Data Structures and Algorithms

Transcription

1 Data Structures an Algorithms CS-0S-9 Connecte Components Davi Galles Department o Computer Science University o San Francisco

2 9-0: Strongly Connecte Graph Directe Path rom every noe to every other noe Strongly Connecte

3 9-: Strongly Connecte Graph Directe Path rom every noe to every other noe Strongly Connecte

4 9-: Connecte Components Subgraph (subset o the vertices) that is strongly connecte.

5 9-: Connecte Components Subgraph (subset o the vertices) that is strongly connecte.

6 9-: Connecte Components Subgraph (subset o the vertices) that is strongly connecte.

7 9-: Connecte Components Subgraph (subset o the vertices) that is strongly connecte.

8 9-6: Connecte Components Connecte components o the graph are the largest possible strongly connecte subgraphs I we put each vertex in its own component each component woul be (trivially) strongly connecte Those woul not be the connecte components o the graph unless there were no larger connecte subgraphs

9 9-: Connecte Components Calculating Connecte Components Two vertices v an v are in the same connecte component i an only i: Directe path rom v to v Directe path rom v to v To in connecte components in irecte paths Use DFS

10 9-8: DFS Revisite We can keep track o the orer in which we visit the elements in a Depth-First Search For any vertex v in a DFS: [v] = Discovery time when the vertex is irst visite [v] = Finishing time when we have inishe with a vertex (an all o its chilren)

11 9-9: DFS Revisite class Ege { public int neighbor; public int next; } voi DFS(Ege G[], int vertex, boolean Visite[], int [], int []) { Ege tmp; Visite[vertex] = true; [vertex] = time++; or (tmp = G[vertex]; tmp!= null; tmp = tmp.next) { i (!Visite[tmp.neighbor]) DFS(G, tmp.neighbor, Visite); } [vertex] = time++; }

12 9-0: DFS Revisite To visit every noe in the graph: TraverseDFS(Ege G[]) { int i; boolean Visite = new boolean[g.length]; int = new int[g.length]; int v = new int[g.length]; time = ; or (i=0; i<g.length; i++) Visite[i] = alse; or (i=0; i<g.length; i++) i (!Visite[i]) DFS(G, i, Visite,, ); }

13 9-: DFS Example

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18 9-6: DFS Example

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20 9-8: DFS Example 6

21 9-9: DFS Example 6

22 9-0: DFS Example 8 6

23 9-: DFS Example 8 9 6

24 9-: DFS Example

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28 9-6: DFS Example

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30 9-8: DFS Example

31 9-9: DFS Example

32 9-0: DFS Example

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37 9-: DFS Example 6

38 9-6: DFS Example 6

39 9-: DFS Example 8 6

40 9-8: DFS Example 9 8 6

41 9-9: DFS Example

42 9-0: DFS Example

43 9-: DFS Example

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47 9-: DFS Example

48 9-6: Using [] & [] Given two vertices v an v, what o we know i [v ] < [v ]?

49 9-: Using [] & [] Given two vertices v an v, what o we know i [v ] < [v ]? Either: Path rom v to v Start rom v Eventually visit v Finish v Finish v

50 9-8: Using [] & [] Given two vertices v an v, what o we know i [v ] < [v ]? Either: Path rom v to v No path rom v to v Start rom v Eventually inish v Start rom v Eventually inish v

51 9-9: Using [] & [] I [v ] < [v ]: Either a path rom v to v, or no path rom v to v I there is a path rom v to v, then there must be a path rom v to v [v ] < [v ] an a path rom v to v v an v are in the same connecte component

52 9-0: Calculating paths Path rom v to v in G i an only i there is a path rom v to v in G T G T is the transpose o G G with all eges reverse I ater DFS, [v ] < [v ] Run secon DFS on G T, starting rom v, an v an v are in the same DFS spanning tree v an v must be in the same connecte component

53 9-: Connecte Components Run DFS on G, calculating [] times Compute G T Run DFS on G T examining noes in inverse orer o inishing times rom irst DFS Any noes that are in the same DFS search tree in G T must be in the same connecte component

54 9-: Connecte Components Eg.

55 9-: Connecte Components Eg

56 9-: Connecte Components Eg

57 9-: Connecte Components Eg

58 9-6: Connecte Components Eg.

59 9-: Connecte Components Eg

60 9-8: Connecte Components Eg

61 9-9: Connecte Components Eg

62 9-60: Topological Sort How coul we use DFS to o a Topological Sort? (Hint Use iscover an/or inish times)

63 9-6: Topological Sort How coul we use DFS to o a Topological Sort? (Hint Use iscover an/or inish times) (What oes it mean i noe x inishe beore noe y?)

64 9-6: Topological Sort How coul we use DFS to o a Topological Sort? Do DFS, computing inishing times or each vertex As each vertex is inishe, a to ront o a linke list This list is a vali topological sort