# Data structure computer graphs

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### Data structure computer graphs

• 2. What is a graph? • A data structure that consists of a set of nodes (vertices) and a set of edges that relate the nodes to each other • The set of edges describes relationships among the vertices
• 3. Formal definition of graphs • A graph G is defined as follows: G=(V,E) V(G): a finite, nonempty set of vertices E(G): a set of edges (pairs of vertices)
• 4. Directed vs. undirected graphs • When the edges in a graph have no direction, the graph is called undirected
• 5. • When the edges in a graph have a direction, the graph is called directed (or digraph) Directed vs. undirected graphs (cont.) E(Graph2) = {(1,3) (3,1) (5,9) (9,11) (5,7) Warning: if the graph is directed, the order of the vertices in each edge is important !!
• 6. • Trees are special cases of graphs!! Trees vs graphs
• 7. Graph terminology • Adjacent nodes: two nodes are adjacent if they are connected by an edge • Path: a sequence of vertices that connect two nodes in a graph • Complete graph: a graph in which every vertex is directly connected to every other vertex 5 is adjacent to 7 7 is adjacent from 5
• 8. • What is the number of edges in a complete directed graph with N vertices? N * (N-1) Graph terminology (cont.)
• 9. • What is the number of edges in a complete undirected graph with N vertices? N * (N-1) / 2 Graph terminology (cont.)
• 10. Graph terminology (cont.) • Multi-graph • Cycle • Loop
• 11. • Weighted graph: a graph in which each edge carries a value Graph terminology (cont.)
• 12. Graph implementation • Array-based implementation – A 1D array is used to represent the vertices – A 2D array (adjacency matrix) is used to represent the edges
• 14. Graph implementation (cont.) • Linked-list implementation – A 1D array is used to represent the vertices – A list is used for each vertex v which contains the vertices which are adjacent from v (adjacency list)
• 16. Adjacency Matrix for Un- Weighted Graphs 1. Adjacency Matrix 2. Matrix for edges 3. Matrix for 2 length paths 4. Matrix for 3 length paths 5. Matrix for n length paths 6. Path Matrix 7. Completely Connected Graph
• 17. Graph searching • Problem: find a path between two nodes of the graph (e.g., Austin and Washington) • Methods: Depth-First-Search (DFS) or Breadth-First-Search (BFS)
• 18. Depth-First-Search (DFS) • What is the idea behind DFS? – Travel as far as you can down a path – Back up as little as possible when you reach a "dead end" (i.e., next vertex has been "marked" or there is no next vertex) • DFS can be implemented efficiently using a stack
• 19. DFS • Algorithm: Starting Node is A 1. Initialize all nodes to ready state(Status=1) 2. Push starting nose A onto Stack and change its status to waiting state (Status=2). 3.Repeat steps 4 & 5 until STACK is empty. 4. Pop Top node N. Process N and change its Status to the processed state (Status=3). 5.Push onto Stack all the neighbors of N that are still in ready state and change their state as waiting state(State=2). 6. Exit.
• 20. Breadth-First-Searching (BFS) • What is the idea behind BFS? – Look at all possible paths at the same depth before you go at a deeper level – Back up as far as possible when you reach a "dead end" (i.e., next vertex has been "marked" or there is no next vertex)
• 21. BFS • Algorithm: Starting Node is A 1. Initialize all nodes to ready state(Status=1) 2. Push starting nose A onto Queue and change its status to waiting state (Status=2). 3.Repeat steps 4 & 5 until Queue is empty. 4. Remove the front node N. Process N and change its Status to the processed state (Status=3). 5.Add to the rear of Queue all the neighbors of N that are still in ready state and change their state to waiting (State=2). 6. Exit.
• 22. • Examples of Both DFS & BFS.
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