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COS 423: Theory of Algorithms
Algorithm. (webster.com)^ n^
A procedure for solving a mathematical problem (as of finding thegreatest common divisor) in a finite number of steps thatfrequently involves repetition of an operation. n^
Broadly: a step-by-step procedure for solving a problem oraccomplishing some end especially by a computer. "Great algorithms are the poetry of computation."Etymology.^ n^
"algos" = Greek word for pain. n^
"algor" = Latin word for to be cold. n^
Abu Ja’far al-Khwarizmi’s = 9th century Arab scholar.^ –
his book "Al-Jabr wa-al-Muqabilah" evolved into today’s highschool algebra text
Theory of Algorithms
3
Imagine:
A World With No Algorithms
Fast arithmetic.^ n^
Cryptography. Quicksort.^ n^
Databases. FFT.^ n^
Signal processing. Huffman codes.^ n^
Data compression. Network flow.^ n^
Routing Internet packets. Linear programming.^ n^
Planning, decision-making.
What is COS 423?
Introduction to design and analysis of computer algorithms.^ n^
Algorithmic paradigms. n^
Analyze running time of programs. n^
Data structures. n^
Understand fundamental algorithmic problems. n^
Intrinsic computational limitations. n^
Models of computation. n^
Critical thinking. Prerequisites.^ n^
COS 226 (array, linked list, search tree, graph, heap, quicksort). n^
COS 341 (proof, induction, recurrence, probability).
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Administrative Stuff
Lectures: (Kevin Wayne)^ n^
Monday, Wednesday 10:00 - 10:50, COS 104. TA’s: (Edith Elkind, Sumeet Sobti)Textbook: Introduction to Algorithms (CLR).Grading:^ n^
Weekly problem sets. n^
Collaboration, no-collaboration. n^
Class participation, staff discretion. n^
Undergrad / grad. Course web site: courseinfo.princeton.edu/courses/COS423_S2001/^ n^
Fill out questionnaire.
Approximate Lecture Outline
Algorithmic paradigms.^ n^
Divide-and-conquer. n^
Greed. n^
Dynamic programming. n^
Reductions. Analysis of algorithms.^ n^
Amortized analysis. Data structures.^ n^
Union find. n^
Search trees and extensions. Graph algorithms.^ n^
Shortest path, MST. n^
Max flow, matching.
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Approximate Lecture Outline
NP completeness.^ n^
More reductions. n^
Approximation algorithms. Other models of computation.^ n^
Parallel algorithms. n^
Randomized algorithms. Miscellaneous.^ n^
Numerical algorithms. n^
Linear programming.
College Admissions
Sample problem.Algorithm.Analysis.
References:
The Stable Marriage Problem by Dan Gusfield and Robert Irving,MIT Press, 1989.Introduction to Algorithms by Jon Kleinberg and
Éva Tardos.
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Stable Matching Problem
Problem: Given N men and N women, find a "suitable" matchingbetween men and women.^ n^
PERFECT MATCHING: everyone is matched monogamously.^ –
each man gets exactly one woman – each woman gets exactly one man n^
STABILITY: no incentive for some pair of participants toundermine assignment by joint action.^ –
in matching M, an unmatched pair (m,w) is UNSTABLE if man mand woman w prefer each other to current partners – unstable pair could each improve by dumping spouses andeloping
STABLE MATCHING = perfect matching with no unstable pairs.(Gale and Shapley, 1962)
Lavender assignment is a perfect matching.Are there any unstable pairs?
Men’s Preference List
Women’s Preference List
ManXavierYanceyZeus
st 1 A B A
nd 2 B A B
rd 3 C C C
WomanAmyBerthaClare
st 1 Y X X
nd 2 X Y Y
rd 3 Z Z Z
Yes. Bertha and Xavier form an unstable pair.They would prefer each other to current partners.
B
X
Example
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Example
Green assignment is a stable matching.
A B A
C^
Y X X
X Y Y
Z Z Z
Men’s Preference List
Women’s Preference List
ManXavierYanceyZeus
st 1
nd 2 B A B
rd 3 C C
WomanAmyBerthaClare
st 1
nd 2
rd 3
Example
Orange assignment is also a stable matching.
A^
X
X Y Y
Z Z
Men’s Preference List
Women’s Preference List
ManXavierYanceyZeus
st 1 A B
nd 2
rd 3 C C
WomanAmyBerthaClare
st 1
nd 2
rd 3
B A B
C
Y X
Z
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Preference List^ B
AdamBobChris
C A B
D D
Doofus
A
B^
D C
Stable Roommate ProblemC A
Not obvious that any stable matching exists.Consider related "stable roommate problem."^ n^
2N people. n^
Each person ranks others from 1 to 2N-1. n^
Assign roommate pairs so that no unstable pairs.
C A
B
D
No stable matching.
All 3 perfect matchings haveunstable pair.E.g., A-C forms unstable pairin lavender matching.
C
st 1 A
nd 2
rd 3
Propose-And-Reject Algorithm
Intuitive method that guarantees to find a stable matching.Initialize each person to be free. while
(some man m is free and hasn’t proposed to every woman) w = first woman on m’s list to whom m has not yet proposed if
(w is free)assign m and w to be engaged else if
(w prefers m to her fiancé m')
assign m and w to be engaged, and m' to be free else
w rejects m
Gale-Shapley Algorithm (men propose)
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Implementation and Running Time Analysis Engagements.^ n^
Maintain two arrays wife[m], and husband[w]; set equal to 0 ifparticipant is free. n^
Store list of free men on a stack (queue). Preference lists.^ n^
For each man, create a linked list of women, ordered from favoriteto worst.^ –
men propose to women at top of list, if rejected goto next n^
For each woman, create a "ranking array" such that m
th^
entry in
array is woman’s ranking of man m.^ –
allows for queries of the form: does woman w prefer m to m’?
Resource consumption.^ n^
Time
=^
Θ(N
n^
Space =
(N
n^
Optimal.
{^
w
m
w
m
to
proposed
has
Men’s Preference List ManVictorWyatt
st 1 A B
nd 2
rd 3 C D
B C
A
XavierYanceyZeus
C D
A B
D A B
C
th 4
th 5 E E
D A
E E
B C D
E
Women’s Preference List ManAmyBertha
st 1 W X
nd 2
rd 3 Y Z
X Y
V
ClareDianeErika
Y Z
V W
Z V W
X
th 4
th 5 V W
Z V
X Y
W X Y
Z
A Worst Case Instance
Number of proposals
n(n-1) + 1.
n^
Algorithm terminates when last woman gets first proposal. Number of proposals = n(n-1) + 1 for following family of instances.
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Understanding the Solution
Fact 4. Gale-Shapley finds WOMAN-PESSIMAL matching.^ n^
Each woman married to worst valid partner.^ –
simultaneously worst for each and every woman. – there is no stable matching in which any single womanindividually does worse
Proof.^ n^
Suppose (Amy, Zeus) matched in S*, but Zeus is not worst validpartner for Amy. n^
There exists stable matching S in which Amy is paired with man,say Yancey, whom she likes less than Zeus. n^
Let Bertha be Zeus’ partner in S. n^
Zeus prefers Amy to Bertha (man optimality). n^
(Amy, Zeus) form unstable pair in S.
Amy-YanceyBertha-Zeus
S
Understanding the Solution
Fact 5. The man-optimal stable matching is weakly Pareto optimal.^ n^
There is no other perfect matching (stable or unstable), whereevery man does strictly better. Proof.^ n^
Let Amy be last woman in some execution of Gale-Shapley (menpropose) algorithm to receive a proposal. n^
No man is rejected by Amy since algorithm terminates when lastwoman receives first proposal. n^
No man matched to Amy will be strictly better off than in man-optimal stable matching.
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Extensions: Unacceptable Partners
Yeah, but in real-world every woman is not willing to marry every man,and vice versa?^ n^
Some participants declare others as "unacceptable."(prefer to be alone than with given partner) n^
Algorithm extends to handle partial preference lists. Matching S unstable if there exists man m and woman w such that:^ n^
m is either unmatched in S, or strictly prefers w to his partner in S n^
w is either unmatched in S, or strictly prefers m to her partner in S. Fact 6. Men and women are each partitioned into two sets:^ n^
those that have partners in all stable matchings; n^
those that have partners in none.
Extensions: Sets of Unequal Size
Also, there may be an unequal number of men and women.^ n^
E.g., |M| = 100 men, |W| = 90 women. n^
Algorithm extends. n^
WLOG, assume |W| < |M|. Matching S unstable if there exists man m and woman w such that:^ n^
m is either unmatched in S, or strictly prefers w to his partner in S; n^
w is either unmatched in S, or strictly prefers m to her partner in S. Fact 7. All women are matched in every stable matching. Men arepartitioned into two subsets:^ n^
men who are matched in every stable matching; n^
men who are matched in none.
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Extensions: Limited Polygamy
What about limited polygamy?^ n^
E.g., Bill wants 3 women. n^
Algorithm extends. Matching S unstable if there exists man m and woman w such that:^ n^
either w is unmatched, or w strictly prefers m to her partner; n^
either m does not have all its "places" filled in the matching, or mstrictly prefers w to at least one of its assigned residents.
Application: Matching Residents to Hospitals Sets of unequal size, unacceptable partners, limited polygamy.Matching S unstable if there exists hospital h and resident r such that:^ n^
h and r are acceptable to each other; n^
either r is unmatched, or r prefers h to her assigned hospital; n^
either h does not have all its places filled in the matching, or hprefers r to at least one of its assigned residents.
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Application: Matching Residents to Hospitals Matching medical school residents to hospitals. (NRMP)^ n^
Hospitals ~ Men (limited polygamy allowed). n^
Residents ~ Women. n^
Original use just after WWII (predates computer usage). n^
Ides of March, 13,000+ residents. Rural hospital dilemma.^ n^
Certain hospitals (mainly in rural areas) were unpopular anddeclared unacceptable by many residents. n^
Rural hospitals were under-subscribed in NRMP matching. n^
How can we find stable matching that benefits "rural hospitals"? Rural Hospital Theorem:
Rural hospitals get exactly same residents in every stablematching!
Deceit: Machiavelli Meets Gale-Shapley
Is there any incentive for a participant to misrepresent his/herpreferences?^ n^
Assume you know men’s propose-and-reject algorithm will be run. n^
Assume that you know the preference lists of all other participants. Fact 8. No, for any man yes, for some women!
A^
X
X Y Y
Z Z
Men’s Preference List
Women’s True Preference List
ManXavierYanceyZeus
st 1 A B
nd 2
rd 3 C C
WomanAmyBerthaClare
st 1
nd 2
rd 3
B A B
C
Y X
Z
X
Z Y Y
X Z
Amy Lies
WomanAmyBerthaClare
st 1
nd 2
rd 3
Y X
Z
X Y
Z
Y X
Z Docsity.com
