1. CSC 213 – Large Scale Programming

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 or why 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 or why 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?

4. Graphs Solve Many Problems

5. Subgraphs  Subgraph of G contains edges & vertices from G Graph

6. Subgraphs  Subgraph of G contains edges & vertices from G Subgraph

7. Subgraphs  Subgraph of G contains edges & vertices from G  Spanning subgraph is subgraph containing all vertices in G Subgraph Graph

8. Subgraphs  Subgraph of G contains edges & vertices from G  Spanning subgraph is subgraph containing all vertices in G Subgraph Spanning subgraph

9. Connected Graphs & Subgraphs  Connected if path exists between all vertex pairs  Requires path, not edge, between vertices Connected Graph

10. Connected Graphs & Subgraphs  Connected if path exists between all vertex pairs not edge  Requires path, not edge, between vertices Disconnected Graph

11. Connected Graphs & Subgraphs  Connected if path exists between all vertex pairs not edge  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  Connected if path exists between all vertex pairs not edge  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 not edge  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.  Connected acyclic graph  Resembles Tree structure  Don’t lose forest for trees  Forest is 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

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

19. 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

20. 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

21. 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

22. 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

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 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 G is connected  Finds connected components if G not connected  Heavily used  Heavily used to solve graph problems

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

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

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

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

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

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

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

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

33. Properties of DFS  Connected component’s vertices & edges visited  Edges visited form spanning tree for component DB A C E

34. Properties of DFS  Connected component’s vertices & edges visited  Edges visited form spanning tree for component DB A C E

35. L0L0 BFS Example DB A C E discovery edge A visited vertex A unexplored vertex unexplored edge F cross edge

36. L1L1 L0L0 BFS Example D B A C E F discovery edge A visited vertex A unexplored vertex unexplored edge cross edge

37. L1L1 L0L0 BFS Example DB A C E F discovery edge A visited vertex A unexplored vertex unexplored edge cross edge

38. L1L1 L0L0 BFS Example DB A C E F discovery edge A visited vertex A unexplored vertex unexplored edge cross edge

39. L1L1 L0L0 BFS Example DB A C E F discovery edge A visited vertex A unexplored vertex unexplored edge cross edge

40. L2L2 L1L1 L0L0 BFS Example DB A C E F discovery edge A visited vertex A unexplored vertex unexplored edge cross edge

41. L2L2 L1L1 L0L0 BFS Example DB A C E F discovery edge A visited vertex A unexplored vertex unexplored edge cross edge

42. L2L2 L1L1 L0L0 BFS Example DB A C E F discovery edge A visited vertex A unexplored vertex unexplored edge cross edge

43. L2L2 L1L1 L0L0 BFS Example DB A C E F discovery edge A visited vertex A unexplored vertex unexplored edge cross edge

44. L2L2 L1L1 L0L0 BFS Example DB A C E F discovery edge A visited vertex A unexplored vertex unexplored edge cross edge

45. Properties of BFS  Some properties similar to DFS:  Visits all vertices & edges in connected component  Component’s spanning tree from discovery edges  For each vertex v in L i  Spanning tree has exactly i edges on path from s to v  All paths in G from s to v have at least i edges L2L2 L1L1 L0L0 B A C D C E F

46. DFS vs. BFS DFSBFS Back edge (v,w)  w is ancestor of v in tree of discovery edges Cross edge (v,w)  v in same or previous level as w in tree of discovery edges L2L2 L1L1 L0L0 B A C D C E F DB A C E

47. For Next Lecture  Weekly assignment available & due Tuesday  Gives you good chance to check your understanding  Next Fri. 1 st 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?