CSE 373: Analysis of Algorithms

CSE 373: Analysis of Algorithms Course Webpage http://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: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 High < Low Get Midpoint Failure Adjust Low Adjust High 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

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.

CSE 373: Analysis of Algorithms