Graph

1. Graph Algorithms Baojian Hua huabj@mail.ustc.edu.cn May 26, 2008

2. Graph • Graph is a basic and simple mathematical abstraction with a long history • Date back at least to Euler • Focus on topological structure (geometry?) • With many important applications • Map, scheduling, transaction, … • In computer science: • Network, compiler, web, circuit, …

3. Representation

4. name name succs succs next next node next /\ node next Adjacency List // a simplified representation (no explicit edges) class Node { String name; Successors succs; Node next; } class Successors { Node node; Successors next; }

5. name edges next from to next /\ Or With Edges class Node { String name; Edges edges; Node next; } class Edges { Node from; Node to; Edegs next; }

6. Depth-first Search (DFS) // a much simplified version dfs (Node n){ visited [n] = true; // memoization :-) for (each successors t of n){ if (visited[t]) {} else dfs (t); } } // with initial call dfs (start).

7. Example B A D C Initial start node: A

8. Example B A D C Visited A

9. Example B A D C Visited B

10. Example B A D C Visited C

11. Example B A D C Visited D

12. Example B A D C All successors of D has been visited. Go back from D.

13. Example B A D C All successors of C has been visited. Go back from C.

14. Example B A D C All successors of B has been visited. Go back from B.

15. Example B A D C All successors of A has been visited. Go back from A.

16. DFS Tree B A D C

17. DFS Tree B A D C

18. DFS Tree B A D C

19. Problems #1 • Simple Path: is there a simple path from node u to v in a digraph g? • Or is u and v connected?

20. Problems #2 • Simple connectivity: is a given undirected graph connected? • Or how many nodes are there in a connected component?

21. Problems #3 • Cycle detection: is there a cycle in a digraph?

22. Problems #4 • Two colorability: is it possible to color all the nodes in a digraph g such that no adjacent nodes with the same color? • Or: is there a cycle with odd length?