CPE 130 Algorithms and Data Structures

1. CPE 130 Algorithms and Data Structures Recursion Asst. Prof. Dr. Nuttanart Facundes

2. Leonardo Fibonacci • In 1202, Fibonacci proposed a problem about the growth of rabbit populations.

3. Leonardo Fibonacci • In 1202, Fibonacci proposed a problem about the growth of rabbit populations.

4. Leonardo Fibonacci • In 1202, Fibonacci proposed a problem about the growth of rabbit populations.

5. The rabbit reproduction model • A rabbit lives forever • The population starts as a single newborn pair • Every month, each productive pair begets a new pair which will become productive after 2 months old • Fn= # of rabbit pairs at the beginning of the nth month

6. The rabbit reproduction model • A rabbit lives forever • The population starts as a single newborn pair • Every month, each productive pair begets a new pair which will become productive after 2 months old • Fn= # of rabbit pairs at the beginning of the nth month

7. The rabbit reproduction model • A rabbit lives forever • The population starts as a single newborn pair • Every month, each productive pair begets a new pair which will become productive after 2 months old • Fn= # of rabbit pairs at the beginning of the nth month

8. The rabbit reproduction model • A rabbit lives forever • The population starts as a single newborn pair • Every month, each productive pair begets a new pair which will become productive after 2 months old • Fn= # of rabbit pairs at the beginning of the nth month

9. The rabbit reproduction model • A rabbit lives forever • The population starts as a single newborn pair • Every month, each productive pair begets a new pair which will become productive after 2 months old • Fn= # of rabbit pairs at the beginning of the nth month

10. The rabbit reproduction model • A rabbit lives forever • The population starts as a single newborn pair • Every month, each productive pair begets a new pair which will become productive after 2 months old • Fn= # of rabbit pairs at the beginning of the nth month

11. The rabbit reproduction model • A rabbit lives forever • The population starts as a single newborn pair • Every month, each productive pair begets a new pair which will become productive after 2 months old • Fn= # of rabbit pairs at the beginning of the nth month

12. Inductive Definition or Recurrence Relation for theFibonacci Numbers Stage 0, Initial Condition, or Base Case: Fib(1) = 1; Fib (2) = 1 Inductive Rule For n>3, Fib(n) = Fib(n-1) + Fib(n-2)

13. Inductive Definition or Recurrence Relation for theFibonacci Numbers Stage 0, Initial Condition, or Base Case: Fib(0) = 0; Fib (1) = 1 Inductive Rule For n>1, Fib(n) = Fib(n-1) + Fib(n-2)

14. Recursion case study: Fibonacci Numbers Fibonacci numbers are a series in which each number is the sum of the previous two numbers: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34.

15. Figure 6-9

16. Algorithm for Fibonacci Numbers The algorithm for calculating Fibonacci numbers: int fib(int n) { if (n < 2) return n; else return fib(n – 1) + fib(n – 2); }

17. Great Pyramid at Gizeh

18. a b The ratio of the altitude of a face to half the base

19. Notre Dame in Paris: Phi

20. Golden Ratio: the divine proportion •  = 1.6180339887498948482045… • “Phi” is named after the Greek sculptor Phidias • How is the Golden Ratio related to Fibonacci numbers?

21. Tower of Hanoi

22. Recursion case study: Tower of Hanoi We need to do 2n – 1 moves to achieve the task, while n = the number of disks.

23. Solution for 2 disks

24. Solution for 3 disks, Part I

25. Solution for 3 disks, Part II

26. Tower of Hanoi: The Algorithm • Move n – 1 disks from source to auxiliary General Case • Move one disk from source to destination Base Case • Move n – 1 disks from auxiliary to destinationGeneral Case