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COMP 170 L2 L11: Recursion, Recurrence, and Induction l Objective n Recursion  A problem solving technique that reduces big problems into smaller ones.

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Presentation on theme: "COMP 170 L2 L11: Recursion, Recurrence, and Induction l Objective n Recursion  A problem solving technique that reduces big problems into smaller ones."— Presentation transcript:

1 COMP 170 L2 L11: Recursion, Recurrence, and Induction l Objective n Recursion  A problem solving technique that reduces big problems into smaller ones n Induction  Proving correctness of recursive programs n Recurrence  Running time for recursive programs Page 1

2 COMP 170 L2 Outline l Recursive algorithm for Towers of Hanoi n Problem and algorithm n Correctness n Running time l Recurrence in general n Example: Number of subsets n Example: Loan repayment l Solving recurrences n One type of First-Order linear recurrences n Another type of First-Order linear recurrences Page 2

3 COMP 170 L2 Page 3

4 COMP 170 L2 Page 4

5 COMP 170 L2 Page 5

6 COMP 170 L2 Recursive Solution of Towers of Hanoi Page 6

7 COMP 170 L2 Recursive Solution of Towers of Hanoi Page 7

8 COMP 170 L2 Recursive Solution of Towers of Hanoi l Task: n Move n disks from peg i to peg j l Algorithm n Recursion base:  When n=1, move one disk from i to j n Recursion Page 8

9 COMP 170 L2 Outline l Recursive algorithm for Towers of Hanoi n Problem and algorithm n Correctness n Running time l Recurrence in general n Example: Number of subsets n Example: Loan repayment l Solving recurrences n Geometric serious n First-Order linear recurrences Page 9

10 COMP 170 L2 Correctness of Algorithm Page 10

11 COMP 170 L2 Outline l Recursive algorithm for Towers of Hanoi n Problem and algorithm n Correctness n Running time l Recurrence in general n Example: Number of subsets n Example: Loan repayment l Solving recurrences n One type of First-Order linear recurrences n Another type of First-Order linear recurrences Page 11

12 COMP 170 L2 Running Time Page 12

13 COMP 170 L2 Solving the Recurrence Page 13

14 COMP 170 L2 Page 14

15 COMP 170 L2 Page 15

16 COMP 170 L2 Outline l Recursive algorithm for Towers of Hanoi n Problem and algorithm n Correctness n Running time l Recurrence in general n Example: Number of subsets n Example: Loan repayment l Solving recurrences n One type of First-Order linear recurrences n Another type of First-Order linear recurrences Page 16

17 COMP 170 L2 Recurrence l Recurrence/recurrence equation is a way to specify functions on the set of integers l It tells us how to n get the n-th value f(n) n from the first n-b values: f(b), f(b+1), …, f(n-1) l Need to give the value for the base case f(b) to complete the description Page 17

18 COMP 170 L2 Number of subsets l S(n): number of subsets of set {1, 2, 3, …., n} of size n l Question: n How to compute S(n) from S(n-1), S(n-2), …? l Consider the case: n=3 n First row: subsets of {1, 2} n Second: subsets of {1, 2}, each adjoined by 3 n So: S(3) = 2 S(2) Page 18

19 COMP 170 L2 Number of Subsets l In general, subsets of set {1, 2, 3, …., n} can be divided into two groups n subsets of set {1, 2, 3, …., n-1} n subsets of set {1, 2, 3, …., n-1}, each adjoined by n. n So: S(n) = 2 S(n-1) l Base case: n S(0) =1 Page 19

20 COMP 170 L2 Page 20

21 COMP 170 L2 Monthly Payment for Loan l Initial loan amount: A l Annual interest rate: p l Monthly Payment: M l T(n): total amount still due after n months Page 21

22 COMP 170 L2 Outline l Recursive algorithm for Towers of Hanoi n Problem and algorithm n Correctness n Running time l Recurrence in general n Example: Number of subsets n Example: Loan repayment l Solving recurrences n One type of First-Order linear recurrences n Another type of First-Order linear recurrences Page 22

23 COMP 170 L2 Page 23

24 COMP 170 L2 One Type of First-Order Linear Recurrence Examples Page 24

25 COMP 170 L2 Iterating the Recurrence/Top-Down Page 25

26 COMP 170 L2 Iterating the Recurrence/Bottom-Up Page 26

27 COMP 170 L2 Page 27

28 COMP 170 L2 One Type of First-Order Linear Recurrence Page 28

29 COMP 170 L2 Page 29

30 COMP 170 L2 Page 30

31 COMP 170 L2 Example Page 31

32 COMP 170 L2 An Application of Theorem 4.1 Geometric Series Page 32

33 COMP 170 L2 Outline l Recursive algorithm for Towers of Hanoi n Problem and algorithm n Correctness n Running time l Recurrence in general n Example: Number of subsets n Example: Loan repayment l Solving recurrences n One type of First-Order linear recurrence n Another type of First-Order linear recurrence Page 33

34 COMP 170 L2 Page 34 Another Type of First-Order Linear Recurrence

35 COMP 170 L2 Page 35

36 COMP 170 L2 Page 36

37 COMP 170 L2 Page 37

38 COMP 170 L2 Page 38

39 COMP 170 L2 Page 39

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