Graph Algorythms

Graph Algorythms - Elementary Graph Algorithms CSE 680...

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Unformatted text preview: Elementary Graph Algorithms CSE 680 Suggested Reading : Appendix B4, Chapter 22 1 Graphs G ( V,E ) V : vertex set; E : edge set. Directed graphs, undirected graphs, weighted graphs. An edge is incident on , incident from , or incident to a vertex. ( u,v ) E iff v is adjacent to u . Degree, in-degree, out-degree of a vertex. A path from a vertex v to a vertex v is a sequence of vertices, ( v ,v 1 ,...,v k ), such that v = v , v = v k and ( v i- 1 ,v i ) E for i = 1 , 2 ,...,k . Length of path: the number of edges in the path. Simple path: all vertices in the path are distinct. Cycle, simple cycle Acyclic graph Graph representations Adjacency matrix Adjacency lists 1 2 Basic Depth-First Search procedure Search ( G = ( V,E )) // Assume V = { 1 , 2 ,...,n } // // global array visited [1 ..n ] // visited [1 ..n ] 0; for i 1 to n if visited [ i ] = 0 then call dfs ( i ) procedure dfs ( v ) visited [ v ] 1; for each node w such that ( v,w ) E do if visited [ w ] = 0 then call dfs ( w ) How to implement the for-loop if an adjacency matrix A is used to represent the graph? Time complexity Using adjacency matrix: O ( n 2 ) Using adjacency lists: O ( | V | + | E | ) 2 3 Connectivity An undirected graph is connected if every pair of vertices are connected by a path. A connected component is a subgraph which is connected and is not contained in any other connected subgraph. A connected component is usually identified by the vertices in that component. Problem : Given an undirected graph, identify all its connected components. procedure Connected Components ( G = ( V,E )) // Assume V = { 1 , 2 ,...,n } // // global array component [1 ..n ] // component [1 ..n ] cn for i 1 to n if component [ i ] = 0 then cn cn + 1 call dfs ( i,cn ) procedure dfs ( v,cn ) component...
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Graph Algorythms - Elementary Graph Algorithms CSE 680...

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