Graduate Algorithms CS F-15 Graphs, BFS, & DFS

Size: px

Start display at page:

Download "Graduate Algorithms CS F-15 Graphs, BFS, & DFS"

Bartholomew Fleming
6 years ago
Views:

1 Grauate Algorithms CS67-206F-5 Graphs, BFS, & DFS Davi Galles Department o Computer Science University o San Francisco

2 5-0: Graphs A graph consists o: A set o noes or vertices (terms are interchangeable) A set o eges or arcs (terms are interchangeable) Eges in graph can be either irecte or unirecte

3 5-: Graphs & Eges Eges can be labele or unlabele Ege labels are typically the cost associate with an ege e.g., Noes are cities, eges are roas between cities, ege label is the length o roa

4 5-2: Graph Representations Ajacency Matrix Represent a graph with a two-imensional array G G[i][j] = i there is an ege rom noe i to noe j G[i][j] = 0 i there is no ege rom noe i to noe j I graph is unirecte, matrix is symmetric Can represent eges labele with a cost as well: G[i][j] = cost o link between i an j I there is no irect link, G[i][j] =

5 5-: Ajacency Matrix Examples:

6 5-4: Ajacency Matrix Examples:

7 5-5: Ajacency Matrix Examples:

8 5-6: Ajacency Matrix Examples:

9 5-7: Graph Representations Ajacency List Maintain a linke-list o the neighbors o every vertex. n vertices Array o n lists, one per vertex Each list i contains a list o all vertices ajacent to i.

10 5-8: Ajacency List Examples: 0 2 2

11 5-9: Ajacency List Examples: Note lists are not always sorte

12 5-0: Sparse vs. Dense Sparse graph relatively ew eges Dense graph lots o eges Complete graph contains all possible eges These terms are uzzy. Sparse in one context may or may not be sparse in a ierent context

13 5-: Breath-First Search Metho or searching a graph Speciy a source noe in the grap Fin all noes reachable rom that noe First in all noes unit away Next in all noes 2 units away... etc

14 5-2: Breath-First Search Auxiliary Data Structures color or each vertex white, black, grey Use to make sure we on t visit vertices more than once Parent o each vertex (Path to source noe) Distance o each vertex rom source

15 5-: Breath-First Search BFS(G, s) or each vertex u in V[G] o color[u] WHITE [u] π[u] nil color[s] GRAY [s] 0 Q {s} while Q not empty o u Q.equeue or each v aj. to u i color[v] = WHITE color[v] GRAY [v] [u]+ π[v] u Q.enqueue(v) color[u] BLACK

16 5-4: Breath-First Search a b c e g h : b

17 5-5: Breath-First Search BFS computes the shortest path rom the start vertex to every other vertex We can run BFS on a irecte or unirecte tree Deines a BFS Tree Parent pointers p[v] BFS Tree is irecte

18 5-6: Breath-First Search a b c e g h :

19 5-7: Breath-First Search BFS Running time: V vertices E eges

20 5-8: Breath-First Search BFS Running time: V vertices E eges Running time Θ(V +E) In terms o just V, O(V 2 ) (why?)

21 5-9: Depth-First Search DFS(G) or each vertex v in G o color[v] WHITE π[v] = nil time 0 or each vertex v in G o i color[v] = WHITE DFS-VISIT(v)

22 5-20: Depth-First Search DFS-VISIT(v, G) color[v] GRAY time time + [v] time or each u ajacent to v in G o i color[u] = WHITE then π[u] v DFS-VISIT(u,G) color[v] BLACK time time + [v] time

23 5-2: Depth-First Search Do DFS, show iscover/inish times & Depth First Forst)

24 5-22: Depth-First Search DFS creates a Depth First Forest We can use DFS to classiy eges: Tree eges eges in the Depth First Forest Back Eges ege (u,v) that connects u to ancestor v in DFF Forwar eges non-tree ege (u,v) that connects u to escenent v in DFF Cross Eges Everything Else

25 5-2: Depth-First Search Labeling eges How coul we label eges (tree/back/orwar/cross) while we are oing DFS?

26 5-24: Depth-First Search Labeling eges How coul we label eges (tree/back/orwar/cross) while we are oing DFS? When examining ege (u,v), i v is: WHITE tree ege GRAY back ege BLACK orwar ege or cross ege

27 5-25: Depth-First Search Labeling eges Can we have cross eges in a DFS o an unirecte graph? Can we have orwar eges in a DFS o an unirecte graph?

28 5-26: Depth-First Search a b c e g Do DFS, show iscover/inish times & Depth First orest)

29 5-27: Depth-First Search a b c e g h i

30 5-28: Topological Sort Directe Acyclic Graph, Vertices v...v n Create an orering o the vertices I there a path rom v i to v j, then v i appears beore v j in the orering Example: Prerequisite chains

31 5-29: Topological Sort How coul we use DFS to o a Topological Sort? (Hint Use iscover an/or inish times)

32 5-0: 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?)

33 5-: 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

34 5-2: Topological Sort Secon metho or oing topological sort: Which noe(s) coul be irst in the topological orering? Noe(s) with no incient (incoming) eges

35 5-: Topological Sort Pick a noe v k with no incient eges A v k to the orering Remove v k an all eges rom v k rom the graph Repeat until all noes are picke.

36 5-4: Topological Sort How can we in a noe with no incient eges? Count the incient eges o all noes

37 5-5: Topological Sort or (i=0; i < NumberOVertices; i++) NumIncient[i] = 0; or(i=0; i < NumberOVertices; i++) each noe k ajacent to i NumIncient[k]++

38 5-6: Topological Sort or(i=0; i < NumberOVertices; i++) NumIncient[i] = 0; or(i=0; i < NumberOVertices; i++) or(tmp=g[i]; tmp!= null; tmp=tmp.next()) NumIncient[tmp.neighbor()]++

39 5-7: Topological Sort Create NumIncient array Repeat Search through NumIncient to in a vertex v with NumIncient[v] == 0 A v to the orering Decrement NumIncient o all neighbors o v Set NumIncient[v] = - Until all vertices have been picke

40 5-8: Topological Sort In a graph with V vertices an E eges, how long oes this version o topological sort take?

41 5-9: Topological Sort In a graph with V vertices an E eges, how long oes this version o topological sort take? Θ(V 2 +E) = Θ(V 2 ) Since E O(V 2 )

42 5-40: Topological Sort Where are we spening extra time

43 5-4: Topological Sort Where are we spening extra time Searching through NumIncient each time looking or a vertex with no incient eges Keep aroun a set o all noes with no incient eges Remove an element v rom this set, an a it to the orering Decrement NumIncient or all neighbors o v I NumIncient[k] is ecremente to 0, a k to the set. How o we implement the set o noes with no incient eges?

44 5-42: Topological Sort Where are we spening extra time Searching through NumIncient each time looking or a vertex with no incient eges Keep aroun a set o all noes with no incient eges Remove an element v rom this set, an a it to the orering Decrement NumIncient or all neighbors o v I NumIncient[k] is ecremente to 0, a k to the set. How o we implement the set o noes with no incient eges? Use a stack

45 5-4: Topological Sort Examples!! Graph Ajacency List NumIncient Stack

46 5-44: More DFS Applications Depth First Search can be use to calculate the connecte components o a irecte graph First, some einitions an examples:

47 5-45: Strongly Connecte Graph Directe Path rom every noe to every other noe Strongly Connecte

48 5-46: Strongly Connecte Graph Directe Path rom every noe to every other noe Strongly Connecte

49 5-47: Connecte Components Subgraph (subset o the vertices) that is strongly connecte

50 5-48: Connecte Components Subgraph (subset o the vertices) that is strongly connecte

51 5-49: Connecte Components Subgraph (subset o the vertices) that is strongly connecte

52 5-50: Connecte Components Subgraph (subset o the vertices) that is strongly connecte

53 5-5: 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

54 5-52: Connecte Components Calculating Connecte Components Two vertices v an v 2 are in the same connecte component i an only i: Directe path rom v to v 2 Directe path rom v 2 to v To in connecte components in irecte paths Use DFS: [v] an [v]

55 5-5: DFS Revisite Recall that we calculate 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)

56 5-54: DFS Example

57 5-55: DFS Example

58 5-56: DFS Example

59 5-57: DFS Example

60 5-58: DFS Example

61 5-59: DFS Example

62 5-60: DFS Example

63 5-6: DFS Example

64 5-62: DFS Example

65 5-6: DFS Example

66 5-64: DFS Example

67 5-65: DFS Example

68 5-66: DFS Example

69 5-67: DFS Example

70 5-68: DFS Example

71 5-69: DFS Example

72 5-70: DFS Example

73 5-7: DFS Example

74 5-72: DFS Example

75 5-7: DFS Example

76 5-74: DFS Example

77 5-75: DFS Example

78 5-76: DFS Example

79 5-77: DFS Example

80 5-78: DFS Example

81 5-79: DFS Example

82 5-80: DFS Example

83 5-8: DFS Example

84 5-82: DFS Example

85 5-8: DFS Example

86 5-84: DFS Example

87 5-85: DFS Example

88 5-86: DFS Example

89 5-87: DFS Example

90 5-88: DFS Example

91 5-89: Using [] & [] Given two vertices v an v 2, what o we know i [v 2 ] < [v ]?

92 5-90: Using [] & [] Given two vertices v an v 2, what o we know i [v 2 ] < [v ]? Either: Path rom v to v 2 Start rom v Eventually visit v 2 Finish v 2 Finish v

93 5-9: Using [] & [] Given two vertices v an v 2, what o we know i [v 2 ] < [v ]? Either: Path rom v to v 2 No path rom v 2 to v Start rom v 2 Eventually inish v 2 Start rom v Eventually inish v

94 5-92: Using [] & [] I [v 2 ] < [v ]: Either a path rom v to v 2, or no path rom v 2 to v I there is a path rom v 2 to v, then there must be a path rom v to v 2 [v 2 ] < [v ] an a path rom v 2 to v v an v 2 are in the same connecte component

95 5-9: Calculating paths Path rom v 2 to v in G i an only i there is a path rom v to v 2 in G T G T is the transpose o G G with all eges reverse I ater DFS, [v 2 ] < [v ] Run secon DFS on G T, starting rom v, an v an v 2 are in the same DFS spanning tree v an v 2 must be in the same connecte component

96 5-94: 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

97 5-95: Connecte Components Eg

98 5-96: Connecte Components Eg

99 5-97: Connecte Components Eg

100 5-98: Connecte Components Eg

101 5-99: Connecte Components Eg

102 5-00: Connecte Components Eg

103 5-0: Connecte Components Eg

104 5-02: Connecte Components Eg

Data Structures and Algorithms

Data Structures and Algorithms Data Structures an Algorithms CS-0S-9 Connecte Components Davi Galles Department o Computer Science University o San Francisco 9-0: Strongly Connecte Graph Directe Path rom every noe to every other noe