequence, called the Fibonacci sequence, and characterized by the fact that every number after the first two is the sum of the two preceding ones. 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597 . . .

6. DEVELOPED TO STUDY RABBIT POPULATIONS

7. RAPID GROWTH

8. RAPID GROWTH

9. RAPID GROWTH

10. RAPID GROWTH

11. RAPID GROWTH

12. FORMULA

13. FORMULA

14. EXAMPLE F20 = 6765

15. EXAMPLE F20 = 6765 F50 = 12586269025

16. EXAMPLE F20 = 6765 F50 = 12586269025 F100 = 354224848179261915075

17. EXAMPLE F20 = 6765 F50 = 12586269025 F100 = 354224848179261915075 F500 = 13942322456169788013972438287040728395007025658769730 7264108962948325571622863290691557658876222521294125

18. COMPUTING FIBONACCI NUMBERS

19. ALGORITHM

20. RUNNING TIME Let T(n) denote the number of lines of code executed by FibRecurs(n) if n ≤ 1 T(n) = 2

21. RUNNING TIME Let T(n) denote the number of lines of code executed by FibRecurs(n) if n ≥ 2 T(n) = 3 + T(n – 1) + T(n – 2)

22. RUNNING TIME Therefore T(n) ≥ Fn T(100) ≈ 1.77∙1021 Takes 56000 years at 1Ghz

23. WHY SO SLOW? Fn Fn-2 Fn-1 Fn-2 Fn-3 Fn-3 Fn-4 Fn-4 Fn-5 Fn-5 Fn-6 Fn-4 Fn-4 Fn-3 Fn-5

24. WHY SO SLOW? Fn Fn-2 Fn-1 Fn-2 Fn-3 Fn-3 Fn-4 Fn-4 Fn-5 Fn-5 Fn-6 Fn-4 Fn-4 Fn-3 Fn-5

25. WHY SO SLOW? Fn Fn-2 Fn-1 Fn-2 Fn-3 Fn-3 Fn-4 Fn-4 Fn-5 Fn-5 Fn-6 Fn-4 Fn-4 Fn-3 Fn-5

26. ANOTHER ALGORITHM Imitate hand computation: 0, 1

27. ANOTHER ALGORITHM Imitate hand computation: 0, 1, 1 0 + 1 = 1

28. ANOTHER ALGORITHM Imitate hand computation: 0, 1, 1, 2 0 + 1 = 1 1 + 1 = 2

29. ANOTHER ALGORITHM Imitate hand computation: 0, 1, 1, 2, 3 0 + 1 = 1 1 + 1 = 2 1 + 2 = 3

30. ANOTHER ALGORITHM Imitate hand computation: 0, 1, 1, 2, 3, 5 0 + 1 = 1 1 + 1 = 2 1 + 2 = 3 2 + 3 = 5

31. ANOTHER ALGORITHM Imitate hand computation: 0, 1, 1, 2, 3, 5, 8 0 + 1 = 1 1 + 1 = 2 1 + 2 = 3 2 + 3 = 5 3 + 5 = 8

32. NEW ALGORITHM T(n) = 2n + 2. So T(100) = 202

33. VISUALIZING ALGORITHMS Recursive Table