Lecture 32: Graph TRAVERSALS

1. CSC 213 – Large Scale Programming Lecture 32:Graph TRAVERSALS

2. Today’s Goals • Make Britney sad through my color choices • Revisit issue of graph terminology and usage • Subgraphs, spanning, & connected graphs, oh my! • Problems solved orwhy you should care about these • Just rats in a maze: using graphs to find the exit • Confident in your decisions: how to travel depth-first • Using bread-first searching to cover all the bases • 2 algorithms both alike in dignity: how do they differ?

3. Today’s Goals • Make Britney sad through my color choices • Revisit issue of graph terminology and usage • Subgraphs, spanning, & connected graphs, oh my! • Problems solved orwhy you should care about these • Just rats in a maze: using graphs to find the exit • Confident in your decisions: how to travel depth-first • Using bread-first searchingto cover all the bases • 2 algorithms both alike in dignity: how do they differ?

4. Graphs Solve Many Problems

5. Subgraphs • Subgraphof G contains edges & vertices from G Graph

6. Subgraphs • Subgraphof G contains edges & vertices from G Subgraph

7. Subgraphs • Subgraphof G contains edges & vertices from G • Spanning subgraphis subgraphcontaining all vertices in G Subgraph Graph

8. Subgraphs • Subgraphof G contains edges & vertices from G • Spanning subgraphis subgraphcontaining all vertices in G Subgraph Spanning subgraph

9. Connected Graphs & Subgraphs • Connectedif path exists between all vertex pairs • Requires path, not edge, between vertices Connected Graph

10. Connected Graphs & Subgraphs • Connectedif path exists between all vertex pairs • Requires path,not edge, between vertices Disconnected Graph

11. Connected Graphs & Subgraphs • Connected if path exists between all vertex pairs • Requires path,not edge, between vertices • Connected component is subgraph containing all connected vertices • Often called “maximally connected” Disconnected Graph Graph

12. Connected Graphs & Subgraphs • Connectedif path exists between all vertex pairs • Requires path,not edge, between vertices • Connected component is subgraph containing all connected vertices • Often called “maximally connected” Disconnected Graph Graph with 2 connected components

13. Connected Graphs & Subgraphs • Connected if path exists between all vertex pairs • Requires path,not edge, between vertices • Connected component is subgraph containing all connected vertices • Often called “maximally connected” Disconnected Graph Graph with 2 connected components

14. Tree • Connected acyclic graph • Resembles Tree structure Graph

15. Tree • Connected acyclic graph • Resembles Tree structure Tree

16. Tree • Connected acyclic graph • Resembles Tree structure • Don’t lose forest for trees • Forestis graph with multiple trees Tree Graph

17. Tree • Connected acyclic graph • Resembles Tree structure • Don’t lose forest for trees • Forest is graph with multiple trees Tree Tree

18. Tree • Connected acyclic graph • Resembles Tree structure • Don’t lose forest for trees • Forest is graph with multiple trees Tree Forest

19. Spanning Tree • Subgraphthat is both spanning subgraph& tree • Contains all vertices in graph spanning subgraph • Tree  connected without any cycles Graph

20. Spanning Tree • Subgraphthat is both spanning subgraph& tree • Contains all vertices in graph spanning subgraph • Tree  connected without any cycles Tree

21. Spanning Tree • Subgraphthat is both spanning subgraph& tree • Contains all vertices in graph spanning subgraph • Tree  connected without any cycles Spanning subgraph

22. Spanning Tree • Subgraphthat is both spanning subgraph& tree • Contains all vertices in graph spanning subgraph • Tree  connected without any cycles Spanning tree

23. Spanning Tree • Subgraphthat is both spanning subgraph& tree • Contains all vertices in graph spanning subgraph • Tree  connected without any cycles Tree Than Spans Royals?

24. Depth- & Breadth-First Search • Common graph traversal algorithms • DFS & BFS traversals have common traits • Visits all vertices and edges in G • Determines whether of not Gis connected • Finds connected components if G not connected • Heavily used to solve graph problems

25. DFS and Maze Traversal • Classic maze strategy • Intersection, corner, & dead endare vertices • Corridors become edges • Traveled corridors marked • Marks trace back to start

26. DFS Example unexplored vertex A visited vertex A unexplored edge A discovery edge back edge B D E C

27. DFS Example unexplored vertex A visited vertex A unexplored edge A discovery edge back edge B D E C

28. DFS Example unexplored vertex A visited vertex A unexplored edge A discovery edge back edge ??? B D E C

29. DFS Example unexplored vertex A visited vertex A unexplored edge A discovery edge back edge B D E C

30. DFS Example unexplored vertex A visited vertex A unexplored edge A discovery edge back edge B D E C

31. DFS Example unexplored vertex A visited vertex A unexplored edge A discovery edge back edge B D E C

32. DFS Example unexplored vertex A visited vertex A unexplored edge A discovery edge back edge B D E C

33. Properties of DFS • Connected component’svertices & edges visited • Edges visited form spanning tree for component A B D E C

34. Properties of DFS • Connected component’svertices & edges visited • Edges visited form spanning tree for component A B D E C

35. BFS Example unexplored vertex A visited vertex A unexplored edge L0 A discovery edge cross edge B D E C F

36. BFS Example unexplored vertex A visited vertex A unexplored edge L0 A discovery edge cross edge L1 B D E C F

37. BFS Example unexplored vertex A visited vertex A unexplored edge L0 A discovery edge cross edge L1 B D E C F

38. BFS Example unexplored vertex A visited vertex A unexplored edge L0 A discovery edge cross edge L1 B D E C F

39. BFS Example unexplored vertex A visited vertex A unexplored edge L0 A discovery edge cross edge L1 B D E C F

40. BFS Example unexplored vertex A visited vertex A unexplored edge L0 A discovery edge cross edge L1 B D E L2 C F

41. BFS Example unexplored vertex A visited vertex A unexplored edge L0 A discovery edge cross edge L1 B D E L2 C F

42. BFS Example unexplored vertex A visited vertex A unexplored edge L0 A discovery edge cross edge L1 B D E L2 C F

43. BFS Example unexplored vertex A visited vertex A unexplored edge L0 A discovery edge cross edge L1 B D E L2 C F

44. BFS Example unexplored vertex A visited vertex A unexplored edge L0 A discovery edge cross edge L1 B D E L2 C F

45. Properties of BFS • Some properties similar to DFS: • Visits all vertices & edgesin connected component • Component’s spanning tree from discovery edges • For each vertex v in Li • Spanning treehas exactlyi edgeson path from s to v • All paths in G from s to v have at least iedges L0 A L1 B D E L2 C C F

46. DFS vs. BFS DFS BFS Back edge (v,w) • w is ancestor of vin tree of discovery edges Cross edge (v,w) • v in same or previous level as win tree of discovery edges A L0 A B D E L1 B D E L2 C C F C

47. For Next Lecture • Weekly assignment available & due Tuesday • Gives you good chance to check your understanding • Next Fri. 1st check-in for program assignment #3 • Planning now saves time later! • Read 13.4.1 – 13.4.3 for class on Monday • When using directed edges, what changes needed? • Other computations possible for directed graphs?