1. S. Raskhodnikova; based on slides by K. Wayne
Discrete Math for CS
LECTURE 10
Last time:
Stable Matching
Propose-and-Reject algorithm
Today:
Analysis of Propose-and-Reject
CMPSC
360
11/29/2018
S. Raskhodnikova; based on slides by K. Wayne

2. An Algorithm for Stable Matching
Propose-and-reject algorithm. [Gale-Shapley 1962]
Input: tables of preferences for men and women
while (there are rejected suitors) {
Start a new day.
Morning: Every man proposes to 1st women remaining on his list.
Afternoon: Every woman says ``Maybe’’ to the man she likes best among the proposals (she now has him on a string) and rejects the rest.
Evening: Each rejected suitor crosses off the woman that rejected him from his list. }
Last day: Each woman marries the man she has on the string.
Output: The set of married couples.
11/29/2018
S. Raskhodnikova; based on slides by E. Demaine, C. Leiserson, A. Smith, K. Wayne

3. Example: Execution of the Algorithm
Men’s preferences
Women’s preferences
Wyatt
Victor
1st B D
2nd A
3rd
Zeus
Yancey
Xavier C E
4th
5th
Bertha
Amy Z X W Y V
Erika
Diane
Clare
stable matching is unique for worst-case instance
11/29/2018

4. Day 1: Morning Men’s preferences Women’s preferences
Wyatt
Victor
1st B D
2nd A
3rd
Zeus
Yancey
Xavier C E
4th
5th
Bertha
Amy Z X W Y V
Erika
Diane
Clare
Victor proposes to Bertha.
Wyatt proposes to Diane.
Xavier proposes to Bertha.
Yancey proposes to Amy.
Zeus proposes to Bertha.
stable matching is unique for worst-case instance
11/29/2018

5. Day 1: Afternoon Men’s preferences Women’s preferences
Wyatt
Victor
1st B D
2nd A
3rd
Zeus
Yancey
Xavier C E
4th
5th
Bertha
Amy Z X W Y V
Erika
Diane
Clare
Amy says ``Maybe’’ to Yancey.
Bertha says ``Maybe’’ to Xavier,
rejects Victor and Zeus.
Diane says ``Maybe’’ to Wyatt.
stable matching is unique for worst-case instance
11/29/2018

6. Day 1: Evening Men’s preferences Women’s preferences
Wyatt
Victor
1st B D
2nd A
3rd
Zeus
Yancey
Xavier C E
4th
5th
Bertha
Amy Z X W Y V
Erika
Diane
Clare
Victor and Zeus cross off Bertha.
stable matching is unique for worst-case instance
11/29/2018

7. Day 2: Morning Men’s preferences Women’s preferences
Wyatt
Victor
1st B D
2nd A
3rd
Zeus
Yancey
Xavier C E
4th
5th
Bertha
Amy Z X W Y V
Erika
Diane
Clare
Victor proposes to Amy.
Wyatt proposes to Diane.
Xavier proposes to Bertha.
Yancey proposes to Amy.
Zeus proposes to Diane.
stable matching is unique for worst-case instance
11/29/2018

8. Day 2: Afternoon Men’s preferences Women’s preferences
Wyatt
Victor
1st B D
2nd A
3rd
Zeus
Yancey
Xavier C E
4th
5th
Bertha
Amy Z X W Y V
Erika
Diane
Clare
Amy says ``Maybe’’ to Victor,
rejects Yancey.
Bertha says ``Maybe’’ to Xavier.
Diane says ``Maybe’’ to Zeus,
rejects Wyatt.
stable matching is unique for worst-case instance
11/29/2018

9. Day 2: Evening Men’s preferences Women’s preferences
Wyatt
Victor
1st B D
2nd A
3rd
Zeus
Yancey
Xavier C E
4th
5th
Bertha
Amy Z X W Y V
Erika
Diane
Clare
Wyatt crosses off Diane.
Yancey crosses off Amy.
stable matching is unique for worst-case instance
11/29/2018

10. Day 3: Morning Men’s preferences Women’s preferences
Wyatt
Victor
1st B D
2nd A
3rd
Zeus
Yancey
Xavier C E
4th
5th
Bertha
Amy Z X W Y V
Erika
Diane
Clare
Victor proposes to Amy.
Wyatt proposes to Bertha.
Xavier proposes to Bertha.
Yancey proposes to Diane.
Zeus proposes to Diane.
stable matching is unique for worst-case instance
11/29/2018

11. Day 3: Afternoon Men’s preferences Women’s preferences
Wyatt
Victor
1st B D
2nd A
3rd
Zeus
Yancey
Xavier C E
4th
5th
Bertha
Amy Z X W Y V
Erika
Diane
Clare
Amy says ``Maybe’’ to Victor..
Bertha says ``Maybe’’ to Xavier,
rejects Wyatt.
Diane says ``Maybe’’ to Zeus,
rejects Yancey.
stable matching is unique for worst-case instance
11/29/2018

12. Day 3: Evening Men’s preferences Women’s preferences
Wyatt
Victor
1st B D
2nd A
3rd
Zeus
Yancey
Xavier C E
4th
5th
Bertha
Amy Z X W Y V
Erika
Diane
Clare
Wyatt crosses off Bertha.
Yancey crosses off Diane.
stable matching is unique for worst-case instance
11/29/2018

13. Day 4: Morning Men’s preferences Women’s preferences
Wyatt
Victor
1st B D
2nd A
3rd
Zeus
Yancey
Xavier C E
4th
5th
Bertha
Amy Z X W Y V
Erika
Diane
Clare
Victor proposes to Amy.
Wyatt proposes to Amy.
Xavier proposes to Bertha.
Yancey proposes to Clare.
Zeus proposes to Diane.
stable matching is unique for worst-case instance
11/29/2018

14. Day 4: Afternoon Men’s preferences Women’s preferences
Wyatt
Victor
1st B D
2nd A
3rd
Zeus
Yancey
Xavier C E
4th
5th
Bertha
Amy Z X W Y V
Erika
Diane
Clare
Amy says ``Maybe’’ to Victor.,
rejects Wyatt.
Bertha says ``Maybe’’ to Xavier.
Clare says ``Maybe’’ to Yancey.
Diane says ``Maybe’’ to Zeus.
stable matching is unique for worst-case instance
11/29/2018

15. Day 4: Evening Men’s preferences Women’s preferences
Wyatt
Victor
1st B D
2nd A
3rd
Zeus
Yancey
Xavier C E
4th
5th
Bertha
Amy Z X W Y V
Erika
Diane
Clare
Wyatt crosses off Amy.
stable matching is unique for worst-case instance
11/29/2018

16. Day 5: Morning Men’s preferences Women’s preferences
Wyatt
Victor
1st B D
2nd A
3rd
Zeus
Yancey
Xavier C E
4th
5th
Bertha
Amy Z X W Y V
Erika
Diane
Clare
Victor proposes to Amy.
Wyatt proposes to Clare.
Xavier proposes to Bertha.
Yancey proposes to Clare.
Zeus proposes to Diane.
stable matching is unique for worst-case instance
11/29/2018

17. Day 5: Afternoon Men’s preferences Women’s preferences
Wyatt
Victor
1st B D
2nd A
3rd
Zeus
Yancey
Xavier C E
4th
5th
Bertha
Amy Z X W Y V
Erika
Diane
Clare
Amy says ``Maybe’’ to Victor.
Bertha says ``Maybe’’ to Xavier.
Clare says ``Maybe’’ to Wyatt,
rejects Yancey.
Diane says ``Maybe’’ to Zeus.
stable matching is unique for worst-case instance
11/29/2018

18. Day 5: Evening Men’s preferences Women’s preferences
Wyatt
Victor
1st B D
2nd A
3rd
Zeus
Yancey
Xavier C E
4th
5th
Bertha
Amy Z X W Y V
Erika
Diane
Clare
Yancey crosses off Clare.
stable matching is unique for worst-case instance
11/29/2018

19. Day 6: Morning Men’s preferences Women’s preferences
Wyatt
Victor
1st B D
2nd A
3rd
Zeus
Yancey
Xavier C E
4th
5th
Bertha
Amy Z X W Y V
Erika
Diane
Clare
Victor proposes to Amy.
Wyatt proposes to Clare.
Xavier proposes to Bertha.
Yancey proposes to Bertha.
Zeus proposes to Diane.
stable matching is unique for worst-case instance
11/29/2018

20. Day 6: Afternoon Men’s preferences Women’s preferences
Wyatt
Victor
1st B D
2nd A
3rd
Zeus
Yancey
Xavier C E
4th
5th
Bertha
Amy Z X W Y V
Erika
Diane
Clare
Amy says ``Maybe’’ to Victor.
Bertha says ``Maybe’’ to Xavier,
rejects Yancey.
Clare says ``Maybe’’ to Wyatt.
Diane says ``Maybe’’ to Zeus.
stable matching is unique for worst-case instance
11/29/2018

21. Day 6: Evening Men’s preferences Women’s preferences
Wyatt
Victor
1st B D
2nd A
3rd
Zeus
Yancey
Xavier C E
4th
5th
Bertha
Amy Z X W Y V
Erika
Diane
Clare
Yancey crosses off Bertha.
stable matching is unique for worst-case instance
11/29/2018

22. Day 7: Morning Men’s preferences Women’s preferences
Wyatt
Victor
1st B D
2nd A
3rd
Zeus
Yancey
Xavier C E
4th
5th
Bertha
Amy Z X W Y V
Erika
Diane
Clare
Victor proposes to Amy.
Wyatt proposes to Clare.
Xavier proposes to Bertha.
Yancey proposes to Erika.
Zeus proposes to Diane.
stable matching is unique for worst-case instance
11/29/2018

23. Day 7: Afternoon Men’s preferences Women’s preferences
Wyatt
Victor
1st B D
2nd A
3rd
Zeus
Yancey
Xavier C E
4th
5th
Bertha
Amy Z X W Y V
Erika
Diane
Clare
Amy says ``Maybe’’ to Victor.
Bertha says ``Maybe’’ to Xavier.
Clare says ``Maybe’’ to Wyatt.
Diane says ``Maybe’’ to Zeus.
Erika says ``Maybe’’ to Yancey.
stable matching is unique for worst-case instance
11/29/2018

24. Day 7: Evening (Happy End)
Men’s preferences
Women’s preferences
Wyatt
Victor
1st B D
2nd A
3rd
Zeus
Yancey
Xavier C E
4th
5th
Bertha
Amy Z X W Y V
Erika
Diane
Clare
Output of our algorithm.
The following couples get married:
Victor and Amy,
Wyatt and Clare,
Xavier and Bertha,
Yancey and Erika,
Zeus and Diane.
stable matching is unique for worst-case instance
11/29/2018

25. I-clicker question (frequency: BC)
True or false?
In the propose-and-reject algorithm, each day some man proposes to a new woman.
True.
False.
11/29/2018

26. I-clicker question (frequency: BC)
True or false?
In the propose-and-reject algorithm, each day some man proposes to a new woman.
True.
False.
Answer: True. If proposals are the same on two consecutive days, it means that nobody got rejected the day before.
11/29/2018

27. Proof of Correctness: Termination
Lemma. Algorithm runs for at most 𝒏 𝟐 days.
Proof. Each day some man proposes to a new woman.
There are 𝒏 men, 𝒏 women. Overall, 𝒏 𝟐 possible proposals. ▪
Wyatt
Victor
1st A B
2nd C D
3rd
Zeus
Yancey
Xavier
4th E
5th
Bertha
Amy W X Y Z V
Erika
Diane
Clare
stable matching is unique for worst-case instance
An instance where n(n-2) + 2 days are required
11/29/2018

28. Analysis of Propose-and-Reject Algorithm
Observation. Men propose to women in decreasing order of preference.
Improvement lemma (``women trade up’’). Once a woman gets a proposal from a man 𝑚, starting from that day, she has somebody on a string that she likes at least as much as 𝑚.
Men’s preferences
Women’s preferences
Wyatt
Victor
1st B D
2nd A
3rd
Zeus
Yancey
Xavier C E
4th
5th
Bertha
Amy Z X W Y V
Erika
Diane
Clare
stable matching is unique for worst-case instance
11/29/2018

29. Proof of Correctness: Perfection
Lemma. All men and women get matched.
Proof: (by contradiction)
Suppose, for the sake of contradiction, some guy, say Zeus, is not matched upon termination of the algorithm.
Then some woman, say Amy, is not matched upon termination.
By Improvement Lemma, Amy was never proposed to.
But Zeus proposes to everyone, since he ends up unmatched.
Contradiction. ▪
11/29/2018

30. Proof of Correctness: Stability
Claim. The final matching has no rogue pairs.
Proof: (by contradiction)
Suppose A-Z is a rogue pair: they prefer each other to their partners in Gale-Shapley matching S*.
Case 1: Z never proposed to A.  Z prefers his GS partner to A.  A-Z is not rogue.
Case 2: Z proposed to A.  A rejected Z (right away or later)  A prefers her GS partner to Z.  A-Z is not rogue.
In either case A-Z is not rogue, a contradiction. ▪
men propose in decreasing order of preference
women only trade up
11/29/2018

31. Summary
Stable matching problem. Given n men and n women, and their preferences, find a stable matching if one exists.
Gale-Shapley algorithm. Guarantees to find a stable matching for every problem instance.
(Also proves that a stable matching always exists.)
Time: It can be implemented to run efficiently.
11/29/2018