CS 5633 -- Spring 2010 Graphs Carola Wenk Slides courtesy of Charles Leiserson with changes and additions by Carola Wenk 3/25/10 CS 5633 Analysis of Algorithms 1
Graphs (review) Definition. A directed graph ( digraph ) G = ( V , E ) is an ordered pair consisting of • a set V of vertices (singular: vertex ), • a set E ⊆ V × V of edges . In an undirected graph G = ( V , E ), the edge set E consists of unordered pairs of vertices. In either case, we have | E | = O (| V| 2 ). Moreover, if G is connected, then | E | ≥ | V | – 1. (Review CLRS, Appendix B.4 and B.5.) 3/25/10 CS 5633 Analysis of Algorithms 2
Adjacency-matrix representation The adjacency matrix of a graph G = ( V , E ), where V = {1, 2, …, n }, is the matrix A [1 . . n , 1 . . n ] given by 1 if ( i , j ) ∈ E, A [ i , j ] = 0 if ( i , j ) ∉ E. A 1 2 3 4 2 1 Θ (| V| 2 ) storage 2 1 1 0 1 1 0 ⇒ dense 2 0 0 1 0 representation. 3 4 3 0 0 0 0 3 4 4 0 0 1 0 3/25/10 CS 5633 Analysis of Algorithms 3
Adjacency-list representation An adjacency list of a vertex v ∈ V is the list Adj [ v ] of vertices adjacent to v . Adj [1] = {2, 3} 2 1 2 1 Adj [2] = {3} Adj [3] = {} 3 4 3 4 Adj [4] = {3} For undirected graphs, | Adj [ v ] | = degree ( v ). For digraphs, | Adj [ v ] | = out-degree ( v ). 3/25/10 CS 5633 Analysis of Algorithms 4
Adjacency-list representation Handshaking Lemma: Every edge is counted twice • For undirected graphs: ∑ v ∈ V degree(v) = 2 | E | • For digraphs: ∑ v ∈ V in-degree(v) + ∑ v ∈ V out-degree(v) = 2 | E | ⇒ adjacency lists use Θ (| V| + |E| ) storage ⇒ a sparse representation ⇒ We usually use this representation, unless stated otherwise 3/25/10 CS 5633 Analysis of Algorithms 5
Graph Traversal Let G =( V , E ) be a (directed or undirected) graph, given in adjacency list representation. | V | = n , | E | = m A graph traversal visits every vertex: • Breadth-first search (BFS) • Depth-first search (DFS) 3/25/10 CS 5633 Analysis of Algorithms 6
Breadth-First Search (BFS) BFS( G= ( V,E )) Mark all vertices in G as “unvisited” // time=0 Initialize empty queue Q for each vertex v ∈ V do if v is unvisited visit v // time++ BFS_iter( G ) Q .enqueue( v ) while Q is non-empty do BFS_iter( G ) v = Q .dequeue() for each w adjacent to v do if w is unvisited visit w // time++ Add edge ( v , w ) to T Q .enqueue( w ) 3/25/10 CS 5633 Analysis of Algorithms 7
Example of breadth-first search a f h a f h d d b g b g e i e i c c Q : 3/25/10 CS 5633 Analysis of Algorithms 8
Example of breadth-first search a f h 0 a f h d d b g b g e i e i c c 0 Q: a 3/25/10 CS 5633 Analysis of Algorithms 9
Example of breadth-first search a f h 0 a f h 2 d d b g 1 b g e i e i c c 1 2 Q: a b d 3/25/10 CS 5633 Analysis of Algorithms 10
Example of breadth-first search a f h 0 a f h 2 d d b g 1 b g e i e i c 4 3 c 2 3 4 Q: a b d c e 3/25/10 CS 5633 Analysis of Algorithms 11
Example of breadth-first search a f h 0 a f h 2 d d b g 1 b g e i e i c 4 3 c 3 4 Q: a b d c e 3/25/10 CS 5633 Analysis of Algorithms 12
Example of breadth-first search a f h 0 a f h 2 d d b g 1 b g e i e i c 4 3 c 4 Q: a b d c e 3/25/10 CS 5633 Analysis of Algorithms 13
Example of breadth-first search a f h 0 a f h 2 d d 5 b g 1 b g e i e i c 4 6 3 c 5 6 Q: a b d c e g i 3/25/10 CS 5633 Analysis of Algorithms 14
Example of breadth-first search 7 a f h 0 a f h 2 d d 5 b g 1 b g e i e i c 4 6 3 c 6 7 Q: a b d c e g i f 3/25/10 CS 5633 Analysis of Algorithms 15
Example of breadth-first search 7 8 a f h 0 a f h 2 d a d 5 a b g 1 b g e i e i c 4 6 3 c 7 8 Q: a b d c e g i f h 3/25/10 CS 5633 Analysis of Algorithms 16
Example of breadth-first search 7 8 a f h 0 a f h 2 d a d 5 a b g 1 b g e i e i c 4 6 3 c 8 Q: a b d c e g i f h 3/25/10 CS 5633 Analysis of Algorithms 17
Example of breadth-first search 7 8 a f h 0 a f h 2 d a d 5 a b g 1 b g e i e i c 4 6 3 c Q: a b d c e g i f h 3/25/10 CS 5633 Analysis of Algorithms 18
Example of breadth-first search 7 8 a f h 0 a f h 2 d a d 5 a b g 1 b g e i e i c 4 6 3 c Q: a b d c e g i f h 3/25/10 CS 5633 Analysis of Algorithms 19
Breadth-First Search (BFS) BFS( G= ( V,E )) Mark all vertices in G as “unvisited” // time=0 O( n ) Initialize empty queue Q O(1) for each vertex v ∈ V do if v is unvisited O( n ) visit v // time++ BFS_iter( G ) without Q .enqueue( v ) while Q is non-empty do BFS_iter BFS_iter( G ) v = Q .dequeue() for each w adjacent to v do if w is unvisited O( m ) visit w // time++ O( deg ( v )) Add edge ( v , w ) to T Q .enqueue( w ) 3/25/10 CS 5633 Analysis of Algorithms 20
BFS runtime • Each vertex is marked as unvisited in the beginning ⇒ O( n ) time • Each vertex is marked at most once, enqueued at most once, and therefore dequeued at most once • The time to process a vertex is proportional to the size of its adjacency list (its degree), since the graph is given in adjacency list representation ⇒ O( m ) time • Total runtime is O( n + m ) = O(|V| + |E|) 3/25/10 CS 5633 Analysis of Algorithms 21
Depth-First Search (DFS) DFS( G= ( V,E )) Mark all vertices in G as “unvisited” // time=0 for each vertex v ∈ V do if v is unvisited DFS_rec( G , v ) DFS_rec( G, v ) visit v // d [ v ]=++time for each w adjacent to v do if w is unvisited Add edge ( v , w ) to tree T DFS_rec( G , w ) // f [ v ]=++time 3/25/10 CS 5633 Analysis of Algorithms 22
Example of depth-first search d / f 0/- a a f h a a f h d d b g b g e i e i c c Store edges in π : a b c d e f g h i predecessor array a - 3/25/10 CS 5633 Analysis of Algorithms 23
Example of depth-first search d / f 0/- a f h a f h d d 1/- b g b g e i e i c c Store edges in π : a b c d e f g h i predecessor array a - b 3/25/10 CS 5633 Analysis of Algorithms 24
Example of depth-first search d / f 0/- a f h a f h d d 1/- b g b g e i e i 2/3 2/- c c Store edges in π : a b c d e f g h i predecessor array a - b 3/25/10 CS 5633 Analysis of Algorithms 25
Example of depth-first search d / f 0/- a f h a f h d d 1/- b g b g e i e i 2/3 c c Store edges in π : a b c d e f g h i predecessor array a - b b 3/25/10 CS 5633 Analysis of Algorithms 26
Example of depth-first search d / f 0/- a f h a f h d d 1/- b g b g e i e i 4/- 2/3 c c Store edges in π : a b c d e f g h i predecessor array a e - b b 3/25/10 CS 5633 Analysis of Algorithms 27
Example of depth-first search d / f 0/- a f h a f h d d 5/- 1/- b g b g e i e i 4/- 2/3 c c Store edges in π : a b c d e f g h i predecessor array a e g - b b 3/25/10 CS 5633 Analysis of Algorithms 28
Example of depth-first search d / f 0/- a f h a f h d d 5/- 1/- b g b g e i 6/- e i 4/- 2/3 c c Store edges in π : a b c d e f g h i predecessor array a e i g - b b 3/25/10 CS 5633 Analysis of Algorithms 29
Example of depth-first search d / f 0/- a f h 7/- 7/8 a f h d d 5/- 1/- b g b g e i 6/- e i 4/- 2/3 c c Store edges in π : a b c d e f g h i predecessor array a e i g - b b 3/25/10 CS 5633 Analysis of Algorithms 30
Example of depth-first search d / f 0/- a f h 7/8 a f h d d 5/- 1/- b g b g e i 6/- 6/9 e i 4/- 2/3 c c Store edges in π : a b c d e f g h i predecessor array a e i g - b b 3/25/10 CS 5633 Analysis of Algorithms 31
Example of depth-first search d / f 0/- a f h 7/8 a f h d d 5/- 1/- b g b g e i 6/9 e i 4/- 2/3 c c Store edges in π : a b c d e f g h i predecessor array g a e i g - b b 3/25/10 CS 5633 Analysis of Algorithms 32
Example of depth-first search d / f 10/- 0/- a f h 7/8 a f h d d 5/- 1/- b g b g e i 6/9 e i 4/- 2/3 c c Store edges in π : a b c d e f g h i predecessor array g a e i g - b f b 3/25/10 CS 5633 Analysis of Algorithms 33
Example of depth-first search d / f 10/- 0/- a f h 7/8 a f h 11/- 11/12 d d 5/- 1/- b g b g e i 6/9 e i 4/- 2/3 c c Store edges in π : a b c d e f g h i predecessor array g a e i g - b f b 3/25/10 CS 5633 Analysis of Algorithms 34
Example of depth-first search d / f 10/- 10/13 0/- a f h 7/8 a f h 11/12 d d 5/- 1/- b g b g e i 6/9 e i 4/- 2/3 c c Store edges in π : a b c d e f g h i predecessor array g a e i g - b f b 3/25/10 CS 5633 Analysis of Algorithms 35
Example of depth-first search d / f 10/13 0/- a f h 7/8 a f h 11/12 d d 5/- 5/14 1/- b g b g e i 6/9 e i 4/- 2/3 c c Store edges in π : a b c d e f g h i predecessor array g a e i g - b f b 3/25/10 CS 5633 Analysis of Algorithms 36
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