Intro to Computer Algorithms Lecture 3

1
Intro to Computer Algorithms Lecture 3

• Phillip G. Bradford
• Computer Science
• University of Alabama

2
Announcements

• Industrial Colloquium Series
• TODAY at 500pm in EE 119
• Herschel Chandler presents
• Being an Independent Computer Consultant

3
Announcements

• Colloquium
• Establishing Wireless Conference Calls Under
  Delay Constraints
• By Dr. Greg Malewicz
• Wednesday, Sept. 3
• 1100 am in HO108

4
Lecture Outline

• New Homework Assignment
• Review last lecture
• Asymptotic Notation
• Recurrences

5
Asymptotic Notation

• Big-O, Omega, Theta
• Little O and limits
• LHoptials rule
• Examples

6
Orders of MagnitudeFrom Levitins Book
7
LHopitals Rule

• Requires indeterminate forms
• Lim f(n)/g(n) ? 0/0
• Lim f(n)/g(n) ? infinity/infinity
• If indeterminate form, then
• lim f(n)/g(n) lim f(n)/g(n)
• Examples

8
An Example Hard Problem

• Towers of Hanoi
• Start Configuration

A
B
C
9
An Example Hard Problem

• Rules
• Move Only One Disk at a Time!
• Smaller Disks Must Be on Top!
• Final Configuration Desired

A
B
C
10
An Example Hard Problem

• What Algorithm Can We Use?

A
B
C
A
B
C
A
B
C
11
An Example Hard Problem

• More of the algorithm

A
B
C
A
B
C
A
B
C
12
An Example Hard Problem

• Finally

A
B
C
13
An Example Hard Problem

• What about 4 Disks?

A
B
C
A
B
C
How Much MORE work than for 3 Disks?
14
How expensive is this?

• T(n) 2T(n-1) 1, ngt 1
• T(1) 1.
• Closed form?
• Claim T(n) 2n -1
• How can we show this?