Week 12 - Monday

 What did we talk about last time?  Topological sort and cycle detection in directed graphs  Connectivity  Strong connectivity

Student Lecture

 An airline has to stop running direct flights between some cities  But, it still wants to be able to reach all the cities that it can now  What’s the set of flights with the lowest total cost that will keep all cities connected?  Essentially, what’s the lowest cost tree that keeps the graph connected?

 This tree is called the minimum spanning tree or MST  It has countless applications in graph problems  How do we find such a thing?

1. Start with two sets, S and V:  S has the starting node in it  V has everything else 2. Find the node u in V that is closest to any node in S 3. Put the edge to u into the MST 4. Move u from V to S 5. If V is not empty, go back to Step 2

A C G E B F D

 Naïve implementation with an adjacency matrix  O(|V| 2 )  Adjacency lists with binary heap  O(|E| log |V|)  Adjacency lists with Fibonacci heap  O(|E| + |V| log |V|)

A L I F B E G C J H K D

 How does Google Maps find the shortest route from California to Elizabethtown?  Graph theory, of course!  It stores a very large graph where locations are nodes and streets (well, parts of streets) are edges

 We use a weighted graph  Weight can represent time, distance, cost: anything, really  The shortest path (lowest total weight) is not always obvious

 Take a moment and try to find the shortest path from A to E.  The shortest path has cost A A B B D D C C E E A B C D E

 On a graph of that size, it isn’t hard to find the shortest path  A Google Maps graph has millions and millions of nodes  How can we come up with an algorithm that will always find the shortest path from one node to another?

 In 1959, Edsger Dijkstra invented an algorithm to find shortest paths  First, a little notation: NotationMeaning sStarting node d(v)d(v) The best distance from s to v found so far d(u, v) The direct distance between nodes u and v S A set which contains the nodes for which we know the shortest path from s V A set which contains the nodes for which we do not yet know the shortest path from s

1. Start with two sets, S and V:  S is empty  V has all the nodes in it 2. Set the distance to all nodes in V to ∞ 3. Set the distance to the starting node s to 0 4. Find the node u in V that is closest to s 5. For every neighbor v of u in V ▪ If d(v) > d(u) + d(u,v) ▪ Set d(v) = d(u) + d(u,v) 6. Move u from V to S 7. If V is not empty, go back to Step 4

Noded(u)d(u) A0 B5 C8 D∞ E16 Noded(u)d(u) A0 B5 C7 D11 E16 Noded(u)d(u) A0 B5 C7 D10 E16 Noded(u)d(u) A0 B5 C7 D10 E14 Noded(u)d(u) A0 B5 C7 D10 E14 Noded(u)d(u) A0 B∞ C∞ D∞ E∞ A A B B D D C C E E Sets SV AB, C, D, E Sets SV A, BC, D, E Sets SV A, B, CD, E Sets SV A, B, C, DE Sets SV A, B, C, D, E Finding the shortest distance from A to all other nodes Sets SV A, B, C, D, E A A

 Always gets the next closest node, so we know there isn’t a better way to get there  Finds the shortest path from a starting node to all other nodes  Works even for directed graphs

 The normal running time for Dijkstra's is O(|V| 2 )  At worst, we may have to update each node in V for each node V that we find the shortest path to  A special data structure called a min-priority queue can implement the process of updating priorities faster  Total running time of O(|E| + |V| log |V|)  Technically faster for sparse graphs  Algorithm wizards Fredman and Tarjan created an implementation called a Fibonacci Heap ▪ Actually slow in practice

A L I F B E G C J H K D

 The algorithms have very similar forms  Here is a graph for which the shortest path tree from A to all other nodes is not an MST A B C D

 Matching on bipartite graphs  Stable marriage  Euler tours

 Start on Project 4  Form teams by Friday  Start on Assignment 6