CSE 373: Analysis of Algorithms Course Webpage

The Course Instructor: Piyush Kumar Office Hours: Mon, Wed 11: :45 PM; Or by appointment (use ) 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 20 th ) + 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 informationCourse Procedure

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.Course Procedure

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 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 … + n = n(n+1) / 2 Basis: If n = 0, then 0 = 0(0+1) / 2 Inductive hypothesis: Assume … + n = n(n+1) / 2 Step (show true for n+1): … + n + n+1 = ( …+ 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 a 0 + a 1 + … + a n = (a n+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.