: Analysis of Algorithms

Course Webpagehttp://www.compgeom.com/~piyush/teach/4531/

And Blackboard

COP4531CGS5427

The Course 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)

Teaching Assistant : TBA (prob. None)

The Course

Grading Policy 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

Prerequisites: COP 4530 w/ grade of C- or better STA 4442 w/ grade of C- or better Either MAD 3107 or MAD 3105

The Course Format

Two lectures/week Homework mostly biweekly

Problem sets One programming assignment 3 Surprise Short Quizzes

(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

Homework

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 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 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

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

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?

Think 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? Attend Class Regularly. Be There or you will

miss on 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?

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?

Don’t be afraid of me! 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

(Don’t wait for others to ask them for you).

Web Pages 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?

An Algorithm is a well-defined computational procedure that transforms inputs into outputs, achieving the desired input-output relationship.

Course Objectives 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 Finiteness Input Output Rigorous, Unambiguous and

Sufficiently Basic at each step

Correctness

A Cookbook Recipe

Add a dash of salt? (Where? How much exactly?)

Toss lightly till the mixture is crumbly?

Wait till the water starts to boil? (Remember: A watched pot never boils)

Resemblances: Input, Output, Finiteness.

What we want?

A ‘good’ Algorithm In some loosely defined aesthetic

sense Time : Faster Space : Smaller Portability/Adaptability to different

machines. Simplicity and Elegance

Applications? WWW and the Internet Computational Biology Scientific Simulation VLSI Design Security Automated Vision/Image Processing Compression of Data Databases Mathematical Optimization

The RAM Model Analysis is performed with respect to a

computational model We will usually use a generic

uniprocessor random-access machine (RAM) All memory equally expensive to access No concurrent operations All reasonable instructions take unit time

Except, of course, function calls Constant word size

Unless we are explicitly manipulating bits

Linear Search I

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 N = 10000

CMP INC

Search I 2000 1000

Search II 1000 1002

Sorting

Input: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

Compare

Adjust High Adjust Low

Failure

Success

High < Low

=

< >

Binary SearchAlgorithm: 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 29

5 19 27 31

Sorted Array: 5, 15, 19, 23, 27, 29, 31Index : 1 2 3 4 5 6 7

Time and Space Complexity

Generally a function of the input size E.g., sorting, multiplication

How we characterize input size depends: Sorting: number of input items Multiplication: total number of bits Graph algorithms: number of nodes & edges Etc

Running Time Number of primitive steps that are

executed Except for time of executing a

function call most statements roughly require the same amount of time

y = m * x + b c = 5 / 9 * (t - 32 ) z = f(x) + g(y)

We can be more exact if need be

Analysis Worst case

Provides an upper bound on running time An absolute guarantee

Average case Provides the expected running time Very useful, but treat with care: what is

“average”? Random (equally likely) inputs Real-life inputs

Binary Search Analysis

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 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 Prove 1 + 2 + 3 + … + n = n(n+1) / 2

Basis: If n = 0, then 0 = 0(0+1) / 2

Inductive hypothesis: 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

Prove a0 + a1 + … + an = (an+1 - 1)/(a -

1)

Read MI from the web.

Homework problem: What’s wrong?

Problem: Prove that x^(k-1) = 1 Proof:

P(1) : x^(1-1) = 1By 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

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

In this course, we care most about asymptotic performance How does the algorithm behave as

the problem size gets very large? Running time Memory/storage requirements Bandwidth/power requirements/logic

gates/etc.