1. CSC 213 – Large Scale Programming

2. 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?

3. 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 849 802 1387 1743 1843 1099 1120 1233 337 2555 142

4. 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 849 802 1387 1743 1843 1099 1120 1233 337 2555 142

5. 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 849 802 1387 1743 1843 1099 1120 1233 337 2555 142

6. 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

7. 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

8. 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 ) = 50 10 e

9. 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 ) = 50 10 e d( z ) = 60

10. 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 ) = 50 10 e d( z ) = 60

11. 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 ) = 50 10 e d( z ) = 60

12. Dijkstra Example CB A E D F 0 428   4 8 7 1 2 5 2 3 9

13. C B A E D F 0 328 5 11 4 8 7 1 2 5 2 3 9

14. Dijkstra Example C B A E D F 0 328 5 8 4 8 7 1 2 5 2 3 9

15. C B A E D F 0 327 5 8 4 8 7 1 2 5 2 3 9

16. CB A E D F 0 327 5 8 4 8 7 1 2 5 2 3 9

17. CB A E D F 0 327 5 8 4 8 7 1 2 5 2 3 9

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

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

20. 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

21. 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 0 4 5 7 5 9 4 8 7 1 2 5 6 0 -8

22. 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 0 4 5 7 5 9 4 8 7 1 2 5 6 0 -8

23. 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

24. 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

25. 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

26. 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

27. 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

28. 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

29. Prim-Jarnik Example B D C A F E 7 4 2 8 5 7 3 9 8 0 7 2 8  

30. B D C A F E 7 4 2 8 5 7 3 9 8 0 7 2 8  

31. B D C A F E 7 4 2 8 5 7 3 9 8 0 7 2 8  7

32. B D C A F E 7 4 2 8 5 7 3 9 8 0 7 2 5  7

33. B D C A F E 7 4 2 8 5 7 3 9 8 0 7 2 5  7

34. B D C A F E 7 4 2 8 5 7 3 9 8 0 7 2 5  7

35. B D C A F E 7 4 2 8 5 7 3 9 8 0 7 2 5  7

36. B D C A F E 7 4 2 8 5 7 3 9 8 0 7 2 5  7

37. B D C A F E 7 4 2 8 5 7 3 9 8 0 7 2 5 4 7

38. B D C A F E 7 4 2 8 5 7 3 9 8 0 7 2 5 4 7

39. B A E 7 4 2 8 5 7 3 9 8 0 3 2 5 4 7 D C F

40. B A E 7 4 2 8 5 7 3 9 8 0 3 2 5 4 7 D C F

41. 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

42. 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?