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CPSC 320: Intermediate AlgorithmDesign and Analysis

July 2, 2014

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Instructor and Course Structure

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InstructorJonatan Schroederjonatan@cs.ubc.caOffice: ICICS/CS 247

Office hours:• Mon/Wed/Fri: 12-1pm• Or by appointment

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Teaching Assistants• Alireza Shafaei• Anupam Srivastava• Jianing Yu• Juyoung Moon• Reza Babanezhad

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Pre-requisites• Listed student numbers should come to me after class• If you fail to act, you may be dropped from the course• Walk-in advising hours:

• Today 4-5pm: ICCS 391 (Paul Carter)• Thu 10:30-11:30am: ICCS 391 (Paul Carter)• Thu 3-4pm: ICCS 307 (George Tsiknis)• Additional hours may be listed in front of CS main office

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Learning Goals• Understand techniques used to design efficient algorithms• Prove (or disprove) that designed algorithms are correct• Prove that designed algorithms are efficient

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Course Outline• Introduction and basic concepts• Asymptotic notation• Greedy algorithms• Graph theory• Amortized analysis• Recursion• Divide-and-conquer algorithms• Randomized algorithms• Dynamic programming algorithms• NP-completeness

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Schedule• Lectures: Mon/Wed/Fri 9:30-12:00• Tutorials:

• T2A: Wed/Fri 1-2pm (Anupam)• T2B: Wed/Fri 2-3pm (Anupam)• T2C: Wed/Fri 3-4pm (Alireza)

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Piazza• All course announcements will be posted on Piazza• All course-related questions should be asked on Piazza

• Instructors and TAs will direct your questions to Piazza when relevant

• Questions including part of a possible solution should be made private

• Instructors may change a private question to public if deemed useful for other students

• Students are encouraged to answer questions as well

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Piazza (continued)• Please take a minute to understand how Piazza works

• Students’ response is for answering questions, not follow-ups

• Remember to mark follow-ups as resolved• If you found the answer on your own, please add an

answer (don’t delete the question)

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Marking scheme• 6 or 7 assignments: 20%• 5 quizzes: 30%• Final exam: 50%

• You must pass the exam to pass the course• You must pass in the average of the quizzes• You must submit at least five of the assignments

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Assignments• 6 or 7 assignments• Electronic submission or physical copy submission?

• Please answer a poll on Piazza today before 6PM!!!• Assignment 1 will be posted some time today

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Quizzes• Quiz 1: Wed, July 9• Quiz 2: Wed, July 16• Quiz 3: Wed, July 23• Quiz 4: Wed, July 30• Quiz 5: Wed, August 6

• Closed book, closed notes, no calculator• Individual component: about 30 minutes• Group component: about 15 minutes

• The group component is to be done in groups of 3-4• Group will solve one of the questions in the individual

component

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Textbook• John Kleinberg and Éva Tardos, Algorithm Design, Addison-

Wesley Publishing company, 2005, ISBN 0-321-29535-8.• Optional: Thomas H. Cormen, Charles E. Leiserson, Ronald L.

Rivest and Clifford Stein, Introduction to Algorithms, 3rd edition, MIT Press, 2009, ISBN 0-262-03384-4.

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Representative Problem

The Stable Matching Problem

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Algorithm• Sequence of steps to solve a specific task or problem• Takes an input, provides a result• Each step depends solely on the input and on previous steps• Stops in finite time

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Algorithm Analysis• Does the algorithm terminate?• How long does it take (as a function of the input size)?• Does it produce a valid result?• Is the produced result correct?• Is the produced result optimal?

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Stable Matching Problem• Given men and women• Each man ranks all women in preferred order and vice-versa• Assign every man to exactly one woman (and vice-versa)• Assignment needs to be stable

• Matching • If there is a pair such that prefers over and prefers

over then matching is unstable

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Example• Assume the input:

• M = {Alvin, Bob, Chuck, Donald}• W = {Janice, Kelly, Linda, Madison}• Preference ranking:

Alvin L J K MBob J M L KChuck K M L JDonald K J M L

Janice A D C BKelly A B C DLinda B D C AMadison C A B D

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Example• Example matching: (A, L), (B, J), (C, K), (D, M)

• Unstable (D prefers J, J prefers D)

Alvin L J K MBob J M L KChuck K M L JDonald K J M L

Janice A D C BKelly A B C DLinda B D C AMadison C A B D

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Example• Example matching: (A, L), (B, M), (C, K), (D, J)

• Stable (no changes give better option)

Alvin L J K MBob J M L KChuck K M L JDonald K J M L

Janice A D C BKelly A B C DLinda B D C AMadison C A B D

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Example• Example matching: (A, L), (B, M), (C, K), (D, J)

• Stable (no changes give better option)

Alvin L J K MBob J M L KChuck K M L JDonald K J M L

Janice A D C BKelly A B C DLinda B D C AMadison C A B D

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Discussion• What strategies can be used to find a matching?• Is there an optimal solution?

• What defines optimal in this case?• Design an algorithm

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Gale-Shapley Algorithmset all and to freewhile some free woman hasn’t proposed to every man do

← the highest-ranking man hasn’t proposed toif is free then

engage else

← s current fiancéeif prefers to then

set to freeengage

return the set of engaged pairs

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What should we do now?• Does the algorithm produce a matching?

• Result includes every woman and every man?• Each woman is assigned to exactly one man?• Each man is assigned to exactly one woman?

• Is the matching stable?• How long (in the worst case) does the algorithm take to

terminate?

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Complexity• Does the algorithm terminate?• What happens to a man m as the algorithm progresses?

• How many times can a man get engaged?• What happens to a woman w as the algorithm progresses?

• How many times can a woman get engaged?• What happens overall as the algorithm progresses?

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Complexity• Theorem: the algorithm terminates in no more than

iterations• Every iteration contains one proposal• Every woman proposes to a man she hasn’t proposed to

before• At most, each woman proposes to men• There are women• Number of iterations:

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Complexity (discussion)• So, does the algorithm run in ?

• How long to find a free woman?• How long to find the preferred man to propose?• How long to decide if a man should switch?• How long to add/replace a pair from the matching set?

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Correctness: Is the result a matching?• Does the resulting matching include every woman?

• Every man, once engaged, is always engaged• If there is a free woman, there is a free man• Free woman will find a free man

• Is each man engaged to one woman only?• If a man is free during proposal, he will be engaged to

one woman• If a man is not free, he may replace one woman for

another• Is each woman engaged to one man only?

• Only free women propose

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Correctness: Is matching stable?• By contradiction, suppose matching is unstable

• and are matched• prefers , prefers

• Three possible cases:• never proposed to • proposed to , but got rejected• proposed to , but got replaced

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Correctness: Is matching stable?• Case 1: never proposed to • proposed to , since they got a matching• Since proposes in order of preference, proposed to first• Contradiction

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Correctness: Is matching stable?• Case 2: proposed to , but got rejected• Men only reject if they are engaged to someone preferred• If rejects when engaged to :

• prefers to , contradiction• If rejects when engaged to someone else ()

• rejects because he prefers over • later engages to because he prefers over • So, prefers over and over , so prefers over ,

contradiction

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Correctness: Is matching stable?• Case 3: proposed to , but got replaced• If replaces with , prefers over , contradiction

• Finally, since all cases generate a contradiction, an unstable matching is never generated by this algorithm

• QED

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Discussion• Is the result optimal?

• What is considered optimal?• Who gets it better, women or men?• How could this be changed?

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Equivalent problems• Matching students and co-op jobs• Matching medical students and hospitals• Matching potential students and college openings

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Variations and Similar Problems• Hospitals can take several students• Number of men and women is different• Restrictions and limited preference lists• Stable roommate problem (matched pairs from same set)• Hospital/student matching with couples• Taxis and customers (reduced cost of pick up)

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Asymptotic Notation

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Machine Model• In all algorithms we will assume:

• Sequential execution• One processor• Memory position can hold arbitrarily large integer

• Unless otherwise specified, we will discuss the worst case• Best case: usually not very useful• Average case: depends on input distribution assumptions

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Upper Bound

• Describes the upper bound of the growth rate of the function

Assume