1. 2620001Data Structures Graph Data Structure

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. Directed vs. undirected graphs (cont.) • When the edges in a graph have a direction, the graph is called directed (or digraph) Warning: if the graph is directed, the order of the vertices in each edge is important !! E(Graph2) = {(1,3) (3,1) (5,9) (9,11) (5,7) (11,1) (9,9)}

6. Trees vs graphs • Trees are special cases of 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. Graph terminology (cont.) • What is the number of edges in a complete directed graph with N vertices?  N * (N-1)

9. Graph terminology (cont.) • What is the number of edges in a complete undirected graph with N vertices?  N * (N-1) / 2

10. Graph terminology (cont.) • Weighted graph: a graph in which each edge carries a value

11. 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

12. Array-based implementation

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)

15. Linked-list implementation

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. Use of a stack • It is very common to use a stack to keep track of: • nodes to be visited next, or • nodes that we have already visited. • Typically, use of a stack leads to a depth-first visit order. • Depth-first visit order is “aggressive” in the sense that it examines complete paths.

20. Graph Traversals • Both take time: O(V+E)

21. start end (initialization)

24. (BFS) • What is the idea behind BFS? • Look at all possible paths at the same depth before you go at a deeper level • Back Breadth-First-Searching up as far as possible when you reach a "dead end" (i.e., next vertex has been "marked" or there is no next vertex) • BFS can be implemented efficiently using a Queue

25. Use of a queue • It is very common to use a queue to keep track of: • nodes to be visited next, or • nodes that we have already visited. • Typically, use of a queue leads to a breadth-first visit order. • Breadth-first visit order is “cautious” in the sense that it examines every path of length i before going on to paths of length i+1.

26. start end (initialization)

27. next:

29. Breadth First Search 2 4 8 s 5 7 3 6 9

30. Breadth First Search Shortest pathfrom s 1 2 2 4 8 0 s 5 7 3 6 9 Undiscovered Queue: s Discovered Top of queue Finished

31. Breadth First Search 1 2 4 8 0 s 5 7 3 3 6 9 1 Undiscovered Queue: s 2 Discovered Top of queue Finished

32. Breadth First Search 1 2 4 8 0 5 s 5 7 1 3 6 9 1 Undiscovered Queue: s 2 3 Discovered Top of queue Finished

33. Breadth First Search 1 2 4 8 0 s 5 7 1 3 6 9 1 Undiscovered Queue: 2 3 5 Discovered Top of queue Finished

34. Breadth First Search 1 2 2 4 4 8 0 s 5 7 1 3 6 9 1 Undiscovered Queue: 2 3 5 Discovered Top of queue Finished

35. Breadth First Search 1 2 2 4 8 5 already discovered:don't enqueue 0 s 5 7 1 3 6 9 1 Undiscovered Queue: 2 3 5 4 Discovered Top of queue Finished

36. Breadth First Search 1 2 2 4 8 0 s 5 7 1 3 6 9 1 Undiscovered Queue: 2 3 5 4 Discovered Top of queue Finished

37. Breadth First Search 1 2 2 4 8 0 s 5 7 1 3 6 9 1 Undiscovered Queue: 3 5 4 Discovered Top of queue Finished

38. Breadth First Search 1 2 2 4 8 0 s 5 7 1 3 6 6 9 1 2 Undiscovered Queue: 3 5 4 Discovered Top of queue Finished

39. Breadth First Search 1 2 2 4 8 0 s 5 7 1 3 6 9 1 2 Undiscovered Queue: 3 5 4 6 Discovered Top of queue Finished

40. Breadth First Search 1 2 2 4 8 0 s 5 7 1 3 6 9 1 2 Undiscovered Queue: 5 4 6 Discovered Top of queue Finished

41. Breadth First Search 1 2 2 4 8 0 s 5 7 1 3 6 9 1 2 Undiscovered Queue: 5 4 6 Discovered Top of queue Finished

42. Breadth First Search 1 2 2 4 8 0 s 5 7 1 3 6 9 1 2 Undiscovered Queue: 4 6 Discovered Top of queue Finished

43. Breadth First Search 1 2 3 2 4 8 8 0 s 5 7 1 3 6 9 1 2 Undiscovered Queue: 4 6 Discovered Top of queue Finished

44. Breadth First Search 1 2 3 2 4 8 0 s 5 7 1 3 6 9 1 2 Undiscovered Queue: 4 6 8 Discovered Top of queue Finished

45. Breadth First Search 1 2 3 2 4 8 0 7 s 5 7 1 3 3 6 9 1 2 Undiscovered Queue: 6 8 Discovered Top of queue Finished

46. Breadth First Search 1 2 3 2 4 8 0 s 5 7 1 3 3 6 9 9 3 1 2 Undiscovered Queue: 6 8 7 Discovered Top of queue Finished

47. Breadth First Search 1 2 3 2 4 8 0 s 5 7 1 3 3 6 9 3 1 2 Undiscovered Queue: 6 8 7 9 Discovered Top of queue Finished

48. Breadth First Search 1 2 3 2 4 8 0 s 5 7 1 3 3 6 9 3 1 2 Undiscovered Queue: 8 7 9 Discovered Top of queue Finished

49. Breadth First Search 1 2 3 2 4 8 0 s 5 7 1 3 3 6 9 3 1 2 Undiscovered Queue: 7 9 Discovered Top of queue Finished

50. Breadth First Search 1 2 3 2 4 8 0 s 5 7 1 3 3 6 9 3 1 2 Undiscovered Queue: 7 9 Discovered Top of queue Finished