comp 170 l2 l11: recursion, recurrence, and induction l objective n recursion a problem solving...

COMP 170 L2

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

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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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Recursive Solution of Towers of HanoiPage 6

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Recursive Solution of Towers of HanoiPage 7

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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

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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

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Correctness of AlgorithmPage 10

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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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Running TimePage 12

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Solving the RecurrencePage 13

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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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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 case f(b) to complete the description

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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)

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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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Monthly Payment for Loan

Initial loan amount: A Annual interest rate: p Monthly Payment: M T(n): total amount still due after n months

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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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One Type of First-Order Linear Recurrence

Examples

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Iterating the Recurrence/Top-DownPage 25

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Iterating the Recurrence/Bottom-UpPage 26

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One Type of First-Order Linear RecurrencePage 28

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ExamplePage 31

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An Application of Theorem 4.1 Geometric Series

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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

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Another Type of First-Order Linear Recurrence

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