L11: Recursion, Recurrence, and Induction
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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 Page 1
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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. Page 2. Outline. - PowerPoint PPT Presentation
Transcript of L11: Recursion, Recurrence, and Induction
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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COMP 170 L2
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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