1. Lecture 10
CSE 331
Sep 21, 2011

2. Online Office Hr Toinght 9:30-10:00

3. Please do let me know if I’m mumbling

4. Gale-Shapley Algorithm
At most n2 iterations
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
O(1) time implementation
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

5. Gale-Shapley Algorithm
Linked list of free women
Intially all men and women are free
Array Next[w]
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
Array Current[m]
If m is free
(m,w) get engaged
Else (m,w’) are engaged
Today
If m prefers w’ to w
w remains free
ManPref[m,j]
WomanPref[w,j]
Else
(m,w) get engaged and w’ is free
Output the engaged pairs as the final output

6. Yes, Firefly again but with numbers1 1 2 2 3 3

7. The data structures so far1 2 3
Man-Pref 1 1 1 2 3 2 2 1 2 3 3 3 1 3
Linked list of free women 2 1 2 3
first 1
Next
Current

8. Implementation Steps (1) How to represent the input?
(2) How do we find a free woman w?
Init: O(n), Query/Update: O(1)
(3) How would w pick her best unproposed man m?
Init: O(n), Query/Update: O(1)
(4) How do we know who m is engaged to?
Init: O(n), Query/Update: O(1)
(5) How do we decide if m prefers w’ to w?

9. Overall running time n2 X ( O(1) + O(1) + O(1)+ Query/Update(5) )
O(n) + O(n) + O(n)+ Init(5)
n2 X ( O(1) + O(1) + O(1)+ Query/Update(5) )

10. Questions?

11. Puzzle
Prove that any algorithm for the SMP takes Ω(n2) time

12. Reading Assignments
Sec 1.1 and Chap. 2 in [KT]

13. Relationship: Mention in other’s program
Graphs
Representation of relationships between pairs of entities/elements
Edge
Entities: News hosts
Relationship: Mention in other’s program
Vertex

14. Graphs are omnipresent
Airline Route maps

15. What does this graph represent?
Internet

16. And this one?
Math articles on Wikipedia

17. And this one?

18. Rest of today’s agenda
Basic Graph definitions