1. CSIE@NUTN 1 Dr. Chow-Sing LinRecursion - CH 10 Problem Solving and Program Design in C Chapter 9 Recursion Chow-Sing Lin

2. CSIE@NUTN Recursive A function that calls itself is said to be recursive. Use recursion as an alternative to iteration (looping). A recursive solution is less efficient than an iterative solution in terms of computation time due to the overhead for the extra function calls. The use of recursion enables us to specify a very natural, simple solution to a problem that would otherwise be very difficult to solve. 2 Dr. Chow-Sing LinRecursion - CH 10

3. CSIE@NUTN 3 Dr. Chow-Sing LinRecursion - CH 10 The Nature of Recursion One or more simple cases of the problem have a straightforward, non-recursive solution The other cases can be redefined in terms of problems that are closer to the simple cases By applying this redefinition process every time the recursive function is called, eventually the problem is reduced entirely to simple cases, which are relatively easy to solve

4. CSIE@NUTN 4 Dr. Chow-Sing LinRecursion - CH 10 Recursive Algorithm The recursive algorithms that we write will generally consist of an if statement –If this is a simple case solve it else redefine the problem using recursive

5. CSIE@NUTN 5 Dr. Chow-Sing LinRecursion - CH 10 Splitting a Problem into Smaller Problems Problem of size n –We can spilt the problem into a problem of size 1 (a simple case which we can solve) and size n-1. –And spilt problem n-1 into size 1 、 size n-2 –…… If we split the problem n-1 times, we will end up with n problems of size 1, all of which we can solve.

6. CSIE@NUTN 6 Dr. Chow-Sing LinRecursion - CH 10 Example 9.1 The problem of multiplying 6 by 3 Can split two problems 1.Multiply 6 by 2 2.Add 6 to the result of problem Multiply 6 by 2 can spilt 1.Multiply 6 by 2 1.1 Multiply 6 by 1 1.2 Add 6 to the result of problem 1.1 2.Add 6 to the result of problem

7. CSIE@NUTN 7 Dr. Chow-Sing LinRecursion - CH 10 Recursive Function multiply

8. CSIE@NUTN 8 Dr. Chow-Sing LinRecursion - CH 10 Example 9.2 Develop a function to count the number of times a particular character appears in a string –Count (‘s’, “Mississippi sassafras”) –Should return the value 8 Can be solved by using a loop!

9. CSIE@NUTN 9 Dr. Chow-Sing LinRecursion - CH 10 Redefined the problem Count s ’s in “ Mississippi sassafras ” –Count s ’s in “ M ” –Count s ’s in “ ississippi sassafras ”

10. CSIE@NUTN 10 Dr. Chow-Sing LinRecursion - CH 10 Recursive Function to Count a Character in a String

11. CSIE@NUTN 11 Dr. Chow-Sing LinRecursion - CH 10 Tracing a Recursive Function Multiply(6, 3) by drawing an activation frame corresponding to each call of the function Activation frame shows the parameter values for each call and summarizes the execution of the call

12. CSIE@NUTN 12 Dr. Chow-Sing LinRecursion - CH 10 Trace of Function multiply

13. CSIE@NUTN 13 Dr. Chow-Sing LinRecursion - CH 10 Example 9.3 Take n words of input and prints them in reverse order reverse_input_words(5) Input –the –course –of –human –events Output will be events human of course the

14. CSIE@NUTN 14 Dr. Chow-Sing LinRecursion - CH 10 Function reverse_input_words

15. CSIE@NUTN 15 Dr. Chow-Sing LinRecursion - CH 10 Trace of reverse_input_words(3) Input “bits” “and” “bytes“ Output “bytes“ “and” “bits”

16. CSIE@NUTN 16 Dr. Chow-Sing LinRecursion - CH 10 Trace of reverse_input_words(3) (cont.)

17. CSIE@NUTN 17 Dr. Chow-Sing LinRecursion - CH 10 Parameter and Local Variable Stacks C uses the stack (FILO)data structure to keep the track of recursive calls. How to keep track of n and word ? –When executing a call to reverse_input_word, push the parameter value associated with the call on the top of the parameter stack, and push a new undefined cell on the top of the stack maintained for the local variable word. –A return from reverse_input_word pops each stack, removing the top value.

18. CSIE@NUTN 18 Dr. Chow-Sing LinRecursion - CH 10 Parameter and Local Variable Stacks (cont.) 3 bits nword Word “bits” is stored in word just before the second call to resverse_input_words 2 3 nword and bits second call to reverse_input_words and “bits” is stored in word 1 2 3 bytes and bits nword third call to reverse_input_words and “bytes” is stored in word

19. CSIE@NUTN 19 Dr. Chow-Sing LinRecursion - CH 10 Parameter and Local Variable Stacks (cont.) 3 bits nword 2 3 nword and bits 1 2 3 bytes and bits nword Output “bytes” After first return Output “and” After second return Output “bits”

20. CSIE@NUTN 20 Dr. Chow-Sing LinRecursion - CH 10 Trace Recursive Function Trace multiply(8, 3) by inserting debugging print statements show entry and exit from the function.

21. CSIE@NUTN 21 Dr. Chow-Sing LinRecursion - CH 10 Trace Recursive Function (cont.) Output trace from multiply(8, 3)

22. CSIE@NUTN Recursive Mathematical Function Many mathematical functions can be defined recursively. N! –0! is 1 –N! is Nx(N-1)!, for N > 0 22 Dr. Chow-Sing LinRecursion - CH 10

23. CSIE@NUTN 23 Dr. Chow-Sing LinRecursion - CH 10 Example 9.4 n!

24. CSIE@NUTN 24 Dr. Chow-Sing LinRecursion - CH 10 Trace of fact = factorial(3)

25. CSIE@NUTN 25 Dr. Chow-Sing LinRecursion - CH 10 Iterative Function factorial

26. CSIE@NUTN 26 Dr. Chow-Sing LinRecursion - CH 10 Example 9.5 Fibonacci number

27. CSIE@NUTN 27 Dr. Chow-Sing LinRecursion - CH 10 Example 9.6 Greatest common divisor –gcd(m,n) Ex: gcd(6,21)=3 Spilt –gcd(m,n) is n if n divides m evenly –gcd(m,n) is gcd(n,remainder of m divided by n) otherwise

28. CSIE@NUTN 28 Dr. Chow-Sing LinRecursion - CH 10 Program Using Recursive Function gcd

29. CSIE@NUTN 29 Dr. Chow-Sing LinRecursion - CH 10 Program Using Recursive Function gcd (cont.)

30. CSIE@NUTN 30 Dr. Chow-Sing LinRecursion - CH 10 Recursive Function with Array and String Parameter Finding Capital Letters in a String –Finding capital letters in” WhosYourDaddy ” WYD Output is “ WYD ”

31. CSIE@NUTN 31 Dr. Chow-Sing LinRecursion - CH 10 Recursive Function with Array and String Parameter (cont.) Data Requirements –Problem Input char *str /* a string from which to extract capital letters */ –Problem Output char *caps /* the capital letters from str */

32. CSIE@NUTN 32 Dr. Chow-Sing LinRecursion - CH 10 Recursive Function with Array and String Parameter (cont.) Algorithm 1. if str is the empty string 2. Store empty string in caps ( a string with no letters certainly has no caps) else 3. if initial letters of str is a capital letter 4. Store in caps this letter and the capital letters from the rest of str else 5. Store in caps the capital letters from the rest of str

33. CSIE@NUTN 33 Dr. Chow-Sing LinRecursion - CH 10 Recursive Function with Array and String Parameter (cont.)

34. CSIE@NUTN 34 Dr. Chow-Sing LinRecursion - CH 10 Trace of Call to Recursive Function find_caps Printf(“Capital letters in JoJo are %s\n”,find_caps(caps,“JoJo”));

35. CSIE@NUTN 35 Dr. Chow-Sing LinRecursion - CH 10 Sequence of Events for Trace find_caps

36. CSIE@NUTN 36 Dr. Chow-Sing LinRecursion - CH 10 Recursive Selection Sort Selection Sort that fills the array from the bottom up

37. CSIE@NUTN 37 Dr. Chow-Sing LinRecursion - CH 10 1. if n is 1 2. the array is sorted else 3.Place the largest array value in last array element 4.Sort the subarray which excludes the last array element (array[0]….array[n-2]) Recursive Algorithm for Selection Sort

38. CSIE@NUTN 38 Dr. Chow-Sing LinRecursion - CH 10 Recursive Selection Sort

39. CSIE@NUTN 39 Dr. Chow-Sing LinRecursion - CH 10 Operations on Sets Develop set operations, , , ,  Also, develop to check that a certain set is valid, to check for the empty set, and to print a set in standard set notation Analysis –Strings can be of varying sizes and can be empty –If characters array that is to hold a set is declared to have one more than the number of characters in the universal set (allow room for the null character) –Then set operations should never produce a string that will overflow the array

40. CSIE@NUTN 40 Dr. Chow-Sing LinRecursion - CH 10 Design Algorithm –Is_empty(set) 1. Is initial character ‘\0’? –Is_element(ele,set) 1. if is_empty(set) /* simple case1 */ 2. Answer is false else if initial character of set matches ele /* simple case1 */ 3. Answer is ture else 4. Answer depends on whether ele /* recursive step */ is in the rest of set

41. CSIE@NUTN 41 Dr. Chow-Sing LinRecursion - CH 10 Design (cont.) Algorithm Is_set(set) 1. if is_empty(set) /* simple case1 */ 2. Answer is true else if is_element(initial set character, rest of set) /* simple case2 */ 3. Answer is false else 4. Answer depends on whether rest of set is a valid set /* recursive step */

42. CSIE@NUTN 42 Dr. Chow-Sing LinRecursion - CH 10 Design (cont.) Algorithm is_subset(sub, set) 1. if is_empty(sub) /* simple case 1 */ 2. Answer is true else if initial character of sub is not an element of set 3. Answer is false /* simple case 2 */ else 4. Answer depends on whether rest of sub is a subset of set /* recursive steps */

43. CSIE@NUTN 43 Dr. Chow-Sing LinRecursion - CH 10 Design (cont.) Algorithm Union of set1 and set2 1. if is empty(set1) /* simple case */ 2. Result is set2 /* recursive steps */ else if initial character of set1 is also an element of set2 3. Result is the union of the rest of set1 with set2 else /* case 1 */ 4. Result includes initial character of set1 and the union of the rest of set1 with set2 /* case 2 */

44. CSIE@NUTN 44 Dr. Chow-Sing LinRecursion - CH 10 Design (cont.) Algorithm Print_set(set) 1.Output a { 2. if set is not empty, print elements separated by commas 3.Output 1}

45. CSIE@NUTN 45 Dr. Chow-Sing LinRecursion - CH 10 Design (cont.) Algorithm print_with_commas(set) 1. if set has exactly one element 2.Print it else 3. Print initial element and comma 4.print_with_commas the rest of set

46. CSIE@NUTN 46 Dr. Chow-Sing LinRecursion - CH 10 Recursive Set Operations on Sets Represented as Character Strings

47. CSIE@NUTN 47 Dr. Chow-Sing LinRecursion - CH 10

48. CSIE@NUTN 48 Dr. Chow-Sing LinRecursion - CH 10

49. CSIE@NUTN 49 Dr. Chow-Sing LinRecursion - CH 10

50. CSIE@NUTN 50 Dr. Chow-Sing LinRecursion - CH 10

51. CSIE@NUTN 51 Dr. Chow-Sing LinRecursion - CH 10

52. CSIE@NUTN 52 Dr. Chow-Sing LinRecursion - CH 10 Towers of Hanoi Move n disks from peg A to peg C using peg B as needed –Only one disk at a time may be moved and this disk must be the top disk on a peg –A larger disk can never be placed on top of a smaller disk

53. CSIE@NUTN 53 Dr. Chow-Sing LinRecursion - CH 10 Towers of Hanoi Analysis 1.Move four disks from peg A to peg B 2.Move disk 5 from peg A to peg C 3.Move four disks from peg B to peg C

54. CSIE@NUTN 54 Dr. Chow-Sing LinRecursion - CH 10 Towers of Hanoi Analysis (cont.) 3.1 Move three disks from peg B to peg A 3.2 Move disk 4 from peg B to peg C 3.3 Move three disks from peg A to peg C Step 3.3 can spilt …..

55. CSIE@NUTN 55 Dr. Chow-Sing LinRecursion - CH 10 Towers of Hanoi Data Requirements Problem Inputs –int n /* the number of disk to be moved */ –char form_peg /* the from peg */ –char to_peg /* the to peg */ –char aux_peg /*the auxiliary peg */ Problem Output A list of individual disk moves

56. CSIE@NUTN 56 Dr. Chow-Sing LinRecursion - CH 10 Towers of Hanoi Design Algorithm 1. if n is 1 then 2. Move disk 1from the from peg to the to peg else 3. Move n-1 disks from the from peg to the auxiliary peg using the to peg 4. Move disk n from the from peg to the to peg 5. Move n-1 disks from the auxiliary peg to the to peg using the from peg

57. CSIE@NUTN 57 Dr. Chow-Sing LinRecursion - CH 10 Recursive Function tower

58. CSIE@NUTN 58 Dr. Chow-Sing LinRecursion - CH 10 Trace of tower ('A', 'C', 'B', 3)

59. CSIE@NUTN 59 Dr. Chow-Sing LinRecursion - CH 10 Output Generated by tower('A', 'C', 'B', 3)

60. CSIE@NUTN 60 Dr. Chow-Sing LinRecursion - CH 10 Common Program Error Most common problem with a recursive function is that it may not terminate properly –Make sure that you identify all simple case and provide a terminating condition for each one –Be sure that each recursive step redefines the problem in terms of arguments that are closer to simple case so that repeated recursive calls will eventually lead to simple cases only

61. CSIE@NUTN 61 Dr. Chow-Sing LinRecursion - CH 10 5 4 3 2 1 A B C