Week 6 - Monday

 What did we talk about last time?  Artificial intelligence  Lab 5

 Vertices (Nodes)  Edges

 Friendships on Facebook  Nodes: People  Edges: Friendship  Routes between cities  Nodes: Cities  Edges: Streets  Steps in a task  Nodes: Subtasks  Edges: Decisions  6 degrees of Kevin Bacon  Nodes: Actors  Edges: Whether or not they've been in a movie together

 Labeled  Weighted ColoredColoredColoredColored  Multigraphs E E A A D D B B F F C C E A D B F C

 When a weighted graph obeys the triangle inequality, the direct route to a node is always fastest

 Some graphs have edges with direction  Example: One way streets  Reachability? ONE WAY

 Often we talk about connected graphs  But, not all graphs have to be connected

 Complete graphs  Every node is connected to every other  How many edges?  |E| = ½(n(n – 1)) = ½(n 2 – n) is O(n 2 )

 We can talk about a part of a graph  For example, what is the largest complete subgraph in this graph? E E A A D D B B F F C C

 A path is a sequence of connected nodes  The cost or weight of the path is usually the sum of the edge weights  This path from A to C costs 5 E E A A D D B B F F C C

 A cycle is a path that starts at a node and comes back to the same node  How many cycles of length 3 does this graph have?  What about 4? 5? 6? E E F F C C A A D D B B A A D D B B

 A tour is a path that visits every node and (usually) returns to its starting node  In other words, it's a cycle that visits every node  This tour costs 24 E E A A D D B B F F C C

 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

A A B B C C D D E E F F A A B B C C D D E E F F X Y

 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

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

A A B B C C D D E E F F A A B B C C D D E E F F X Y Anna Becky CaitlinDaisyErinFiona Adam Ben CarlosDanEvanFred

 Stable marriage  Euler paths and tours  Minimum spanning trees  Lab 6

 Lab is Wednesday this week!  Start working on Project 2  Read Python Chapter 5  Think about what you want to do for your Final Project  Proposal due by 10/28