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

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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

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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.)
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 .

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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.
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}

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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.)
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 ) , ( ) , ( ) ( .

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

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