Transcript CS 615: Design & Analysis of Algorithms

CS 615:
Design & Analysis of Algorithms
Chapter 1: Introduction, Algorithmic
Notation and Flowcharts
Course Content
1. Introduction, Algorithmic Notation and Flowcharts
(Brassard & Bratley, Chap. 3)
2. Efficiency of Algorithms (Brassard & Bratley, Chap
2)
3. Basic Data Structures (Brassard & Bratley, Chap. 5)
4. Sorting (Weiss, Chap. 7)
5. Searching (Brassard & Bratley, Chap. 9)
6. Graph Algorithms (Weiss, Chap. 9)
7. Randomized Algorithms (Weiss, Chap. 10)
8. String Searching (Sedgewick, Chap. 19)
9. NP Completeness (Sedgewick, Chap. 40)
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Content
Text books,
Lectures
Assessments
Asymptotic Notation
Notation for “the order of”
Maximum Rule
Flowcharts
Basic Flowchart Constructions
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Instructor
Yusuf Altunel
Email:
[email protected]
Web:
http://web.iku.edu.tr/~yaltunel
Tel:
(212) 498 42 10
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Text Books
Main Resources
Data Structures and Algorithm Analysis in C, 2nd Edition; Mark Allen
Weiss; Addison Wesley Longman, 1997
Fundamentals of Algorithmics; Gilles Brassard & Paul Bratley; PrenticeHall, 1996.
Data Structures, Algorithms & Software Principles, Thomas A. Standish,
Addison Wesley, 1995.
Other Resources
Algorithms, Richard Johnsonbaugh and Marcus Schaefer,
Pearson, 2004, ISBN 0-02-360692-4.
http://condor.depaul.edu/~rjohnson/algorithm/index.html
Algorithm Design: Foundations, Analysis, and Internet
Examples; Michael T. Goodrich & Roberto Tamassia; Wiley,
2002, ISBN 978-0-471-38365-9.
http://ww3.algorithmdesign.net/
Algorithms, Robert Sedgewick, Addison-Wesley 1983, e-book.
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Lectures
Courses and labs
Day
Hours
Location
Theory
Thu
11:00-12:45 (2 Hours)
Lab 251
Lab
Fri
11:00-12:45 (2 Hours)
Lab 251
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Follow up the course
Use the web page of the course
http://web.iku.edu.tr/~yaltunel/Spring2005/CS615
Use the email group of the course
Send an empty email to subscribe:
[email protected]
To send a message:
[email protected]
To access on web:
http://groups.yahoo.com/group/CS_615
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Assessment
Participation
%10
Project
%30
MidTerm
%30
Final
•Please take care of the rules and regulations of the University.
•Otherwise you might be reported to the faculty, as well as
lose all bonus options of this course.
%40
Bonus
Grading: %110
%10 Bonus
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Project
A real world application with dynamically
constructed graph should be implemented
until the end of the semester!
Details will be announced later!
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Algorithmic Notation and
Flowcharts (Brassard & Bratley
Chp: Chapter 3)
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Asymptotatic Notation
Used to express
The time taken by an algorithm
Within a multiplicative constant
Permits
Simplification in measuring other tangible things
Deals with
The behavior of functions in the limits
For sufficiently large parameters
An asymptotically superior algorithm is
very often preferable
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Notation for “the order of”
f:NR0 be an arbitrary function
from the natural numbers
to the nonnegative numbers
The mathematical symbol order of is
denoted by O(f(n))
the set of all functions t: NR0 such that
t(n) cf(n) for all nn0
O(f(n))={t: NR0 |(cR+)(nN)[t(n) cf(n)]}
Examples
f(n)=n3-3n2-n-8  O(n3)
f(n)= n3 O(n3)
Maximum Rule:
Let f,g: NR0 be two arbitrary functions
O(f(n),g(n))=O(max(f(n),g(n)))
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Example for Maximum Rule
An algorithm that proceeds in three steps:
Initialization
Take time in O(n2)
Processing
Take time in O(n3)
Finalization
Take time in O(nlogn)
The complete algorithm takes time in
O(n2 + n3 + nlogn)
= O(max(n2,n3,nlogn))
=O(n3)
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Results of Maximum Rule
If t(n) is a complicated function
the most significant term of t(n) gives
the order of function discarding its coefficient
Example:
f(n)= 12n3- 5n2 + logn + 36
O(f(n))=O(n3)
f(n)= 999999n94 - 345n35 + 1
O(f(n))=O(n94)
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Scale of Strength for O-Notation
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More efficient
O(1) < O(logn) < O(n) <
O(nlogn) < O(n2) < O(n3)
< O(2n) < O(10n)
O(1)
O(logn)
O(n)
O(nlogn)
O(n2)
O(n3)
O(2n)
O(10n)
CS 615 Design & Analysis of Algorithms
Higher in order
the relationship
between order of
functions
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Examples
What is the order of 18nlog2n + 19n + 3
=O(18nlog2n + 19n + 3)
=O(max (nlogn, n, 3) )
=O(nlogn)
What is the order of n4 + 2n + 3nlogn + 3
=O(n4 + 2n + 3nlogn + 3)
=O(max (n4, 2n, nlogn , 3) )
=O(2n)
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Flowcharts
Process
Data
Decision
Process
Display
Terminator
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Basic Flowchart Constructions
First
Task
Next
Task
F
T
Cond
Else
Part
Then
Part
F
Cond
T
Sequence
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If Then Else
CS 615 Design & Analysis of Algorithms
Repeat-Until
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Basic Flowchart Constructions
T
Cond1
F
T
Cond
F
Cond2
...
...
T
T
Condn
F
Selection (Case)
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While-Do
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Basic Flowchart Constructions
initialization
F
Cond
T
for (i=1; i<n; i++) {..}
for (<initialization>; <condition>; <post statement>) <statement>
statement
post statement
For statement
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Data Flow: Example
Start
Task1
if Cond1
then
Start
Task1
T
T
Cond1
F
End
Task4
Cond2
Task2
F
if Cond2
then Task2
else Task3
else repeat
Task4
until Cond3
Task3
Cond3
F
T
End
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Example find Maximum
Start
max= infinitive
Start
max= minimum
do
get a number
if (max<number)
then max = number
while more numbers
show max
get a number
T
F
max<number
max=number
T
More numbers?
F
T
End
Show max
end
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End of Chapter 1
Introduction, Algorithmic
Notation and Flowcharts
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