1. Induction and Recursion CSC-2259 Discrete Structures Konstantin Busch - LSU1

2. Induction Konstantin Busch - LSU2 Induction is a very useful proof technique In computer science, induction is used to prove properties of algorithms Induction and recursion are closely related Recursion is a description method for algorithms Induction is a proof method suitable for recursive algorithms

3. Konstantin Busch - LSU3 Use induction to prove that a proposition is true: Inductive Basis: Inductive Hypothesis: Inductive Step: Prove that is true Assume is true Prove that is true (for any positive integer k)

4. Konstantin Busch - LSU4 Inductive Step:Prove that is true Inductive Hypothesis:Assume is true In other words in inductive step we prove: (for any positive integer k) for every positive integer k

5. Konstantin Busch - LSU5 True Inductive basis Inductive Step Proposition true for all positive integers

6. Konstantin Busch - LSU6 Induction as a rule of inference:

7. 7 Inductive Basis: Inductive Hypothesis: Inductive Step: K. Busch - LSU Theorem: Proof: We will prove assume that it holds

8. Konstantin Busch - LSU8 Inductive Step: (inductive hypothesis) End of Proof

9. Konstantin Busch - LSU9 Harmonic numbers Example:

10. Konstantin Busch - LSU10 Theorem: Proof: Inductive Basis:

11. Konstantin Busch - LSU11 Inductive Hypothesis: Suppose it holds: Inductive Step: We will show:

12. Konstantin Busch - LSU12 End of Proof from inductive hypothesis

13. Konstantin Busch - LSU13 Theorem: Proof: Inductive Basis:

14. Konstantin Busch - LSU14 Inductive Hypothesis: Suppose it holds: Inductive Step: We will show:

15. Konstantin Busch - LSU15 End of Proof from inductive hypothesis

16. Konstantin Busch - LSU16 We have shown: It holds that:

17. Konstantin Busch - LSU17 hole Triominos

18. Konstantin Busch - LSU18 Theorem:Every checkerboard with one square removed can be tiled with triominoes Proof: Inductive Basis: hole

19. Konstantin Busch - LSU19 Inductive Hypothesis: Hole can be anywhere Assume that a checkerboard can be tiled with the hole anywhere

20. Konstantin Busch - LSU20 Inductive Step:

21. Konstantin Busch - LSU21 By inductive hypothesis squares with a hole can be tiled add three artificial holes

22. Konstantin Busch - LSU22 2 3 x 2 3 case:

23. Konstantin Busch - LSU23 Replace the three holes with a triomino Now, the whole area can be tiled

24. Konstantin Busch - LSU24 End of Proof 2 3 x 2 3 case:

25. Strong Induction Konstantin Busch - LSU25 Inductive Basis: Inductive Hypothesis: Inductive Step: Prove that is true Assume is true Prove that is true To prove :

26. Konstantin Busch - LSU26 Theorem: Every integer is a product of primes (at least one prime in the product) Proof: Inductive Basis: Number 2 is a prime Inductive Hypothesis: Suppose that every integer between and is a product of primes (Strong Induction)

27. Konstantin Busch - LSU27 Inductive Step: If is prime then the proof is finished If is not a prime then it is composite:

28. Konstantin Busch - LSU28 By the inductive hypothesis: primes End of Proof

29. Konstantin Busch - LSU29 Theorem: Every postage amount can be generated by using 4-cent and 5-cent stamps Proof: Inductive Basis: We examine four cases (because of the inductive step) (Strong Induction)

30. Konstantin Busch - LSU30 Inductive Hypothesis: Assume that every postage amount between and can be generated by using 4-cent and 5-cent stamps Inductive Step: If then the inductive step follows directly from inductive basis

31. Consider: Konstantin Busch - LSU31 Inductive hypothesis End of Proof

32. Factorial function Recursion Konstantin Busch - LSU32 Recursion is used to describe functions, sets, algorithms Example: Recursive Step: Recursive Basis:

33. Konstantin Busch - LSU33 factorial( ) { if then return else return } //recursive basis //recursive step Recursive algorithm for factorial

34. Konstantin Busch - LSU34 Fibonacci numbers Recursive Basis: Recursive Step:

35. Konstantin Busch - LSU35

36. Konstantin Busch - LSU36 fibonacci( ) { if then return else return } //recursive basis //recursive step Recursive algorithm for Fibonacci function

37. Konstantin Busch - LSU37 fibonacci( ) { if then else { for to do { } return } Iterative algorithm for Fibonacci function

38. Konstantin Busch - LSU38 Theorem: (golden ratio) for Proof:Proof by (strong) induction Inductive Basis:

39. We will prove for Konstantin Busch - LSU39 Inductive Hypothesis: Inductive Step: Suppose it holds

40. Konstantin Busch - LSU40 is the solution to equation End of Proof induction hypothesis

41. Konstantin Busch - LSU41 Greatest common divisor Recursive Step: Recursive Basis:

42. Konstantin Busch - LSU42 gcd( ) { if then return else return } //recursive basis //recursive step Recursive algorithm for greatest common divisor //assume a>b

43. Konstantin Busch - LSU43 Lames Theorem: The Euclidian algorithm for, uses at most divisions (iterations) Proof: We show that there is a Fibonacci relation in the divisions of the algorithm

44. Konstantin Busch - LSU44 first zeroresult divisions remainder

45. Konstantin Busch - LSU45 This holds since and is integer

46. Konstantin Busch - LSU46 This holds since

47. Konstantin Busch - LSU47 End of Proof

48. Konstantin Busch - LSU48 Algorithm Mergesort 8 2 4 6 9 7 10 1 5 3 8 2 4 6 97 10 1 5 3 2 4 6 8 9 1 3 5 7 10 1 2 3 4 5 6 7 8 9 10 split sort merge

49. Konstantin Busch - LSU49 sort( ) { if then { return } else return }

50. Konstantin Busch - LSU50 8 2 4 6 9 7 10 1 5 3 Input values of recursive calls 8 2 4 6 97 10 1 5 3 8 2 46 97 10 15 3 4 8 269 82 7 101 710 53

51. Konstantin Busch - LSU51 1 2 3 4 5 6 7 8 9 10 Input and output values of merging 2 4 6 8 91 3 5 7 10 2 4 86 97 10 15 3 4 2 869 82 7 101 710 53

52. Konstantin Busch - LSU52 merge( ) { while do { Remove smaller first element of from its list and insert it to } if then { append remaining elements to } return } //two sorted lists

53. Konstantin Busch - LSU53 2 4 6 8 9 2 4 86 9 merging Comparison 24 8 4 8 8 69 9 2 2 4 24 6 34 6 8 2 4 6 8 9 2<6 4<6 6<8 8<9

54. Konstantin Busch - LSU54 The total number of comparisons to merge two lists is at most: Length of ALength of B Merged size

55. Konstantin Busch - LSU55 Recursive invocation tree Assume

56. Konstantin Busch - LSU56 Elements per list #levels of tree = Assume Recursive invocation tree

57. Konstantin Busch - LSU57 merging tree

58. Konstantin Busch - LSU58 merging tree Elements per list

59. Konstantin Busch - LSU59 Merges per level Total cost: merging tree Comparisons per level Elements per merge

60. If the number of comparisons is at most Konstantin Busch - LSU60 If the number of comparisons is at most Time complexity of merge sort: