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