1. Lecture 6
CSE 331
Sep 12, 2011

2. Grading Rubric for HW 1

3. ≤ n2 iterations of the while loop
Intially all men and women are free
Each proposal is new
While there exists a free woman who can propose
Let w be such a woman and m be the best man she has not proposed to
w proposes to m
Each iteration has one proposal
If m is free
(m,w) get engaged
Else (m,w’) are engaged
# iterations ≤
# proposals
# pairs (m,w) n2
If m prefers w’ to w
w remains free
Else
(m,w) get engaged and w’ is free
Output the engaged pairs as the final output

4. Why do all the formal progress stuff?
Notion of proof by progress will be useful later
Example of writing a proof idea/ formal proof

5. ≤ n2 iterations of the while loop
Intially all men and women are free
While there exists a free woman who can propose
Let w be such a woman and m be the best man she has not proposed to
w proposes to m
Can make GS execute very close to n2 iterations
If m is free
(m,w) get engaged
Else (m,w’) are engaged
If m prefers w’ to w
w remains free
Else
(m,w) get engaged and w’ is free
Output the engaged pairs as the final output

6. Today’s lecture Perfect Macthing No instability
GS algorithms always outputs a stable marriage
Perfect Macthing
No instability

7. Once a man gets engaged, he remains engaged (to “better” women)
Observation 1
Intially all men and women are free
While there exists a free woman who can propose
Let w be such a woman and m be the best man she has not proposed to
w proposes to m
If m is free
(m,w) get engaged
Once a man gets engaged, he remains engaged (to “better” women)
Else (m,w’) are engaged
If m prefers w’ to w
w remains free
Else
(m,w) get engaged and w’ is free
Output the engaged pairs as the final output

8. If w proposes to m after m’, then she prefers m’ to m
Observation 2
Intially all men and women are free
While there exists a free woman who can propose
Let w be such a woman and m be the best man she has not proposed to
w proposes to m
If m is free
If w proposes to m after m’, then she prefers m’ to m
(m,w) get engaged
Else (m,w’) are engaged
If m prefers w’ to w
w remains free
Else
(m,w) get engaged and w’ is free
Output the engaged pairs as the final output

9. All engaged pairs form a matching
Observation 3
Intially all men and women are free
While there exists a free woman who can propose
Let w be such a woman and m be the best man she has not proposed to
w proposes to m
If m is free
(m,w) get engaged
All engaged pairs form a matching
Else (m,w’) are engaged
If m prefers w’ to w
w remains free
Else
(m,w) get engaged and w’ is free
Output the engaged pairs as the final output

10. Questions/Comments?

11. Proof technique de jour
Proof by contradiction
Assume the negation of what you want to prove
After some reasoning
Source: 4simpsons.wordpress.com