CS3000:Algorithms&Data PaulHand

Lecture2:• FinishInduction• StableMatching:theGale-ShapleyAlgorithm

Jan9,2019

CourseWebsite

http://www.ccs.neu.edu/home/hand/teaching/cs3000-spring-2018/

• HomeworkwillbesubmittedonGradescope!• MoredetailsonWednesday• EntryCode:MKKEW2• https://www.gradescope.com/courses/36055

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CourseStructureEnd4/17

Start1/7

DivideandConquer

DynamicProgramming

Graphs NetworkFlow

Textbook:AlgorithmDesignbyKleinbergandTardos

Moreresourcesonthecoursewebsite

FinalTBD

MidtermI2/20

MidtermII3/27

ProofsbyInduction

• Claim:Forevery,

• ProofbyInduction:

Exercise

n 2 N,n�1X

i=0

2i = 2n � 1

100000,0

Base cases n L 42 2 1 1

General case n I

Assume 02 I I

o2i f j2 it 27 271 2

2Mt I

StableMatchingProblemandtheGale-ShapleyAlgorithm

Processforsolvingcomputationalproblemswithalgorithms

• Formulateproblemandquestions• Playaround• Devisealgorithm• Determinehowlongittakestorun• Determineifalgorithmiscorrect• Determineappropriatedatastructures

StableMatchingProblem

• Manyjobcandidates(eg.doctors).Manyjobs(eg.residencyprograms).Youaretoassigncandidatestojobs.Howshouldyoudoit?

inputs to alg

Candidates G G Cnjobs I j2 Jn3

pair Ci Jsfirst isthjobperson

somorppl havea job

not EVERYone

itbelongsto

any of the followingprefers

note proofs of jd dont matter

Find a

c painthat both prefersEach other

0 UC 2732 c B

M l B Ca A 13,4Mz HA B 12,93

Gale-ShapleyAlgorithm

• Let M be empty • While (some job j is unmatched):

• If (j has offered a job to everyone): break • Else: let c be the highest-ranked candidate to which j has not yet offered a job

• j makes an offer to c: • If (c is unmatched):

• c accepts, add (c,j) to M • ElseIf (c is matched to j’ & c: j’ > j):

• c rejects, do nothing • ElseIf (c is matched to j’ & c: j > j’):

• c accepts, remove (c,j’) from M and add (c,j) to M

• Output M

Icoording to j

Gale-ShapleyDemo

1st 2nd 3rd 4th 5th

MGH Bob Alice Dorit Ernie Clara

BW Dorit Bob Alice Clara Ernie

BID Bob Ernie Clara Dorit Alice

MTA Alice Dorit Clara Bob Ernie

CH Bob Dorit Alice Ernie Clara

1st 2nd 3rd 4th 5th

Alice CH MGH BW MTA BID

Bob BID BW MTA MGH CH

Clara BW BID MTA CH MGH

Dorit MGH CH MTA BID BW

Ernie MTA BW CH BID MGH

D

XO

Activity:Whatarethefirst4stepsofG-Salgorithm?

• Jobs:1,2,3,4• Candidates:A,B,C,D

Jobs’Preferences Candidates’Preferences

(Assumeitstepsthroughjobsinorder1-4,afterwardsstartingoverwith1ifnecessary)

Observations• Atallsteps,thestateofthealgorithmisamatching

• Jobsmakeoffersindescendingorder

• Candidatesthatgetajobneverbecomeunemployed

• Candidatesacceptoffersinascendingorder

Gale-ShapleyAlgorithm

• QuestionsabouttheGale-ShapleyAlgorithm:• Willthisalgorithmterminate?Afterhowlong?• Doesitoutputaperfectmatching?• Doesitoutputastablematching?• Howdoweimplementthisalgorithmefficiently?

GSAlgorithm:Termination

• Claim:TheGSalgorithmterminatesaftern2iterationsofthemainloop,wherenisnumberofcandidates/jobs.

af most

At most n2 possible offasAt Each iter an offer ismade None repeated

So f n iteratus

GSAlgorithm:PerfectMatching

• Claim:TheGSalgorithmreturnsaperfectmatching(alljobs/candidatesarematched)

GSAlgorithm:StableMatching

• Stability:GSalgorithmoutputsastablematching• Proofbycontradiction:

• Supposethereisaninstability

GSAlgorithm:RunningTime

• RunningTime:• Astraightforwardimplementationrequiresatoperations,space(memory).

⇡ n3

⇡ n2

GSAlgorithm:RunningTime

• Let M be empty • While (some job j is unmatched):

• If (j has offered a job to everyone): break • Else: let c be the highest-ranked candidate to which j has not yet offered a job

• j makes an offer to c: • If (c is unmatched):

• c accepts, add (c,j) to M • ElseIf (c is matched to j’ & c: j’ > j):

• c rejects, do nothing • ElseIf (c is matched to j’ & c: j > j’):

• c accepts, remove (c,j’) from M and add (c,j) to M

• Output M

GSAlgorithm:RunningTime

• RunningTime:• Acarefulimplementationrequiresjusttimeandspace

⇡ n2

⇡ n2

GSAlgorithm:RunningTime

• RunningTime:• Acarefulimplementationrequiresjusttimeandspace

1st 2nd 3rd 4th 5th

Alice CH MGH BW MTA BID

Bob BID BW MTA MGH CH

Clara BW BID MTA CH MGH

Dorit MGH CH MTA BID BW

Ernie MTA BW CH BID MGH

MGH BW BID MTA CH

Alice 2nd 3rd 5th 4th 1st

Bob 4th 2nd 1st 3rd 5th

Clara 5th 1st 2nd 3rd 4th

Dorit 1st 5th 4th 3rd 2nd

Ernie 5th 2nd 4th 1st 3rd

GSAlgorithm:RunningTime

• RunningTime:• Acarefulimplementationrequiresjusttimeandspace

⇡ n2

⇡ n2

NotesforinstructorStudentsmayignorebecausetheyarerepeatedelsewhere