CSE 373: Analysis of Algorithms

Course Webpagehttp://www.ams.sunysb.edu/~piyush/teach/373/

The Course

Instructor: Piyush Kumar email: piyush@cs.sunysb.edu Office Hours: Mon, Wed 11:45 - 12:45 PM;

Or by appointment (use email)

Teaching Assistants : TBA

The Course

Grading Policy Homework: 18% Mid Term: 35% Final: 47%

The Course

Prerequisites: MAT 211 w/ grade of C- or better AMS 211 w/ grade of C- or better CSE 214 w/ grade of C- or better Programming in [C++/C/Java]

The Course

Format Three lectures/week Homework mostly biweekly

Problem sets Maybe occasional programming

assignments One MidTerm (Oct 20th) + final exam FINAL EXAM is on DEC 15th, 8:00am to 10:30pm. Venue: TBA

Homework

Write problems beginning with a new page.

Only hard-copy (paper) submissions are allowed.

No late assignments Look at the Course Procedure

webpage 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.

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 Procedure 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, 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%

Algorithm: What is it?

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

Algorithm Characteristics Finiteness Input Output Rigorous, Unambiguous and

Sufficiently Basic at each step

Correctness

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

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.

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

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

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 Mathematical Induction from BB

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.