1. Graph Traversal C and Data Structures Baojian Hua bjhua@ustc.edu.cn

2. Traversal A systematic way to visit vertices in a graph Two general approaches: breath first searching (BFS) start from one vertex, first visit all the adjacency vertices depth first searching (DFS) eager method These slides assume the adjacency list representation

3. “ graph ” ADT in C: Interface // in file “graph.h” #ifndef GRAPH_H #define GRAPH_H #define T Graph_t typedef struct T *T; typedef void (*tyVisit)(poly); T Graph_new (); void Graph_insertVertex (T g, poly data); void Graph_insertEdge (T g, poly from, poly to); void Graph_dfs (T g, poly start, tyVisit visit); void Graph_bfs (T g, poly start, tyVisit visit); // we’d see more later … #endif

4. Sample Graph 1 4 2 65 3

5. Sample Graph BFS 1 4 2 65 3 bfs (g, 1, Int_print);

6. Sample Graph BFS 1 4 2 65 3 bfs (g, 1, Int_print); print 1;

7. Sample Graph BFS 1 4 2 65 3 bfs (g, 1, Int_print); print 1; // a choice print 2;

8. Sample Graph BFS 1 4 2 65 3 bfs (g, 1, Int_print); print 1; // a choice print 2; print 4;

9. Sample Graph BFS 1 4 2 65 3 bfs (g, 1, Int_print); print 1; // a choice print 2; print 4; print 5;

10. Sample Graph BFS 1 4 2 65 3 bfs (g, 1, Int_print); print 1; // a choice print 2; print 4; print 5; // a choice print 3;

11. Sample Graph BFS 1 4 2 65 3 bfs (g, 1, Int_print); print 1; // a choice print 2; print 4; print 5; // a choice print 3; print 6;

12. BFS Algorithm void bfsOneVertex (Tv start, tyVisit visit) { Queue_t q = Queue_new (); Queue_en (q, start); while (q not empty) { Tv current = Queue_de (q); visit (current); for (each adjacent vertex u of “current”) if (not visited u) Queue_en (q, u); } }

13. BFS Algorithm void bfs (T g, poly start, tyVisit visit) { Tv startV = searchVertex (g, start); bfsOneVertex (startV, visit); for (each vertex u in graph g) if (not visited u) bfsOneVertex (q, u); }

14. Sample Graph BFS 1 4 2 65 3 bfs (g, 1, Int_output); Queue: 1

15. Sample Graph BFS 1 4 2 65 3 bfs (g, 1, Int_output); print 1; Queue: 2, 4 Queue: 1

16. Sample Graph BFS 1 4 2 65 3 bfs (g, 1, Int_print); print 1; // a choice print 2; Queue: 2, 4 Queue: 1 Queue: 4, 5

17. Sample Graph BFS 1 4 2 65 3 bfs (g, 1, Int_print); print 1; // a choice print 2; print 4; Queue: 5 Queue: 2, 4 Queue: 1 Queue: 4, 5

18. Sample Graph BFS 1 4 2 65 3 bfs (g, 1, Int_print); print 1; // a choice print 2; print 4; print 5; Queue: Queue: 5 Queue: 2, 4 Queue: 1 Queue: 4, 5 Queue: 3

19. Sample Graph BFS 1 4 2 65 3 bfs (g, 1, Int_print); print 1; // a choice print 2; print 4; print 5; // a choice print 3; Queue: Queue: 5 Queue: 2, 4 Queue: 1 Queue: 4, 5 Queue: 3 Queue: 6

20. Sample Graph BFS 1 4 2 65 3 bfs (g, 1, Int_print); print 1; // a choice print 2; print 4; print 5; // a choice print 3; print 6; Queue: Queue: 5 Queue: 2, 4 Queue: 1 Queue: 4, 5 Queue: 3 Queue: 6 Queue:

21. Moral BFS is a generalization like the level- order traversal on trees Also maintain internally a queue to control the visit order Obtain a BFS forest when finished

22. Sample Graph DFS 1 4 2 65 3 dfs (g, 1, Int_print);

23. Sample Graph DFS 1 4 2 65 3 dfs (g, 1, Int_print); print 1;

24. Sample Graph DFS 1 4 2 65 3 dfs (g, 1, Int_print); print 1; // a choice print 2;

25. Sample Graph DFS 1 4 2 65 3 dfs (g, 1, Int_print); print 1; // a choice print 2; print 5;

26. Sample Graph DFS 1 4 2 65 3 dfs (g, 1, Int_print); print 1; // a choice print 2; print 5; print 4;

27. Sample Graph DFS 1 4 2 65 3 dfs (g, 1, Int_print); print 1; // a choice print 2; print 5; print 4; // a choice print 3;

28. Sample Graph DFS 1 4 2 65 3 dfs (g, 1, Int_print); print 1; // a choice print 2; print 5; print 4; // a choice print 3; print 6;

29. DFS Algorithm void dfsOneVertex (Tv start, tyVisit visit) { visit (start); for (each adjacent vertex u of “start”) if (not visited u) dfsOneVertex (u, visit); }

30. DFS Algorithm void dfs (T g, poly start, tyVisit visit) { Tv startV = searchVertex (g, start); dfsOneVertex (startV, visit); for (each vertex u in graph g) if (not visited u) dfsOneVertex (u, visit); }

31. Sample Graph DFS 1 4 2 65 3 dfs (g, 1, Int_print); print 1; dfs(1)

32. Sample Graph DFS 1 4 2 65 3 dfs (g, 1, Int_print); print 1; // a choice print 2; dfs(1) => dfs(2)

33. Sample Graph DFS 1 4 2 65 3 dfs (g, 1, Int_print); print 1; // a choice print 2; print 5; dfs(1) => dfs(2) => dfs(5)

34. Sample Graph DFS 1 4 2 65 3 dfs (g, 1, Int_print); print 1; // a choice print 2; print 5; print 4; dfs(1) => dfs(2) => dfs(5) => dfs(4)

35. Sample Graph DFS 1 4 2 65 3 dfs (g, 1, Int_print); print 1; // a choice print 2; print 5; print 4; dfs(1) => dfs(2) => dfs(5) => dfs(4) => dfs(2)???

36. Sample Graph DFS 1 4 2 65 3 dfs (g, 1, Int_print); print 1; // a choice print 2; print 5; print 4; dfs(1) => dfs(2) => dfs(5)

37. Sample Graph DFS 1 4 2 65 3 dfs (g, 1, Int_print); print 1; // a choice print 2; print 5; print 4; dfs(1) => dfs(2)

38. Sample Graph DFS 1 4 2 65 3 dfs (g, 1, Int_print); print 1; // a choice print 2; print 5; print 4; dfs(1)

39. Sample Graph DFS 1 4 2 65 3 dfs (g, 1, Int_print); print 1; // a choice print 2; print 5; print 4; dfs(1) =>dfs(4)???

40. Sample Graph DFS 1 4 2 65 3 dfs (g, 1, Int_print); print 1; // a choice print 2; print 5; print 4; dfs(1)

41. Sample Graph DFS 1 4 2 65 3 dfs (g, 1, Int_print); print 1; // a choice print 2; print 5; print 4; empty!

42. Sample Graph DFS 1 4 2 65 3 dfs (g, 1, Int_print); print 1; // a choice print 2; print 5; print 4; // a choice print 3; dfs(3)

43. Sample Graph DFS 1 4 2 65 3 dfs (g, 1, Int_print); print 1; // a choice print 2; print 5; print 4; // a choice print 3; print 6; dfs(3) =>dfs(6)

44. Sample Graph DFS 1 4 2 65 3 dfs (g, 1, Int_print); print 1; // a choice print 2; print 5; print 4; // a choice print 3; print 6; dfs(3) =>dfs(6) =>dfs(6)???

45. Sample Graph DFS 1 4 2 65 3 dfs (g, 1, Int_print); print 1; // a choice print 2; print 5; print 4; // a choice print 3; print 6; dfs(3) =>dfs(6)

46. Sample Graph DFS 1 4 2 65 3 dfs (g, 1, Int_print); print 1; // a choice print 2; print 5; print 4; // a choice print 3; print 6; dfs(3)

47. Sample Graph DFS 1 4 2 65 3 dfs (g, 1, Int_print); print 1; // a choice print 2; print 5; print 4; // a choice print 3; print 6; dfs(3) =>dfs(5)???

48. Sample Graph DFS 1 4 2 65 3 dfs (g, 1, Int_print); print 1; // a choice print 2; print 5; print 4; // a choice print 3; print 6; dfs(3)

49. Sample Graph DFS 1 4 2 65 3 dfs (g, 1, Int_print); print 1; // a choice print 2; print 5; print 4; // a choice print 3; print 6; empty!

50. Moral DFS is a generalization of the pre-order traversal on trees Maintain internally a stack to control the visit order for recursion function, machine maintain an implicit stack Obtain a DFS forest when finished

51. Edge Classification Once we obtain the DFS (or BFS) spanning trees (forests), the graph edges could be classified according to the trees: tree edges: edges in the trees forward edges: ancestors to descants back edges: descants to ancestors cross edges: others

52. Edge Classification Example 1 4 2 65 3 tree edges: 1->2, 2->5, 5->4, 3->6 forward edges: 1->4 back edges: 4->2, 6->6 cross edges: 3->5