lecture_13

lecture_13 - Introduction to Algorithms 6.046J/18.401J...

Info iconThis preview shows pages 1–10. Sign up to view the full content.

View Full Document Right Arrow Icon
Introduction to Algorithms 6.046J/18.401J Prof. Charles E. Leiserson L ECTURE 13 Graph algorithms Graph representation Minimum spanning trees Greedy algorithms Optimal substructure Greedy choice Prim’s greedy MST algorithm
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Introduction to Algorithms October 27, 2004 L13.2 © 2001–4 by Charles E. Leiserson 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 , which implies that lg | E | = Θ (lg V ) . (Review CLRS, Appendix B.)
Background image of page 2
Introduction to Algorithms October 27, 2004 L13.3 © 2001–4 by Charles E. Leiserson 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 A [ i , j ] = 1 if ( i , j ) E , 0 if ( i , j ) E .
Background image of page 3

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Introduction to Algorithms October 27, 2004 L13.4 © 2001–4 by Charles E. Leiserson 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 A [ i , j ] = 1 if ( i , j ) E , 0 if ( i , j ) E . 2 2 1 1 3 3 4 4 A 1234 1 2 3 4 0110 0010 0000 Θ ( V 2 ) storage dense representation.
Background image of page 4
Introduction to Algorithms October 27, 2004 L13.5 © 2001–4 by Charles E. Leiserson Adjacency-list representation An adjacency list of a vertex v V is the list Adj [ v ] of vertices adjacent to v . 2 2 1 1 3 3 4 4 Adj [1] = {2, 3} Adj [2] = {3} Adj [3] = {} Adj [4] = {3}
Background image of page 5

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Introduction to Algorithms October 27, 2004 L13.6 © 2001–4 by Charles E. Leiserson Adjacency-list representation An adjacency list of a vertex v V is the list Adj [ v ] of vertices adjacent to v . 2 2 1 1 3 3 4 4 Adj [1] = {2, 3} Adj [2] = {3} Adj [3] = {} Adj [4] = {3} For undirected graphs, | Adj [ v ] | = degree ( v ) . For digraphs, | Adj [ v ] | = out-degree ( v ) .
Background image of page 6
Introduction to Algorithms October 27, 2004 L13.7 © 2001–4 by Charles E. Leiserson Adjacency-list representation An adjacency list of a vertex v V is the list Adj [ v ] of vertices adjacent to v . 2 2 1 1 3 3 4 4 Adj [1] = {2, 3} Adj [2] = {3} Adj [3] = {} Adj [4] = {3} For undirected graphs, | Adj [ v ] | = degree ( v ) . For digraphs, | Adj [ v ] | = out-degree ( v ) . Handshaking Lemma: v V = 2 | E | for undirected graphs adjacency lists use Θ ( V + E ) storage — a sparse representation (for either type of graph).
Background image of page 7

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Introduction to Algorithms October 27, 2004 L13.8 © 2001–4 by Charles E. Leiserson Minimum spanning trees Input: A connected, undirected graph G = ( V , E ) with weight function w : E R . For simplicity, assume that all edge weights are distinct. (CLRS covers the general case.)
Background image of page 8
Introduction to Algorithms October 27, 2004 L13.9 © 2001–4 by Charles E. Leiserson Minimum spanning trees Input: A connected, undirected graph G = ( V , E ) with weight function w : E R . For simplicity, assume that all edge weights are distinct. (CLRS covers the general case.) = T v u v u w T w ) , ( ) , ( ) ( .
Background image of page 9

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Image of page 10
This is the end of the preview. Sign up to access the rest of the document.

Page1 / 52

lecture_13 - Introduction to Algorithms 6.046J/18.401J...

This preview shows document pages 1 - 10. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online