L11: Recursion, Recurrence, and Induction

1. Page 1 L11: Recursion, Recurrence, and Induction • Objective • Recursion • A problem solving technique that reduces big problems into smaller ones • Induction • Proving correctness of recursive programs • Recurrence • Running time for recursive programs

2. Page 2 Outline • Recursive algorithm for Towers of Hanoi • Problem and algorithm • Correctness • Running time • Recurrence in general • Example: Number of subsets • Example: Loan repayment • Solving recurrences • One type of First-Order linear recurrences • Another type of First-Order linear recurrences

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6. Page 6 Recursive Solution of Towers of Hanoi

7. Page 7 Recursive Solution of Towers of Hanoi

8. Page 8 Recursive Solution of Towers of Hanoi • Task: • Move n disks from peg i to peg j • Algorithm • Recursion base: • When n=1, move one disk from i to j • Recursion

9. Page 9 Outline • Recursive algorithm for Towers of Hanoi • Problem and algorithm • Correctness • Running time • Recurrence in general • Example: Number of subsets • Example: Loan repayment • Solving recurrences • Geometric serious • First-Order linear recurrences

10. Page 10 Correctness of Algorithm

11. Page 11 Outline • Recursive algorithm for Towers of Hanoi • Problem and algorithm • Correctness • Running time • Recurrence in general • Example: Number of subsets • Example: Loan repayment • Solving recurrences • One type of First-Order linear recurrences • Another type of First-Order linear recurrences

12. Page 12 Running Time

13. Page 13 Solving the Recurrence

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16. Page 16 Outline • Recursive algorithm for Towers of Hanoi • Problem and algorithm • Correctness • Running time • Recurrence in general • Example: Number of subsets • Example: Loan repayment • Solving recurrences • One type of First-Order linear recurrences • Another type of First-Order linear recurrences

17. Page 17 Recurrence • Recurrence/recurrence equation is a way to specify functions on the set of integers • It tells us how to • get the n-th value f(n) • from the first n-b values: f(b), f(b+1), …, f(n-1) • Need to give the value for the base casef(b) to complete the description

18. Page 18 Number of subsets • S(n): number of subsets of set {1, 2, 3, …., n} of size n • Question: • How to compute S(n) from S(n-1), S(n-2), …? • Consider the case: n=3 • First row: subsets of {1, 2} • Second: subsets of {1, 2}, each adjoined by 3 • So: S(3) = 2 S(2)

19. Page 19 Number of Subsets • In general, subsets of set {1, 2, 3, …., n} can be divided into two groups • subsets of set {1, 2, 3, …., n-1} • subsets of set {1, 2, 3, …., n-1}, each adjoined by n. • So: S(n) = 2 S(n-1) • Base case: • S(0) =1

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21. Page 21 Monthly Payment for Loan • Initial loan amount: A • Annual interest rate: p • Monthly Payment: M • T(n): total amount still due after n months

22. Page 22 Outline • Recursive algorithm for Towers of Hanoi • Problem and algorithm • Correctness • Running time • Recurrence in general • Example: Number of subsets • Example: Loan repayment • Solving recurrences • One type of First-Order linear recurrences • Another type of First-Order linear recurrences

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24. Page 24 One Type of First-Order Linear Recurrence Examples

25. Page 25 Iterating the Recurrence/Top-Down

26. Page 26 Iterating the Recurrence/Bottom-Up

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28. Page 28 One Type of First-Order Linear Recurrence

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31. Page 31 Example

32. Page 32 An Application of Theorem 4.1 Geometric Series

33. Page 33 Outline • Recursive algorithm for Towers of Hanoi • Problem and algorithm • Correctness • Running time • Recurrence in general • Example: Number of subsets • Example: Loan repayment • Solving recurrences • One type of First-Order linear recurrence • Another type of First-Order linear recurrence

34. Another Type of First-Order Linear Recurrence

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