1. APS105 Recursion

2. A function is allowed to call itself! –Example:. A function calling itself is called “recursion” Such a function is called a “recursive” function –Why would you want to do this?

3. Recursion: Motivation Sometimes a prob has self-similar sub-probs –E.g., problems-within-a-problem –“inner” problems have same description as outer but inner problems are smaller in some way –inner problems themselves have inner problems Recursion continues until: –some small indivisible problem reached –or some other stopping condition is reached –called the “base case” or “terminating case” Note: –sometimes recursion can be expressed as a loop –or vice-versa

4. Recursion: Walking Across the Room Loop-based solution basic idea: –while not at the wall take another step Recursive solution basic idea:.

5. Recursion: Cake Cutting Cut a jelly-roll into equal parts < 100g each.

6. Reading/Evaluating Expressions Recall: BEDMAS: order of evaluation –Brackets –Exponents –Division & Multiplication –Addition & Subtraction Apply BEDMAS to every expression –and to every sub-expression within the expression

7. Example: Evaluating Expressions BEDMAS Example:.

8. Other Examples Finding your way out of a maze Fractals Solving a sudoku

9. Recursive Math Functions

10. Example: f(3):. Recursive Math Functions

11. . Implementing Recursion in C

12. . Factorial Using a Loop

13. . Factorial Using Recursion

14. . Factorial Using Recursion (again)

15. Printing Patterns with Recursion

16. . Print a Row of n Stars (recursively)

17. . Print a Row of n Stars (attempt2)

18. **** *** ** *. Print a Triangle of Stars

19. * ** *** ****. Print an Inverted Triangle of Stars

20. . What Will This Print?

21. Recursion and Strings

22. Can think of strings using a recursive definition –Example: one of these recursive definitions: –a string is a character followed by a string –a string is a character preceded by a string –a string is two characters with a string in the middle

23. Palindromes Palindrome: –a string that is the same backwards and forwards –Examples: racecar, level, civic, madam, noon Write a function to determine if a palindrome

24. . Function to test Palindrome

25. . Palindrome tester with a Helper

26. Greatest Common Divisor (GCD)

27. GCD Algorithm GCD of two numbers is –if the numbers are the same: the GCD is either number –if the numbers are different: the GCD of the smaller of the two and the difference between the two Example: GCD(20,8) = GCD(8,12) = GCD(8,4) = GCD(4,4) The GCD is 4

28. Formalized GCD Alg., and Code..

29. Determining Powers Efficiently

30. Determining Powers Computing powers the easy (but slow) way: X 5 = X*X*X*X*X X 20 = X*X*X*X*X*X*X*X*X*X.... The more efficient way:.

31. Formula for Determining Powers.

32. . Recursively Determining Power

33. Ackermann’s Function and Maze Routing

34. Ackermann’s Function.

35. Maze Routing Basic idea (see Ch8).

36. Towers of Hanoi

37. Move all disks to another rod, but: –Only one disk may be moved at a time. –No disk may be placed on a smaller disk. Initial:Goal:

38. Towers of Hanoi: Outer Problem =

39. Towers of Hanoi: 1 st Inner Problem =

40. Towers of Hanoi: Ex. base problem =

41. Towers of Hanoi Write a program that prints a solution –assume rods 1,2,3 –assume some number of discs given by height

42. . Towers of Hanoi

43. Towers Algorithm Legend: –Monks found 64 disk tower-game –Universe will end when they finish the game number of moves is 2 n -1 –for n=3: 7 moves –for n=20: 1,048,575 moves –for n=64: 1.8*10 19 moves == 585billion years at one move per second note: 14billion years since big bang The algorithm is “exponential” –roughly X n moves where n is size of problem –exponential algorithms are impractical!