1. Lecture 11
CSE 331
Sep 20, 2013

2. HW 2 due today I will not accept HWs after 1:15pm
Place Q1, Q2 and Q3 in separate piles
I will not accept HWs after 1:15pm

3. Other HW related stuff HW 3 has been posted online: see piazza
Solutions to HW 2 at the END of the lecture
Graded HW 1 available from Monday onwards

4. My New Monday OH
3:00-3:50pm in Davis 319

5. 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

6. 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

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

8. 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

9. 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?

10. 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) )

11. HW 2 due today I will not accept HWs after 1:15pm
Place Q1, Q2 and Q3 in separate piles
I will not accept HWs after 1:15pm

12. Questions?

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

14. Main Steps in Algorithm Design
Problem Statement
Problem Definition n!
Algorithm
“Implementation”
Analysis
Correctness Analysis

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

16. A generic tool to abstract out problems
Up Next….
Problem Statement
A generic tool to abstract out problems
Problem Definition
Algorithm
“Implementation”
Analysis

17. 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/Node

18. Graphs are omnipresent
Airline Route maps

19. What does this graph represent?
Internet

20. And this one?
Math articles on Wikipedia

21. And this one?

22. Rest of today’s agenda
Basic Graph definitions

23. Paths , , Sequence of vertices connected by edges Connected
Path length 3 ,

24. Connectivity
u and w are connected iff there is a path between them
A graph is connected iff all pairs of vertices are connected

25. Connected Graphs
Every pair of vertices has a path between them

26. Cycles
Sequence of k vertices connected by edges, first k-1 are distinct ,

28. Formally define everything

29. Tree
Connected undirected graph with no cycles

30. Rooted Tree

31. How many rooted trees can an n vertex tree have?
A rooted tree
How many rooted trees can an n vertex tree have?
AC’s child=SG
Pick any vertex as root
SG’s parent=AC
Let the rest of the tree hang under “gravity”

32. Rest of Today’s agenda Prove n vertex tree has n-1 edges
Algorithms for checking connectivity