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

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

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COMP 170 L2 Recursive Solution of Towers of Hanoi Page 6

COMP 170 L2 Recursive Solution of Towers of Hanoi Page 7

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

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

COMP 170 L2 Correctness of Algorithm Page 10

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

COMP 170 L2 Running Time Page 12

COMP 170 L2 Solving the Recurrence Page 13

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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

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

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

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

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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

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

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

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

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

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

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COMP 170 L2 Example Page 31

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

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

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

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