CS221Week 11 - Friday

Last time

What did we talk about last time? Exam 2 post mortem Cycle detection Connectivity

Questions?

Spellchecker

Project 3

Assignment 6

Connectivity

Connected graph?

Connected for an undirected graph:There is a path from every node to every other node

How can we determine if a graph is connected?

DFS to the rescue again!

Startup1. Set the number of all nodes to 02. Pick an arbitrary node u and run DFS( u, 1 )

DFS( node v, int i )1. Set number(v) = i++2. For each node u adjacent to v

If number(u) is 0DFS( u, i )

If any node has a number of 0, the graph is not connected

Directed connectivity

Weakly connected directed graph:If the graph is connected when you make all the edges undirected

Strongly connected directed graph:If for every pair of nodes, there is a path between them in both directions

Short of strong connectivity Components of a directed graph can be

strongly connected A strongly connected component is a

subgraph such that all its nodes are strongly connected

To find strongly connected components, we can use a special DFS

It includes the notion of a predecessor node, which is the lowest numbered node in the DFS that can reach a particular node

DFS used for strong connectivity StrongDFS( node v, int i )

1. Set number(v) = i++2. Set predecessor(v) to number(v)3. Push(v)4. For each node u adjacent to v

If number(u) is 0StrongDFS( u, i )predecessor(v) = min(predecessor(v), predecessor(u))

Else if number(u) < number(v) and u is 0n the stackpredecessor(v) = min(predecessor(v), number(u))

5. If predecessor(v) is number(v)w = pop()while w is not v

output ww = pop()

output w

Full strongly connected component algorithm

StronglyConnected( Vertices V ) Set i = 1 For each node v in V

Set number(v) = 0 For each node v in V

If number(v) is 0StrongDFS( v, i )

Find the strongly connected components

A

B

I

D

F

H

C

E

G

Topological SortStudent Lecture

Topological Sort

Directed acyclic graph

A directed acyclic graph (DAG) is a directed graph without cycles in it Well, obviously

These can be used to represent dependencies between tasks

An edge flows from the task that must be completed first to a task that must come after

This is a good model for course sequencing Especially during advising

A cycle in such a graph would mean there was a circular dependency

Topological sort

A topological sort gives an ordering of the tasks such that all tasks are completed in dependency ordering

In other words, no task is attempted before its prerequisite tasks have been done

There are usually multiple legal topological sorts for a given DAG

Topological sort

Give a topological sort for the following DAG:

A F I C G K D J E H

A

HE

JD

KG

C

F I

Topological sort algorithm Create list L Add all nodes with no incoming edges into set S While S is not empty

Remove a node u from S Add u to L For each node v with an edge e from u to v▪ Remove edge e from the graph▪ If v has no other incoming edges, add v to S

If the graph still has edges Print "Error! Graph has a cycle"

Otherwise Return L

Matching

Bipartite graphs

A bipartite graph is one whose nodes can be divided into two disjoint sets X and Y

There can be edges between set X and set Y

There are no edges inside set X or set Y

A graph is bipartite if and only if it contains no odd cycles

Bipartite graph

A B C D E F

A B C D E F

X

Y

Maximum matching

A perfect matching is when every node in set X and every node in set Y is matched

It is not always possible to have a perfect matching

We can still try to find a maximum matching in which as many nodes are matched up as possible

Matching algorithm

1. Come up with a legal, maximal matching

2. Take an augmenting path that starts at an unmatched node in X and ends at an unmatched node in Y

3. If there is such a path, switch all the edges along the path from being in the matching to being out and vice versa

4. If there is another augmenting path, go back to Step 2

Match the graph

A B C D E F

A B C D E F

X

Y

Anna

Becky

Caitlin

Daisy Erin Fiona

Adam

Ben Carlos

Dan Evan Fred

Stable Marriage

Imagine n men and n women

All 2n people want to get married Each woman has ranked all n men in

order of preference Each man has ranked all n women in

order of preference We want to match them up so that

the marriages are stable

Stability

Consider two marriages: Anna and Bob Caitlin and Dan

This pair of marriages is unstable if Anna likes Dan more than Bob and Dan

likes Anna more than Caitlin or Caitlin likes Bob more than Dan and Bob

likes Caitlin more than Anna We want to arrange all n marriages

such that none are unstable

Gale-Shapley algorithm

n rounds In each round, every unengaged man

proposes to the woman he likes best (who he hasn’t proposed to already)

An unengaged woman must accept the proposal from the one of her suitors she likes best

If a woman is already engaged, she accepts the best suitor only if she likes him better than her fiance

Lessons from Stable Marriage It’s a graph problem where we never look at

a graph The algorithm is guaranteed to terminate

A woman can’t refuse to get engaged If a man gets dumped, he will eventually

propose to everyone The algorithm is guaranteed to find stable

marriages Unfortunately, it’s male-optimal and female-

pessimal The lesson: Women should propose to men

Quiz

Upcoming

Next time…

Finish matching and stable marriage Euler paths and tours

Reminders

Finish Project 3 Due tonight before midnight

Start Assignment 6 Due next Friday

Keep reading Chapter 8