Properties of matching algorithm in college admission problem with variable capacity quotas. Robert Golański Warsaw School of Economics
|
|
- Naomi Green
- 6 years ago
- Views:
Transcription
1 Properties of matching algorithm in college admission problem with variable capacity quotas Robert Golański Warsaw School of Economics Social Choice and Welfare Moscow, July
2 College admission problem: S = {s 1,...,s n } a set of students C = {c 1,...,c m } a set of colleges R =(R s1,..., R sn, R c1,..., R cm ) a list of preferences of students and colleges q =(q 1,...,q m ) a vector of college capacities C, S finite, disjoint, non-empty sets 2
3 Gale-Shapley student-optimal college admission algorithm: Step 1: Each student proposes to her first choice. Each college tentatively assigns its seats to its proposers one at a time following their priority order. Any remaining proposers are rejected. 3
4 Gale-Shapley student-optimal college admission algorithm: Step 1: Each student proposes to her first choice. Each college tentatively assigns its seats to its proposers one at a time following their priority order. Any remaining proposers are rejected. Step k: Each student who was rejected in the previous step proposes to her next choice. Each school considers the students it has been holding together with its new proposers and tentatively assigns its seats to these students one at a time following their priority order. Any remaining proposers are rejected. 4
5 Example: 3 colleges (A, B, C), 6 students Students preferences: 1 A B C 2 A C B 3 A B C 4 B A C 5 B C A 6 C A B Each college prefers 1 to 2 to to 6. Each quota set to 2. 5
6 Step 1 1, 2, 3 apply to A 4 and 5 apply to B 6 applies to C. All but 3 are tentatively accepted Step 2 3 applies to B (his second choice) as a result B rejects student 5 (who was already tentatively applied); Step 3 5 applies to C and is accepted. 6
7 Properties: - individual rationality; - stability (no blocking pair); - (group-)strategy proofness of student-optimal mechanism (Dubins- Freedman 1981) 7
8 Modification preference-induced quotas Why it matters student assignment problems In what follows we ll assume that each college has the same preferences over students + no indifference in preferences. So the timing of the problem is as follows: - the problem is defined by (C, S, R) only; - each students submits the list of preferences; - based on the students preferences, the quotas are determined; - given the quotas the students are assigned to colleges. 8
9 In what follows we ll assume responsiveness in college preferences and that having any student is preferred to not having them. Different possibilities of choosing quotas based on preferences. Quota = number of top preferences not very interesting (everybody will get assigned to their optimal college; equivalent to setting q i = S in the original problem, so clearly stable, optimal and non-manipulable) 9
10 We ll have a look at two different rules: - quotas decided based on top preferences with restrictions; - quotas based on scoring rules (Borda). 10
11 We ll have a look at two different rules: - quotas decided based on top preferences with restrictions; - quotas based on scoring rules (Borda) (in both cases we need some additional rules on rounding and tiebreaking. Consider for example the case in which top preferences for students are A for 7, B for 2 and C for 2 and the maximum capacity is set to 6. The natural assignment of quotas would be to assign 6 to A (since it already exceeds the maximum capacity) and equal number to B and C since they have equal numbers of students top choices. That would result in the quotas of (6; 2.5; 2.5) which is clearly not feasible). 11
12 Top preferences with restrictions: Consider the following problem: 3 colleges (A, B, C), 11 students with following preferences: 1 A to B to C 6 A to B to C 2 A to B to C 7 B to A to C 3 A to B to C 8 B to A to C 4 A to B to C 9 A to B to C 5 A to B to C 10 C to A to B 11 C to A to B 12
13 Set the maximum capacity to 5. The top-preference induced quotas will thus be q A = 5; q B = 3; q C = 3. 1 A to B to C 6 A to B to C 2 A to B to C 7 B to A to C 3 A to B to C 8 B to A to C 4 A to B to C 9 A to B to C 5 A to B to C 10 C to A to B 11 C to A to B 13
14 Set the maximum capacity to 5. The top-preference induced quotas will thus be q A = 5; q B = 3; q C = 3. 1 A to B to C 6 A to B to C 2 A to B to C 7 B to A to C 3 A to B to C 8 B to A to C 4 A to B to C 9 A to B to C 5 A to B to C 10 C to A to B 11 C to A to B 14
15 Set the maximum capacity to 5. The top-preference induced quotas will thus be q A = 5; q B = 3; q C = 3. 1 A to B to C 6 A to B to C 2 A to B to C 7 B to A to C 3 A to B to C 8 B to A to C 4 A to B to C 9 A to B to C 5 A to B to C 10 C to A to B 11 C to A to B Student optimal assignment is 1 through 5 to college A, 6 through 8 to college B, 9 through 11 to college C. 15
16 Stability needs redefining (is having one more student feasible?) Optimality holds if all colleges have the same preferences. 16
17 Stability needs redefining (is having one more student feasible?) Optimality holds if all colleges have the same preferences. Bad news the rule is manipulable. Suppose student 9 (with preferences A B C; assigned to college C) submits the preference ordering of B A C. In the standard Gale-Shapley problem nothing changes (9 is now tentatively assigned to college B in step 2, but eventually ends up in C anyway) 17
18 Now quotas change: With the falsely submitted preferences we now have: - 6 students with preferences A B C, - 3 students with preferences B A C, - 2 students with preferences C A B. Top-preference restricted quotas should thus be (5; 3.6; 2.4), rounded to (5; 4; 2) rather than (5; 3; 3). With new quotas the only change in the final assignment is that student 9 is now assigned to B (preferred to originally received C). 18
19 Does breaking the problem into two rounds of voting help? - in the second round the quotas are set and decided => standard Gale- Shapley problem (so stability and strategy-proofness follow); - in the first round still potential for manipulability 19
20 Does breaking the problem into two rounds of voting help? - in the second round the quotas are set and decided => standard Gale- Shapley problem (so stability and strategy-proofness follow); - in the first round still potential for manipulability Consider the previous example given others behave truthfully, student 9 still has an incentive to falsely present his preference in the quota-setting stage to increase q B. 20
21 Scoring rule (Borda) Again, we can show Pareto-optimality but the rule is manipulable 21
22 Scoring rule (Borda) Again, we can show Pareto-optimality but the rule is manipulable 1 C to A to B 6 B to A to C 2 C to A to B 7 B to A to C 3 C to B to A 8 B to A to C 4 C to B to A 9 A to B to C 5 B to C to A 10 A to C to B Borda scores A 9; B 11, C 10 Proportional quotas A 3; B 3.(6); C 3.(3) Rounded to q A = 3; q B = 4; q C = 3 22
23 Scoring rule (Borda) Again, we can show Pareto-optimality but the rule is manipulable 1 C to A to B 6 B to A to C 2 C to A to B 7 B to A to C 3 C to B to A 8 B to A to C 4 C to B to A 9 A to B to C 5 B to C to A 10 A to C to B Borda scores A 9; B 11, C 10 Proportional quotas A 3; B 3.(6); C 3.(3) Rounded to q A = 3; q B = 4; q C = 3 23
24 Suppose student 8 misrepresents his preferences as B C A: 1 C to A to B 6 B to A to C 2 C to A to B 7 B to A to C 3 C to B to A 8 B to C to A 4 C to B to A 9 A to B to C 5 B to C to A 10 A to C to B 24
25 Suppose student 8 misrepresents his preferences as B C A: 1 C to A to B 6 B to A to C 2 C to A to B 7 B to A to C 3 C to B to A 8 B to C to A 4 C to B to A 9 A to B to C 5 B to C to A 10 A to C to B New Borda scores are A 8; B 11, C 11 25
26 Suppose student 8 misrepresents his preferences as B C A: 1 C to A to B 6 B to A to C 2 C to A to B 7 B to A to C 3 C to B to A 8 B to C to A 4 C to B to A 9 A to B to C 5 B to C to A 10 A to C to B New Borda scores are A 8; B 11, C 11 New quotas: q A = 2; q B = 4; q C = 4 26
27 Suppose student 8 misrepresents his preferences as B C A: Final assignment becomes: 1 C to A to B 6 B to A to C 2 C to A to B 7 B to A to C 3 C to B to A 8 B to C to A 4 C to B to A 9 A to B to C 5 B to C to A 10 A to C to B New Borda scores are A 8; B 11, C 11 New quotas: q A = 2; q B = 4; q C = 4 27
28 Suppose student 8 misrepresents his preferences as B C A: Final assignment becomes: 1 C to A to B 6 B to A to C 2 C to A to B 7 B to A to C 3 C to B to A 8 B to C to A 4 C to B to A 9 A to B to C 5 B to C to A 10 A to C to B Now every student gets their best college! Student 8 gets to college B even though q B didn t change. (It s not a Pareto improvement though college A lost) 28
29 So even more problems manipulability now occurs even without setting additional maximum capacity rules. 29
30 Do stable non-manipulable rules exist? For some quota-setting rules they may (unrestricted first preferences; scoring rules with equal weights = always assigns equal quotas to each college). 30
The College Admissions Problem Reconsidered
The College Admissions Problem Reconsidered Zhenhua Jiao School of Economics Shanghai University of Finance and Economics Shanghai, 200433, China Guoqiang Tian Department of Economics Texas A&M University
More informationPairwise stability and strategy-proofness for college admissions with budget constraints
Pairwise stability and strategy-proofness for college admissions with budget constraints Azar Abizada University of Rochester First draft: October 22, 2009 This version: January 5, 2012 Abstract We study
More informationDEFERRED ACCEPTANCE 2
DEFERRED ACCEPTANCE The point of this note is to take you through the logic behind the deferred acceptance algorithm. Please read the papers associated with the links on the web page for motivation and
More informationEfficiency and Respecting Improvements in College Admissions
Efficiency and Respecting Improvements in College Admissions Zhenhua Jiao School of Economics Shanghai University of Finance and Economics Shanghai, 200433, China Guoqiang Tian Department of Economics
More informationSchool Choice with Controlled Choice Constraints: Hard Bounds versus Soft Bounds
School Choice with Controlled Choice Constraints: Hard Bounds versus Soft Bounds Lars Ehlers Isa E. Hafalir M. Bumin Yenmez Muhammed A. Yildirim September 2012 Abstract Controlled choice over public schools
More informationAssortment Planning in School Choice [Preliminary Draft]
Assortment Planning in School Choice [Preliminary Draft] Peng Shi MIT Operations Research Center, Cambridge, MA 02139, pengshi@mit.edu Draft Date: Jan. 2016 In many public school systems across the US,
More informationCOMPSCI 311: Introduction to Algorithms First Midterm Exam, October 3, 2018
COMPSCI 311: Introduction to Algorithms First Midterm Exam, October 3, 2018 Name: ID: Answer the questions directly on the exam pages. Show all your work for each question. More detail including comments
More information1 The Arthur-Merlin Story
Comp 260: Advanced Algorithms Tufts University, Spring 2011 Prof. Lenore Cowen Scribe: Andrew Winslow Lecture 1: Perfect and Stable Marriages 1 The Arthur-Merlin Story In the land ruled by the legendary
More informationFinite Termination of Augmenting Path Algorithms in the Presence of Irrational Problem Data
Finite Termination of Augmenting Path Algorithms in the Presence of Irrational Problem Data Brian C. Dean Michel X. Goemans Nicole Immorlica June 28, 2006 Abstract This paper considers two similar graph
More informationAnarchy, Stability, and Utopia: Creating Better Matchings
Noname manuscript No. (will be inserted by the editor) Anarchy, Stability, and Utopia: Creating Better Matchings Elliot Anshelevich Sanmay Das Yonatan Naamad September 20 Abstract Historically, the analysis
More information18 Spanning Tree Algorithms
November 14, 2017 18 Spanning Tree Algorithms William T. Trotter trotter@math.gatech.edu A Networking Problem Problem The vertices represent 8 regional data centers which need to be connected with high-speed
More informationCopyright 2007 Pearson Addison-Wesley. All rights reserved. A. Levitin Introduction to the Design & Analysis of Algorithms, 2 nd ed., Ch.
Iterative Improvement Algorithm design technique for solving optimization problems Start with a feasible solution Repeat the following step until no improvement can be found: change the current feasible
More informationAPPLIED MECHANISM DESIGN FOR SOCIAL GOOD
APPLIED MECHANISM DESIGN FOR SOCIAL GOOD JOHN P DICKERSON Lecture #8 09/22/2016 CMSC828M Tuesdays & Thursdays 12:30pm 1:45pm LET S TALK ABOUT PROJECTS 2 THIS CLASS: MATCHING & NOT THE NRMP 3 OVERVIEW OF
More informationCoalitional Permutation Manipulations in the Gale-Shapley Algorithm
Coalitional Permutation Manipulations in the Gale-Shapley Algorithm Yuan Deng Duke University ericdy@cs.duke.edu Weiran Shen IIIS, Tsinghua University emersonswr@gmail.com Pingzhong Tang IIIS, Tsinghua
More informationMinimum Maximal Matching Is NP-Hard in Regular Bipartite Graphs
Minimum Maximal Matching Is NP-Hard in Regular Bipartite Graphs M. Demange 1 and T. Ekim 2, 1 ESSEC Business School, Avenue Bernard HIRSH, BP 105, 95021 Cergy Pontoise cedex France demange@essec.fr 2 Boğaziçi
More informationPackage matchingr. January 26, 2018
Type Package Title Matching Algorithms in R and C++ Version 1.3.0 Date 2018-01-26 Author Jan Tilly, Nick Janetos Maintainer Jan Tilly Package matchingr January 26, 2018 Computes matching
More informationLecture 7: Bipartite Matching
Lecture 7: Bipartite Matching Bipartite matching Non-bipartite matching What is a Bipartite Matching? Let G=(N,A) be an unrestricted bipartite graph. A subset X of A is said to be a matching if no two
More informationUse the following preference schedule to answer questions #1 - #4. Number of Voters st choice 2 nd Choice 3 rd choice 4 th choice
M130 Sample Final Exam Name TA Name Calculators are allowed, but cell phones or palm pilots are NOT acceptable. Mark Version A on your scantron. MULTIPLE CHOICE. Choose the one alternative that best completes
More informationCHAPTER 13: FORMING COALITIONS. Multiagent Systems. mjw/pubs/imas/
CHAPTER 13: FORMING COALITIONS Multiagent Systems http://www.csc.liv.ac.uk/ mjw/pubs/imas/ Coalitional Games Coalitional games model scenarios where agents can benefit by cooperating. Issues in coalitional
More informationSolutions to Math 381 Quiz 2
Solutions to Math 381 Quiz 2 November 7, 2018 (1) In one sentence, what is the goal of your class project? Each team has a different answer here. (2) Write a function in Python which computes a linear
More informationChoice under Social Constraints
Choice under Social Constraints T.C.A Madhav Raghavan January 28, 2011 Abstract Agents choose a best element from the available set. But the available set may itself depend on the choices of other agents.
More informationM130 Sample Final Exam
M130 Sample Final Exam Use the following preference schedule to answer questions #1 - #4. Number of Voters 71 22 15 63 1 st choice 2 nd Choice 3 rd choice 4 th choice A B C D D C B A D B C A C A B D 1)
More informationCPSC W1: Midterm 1 Sample Solution
CPSC 320 2017W1: Midterm 1 Sample Solution January 26, 2018 Problem reminders: EMERGENCY DISTRIBUTION PROBLEM (EDP) EDP's input is an undirected, unweighted graph G = (V, E) plus a set of distribution
More informationCardinality of Sets MAT231. Fall Transition to Higher Mathematics. MAT231 (Transition to Higher Math) Cardinality of Sets Fall / 15
Cardinality of Sets MAT Transition to Higher Mathematics Fall 0 MAT (Transition to Higher Math) Cardinality of Sets Fall 0 / Outline Sets with Equal Cardinality Countable and Uncountable Sets MAT (Transition
More informationGraph Theory II. Po-Shen Loh. June edges each. Solution: Spread the n vertices around a circle. Take parallel classes.
Graph Theory II Po-Shen Loh June 009 1 Warm-up 1. Let n be odd. Partition the edge set of K n into n matchings with n 1 edges each. Solution: Spread the n vertices around a circle. Take parallel classes..
More informationSources for this lecture. 3. Matching in bipartite and general graphs. Symmetric difference
S-72.2420 / T-79.5203 Matching in bipartite and general graphs 1 3. Matching in bipartite and general graphs Let G be a graph. A matching M in G is a set of nonloop edges with no shared endpoints. Let
More informationProblem set 2. Problem 1. Problem 2. Problem 3. CS261, Winter Instructor: Ashish Goel.
CS261, Winter 2017. Instructor: Ashish Goel. Problem set 2 Electronic submission to Gradescope due 11:59pm Thursday 2/16. Form a group of 2-3 students that is, submit one homework with all of your names.
More informationMatchings, Ramsey Theory, And Other Graph Fun
Matchings, Ramsey Theory, And Other Graph Fun Evelyne Smith-Roberge University of Waterloo April 5th, 2017 Recap... In the last two weeks, we ve covered: What is a graph? Eulerian circuits Hamiltonian
More informationDefining Project Requirements
Defining Project Requirements SWEN-610 Foundations of Software Engineering Department of Software Engineering Rochester Institute of Technology 1 There are functional and non-functional requirements. Functional
More informationBroadcast: Befo re 1
Broadcast: Before 1 After 2 Spanning Tree ffl assume fixed spanning tree ffl asynchronous model 3 Processor State parent terminated children 4 Broadcast: Step One parent terminated children 5 Broadcast:Step
More informationTaibah University College of Computer Science & Engineering Course Title: Discrete Mathematics Code: CS 103. Chapter 2. Sets
Taibah University College of Computer Science & Engineering Course Title: Discrete Mathematics Code: CS 103 Chapter 2 Sets Slides are adopted from Discrete Mathematics and It's Applications Kenneth H.
More informationAutomatic Web Forms II
Automatic Web Forms II for ACT! 2011 and up User s manual part 6 How to use forms to validate your ACT! data The process explained... 2 Choosing a URL address field for the unique Web address... 2 Preventing
More informationEpisode 5. Scheduling and Traffic Management
Episode 5. Scheduling and Traffic Management Part 2 Baochun Li Department of Electrical and Computer Engineering University of Toronto Outline What is scheduling? Why do we need it? Requirements of a scheduling
More informationTHE TOPOLOGICAL APPROACH TO SOCIAL CHOICE
THE TOPOLOGICAL APPROACH TO SOCIAL CHOICE WEI HAN CHIA Abstract. Social Choice Theory is a field in economics which studies the possibility of fair aggregation maps of choices. One particularly famous
More informationEfficient Random Assignment with Constrained Rankings
Efficient Random Assignment with Constrained Rankings Gabriel Carroll Department of Economics, Massachusetts Institute of Technology E52-391, 50 Memorial Drive, Cambridge MA 02142, USA gdc@mit.edu January
More informationScheduling Algorithms for Input-Queued Cell Switches. Nicholas William McKeown
Scheduling Algorithms for Input-Queued Cell Switches by Nicholas William McKeown B.Eng (University of Leeds) 1986 M.S. (University of California at Berkeley) 1992 A thesis submitted in partial satisfaction
More informationPossibilities of Voting
Possibilities of Voting MATH 100, Survey of Mathematical Ideas J. Robert Buchanan Department of Mathematics Summer 2018 Introduction When choosing between just two alternatives, the results of voting are
More informationCHAPTER 8. Copyright Cengage Learning. All rights reserved.
CHAPTER 8 RELATIONS Copyright Cengage Learning. All rights reserved. SECTION 8.3 Equivalence Relations Copyright Cengage Learning. All rights reserved. The Relation Induced by a Partition 3 The Relation
More informationAlgorithms, Games, and Networks March 28, Lecture 18
Algorithms, Games, and Networks March 28, 2013 Lecturer: Ariel Procaccia Lecture 18 Scribe: Hanzhang Hu 1 Strategyproof Cake Cutting All cake cutting algorithms we have discussed in previous lectures are
More informationUse the following preference schedule to answer questions #1 - #4. Number of Voters st choice 2 nd Choice 3 rd choice 4 th choice
M130 Sample Final Exam Name TA Name Calculators are allowed, but cell phones or palm pilots are NOT acceptable. Mark Version A on your scantron. MULTIPLE CHOICE. Choose the one alternative that best completes
More informationCS6450: Distributed Systems Lecture 13. Ryan Stutsman
Eventual Consistency CS6450: Distributed Systems Lecture 13 Ryan Stutsman Material taken/derived from Princeton COS-418 materials created by Michael Freedman and Kyle Jamieson at Princeton University.
More informationLet s Talk About Logic
Let s Talk About Logic Jan van Eijck CWI & ILLC, Amsterdam Masterclass Logica, 2 Maart 2017 Abstract This lecture shows how to talk about logic in computer science. To keep things simple, we will focus
More informationCP nets: representing and reasoning with preferences of multiple agents
CP nets: representing and reasoning with preferences of multiple agents F. Rossi and K. B. Venable Dept. of Mathematics University of Padova Padova Italy frossikvenable @math.unipd.it Abstract We introduce
More informationOn Galvin s Theorem and Stable Matchings. Adam Blumenthal
On Galvin s Theorem and Stable Matchings by Adam Blumenthal A thesis submitted to the Graduate Faculty of Auburn University in partial fulfillment of the requirements for the Degree of Master of Science
More informationSection 1-3A: Creating a hierarchically numbered list style
Section 1-3A: Creating a hierarchically numbered list style The key to making automatically numbered headings is to create a numbered list style and then link it to the relevant Heading styles. Here we
More informationSeptember 4, Input: two strings s 1 and s 2 of length n. s 1 = AGGCTACC s 2 = CAGGCTAC
COMPSCI 311: Introduction to Algorithms COMPSCI 311: Introduction to Algorithms Marius Minea marius@cs.umass.edu University of Massachusetts Amherst slides credit: Akshay Krishnamurthy, Andrew McGregor,
More information学校選択制. Spread of school choice around the globe
学校選択制 Spread of school choice around the globe 学校選択制 Spread of school choice around the globe School authorities take into account preferences of students/parents National Governors Association Report
More informationTheorem 3.1 (Berge) A matching M in G is maximum if and only if there is no M- augmenting path.
3 Matchings Hall s Theorem Matching: A matching in G is a subset M E(G) so that no edge in M is a loop, and no two edges in M are incident with a common vertex. A matching M is maximal if there is no matching
More informationUtility Maximization
Utility Maximization Mark Dean Lecture Notes for Spring 2015 Behavioral Economics - Brown University 1 Lecture 1 1.1 Introduction The first topic we are going to cover in the course isn t going to seem
More informationSoftware Design and Analysis for Engineers
Software Design and Analysis for Engineers by Dr. Lesley Shannon Email: lshannon@ensc.sfu.ca Course Website: http://www.ensc.sfu.ca/~lshannon/courses/ensc251 Simon Fraser University Slide Set: 9 Date:
More information9.5 Equivalence Relations
9.5 Equivalence Relations You know from your early study of fractions that each fraction has many equivalent forms. For example, 2, 2 4, 3 6, 2, 3 6, 5 30,... are all different ways to represent the same
More informationRecord Transfer from AMS to Colleague (User Manual) V 1.0
Record Transfer from AMS to Colleague (User Manual) V 1.0 Index Introduction 1.0 Manual Transfer 1.1 Automatic Transfer 1.2 Transfer Rules 1.3 Fields to Transfer 1.4 Colleague Manual Process 1.5 Help 1.6
More informationGame Theory & Networks
Game Theory & Networks (an incredibly brief overview) ndrew Smith ECS 253/ME 289 May 10th, 2016 Game theory can help us answer important questions for scenarios where: players/agents (nodes) are autonomous
More informationOn the Efficiency of Negligence Rule
Jawaharlal Nehru University From the SelectedWorks of Satish K. Jain 2009 On the Efficiency of Negligence Rule Satish K. Jain, Jawaharlal Nehru University Available at: https://works.bepress.com/satish_jain/2/
More informationReview implementation of Stable Matching Survey of common running times. Turn in completed problem sets. Jan 18, 2019 Sprenkle - CSCI211
Objectives Review implementation of Stable Matching Survey of common running times Turn in completed problem sets Jan 18, 2019 Sprenkle - CSCI211 1 Review: Asymptotic Analysis of Gale-Shapley Alg Not explicitly
More informationMaterial from Recitation 1
Material from Recitation 1 Darcey Riley Frank Ferraro January 18, 2011 1 Introduction In CSC 280 we will be formalizing computation, i.e. we will be creating precise mathematical models for describing
More informationSpring 2007 Midterm Exam
15-381 Spring 2007 Midterm Exam Spring 2007 March 8 Name: Andrew ID: This is an open-book, open-notes examination. You have 80 minutes to complete this examination. Unless explicitly requested, we do not
More informationMyELT STUDENT QUICK START GUIDE
MyELT STUDENT QUICK START GUIDE TABLE OF CONTENTS ABOUT THIS GUIDE... 2 GETTING STARTED... 3 HOW DO I REGISTER FOR AN INSTRUCTOR- LED COURSE?... 3 HOW DO I CREATE A SELF- STUDY ACCOUNT?... 7 WHAT SHOULD
More informationSection Sets and Set Operations
Section 6.1 - Sets and Set Operations Definition: A set is a well-defined collection of objects usually denoted by uppercase letters. Definition: The elements, or members, of a set are denoted by lowercase
More informationWhat Every Programmer Should Know About Floating-Point Arithmetic
What Every Programmer Should Know About Floating-Point Arithmetic Last updated: October 15, 2015 Contents 1 Why don t my numbers add up? 3 2 Basic Answers 3 2.1 Why don t my numbers, like 0.1 + 0.2 add
More informationIntroduction to Algorithms / Algorithms I Lecturer: Michael Dinitz Topic: Algorithms and Game Theory Date: 12/3/15
600.363 Introduction to Algorithms / 600.463 Algorithms I Lecturer: Michael Dinitz Topic: Algorithms and Game Theory Date: 12/3/15 25.1 Introduction Today we re going to spend some time discussing game
More informationNetwork Flow. November 23, CMPE 250 Graphs- Network Flow November 23, / 31
Network Flow November 23, 2016 CMPE 250 Graphs- Network Flow November 23, 2016 1 / 31 Types of Networks Internet Telephone Cell Highways Rail Electrical Power Water Sewer Gas... CMPE 250 Graphs- Network
More informationSet and Set Operations
Set and Set Operations Introduction A set is a collection of objects. The objects in a set are called elements of the set. A well defined set is a set in which we know for sure if an element belongs to
More informationMath and Marriage - Don t Call a Lawyer Yet!
Math and Marriage - Don t Call a Lawyer Yet! Ron Gould Emory University Supported by the Heilbrun Distinguished Emeritus Fellowship October 7,27 Philip Hall, 935 Definition By a system of distinct representatives
More informationCS510 \ Lecture Ariel Stolerman
CS510 \ Lecture02 2012-10-03 1 Ariel Stolerman Midterm Evan will email about that after the lecture, at least 2 lectures from now. The exam will be given in a regular PDF (not an online form). We will
More informationMATH 1340 Mathematics & Politics
MATH 1340 Mathematics & Politics Lecture 5 June 26, 2015 Slides prepared by Iian Smythe for MATH 1340, Summer 2015, at Cornell University 1 An example (Exercise 2.1 in R&U) Consider the following profile
More informationName: UW CSE 473 Midterm, Fall 2014
Instructions Please answer clearly and succinctly. If an explanation is requested, think carefully before writing. Points may be removed for rambling answers. If a question is unclear or ambiguous, feel
More information15-780: Graduate AI Homework Assignment #2 Solutions
15-780: Graduate AI Homework Assignment #2 Solutions Out: February 12, 2015 Due: February 25, 2015 Collaboration Policy: You may discuss the problems with others, but you must write all code and your writeup
More informationDiscrete Optimization Lecture-11
Discrete Optimization Lecture-11 Ngày 2 tháng 11 năm 2011 () Discrete Optimization Lecture-11 Ngày 2 tháng 11 năm 2011 1 / 12 A scheduling problem Discussion A company owns 7 ships. During a 12 days period
More informationCS 331: Artificial Intelligence Informed Search. Informed Search
CS 331: Artificial Intelligence Informed Search 1 Informed Search How can we make search smarter? Use problem-specific knowledge beyond the definition of the problem itself Specifically, incorporate knowledge
More informationClasses, interfaces, & documentation. Review of basic building blocks
Classes, interfaces, & documentation Review of basic building blocks Objects Data structures literally, storage containers for data constitute object knowledge or state Operations an object can perform
More informationhard to perform, easy to verify
Proof of Stake The Role of PoW Bitcoin, Ethereum and similar systems are open, permissionless networks Anyone can participate The system must agree on some canonical order of transactions Think of this
More information1 of 5 3/28/2010 8:01 AM Unit Testing Notes Home Class Info Links Lectures Newsgroup Assignmen [Jump to Writing Clear Tests, What about Private Functions?] Testing The typical approach to testing code
More informationSearch Engines. Provide a ranked list of documents. May provide relevance scores. May have performance information.
Search Engines Provide a ranked list of documents. May provide relevance scores. May have performance information. 3 External Metasearch Metasearch Engine Search Engine A Search Engine B Search Engine
More informationEpisode 5. Scheduling and Traffic Management
Episode 5. Scheduling and Traffic Management Part 2 Baochun Li Department of Electrical and Computer Engineering University of Toronto Keshav Chapter 9.1, 9.2, 9.3, 9.4, 9.5.1, 13.3.4 ECE 1771: Quality
More informationCS 331: Artificial Intelligence Informed Search. Informed Search
CS 331: Artificial Intelligence Informed Search 1 Informed Search How can we make search smarter? Use problem-specific knowledge beyond the definition of the problem itself Specifically, incorporate knowledge
More informationTopology Homework 3. Section Section 3.3. Samuel Otten
Topology Homework 3 Section 3.1 - Section 3.3 Samuel Otten 3.1 (1) Proposition. The intersection of finitely many open sets is open and the union of finitely many closed sets is closed. Proof. Note that
More informationDynamics, stability, and foresight in the Shapley-Scarf housing market. Discussion Paper No
Dynamics, stability, and foresight in the Shapley-Scarf housing market Yoshio Kamijo Ryo Kawasaki Discussion Paper No. 2008-12 October 2008 Dynamics, stability, and foresight in the Shapley-Scarf housing
More information2. Matching. Maximum bipartite matchings. Stable marriage problem. Matching 2-1
Matching 2-1 2. Matching Maximum bipartite matchings Stable marriage problem Matching 2-2 Matching in graphs given a graph G = (V ; E), a matching M = fe 1; : : :g E is a subset of edges such that no two
More informationTop-Trading-Cycles Mechanisms with Acceptable Bundles
Top-Trading-Cycles Mechanisms with Acceptable Bundles Sujoy Sikdar Department of Computer Science Rensselaer Polytechnic Institute sikdas@rpi.edu Sibel Adalı Department of Computer Science Rensselaer Polytechnic
More informationREU 2006 Discrete Math Lecture 5
REU 2006 Discrete Math Lecture 5 Instructor: László Babai Scribe: Megan Guichard Editors: Duru Türkoğlu and Megan Guichard June 30, 2006. Last updated July 3, 2006 at 11:30pm. 1 Review Recall the definitions
More informationHow Bitcoin achieves Decentralization. How Bitcoin achieves Decentralization
Centralization vs. Decentralization Distributed Consensus Consensus without Identity, using a Block Chain Incentives and Proof of Work Putting it all together Centralization vs. Decentralization Distributed
More informationThe Geometry of Carpentry and Joinery
The Geometry of Carpentry and Joinery Pat Morin and Jason Morrison School of Computer Science, Carleton University, 115 Colonel By Drive Ottawa, Ontario, CANADA K1S 5B6 Abstract In this paper we propose
More informationANSWER everything on page
APPLY TEXAS SESSION ANSWER everything on page DO NOT use school email use your personal one USE A Valid Email NCTC WILL use this to contact you! One you check regularly preferred If this is a PO BOX
More informationTimesheetX helps schools automate the time sheet submission and approval process for employees, employers, and administrators.
Employee Training = Total Solution TimesheetX helps schools automate the time sheet submission and approval process for employees, employers, and administrators. TimesheetX is seamlessly integrated with
More information2018 Pummill Relay problem statement
2018 Pummill Relays CS Problem: Minimum Spanning Tree Missouri State University For information about the Pummill Relays CS Problem, please contact: KenVollmar@missouristate.edu, 417-836-5789 Suppose there
More informationCS 580: Algorithm Design and Analysis. Jeremiah Blocki Purdue University Spring 2018
CS 580: Algorithm Design and Analysis Jeremiah Blocki Purdue University Spring 2018 Recap: Stable Matching Problem Definition of a Stable Matching Stable Roomate Matching Problem Stable matching does not
More informationSets MAT231. Fall Transition to Higher Mathematics. MAT231 (Transition to Higher Math) Sets Fall / 31
Sets MAT231 Transition to Higher Mathematics Fall 2014 MAT231 (Transition to Higher Math) Sets Fall 2014 1 / 31 Outline 1 Sets Introduction Cartesian Products Subsets Power Sets Union, Intersection, Difference
More informationCSE 143, Winter 2009 Final Exam Thursday, March 19, 2009
CSE 143, Winter 2009 Final Exam Thursday, March 19, 2009 Personal Information: Name: Section: Student ID #: TA: You have 110 minutes to complete this exam. You may receive a deduction if you keep working
More informationPlan. Design principles: laughing in the face of change. What kind of change? What are we trying to achieve?
Plan Design principles: laughing in the face of change Perdita Stevens School of Informatics University of Edinburgh What are we trying to achieve? Review: Design principles you know from Inf2C-SE Going
More informationReasoning About Imperative Programs. COS 441 Slides 10
Reasoning About Imperative Programs COS 441 Slides 10 The last few weeks Agenda reasoning about functional programming It s very simple and very uniform: substitution of equal expressions for equal expressions
More informationGeorgia Institute of Technology College of Engineering School of Electrical and Computer Engineering
Georgia Institute of Technology College of Engineering School of Electrical and Computer Engineering ECE 8832 Summer 2002 Floorplanning by Simulated Annealing Adam Ringer Todd M c Kenzie Date Submitted:
More informationAvailability versus consistency. Eventual Consistency: Bayou. Eventual consistency. Bayou: A Weakly Connected Replicated Storage System
Eventual Consistency: Bayou Availability versus consistency Totally-Ordered Multicast kept replicas consistent but had single points of failure Not available under failures COS 418: Distributed Systems
More informationTri-C College Credit Plus Application Instructions
Tri-C College Credit Plus Application Instructions 1. Go to the website: http://www.tri-c.edu/college-credit-plus/. This website provides you with enrollment information for prospective and current CCP
More informationThe Game Chromatic Number of Some Classes of Graphs
The Game Chromatic Number of Some Classes of Graphs Casper Joseph Destacamento 1, Andre Dominic Rodriguez 1 and Leonor Aquino-Ruivivar 1,* 1 Mathematics Department, De La Salle University *leonorruivivar@dlsueduph
More informationTo illustrate what is intended the following are three write ups by students. Diagonalization
General guidelines: You may work with other people, as long as you write up your solution in your own words and understand everything you turn in. Make sure to justify your answers they should be clear
More informationCPHC (Certified Passive House Consultant) Training Terms & Conditions
CPHC (Certified Passive House Consultant) Training Terms & Conditions To help building professionals attain professional accreditation for the passive building energy sector, the Passive House Institute
More informationInfinity part I. High School Circle I. June 4, 2017
Infinity part I High School Circle I June 4, 2017 1. For the next few classes, we are going to talk about infinite sets. When we talk about infinite sets, we have to use a very special language, and set
More informationInteractive Geometry for Surplus Sharing in Cooperative Games
Utah State University DigitalCommons@USU Applied Economics Faculty Publications Applied Economics 2006 Interactive Geometry for Surplus Sharing in Cooperative Games Arthur J. Caplan Utah State University
More informationFundamentals of Queueing Models
Fundamentals of Queueing Models Michela Meo Maurizio M. Munafò Michela.Meo@polito.it Maurizio.Munafo@polito.it TLC Network Group - Politecnico di Torino 1 Modeling a TLC network Modelization and simulation
More information