CSC 213 – Large Scale Programming

Today’s Goals  Discuss what is meant by weighted graphs  Where weights placed within Graph  How to use Graph ’s weights to model problems  How to solve problems once Graph is set up  Learn about myth & legend of Edsgar Dijkstra  Who was he? Why should we care? How is it related?  What was his largest contribution to graph theory?  How does Dijkstra’s algorithm find smallest path?

Weighted Graphs  Edge’s weight is cost of using edge  Distance, cost, travel time, &c. usable as the weight  Weights below are distance in miles ORD PVD MIA DFW SFO LAX LGA HNL

Cheapest Path Problem  Find path with min. weight between 2 vertices  Sum of edge weights is the path weight  Consider the cheapest path from PVD to HNL  None of edges is cheapest in this example ORD PVD MIA DFW SFO LAX LGA HNL

Cheapest Path Problem  Subpath on shortest path is shortest path also  Otherwise we would use shorter subpath  Tree made by all shortest paths from vertex  Consider all shortest paths from PVD ORD PVD MIA DFW SFO LAX LGA HNL

Dijkstra’s Algorithm  Finds cheapest paths from single vertex  Normally, computes cheapest path to all vertices  Stop once vertex computed for single target vertex  Makes several fundamental assumptions  Connected graph needed when targeting all vertices edge weights must be nonnegative  Only works if edge weights must be nonnegative

Dijkstra’s Algorithm  Grows cloud of vertices as it goes  Cloud starts with source vetex  Add vertex to cloud with each step  Tracks distances to each vertex not in cloud  For each vertex, considers only cheapest path  Only uses 1 edge from cloud to vertex not in cloud  Each step uses vertex with smallest distance  Adds this vertex to cloud, if not done yet  Checks if creates smaller path to any vertices

Edge Relaxation  Consider e from u to z  When u added to cloud  Check adjacent vertices  Assume z not in cloud  Found faster path!  Update via relaxation  New minimum selected: d( z ) = 75 z s u d( u ) = e

Edge Relaxation  Consider e from u to z  When u added to cloud  Check adjacent vertices  Assume z not in cloud  Found faster path!  Update via relaxation  New minimum selected: d( z ) = 75 z s u d( u ) = e d( z ) = 60

Edge Relaxation  Consider e from u to z  When u added to cloud  Check adjacent vertices  Assume z not in cloud  Found faster path!  Update via relaxation  New minimum selected: d( z ) = 75 z s u d( u ) = e d( z ) = 60

Edge Relaxation  Consider e from u to z  When u added to cloud  Check adjacent vertices  Assume z not in cloud  Found faster path!  Update via relaxation  New minimum selected: z s u d( u ) = e d( z ) = 60

Dijkstra Example CB A E D F  

C B A E D F

Dijkstra Example C B A E D F

C B A E D F

CB A E D F

CB A E D F

Why Dijkstra’s Algorithm Works  Ultimately, Dijkstra was smart  Smarter than me, if that is possible

Why Dijkstra’s Algorithm Works  Ultimately, Dijkstra was smart  Smarter than me, if that is possible

Why Dijkstra’s Algorithm Works  Ultimately, Dijkstra was smart  Smarter than me, if that is possible  Example of a greedy algorithm  Takes best choice at each point in time  Vertices added in increasing distance  Brings vertices closer at each step  Stops when vertex cannot move closer

Why No Negative-Weight Edges?  Assume edge has negative weight  Greedily chose vertex before finding edge  Cloud will include only one endpoint  Negative weight changes everything, however  Vertices not added in order  Negative weight cycles?  Repeat cycle to optimize CB A E D F

Why No Negative-Weight Edges?  Assume edge has negative weight  Greedily chose vertex before finding edge  Cloud will include only one endpoint  Negative weight changes everything, however  Vertices not added in order  Negative weight cycles?  Repeat cycle to optimize C added when distance was 5, but cheapest distance is 1! CB A E D F

Spanning Tree spanning subgraph tree  Subgraph that is both spanning subgraph & tree  Contains all vertices in graph  spanning subgraph  Tree  connected without any cycles Graph

Spanning Tree spanning subgraph tree  Subgraph that is both spanning subgraph & tree  Contains all vertices in graph  spanning subgraph  Tree  connected without any cycles Tree

Spanning Tree spanning subgraph tree  Subgraph that is both spanning subgraph & tree  Contains all vertices in graph  spanning subgraph  Tree  connected without any cycles Spanning subgraph

Spanning Tree spanning subgraph tree  Subgraph that is both spanning subgraph & tree  Contains all vertices in graph  spanning subgraph  Tree  connected without any cycles Spanning tree

Prim-Jarnik’s Algorithm  Similar to Dijkstra’s algorithm but for MST  Processing must start with some vertex s  Grow MST using “cloud” of vertices  Label vertices with least Edge weight to cloud  At each step:  Find and add vertex closest to cloud  Update adjacent vertices to vertex just added

Prim-Jarnik’s Algorithm  Priority queue stores vertices outside of cloud  You all should be reminded of Dijkstra's algorithm  Three decorations used for each Vertex  Distance from cloud  Edge connecting vertex to cloud  Entry for Vertex in the priority queue

Prim-Jarnik Example B D C A F E  

B D C A F E  

B D C A F E  7

B D C A F E  7

B D C A F E  7

B D C A F E  7

B D C A F E  7

B D C A F E  7

B D C A F E

B D C A F E

B A E D C F

B A E D C F

Prim-Jarnik’s Analysis  Each connected vertex is:  Decorated O (deg( v )) times going through algorithm  Priority queue will have added & removed once  Takes O (( n + m ) log n ) time using adjacency list  Each operation on priority queue takes O (log n ) time  Takes O (log n ) time to decorate Vertex each time

For Next Lecture  Weekly assignment available on Angel before next Monday’s quiz  Due at special time: before next Monday’s quiz  Programming assignment #3 designs due Friday  Reading on more cheap paths for Friday  Why does everything need to be connected?  Algorithms for the uptight who do not want to relax?