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GSU CSC 2320 - Graph

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GraphGraph AlgorithmsWhat can graphs model?What is a Graph?Directed GraphSlide 6Undirected GraphSlide 8DegreeSlide 10Slide 11Weighted GraphSlide 13Implementation of a GraphSlide 15Slide 16Adjacency listsAdjacency-matrix-representationSlide 19Slide 20Slide 21GraphDr. Bernard Chen Ph.D.University of Central ArkansasFall 2008Graph AlgorithmsGraphs and Theorems about GraphsGraph ADT and implementationGraph AlgorithmsShortest pathsminimum spanning treeWhat can graphs model?Cost of wiring electronic components together.Shortest route between two cities.Finding the shortest distance between all pairs of cities in a road atlas.What is a Graph?Informally a graph is a set of nodes joined by a set of lines or arrows.1123445 56 62 3Directed GraphA directed graph is a pair ( V, E ), where the set V is a finite set and E is a binary relation on V . The set V is called the vertex set of G and the elements are called vertices. The set E is called the edge set of G and the elements are edgesDirected Graph12345 6V = { 1, 2, 3, 4, 5, 6, 7 }| V | = 7E = { (1,2), (2,2), (2,4), (4,5), (4,1), (5,4),(6,3) }| E | = 7Self loop7Isolated nodeUndirected GraphAn undirected graph G = ( V , E ) , but unlike a digraph the edge set E consist of unordered pairs. We use the notation (a, b ) to refer to a directed edge, and { a, b } for an undirected edge.Undirected GraphADEFBCV = { A, B, C, D, E, F } |V | = 6E = { {A, B}, {A,E}, {B,E}, {C,F} } |E | = 4Some texts use (a, b) also for undirected edges. So ( a, b ) and ( b, a ) refers to the same edge.DegreeDegree of a Vertex in an undirected graph is the number of edges incident on it. In a directed graph , the out degree of a vertex is the number of edges leaving it and the in degree is the number of edges entering it.DegreeADEFBCThe degree of B is 2.Degree1245The in degree of 2 is 2 andthe out degree of 2 is 3.Weighted GraphA weighted graph is a graph for which each edge has an associated weight, usually given by a weight function w: E  R.12345 6.51.2.2.51.5.3Weighted GraphImplementation of a GraphAdjacency-list representation of a graph G = ( V, E ) consists of an array ADJ of |V | lists, one for each vertex in V. For each u  V , ADJ [ u ] points to all its adjacent vertices.Implementation of a Graph15122544332 51 5 3 42 4245132Implementation of a Graph15122544332 55 3 4455Adjacency listsAdvantage: Saves space for sparse graphs. Most graphs are sparse.“Visit” edges that start at vMust traverse linked list of vSize of linked list of v is degree(v)(degree(v))Adjacency-matrix-representationAdjacency-matrix-representation of a graph G = ( V, E) is a |V | x |V | matrix A = ( aij ) such that aij = 1 (or some Object) if (i, j ) E 0 (or null) otherwise.Adjacency-matrix-representation04132 0 1 2 3 4 012340 1 0 0 11 0 1 1 10 1 0 1 00 1 1 0 11 1 0 1 0Adjacency-matrix-representation0 1 0 0 10 0 1 1 10 0 0 1 00 0 0 0 10 0 0 0 0 0 1 2 3 4 0123404132Advantage:Saves space on pointers for dense graphs, and onsmall unweighted graphs using 1 bit per edge.Check for existence of an edge (v, u) (adjacency [i] [j]) == true?)So


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GSU CSC 2320 - Graph

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