1. Graph Search Methods Spring 2007 CSE, POSTECH

2. Graph Search Methods A vertex u is reachable from vertex v iff there is a path from v to u. A search method starts at a given vertex v and visits/labels/marks every vertex that is reachable from v. 2 3 5 4 1 8 9 6 7 11 10

3. Graph Search Methods Many graph problems solved by a search method – Finding a path from one vertex to another. – Determining whether a graph is connected – Find a spanning tree – Finding a minimum-cost path/spanning tree Commonly used search methods – Breadth-first search (BFS) – Depth-first search (DFS)

4. Breadth-First Search (BFS) Algorithm BFS algorithm is the method of starting at a vertex and identifying all vertices reachable from it Read Section 16.8.1 and the pseudo-code of BFS in Figure 16.7 Similar to the level-order traversal of a binary tree

5. Breadth-First Search (BFS) Algorithm Visit the starting vertex and put into a FIFO queue Repeatedly remove a vertex from the queue, visit its unvisited adjacent vertices, put newly visited vertices into the queue All vertices reachable from the start vertex (including the start vertex) are visited When the queue becomes empty, the search is terminated

6. Breadth-First Search Example Start search at vertex 1. 2 3 5 4 1 8 9 6 7 11 10

7. Breadth-First Search Example Mark/label start vertex and put in a FIFO queue. Remove 1 from Queue; Visit adjacent unvisited vertices; Put the visited vertices in Queue. 2 3 5 4 1 8 9 6 7 11 10 FIFO queue 1

8. Breadth-First Search Example Remove 2 from Queue; Visit adjacent unvisited vertices; Put the visited vertices in Queue. Are there any newly visited vertices? 2 3 5 4 1 8 9 6 7 11 10 FIFO queue 2 4

9. Breadth-First Search Example Remove 4 from Queue; Visit adjacent unvisited vertices; Put the visited vertices in Queue. Are there any newly visited vertices? 2 3 5 4 1 8 9 6 7 11 10 FIFO queue 4 5 3 6

10. Breadth-First Search Example Remove 5 from Queue; Visit adjacent unvisited vertices; Put the visited vertices in Queue. Are there any newly visited vertices? 2 3 5 4 1 8 9 6 7 11 10 FIFO queue 5 3 6

11. Breadth-First Search Example Remove 3 from Queue; Visit adjacent unvisited vertices; Put the visited vertices in Queue. Are there any newly visited vertices? 2 3 5 4 1 8 9 6 7 11 10 FIFO queue 3 6 9 7

12. Breadth-First Search Example Remove 6 from Queue; Visit adjacent unvisited vertices; Put the visited vertices in Queue. Are there any newly visited vertices? 2 3 5 4 1 8 9 6 7 11 10 FIFO queue 6 9 7

13. Breadth-First Search Example Remove 9 from Queue; Visit adjacent unvisited vertices; Put the visited vertices in Queue. Are there any newly visited vertices? 2 3 5 4 1 8 9 6 7 11 10 FIFO queue 9 7

14. Breadth-First Search Example Remove 7 from Queue; Visit adjacent unvisited vertices; Put in Queue. 2 3 5 4 1 8 9 6 7 11 10 FIFO queue 7 8

15. Breadth-First Search Example 2 3 5 4 1 8 9 6 7 11 10 FIFO queue 8 Remove 8 from Queue; Visit adjacent unvisited vertices; Put in Queue.

16. Breadth-First Search Example Queue is empty, thus the search terminates. 2 3 5 4 1 8 9 6 7 11 10 FIFO queue

17. Breadth-First Search Example What was the order of visited vertices?  1, 2, 4, 5, 3, 6, 9, 7, 8 What is the subgraph formed by the edges used to reach new vertices during the BFS? 2 3 5 4 1 8 9 6 7

18. Time Complexity Each visited vertex is put on the queue exactly once. When a vertex is removed from the queue, we examine its adjacent vertices. – O(n) if adjacency matrix used. – O(vertex degree) if adjacency lists used. Total time – O(n 2 ) if adjacency matrix used – O(n+e) if adjacency lists used where e is the sum of vertex degrees and therefore e is also the number of edges.

19. Path Finding Problem from vertex u to vertex v Start a breadth-first search at vertex u. Terminate if either of the following occurs – successfully when vertex v is visited or – unsuccessfully when queue becomes empty but v is not visited Time – O(n 2 ) when adjacency matrix used – O(n+e) when adjacency lists used (e is number of edges)

20. Is a Graph Connected? Start a breadth-first search at any vertex of the graph. A graph is connected iff all n vertices are visited. Time – O(n 2 ) when adjacency matrix used – O(n+e) when adjacency lists used (e is number of edges)

21. Connected Components Start a breadth-first search at any unvisited vertex of the graph. Newly visited vertices (plus edges between them) define a component. Repeat until all vertices are visited. Time – O(n 2 ) when adjacency matrix used – O(n+e) when adjacency lists used (e is number of edges)

22. Connected Components 2 3 5 4 1 8 9 6 7 11 10

23. Breath-First Spanning Tree Start a breadth-first search at any vertex of the graph. If a graph is connected, the n-1 edges used to get to unvisited vertices define a spanning tree  breadth-first spanning tree Time – O(n 2 ) when adjacency matrix used – O(n+e) when adjacency lists used (e is number of edges)

24. Breadth-First Spanning Tree Digraph ExampleBF Spanning Tree of Digraph (a)

25. Depth-First Search (DFS) Algorithm DFS algorithm is an alternative to BFS Similar to the pre-order traversal of a binary tree See Figure 16.18 for the pseudo-code of DFS

26. Visit the starting vertex v and mark it as reached Select an unreached vertex u adjacent from v If such a vertex does not exist, the search terminates, otherwise a depth-first search from u is initiated When this search is completed, we select another unreached vertex adjacent to from v If such a vertex does not exist, then the search terminates, otherwise a depth-first search starts from this vertex and so on….. Depth-First Search (DFS) Algorithm

27. Start search at vertex 1. Label vertex 1 as reached and do a depth-first search from either 2 or 4. 2 3 5 4 1 8 9 6 7 11 10 Depth-First Search Example

28. Label vertex 2 and do a depth-first search from either 3, 5 or 6. 2 3 5 4 1 8 9 6 7 11 10 Depth-First Search Example

29. Label vertex 5 and do a depth-first search from either 3, 7 or 9. 2 3 5 4 1 8 9 6 7 11 10 Depth-First Search Example

30. Label vertex 9 and do a depth-first search from either 6 or 8. 2 3 5 4 1 8 9 6 7 11 10 Depth-First Search Example

31. Label vertex 8 and return to vertex 9. From vertex 9 do a DFS(6). 2 3 5 4 1 8 9 6 7 11 10 Depth-First Search Example

32. Label vertex 6 and do a depth-first search from either 4 or 7. 2 3 5 4 1 8 9 6 7 11 10 Depth-First Search Example

33. Label vertex 4 and return to 6. From vertex 6 do a DFS(7). 2 3 5 4 1 8 9 6 7 11 10 Depth-First Search Example

34. Label vertex 7 and return to 6. Return to 9. 2 3 5 4 1 8 9 6 7 11 10 Depth-First Search Example

35. Return to 5. 2 3 5 4 1 8 9 6 7 11 10 Depth-First Search Example

36. Do a DFS(3). 2 3 5 4 1 8 9 6 7 11 10 Depth-First Search Example

37. Label 3 and return to 5. Return to 2. Return to 1. 2 3 5 4 1 8 9 6 7 11 10 Depth-First Search Example

38. DFS is done and return to invoking method. 2 3 5 4 1 8 9 6 7 11 10 Depth-First Search Example

39. What was the order of visited vertices?  1, 2, 5, 9, 8, 6, 4, 7, 3 What is the subgraph formed by the edges used to reach new vertices during the BFS? 2 3 5 4 1 8 9 6 7

40. Same complexity as BFS. Same properties with respect to path finding, connected components, and spanning trees. Edges used to reach unlabeled vertices define a depth-first spanning tree when the graph is connected. Depth-First Search Properties

41. 1 2 6 4 5 37 10 28 14 16 25 24 22 18 12 Exercise 16.41 – for graph of Figure 16.4(a) Draw a linked adjacency-list representation Starting from vertex 4, the order of visited vertices using BFS are? Show the subgraph formed in part (b) Redo parts (b) & (c) using DFS

42. (a) Draw a linked adjacency- list representation Exercise 16.41 – solution (a) & (b) (b) Starting from vertex 4, the order of visited vertices are  4,3,5,7,2,6,1

43. (c) Show the subgraph formed in part (b) Exercise 16.41 – solution (c) & (d) (d) Redo parts (b) & (c) using DFS  DFS order: 4,3,2,1,6,5,7

44. Do Exercise 16.41 using Figure 16.16(a) starting from vertex 1 Read Chapter 16 READING