1. CISC220 Fall 2009 James Atlas Nov 13: Graphs, Line Intersections

2. Graph Representations How do we represent a graph? A B E D C

3. List Structures V = {A, B, C, D, E} E = {{A, B}, {A, D}, {C, E}, {D, E}} Incidence List –E = {{A, B}, {A, D}, {C, E}, {D, E}} Adjacency List –L = [A={B, D}, B={A}, C={E}, D={A, E}, E={C, D}] A B E D C

4. Matrix Structures V = {A, B, C, D, E} E = {{A, B}, {A, D}, {C, E}, {D, E}} Adjacency Matrix A B E D C ABCDE A11010 B11000 C00101 D10011 E00111

5. A Directed Graph V = {A, B, C, D, E} E = {(A, B), (B, A), (B, E), (D, A), (E, A), (E, D), (E, C)}

6. Weighted Graph ABCD A0260  148 B2600  C  0  D148  0

7. Disconnected Graph

8. Equivalence of Graphs The numbering of the vertices, and their physical arrangement are not important. The following is the same graph as the previous slide.

9. Path Definition Adjacency Path –A sequence of vertices in which each successive vertex is adjacent to its predecessor Connected Graph –A path from every vertex to every other vertex Fully Connected Graph –Every vertex is adjacent to every other vertex

10. Example of a Path

11. Example of a Cycle

12. Relationship Between Graphs and Trees A tree is a special case of a graph. Any connected graph which does not contain cycles can be considered a tree. Any node can be chosen to be the root.

13. What kinds of things do we do with graphs?

14. Breadth First Search Starting at a source vertex Systematically explore the edges to “discover” every vertex reachable from s. Produces a “breadth-first tree” –Root of s –Contains all vertices reachable from s –Path from s to v is the shortest path

15. Algorithm 1.Take a start vertex, mark it identified (color it gray), and place it into a queue. 2.While the queue is not empty 1.Take a vertex, u, out of the queue (Begin visiting u) 2.For all vertices v, adjacent to u, 1.If v has not been identified or visited 1.Mark it identified (color it gray) 2.Place it into the queue 3.Add edge u, v to the Breadth First Search Tree 3.We are now done visiting u (color it black)

16. Example

18. Trace of Breadth First Search

19. Breadth-First Search Tree

20. Maze as a Graph

21. Breadth First Search Tree

22. Maze Solution

23. Find all intersections