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# lec9 - CS575 Parallel Processing Lecture nine Graph...

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CS575 lecture 9 2 Definitions Graph G = (V, E) V : set of nodes or vertices E : set of edges (pairs of nodes) In an undirected graph, edges are unordered pairs of nodes In a directed graph edges are ordered pairs of nodes Path : sequence of nodes (v 0 ..v n ) s.t. 2200 i: (v i ,v i+1 ) is an edge Path length: number of edges in the path Simple path: all nodes distinct Cycle: path with first and last node equal Acyclic graph: graph without cycles
CS575 lecture 9 3 Definitions cont’ Adjacency Two nodes are adjacent if there is a path of length 1 between them Complete graph all nodes in the graph are adjacent Undirected graph is connected if for all nodes v i and v j there is a path from v i to v j Sub-graph G’(V’, E’) is a sub-graph of G(V,E) if V’ V and E’ E Sub-graph of G induced by V’ has all the edges (u,v) E such that u V’, v V’ Weighted graph edges have a weight (cost, length,. .) associated with them

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CS575 lecture 9 4 Representations Nodes: structures with pointers to other nodes Good representation when nodes have limited degree Adjacency matrix A ij = 1 if (v i ,v j ) is an edge, 0 otherwise Alternative: A ij = weight(v i ,v j ) or if no edge, A ii = 0 Good representation for dense graphs (many edges) Adjacency lists n lists (one for each node) Lists contain (representations of) the adjacent nodes Good representation for sparse graphs (few edges)
CS575 lecture 9 5 Minimal Spanning Tree (MST) Spanning tree of an undirected graph G A tree that is a sub-graph of G containing ALL vertices Minimal spanning tree of a weighted graph G Spanning tree with minimal total weight G must be a connected graph Applications Lowest cost set of roads connecting a set of towns Shortest cable connecting a set of computers

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lec9 - CS575 Parallel Processing Lecture nine Graph...

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