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Graph Concepts

Graph Data Structure

Directed/undirected Graph

Graph Relationship

Graph implementation

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- 1. Lecture 8 Graph Data Structure (CS221) Abdisalam Issa-Salwe Taibah University College of Computer Science & Engineering Computer Science DepartmentOutline Graph Data Type Graph Concepts Graph Data Structure Directed/undirected Graph Graph Relationship Graph implementation 2 1
- 2. Learning Outcomes Describe a Graph ADT Understand how to implement a Graph ADT Describe algorithms to traverse a graph 3The Graph ADT A graph is a collection of nodes connected by edges If an edge e connects vertex u and v, we denote it as (u; v).if u = v, e is called a loop 4 2
- 3. Graph Concepts A tree is an example of a graph In a graph a node may be it’s own ancestor In general a graph has no root node Formally, a graph is an ordered pair of sets (N,E) Both nodes and edges may have data values associated 5Graph Data Structure (cont…) Graph is a data structure that consists of a set of nodes and a set of edges that relate the nodes to one another It is made up of a set of nodes called vertices and a set of lines called edges (or arcs) that connect the nodes. If the edges in the graph have no direction it is called an undirected graph. A graph whose edges are directed from one vertex to another is called a directed graph, or digraph 6 3
- 4. Graphs Data Structure (cont…)More definition:• Like a tree, is a NON-Linear structure consisting of nodes and links between the nodes Types: Undirected graph Directed graphGraphs are often used to solve problems 7Undirected Graph • An undirected graph is a set of nodes and a set of links between the nodes. • Each NODE is called a VERTEX V0 V1 • Each LINK is called an EDGE E0 • The order of the two connected vertices is unimportant ex: whether “V0 connects V1” or “V1 connects V0” means the same thing 8 4
- 5. Undirected Graph examples 5 vertices 6 edges V0 E0 E1 V2 E5 V1 E4 E2 V4 V3 E3 9Undirected Graph examples 5 vertices 6 edges V3 E4 V2 E2 E1 E0 V0 V1 E3 V4 E5 In drawing, the placement of the vertices is UNIMPORTANT same as previous graph 10 5
- 6. Directed GraphA directed graph is a finite set of verticestogether with a finite set of edges. If bothare empty then we have an empty graphV1 SOURCE V0 V2 V1 V4V3 TARGET V3 11 Directed Graph• Loops: a loop connects a vertex to itself LOOP Vertex• Path: is a sequence of vertices P0, P1, P2,… Pn , each adjacent pairs of vertices Pi and Pi+1 are connected by an edge path V1 V3 SOURCE TARGET 12 6
- 7. Directed Graph• Multiple edges: a graph may have two or more edges connecting the same two vertices in the same direction Directed Undirected V0 V0 V2 V2 V1V1 V4 V3 V3 V4 13 Directed Graph• Simple graph: • No loops • No multiple edges • Directed • Undirected• Directed graph representation: • Using a 2-Dim array • Using a Linked List for each vertex • Using the set class from the C++ Standard Library ( edge sets ) 14 7
- 8. Graph Relationship Adjacent vertices: Two vertices in a graph that are connected by an edge Path: A sequence of vertices that connects two nodes in a graph Complete graph: A graph in which every vertex is directly connected to every other vertex Weighted graph: A graph in which each edge carries a value 15Graph Relationship (cont…) In a complete graph, every vertex is adjacent to every other vertex. If there are N vertices, there will be N * (N - 1) edges in a complete directed graph and N * (N - 1) / 2 edges in a complete undirected graph. 16 8
- 9. Weighted graphs can be used to representWeighted graph applications in which the value of the connection between the vertices is important, not just the existence of a connection. 17Complete graph 18 9
- 10. Degree In an undirected graph, the degree of a vertex u is the number of edges connected to u. In a directed graph, the out-degree of a vertex u is the number edges leaving u, and its in-degree is the number of edges ending at u Each vertex is associated with a linked list that enumerates all the vertices that are connected to u 19Graph Traversals (cont…)In general:•Both traversal algorithms 1. Start at one vertex of a graph 2. Process the information contained at that vertex 3. Move along an edge to process a neighbor 20 10
- 11. Graph Traversals (cont…)• The traversal algorithm should be careful that it doesn’t enter a “Repetitive Cycle”. Therefore we will mark each vertex as it is processed..• When the traversal finishes, all of the vertices that can be reached from the start vertex have been processed.• Some of the vertices may not be processed because there is NO PATH from the START VERTEX 21 Solving a simple problem using a graphA network of computers can be representedBy a graph, with• each vertex representing One of the machines in the network, and• an Edge representing a communication wire Between the two machines.The question of whether one machine can senda message to another machine boils down towhether the corresponding vertices areConnected by a path 22 11
- 12. Solving a simple problem using a graph V0 6 1 V0 to V3 = 1 3 V1 V3 V3 to V1 = 3 2 V1 to V2 = 2 7 V2 Total 6 6• Each edge has a NON-Negative Integer valueattached to it called the weight or cost of the edgeThe path with the lowest cost is called theShortest path 23 Node ADT Essential Operations create: Element → Node changeElement: Node´Element → Node getElement: Node → Element 24 12
- 13. Edge ADT Essential Operations • create: Node´Node´Attribute → Edge • changeAttribute: Edge´Attribute → Edge • getAttribute: Edge → Attribute • getNodes: Edge → Node´Node • getEdge: Node´Node → Edge 25Digraph ADTcreate: → GraphaddNode: Node´Graph → GraphaddEdge: Edge´Graph → GraphcontainsNode: Node´Graph → BooleancontainsEdge: Node´Node´Graph → BooleanremoveNode: Node´Graph → GraphremoveEdge: Node´Node´Graph → GraphnumNodes: Graph → IntegernumEdges: Graph → Integer 26 13
- 14. Digraph ADT Other helpful operations an iterator over all nodes in the graph; an iterator over all edges in the graph; for each node, an iterator over all nodes to which it has an edge. 27Digraph Implementations Edge Set implementation Set of Nodes Set of Edges - (node, node, attribute) Standard method of representing sets e.g. hash table, BSTree 28 14
- 15. Digraph Implementations (cont…) Adjacency Set (or Adjacency List) implementation Set of Nodes Data at each node includes list of adjacent nodes and edge attributes Standard method of representing sets e.g. hash table, BSTree 29 15

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