L11: Recursion, Recurrence, and Induction

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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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