Graph Operations And RepresentationSample Graph ProblemsPath FindingAnother Path Between 1 and 8Example Of No PathConnected GraphExample Of Not ConnectedConnected Graph ExampleConnected ComponentsConnected ComponentNot A ComponentCommunication NetworkCommunication Network ProblemsCycles And ConnectednessSlide 15TreeSpanning TreeMinimum Cost Spanning TreeA Spanning TreeSlide 20A Wireless Broadcast TreeGraph RepresentationAdjacency MatrixAdjacency Matrix PropertiesAdjacency Matrix (Digraph)Slide 26Adjacency ListsLinked Adjacency ListsArray Adjacency ListsWeighted GraphsNumber Of Java Classes NeededAbstract Class GraphAbstract Methods Of GraphGraph Operations And RepresentationSample Graph Problems•Path problems.•Connectedness problems.•Spanning tree problems.Path FindingPath between 1 and 8.238101459116748667524453Path length is 20.Another Path Between 1 and 8238101459116748667524453Path length is 28.Example Of No PathNo path between 2 and 9.2381014591167Connected Graph•Undirected graph.•There is a path between every pair of vertices.Example Of Not Connected2381014591167Connected Graph Example2381014591167Connected Components2381014591167Connected Component•A maximal subgraph that is connected.Cannot add vertices and edges from original graph and retain connectedness.•A connected graph has exactly 1 component.Not A Component2381014591167Communication NetworkEach edge is a link that can be constructed (i.e., a feasible link).2381014591167Communication Network Problems•Is the network connected?Can we communicate between every pair of cities?•Find the components.•Want to construct smallest number of feasible links so that resulting network is connected.Cycles And Connectedness2381014591167Removal of an edge that is on a cycle does not affect connectedness.Cycles And Connectedness2381014591167Connected subgraph with all vertices and minimum number of edges has no cycles.Tree•Connected graph that has no cycles.•n vertex connected graph with n-1 edges.Spanning Tree•Subgraph that includes all vertices of the original graph.•Subgraph is a tree.If original graph has n vertices, the spanning tree has n vertices and n-1 edges.Minimum Cost Spanning Tree•Tree cost is sum of edge weights/costs.23810145911674866752445382A Spanning TreeSpanning tree cost = 51.23810145911674866752445382Minimum Cost Spanning TreeSpanning tree cost = 41.23810145911674866752445382A Wireless Broadcast TreeSource = 1, weights = needed power.Cost = 4 + 8 + 5 + 6 + 7 + 8 + 3 = 41.23810145911674866752445382Graph Representation•Adjacency Matrix•Adjacency ListsLinked Adjacency ListsArray Adjacency ListsAdjacency Matrix•0/1 n x n matrix, where n = # of vertices•A(i,j) = 1 iff (i,j) is an edge231451 2 3 4 5123450 1 0 1 01 0 0 0 10 0 0 0 11 0 0 0 10 1 1 1 0Adjacency Matrix Properties231451 2 3 4 5123450 1 0 1 01 0 0 0 10 0 0 0 11 0 0 0 10 1 1 1 0•Diagonal entries are zero.•Adjacency matrix of an undirected graph is symmetric. A(i,j) = A(j,i) for all i and j.Adjacency Matrix (Digraph)231451 2 3 4 5123450 0 0 1 01 0 0 0 10 0 0 0 00 0 0 0 10 1 1 0 0•Diagonal entries are zero.•Adjacency matrix of a digraph need not be symmetric.Adjacency Matrix•n2 bits of space•For an undirected graph, may store only lower or upper triangle (exclude diagonal). (n-1)n/2 bits•O(n) time to find vertex degree and/or vertices adjacent to a given vertex.Adjacency Lists•Adjacency list for vertex i is a linear list of vertices adjacent from vertex i.•An array of n adjacency lists.23145aList[1] = (2,4)aList[2] = (1,5)aList[3] = (5)aList[4] = (5,1)aList[5] = (2,4,3)Linked Adjacency Lists•Each adjacency list is a chain.23145aList[1]aList[5][2][3][4]2 41 555 12 4 3Array Length = n# of chain nodes = 2e (undirected graph)# of chain nodes = e (digraph)Array Adjacency Lists•Each adjacency list is an array list.23145aList[1]aList[5][2][3][4]2 41 555 12 4 3Array Length = n# of list elements = 2e (undirected graph)# of list elements = e (digraph)Weighted Graphs•Cost adjacency matrix.C(i,j) = cost of edge (i,j)•Adjacency lists => each list element is a pair (adjacent vertex, edge weight)Number Of Java Classes Needed•Graph representationsAdjacency MatrixAdjacency ListsLinked Adjacency ListsArray Adjacency Lists3 representations•Graph typesDirected and undirected.Weighted and unweighted.2 x 2 = 4 graph types•3 x 4 = 12 Java classesAbstract Class Graphpackage dataStructures;import java.util.*;public abstract class Graph{ // ADT methods come here // create an iterator for vertex i public abstract Iterator iterator(int i); // implementation independent methods come here}Abstract Methods Of Graph // ADT methods public abstract int vertices(); public abstract int edges(); public abstract boolean existsEdge(int i, int j); public abstract void putEdge(Object theEdge); public abstract void removeEdge(int i, int j); public abstract int degree(int i); public abstract int inDegree(int i); public abstract int outDegree(int