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Unformatted text preview: UMass Lowell Computer Science 91.404 Analysis of Algorithms Prof. Karen Daniels Fall, 2011 Chapter 22: Graph Algorithms Chapter 23: Minimum Spanning Trees Chapter 24: Shortest Paths [Source: Cormen et al . textbook except where noted] Overview: Graph Algorithms Chapter 22: Elementary Graph Algorithms Introductory Concepts Graph Traversals: DepthFirst Search BreadthFirst Search Topological Sort Chapter 23: Minimum Spanning Trees Kruskal Prim Chapter 24: Shortest Paths Dijkstra Chapter 22 Graph Algorithms Introductory Concepts DepthFirst Search BreadthFirst Search Topological Sort [Source: Cormen et al. textbook except where noted] Introductory Graph Concepts G= (V,E) Vertex Degree SelfLoops Edges may have weights (see Chapters 2324) B E C F D A B E C F D A Directed Graph (digraph) Degree: in/out SelfLoops allowed Undirected Graph No SelfLoops Adjacency is symmetric This treatment follows 91.503 textbook Cormen et al. Some definitions differ slightly from other graph literature. Introductory Graph Concepts: Motivation Networks are often modeled using graphs. Introductory Graph Concepts: Representations B E C F D A B E C F D A Undirected Graph Directed Graph (digraph) 1 1 1 1 1 1 1 1 1 A B C D E F A B C D E F 1 1 1 1 1 1 1 1 1 1 1 1 1 1 A B C D E F A B C D E F A BC B ACEF C AB D E E BDF F BE A BC B CEF C D D E BD F E Adjacency Matrix Adjacency List Adjacency Matrix Adjacency List This treatment follows textbook Cormen et al. Some definitions differ slightly from other graph literature. Introductory Graph Concepts: Paths, Cycles Path: length: number of edges simple: all vertices distinct Cycle: Directed Graph: <v ,v 1 ,...,v k > forms cycle if v =v k and k>=1 simple cycle: v 1 ,v 2 ..,v k also distinct selfloop is cycle of length 1 Undirected Graph: <v ,v 1 ,...,v k > forms (simple) cycle if v =v k and k>=3 simple cycle: v 1 ,v 2 ..,v k also distinct B E C F D A path <A,B,F> B E C F D A simple cycle <E,B,F,E> This treatment follows 91.503 textbook Cormen et al. Some definitions differ slightly from other graph literature. B E C F D A simple cycle <A,B,C,A>= <B,C,A,B> Introductory Graph Concepts: Connectivity Undirected Graph: connected every pair of vertices is connected by a path one connected component connected components: equivalence classes under “is reachable from” relation Directed Graph: strongly connected every pair of vertices is reachable from each other one strongly connected component strongly connected components: equivalence classes under “mutually reachable” relation maximal sized vertex subset inducing strongly connected subgraph B E C F D A B E C F D A connected 2 connected components not strongly connected strongly connected component B E C F D A B E C F D A This treatment follows 91.503 textbook Cormen et al. Some definitions differ slightly from other graph literature. DepthFirst Search (DFS)...
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 Fall '09
 DR.KARENDANIELS
 Algorithms, Graph Theory, Adjacency list

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