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

2. Induction 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 Konstantin Busch - LSU

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

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

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

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

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

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

9. Harmonic numbers Example: Konstantin Busch - LSU

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

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

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

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

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

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

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

17. Triominos hole hole hole Konstantin Busch - LSU

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

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

20. Inductive Step: Konstantin Busch - LSU

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

22. 23 x 23 case: Konstantin Busch - LSU

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

24. 23 x 23 case: End of Proof Konstantin Busch - LSU

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

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

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

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

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

30. 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 Konstantin Busch - LSU

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

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

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

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

35. Konstantin Busch - LSU

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

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

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

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

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

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

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

43. 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 Konstantin Busch - LSU

44. remainder divisions first zero result Konstantin Busch - LSU

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

46. This holds since Konstantin Busch - LSU

47. End of Proof Konstantin Busch - LSU

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

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

50. Input values of recursive calls 8 2 4 6 9 7 10 1 5 3 8 2 4 6 9 7 10 1 5 3 8 2 4 6 9 7 10 1 5 3 8 2 6 9 7 10 1 5 3 4 8 2 7 10 Konstantin Busch - LSU