COP4531
CGS5427
: Analysis of Algorithms
Course Webpage
http://www.compgeom.com/~piyush/teach/4531/
And Blackboard
The Course
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Instructor: Piyush Kumar (piyush@acm.org)
Office: Love 105B
Phone: 850-645-2355
Office Hours: Tuesday 5:00pm - 6:00 pm;
Thursday 4:00pm – 5:00pm;
Or by appointment (use email)
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Teaching Assistant : TBA (prob. None)
The Course
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Grading Policy
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Homework + Programming Assignment +
Quiz : 30% = h score
Mid Terms + Finals: 70% = f score
To Pass: h >= 16 and f >= 40
Letter Grades based on sorted (h+f) scores
provided you pass.
The Course
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Prerequisites:
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COP 4530 w/ grade of C- or better
STA 4442 w/ grade of C- or better
Either MAD 3107 or MAD 3105
The Course
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Format
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Two lectures/week
Homework mostly biweekly
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Problem sets
One programming assignment
3 Surprise Short Quizzes
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(Mostly will depend on what I did in the past week)
( + One extra credit programming assignment )
Two MidTerms (Sep 28th, Nov 4th)
 FINAL EXAM is on DEC 10th,
3:00pm to 5:00pm. Venue: In Class, Love 301
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Homework
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Write problems beginning with a new
page.
Only hard-copy (paper) submissions are
allowed.
No late assignments
Look at the Course Information pages
for more information
Homework Policy
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If you ask to re-grade your homework please
write out the basis of your request.
If the grader finds no basis for your
complaint, then 10% points will be deducted
from your original grade unless the grade is
changed.
Note: This is not to discourage you from
disputing your grade, but rather we
encourage you to read and understand the
posted solutions on the web before you ask
your solutions to be re-graded
Homework Policy
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Under no circumstances should you be
copying others or modifying other
people’s work.
It is fine to discuss problems with others, but
all of the writing should be done without any
collaboration. Make sure you read the
Course Information webpage.
Homework Policy
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You can work in a pair or alone
If you work in a pair, You are both
supposed to write the solutions
independently and staple before you
submit.
Only one solution from a pair will be
graded (The one on top).
Exam Policy
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If you say “I don’t know” in any
question in the exam/hw, you get 25%
marks for that question/sub-question.
In case you don’t know the answer its
better to leave it than filling the answer
sheet with ‘crap’ because you might
even loose that 25%
How to do well in this class?
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Think
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Keep a few minutes everyday for thinking
about homework problems
Do all homework yourself, even if you
have a partner. If you did not think on
homework, be prepared to get burnt on
exams.
How to do well in this class?
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Attend Class Regularly. Be There or you will miss on
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The explanations
The demos
In-Class Quizzes
Announcements
Hints on homework and exams
It’s not the same to have notes from the web or
friends. Nothing can replace the experience of being
there.
Read Before Class.
How to do well in Class?
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Plan to spend a few hours every week
for Reading Assignments and doing
homework. ( 3 hours per class? )
Check web-pages/blackboard
frequently.
How to do well in Class?
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Don’t be afraid of me!
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Ask Questions in Class.
Attend office hours.
Feel free to visit me anytime (105b).
Feel free to email me.
piyush@acm.org
Feel free to call me.
850-645-2355
Again: Ask Questions in Class
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(Don’t wait for others to ask them for you).
Web Pages
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http://www.compgeom.com/~piyush/teach/4531/ : Course
Calendar, Instructor Information, Scribing Material,
Course Information, Grades.
Blackboard: Announcements, Homework,
Discussion, Reading Assignments, Programming
Projects.
Make Sure you check both these pages frequently.
Make Sure you check the email that is listed on the blackboard.
My email address for the course is piyush@acm.org
Algorithm: What is it?
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An Algorithm is a well-defined
computational procedure that
transforms inputs into outputs,
achieving the desired input-output
relationship.
Course Objectives
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design new algorithms.
analyze a given algorithm.
read and understand algorithms published in
journals.
develop writing skills to present your own
algorithms.
collaborate and work together with other
people to design new algorithms.
Crack job interviews.
Algorithm Characteristics
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Finiteness
Input
Correctness
Output
Rigorous, Unambiguous and Sufficiently
Basic at each step
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A Cookbook Recipe
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Add a dash of salt?
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Toss lightly till the mixture is crumbly?
Wait till the water starts to boil?
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(Where? How much exactly?)
(Remember: A watched pot never boils)
Resemblances: Input, Output,
Finiteness.
What we want?
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A ‘good’ Algorithm
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In some loosely defined aesthetic sense
Time
: Faster
Space
: Smaller
Portability/Adaptability to different
machines.
Simplicity and Elegance
Applications?
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WWW and the Internet
Computational Biology
Scientific Simulation
VLSI Design
Security
Automated Vision/Image Processing
Compression of Data
Databases
Mathematical Optimization
The RAM Model
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Analysis is performed with respect to a
computational model
We will usually use a generic uniprocessor
random-access machine (RAM)
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All memory equally expensive to access
No concurrent operations
All reasonable instructions take unit time
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Except, of course, function calls
Constant word size
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Unless we are explicitly manipulating bits
Linear Search I
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For I = 1 to N
if (a[I] == x) return true;
return false;
Linear Search II
A[N+1] = x; I = 1;
While A[I] != x
++I;
if I == N+1 return false;
Else return true;
A Rough Analysis
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N = 10000
CMP
INC
Search I
2000
1000
Search II
1000
1002
Sorting
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Input:
Array A[1...n], of elements in
arbitrary order
Output: Array A[1...n] of the same
elements, but in increasing order
Given a teacher find all his/her students.
Given a student find all his/her teachers.
Binary Search
Initialize
Get Midpoint
High < Low
Adjust Low
Adjust High
<
Failure
>
Compare
=
Success
Binary Search
Algorithm:
Low= 1; High = n;
while Low < High {
m = floor( (Low+High)/2 );
if k <= A[m]
then High = m - 1
else Low = m + 1
}
if A[Low] = k then j = Low else j = 0
An Illustration
23
15
5
29
19
27
31
Sorted Array: 5, 15, 19, 23, 27, 29, 31
Index
:1
2
3
4
5
6 7
Time and Space Complexity
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Generally a function of the input size
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E.g., sorting, multiplication
How we characterize input size depends:
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Sorting: number of input items
Multiplication: total number of bits
Graph algorithms: number of nodes & edges
Etc
Running Time
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Number of primitive steps that are
executed
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Except for time of executing a function call
most statements roughly require the same
amount of time
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y=m*x+b
c = 5 / 9 * (t - 32 )
z = f(x) + g(y)
We can be more exact if need be
Analysis
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Worst case
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Provides an upper bound on running time
An absolute guarantee
Average case
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Provides the expected running time
Very useful, but treat with care: what is
“average”?
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Random (equally likely) inputs
Real-life inputs
Binary Search Analysis
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Order Notation
Upper Bounds
Search Time = ??
A better way to look at it,
Binary Search Trees
Searching
A bad king has a cellar of 1000 bottles of delightful and
very expensive wine. a neighbouring queen plots to kill
the bad king and sends a servant to poison the wine.
(un)fortunately the bad king's guards catch the servant
after he has only poisoned one bottle. alas, the guards
don't know which bottle but know that the poison is so
strong that even if diluted 1,000,000 times it would still
kill the king. furthermore, it takes one month to have an
effect. the bad king decides he will get some of the
prisoners in his vast dungeons to drink the wine. being
a clever bad king he knows he needs to murder no more
than 10 prisoners - believing he can fob off such a low
death rate - and will still be able to drink the rest of the
wine at his anniversary party in 5 weeks time. Explain
how...
Solution
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Number each bottle in binary digits
Feed each prisoner one column of the
list of the binary digits where 1 means
the bottle is tasted and zero means its
not
Convert the death of the 10 prisoners
into a decimal number, That’s the bottle
we are looking for.
Induction
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Prove 1 + 2 + 3 + … + n = n(n+1) / 2
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Basis:
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Inductive hypothesis:
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If n = 0, then 0 = 0(0+1) / 2
Assume 1 + 2 + 3 + … + n = n(n+1) / 2
Step (show true for n+1):
1 + 2 + … + n + n+1 = (1 + 2 + …+ n) + (n+1)
= n(n+1)/2 + n+1 = [n(n+1) + 2(n+1)]/2
= (n+1)(n+2)/2 = (n+1)(n+1 + 1) / 2
Induction: A Fine example
Practice Problem
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Prove
a0 + a1 + … + an = (an+1 - 1)/(a - 1)
Read MI from the web.
Homework problem: What’s wrong?
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Problem: Prove that x^(k-1) = 1
Proof:
P(1) : x^(1-1) = 1
By Induction assume P(1),P(2)..P(n) are true.
P(n+1):
x^{(n+1)-1} = x^n
= x^{n-1} * x^{n-1} / x^{n-2}
=1*1/1=1!
Assignments
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Go thru the slides for the first lecture.
Read chapter 1 of the text book
'Suggested' Exercises: 1.1-5, 1.2-3
(Solutions not to be Submitted/Graded)
Read the course Information Sheet
Submit: Why the proof is wrong on the
previous slide.
Next Time
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In this course, we care most about
asymptotic performance
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How does the algorithm behave as the
problem size gets very large?
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Running time
Memory/storage requirements
Bandwidth/power requirements/logic gates/etc.