1. Graph Representations And Traversals

2. Graphs Graph : – Set of Vertices (Nodes) – Set of Edges connecting vertices (u, v) : edge connecting Origin: u Destination: v

3. Graphs Examples:

4. Terms Undirected graph – Edges go both ways

5. Terms Digraph : Directed graph – Edges can be one direction

6. Graph Definitions and Notations Adjacent: there is an edge from one vertex to the other Incident: if edge e = (u, v) then e is incident on u and v – Loop: edge incident on a single vertex

7. Graph Definitions and Notations Parallel edges: Edges between same vertices Simple graph: no loops or parallel edges

8. Graph Definitions and Notations Path: sequence of vertices u 1, u 2,..., u n such that exist edges (u 1, u 2 ), (u 2, u 3 )... (u n, u n-1 ) Connected vertices u and v: there is a path from u to v Simple path: path where all vertices but first and last are distinct

9. Graph Definitions and Notations Connected : there is an undirected path between any 2 nodes Strongly connected : directed path exists between any two nodes

10. Terms Spanning Tree : – Tree covering every vertex made with subset of edges Cycle : – Simple path back to starting node Clique : – Group of completely connected nodes

11. Everything is a digraph Every pointer based data structure is a digraph – Trees & Linked lists = acyclical digraphs

12. Representation Can represent as objects – Nodes as objects – Edges as pointers – Node maintains vector of pointers But, some graph algorithms benefit from random access to vertices…

13. Representation Adjacency matrix – 2D array indicating edges

14. Representation Undirected graph produces symmetric matrix

15. Efficiency Graph operations described in terms of – E : num edges – V : num verticies Adjacency Matrix – Edges are cheap – Vertices expensive Best for dense graphs Adjacency matrix Storage O(V 2 ) Add vertex O(V 2 ) Add edge O(1) Remove vertex O(V 2 ) Remove edge O(1) Query: are vertices u, v adjacent? (Assuming that the storage positions for u, v are known) O(1)

16. Adjacency Matrix Power of adj matrix shows # of paths of that length

17. Representation Adjacency list : – Store a list of all edges starting from given vertex:

18. Efficiency Adjacency List – Cost based more on edges – Best for sparse graphs Adjacency ListAdjacency matrix Storage O(V + E)O(V 2 ) Add vertex O(1)O(V 2 ) Add edge O(1) Remove vertex O(V + E)O(V 2 ) Remove edge O(E)O(1) Query: are vertices u, v adjacent? (Assuming that the storage positions for u, v are known) O(V) O(1)

19. Other Options Incident Based – Incident Matrix : V rows x E cols Adjacency set – BST for edges related to a vertex

20. BFS BFS : Breadth First Search

21. BFS BFS - Spreads out in all directions – 1 Hop

22. BFS BFS - Spreads out in all directions – 1 Hop

23. BFS BFS - Spreads out in all directions – 1 Hop

24. BFS BFS - Spreads out in all directions – 1 Hop

25. BFS BFS - Spreads out in all directions – 1 Hop

26. BFS BFS - Spreads out in all directions – 1 Hop – 2 Hops

27. BFS BFS - Spreads out in all directions – 1 Hop – 2 Hops

28. BFS BFS - Spreads out in all directions – 1 Hop – 2 Hops

29. BFS BFS - Spreads out in all directions – 1 Hop – 2 Hops

30. BFS BFS - Spreads out in all directions – 1 Hop – 2 Hops – 3 Hops

31. BFS BFS - Spreads out in all directions – 1 Hop – 2 Hops – 3 Hops

32. BFS BFS - Spreads out in all directions – 1 Hop – 2 Hops – 3 Hops

33. BFS BFS - Spreads out in all directions – 1 Hop – 2 Hops – 3 Hops – 4 Hops

34. DFS DFS – Depth First Search

35. DFS DFS -

36. DFS DFS -

37. DFS DFS -

38. DFS DFS -

39. DFS DFS -

40. DFS DFS -

41. DFS DFS -

42. DFS DFS -

43. DFS DFS -

44. DFS DFS -

45. DFS DFS -

46. DFS DFS – Backtrack when stuck – Chain again as soon as a new path available

47. DFS DFS – Backtrack when stuck – Chain again as soon as a new path available

48. DFS DFS – Backtrack when stuck – Chain again as soon as a new path available

49. DFS DFS – Backtrack when stuck – Chain again as soon as a new path available

50. DFS DFS – Backtrack when stuck – Chain again as soon as a new path available

51. DFS DFS – Backtrack when stuck – Chain again as soon as a new path available

52. DFS DFS – Backtrack when stuck – Chain again as soon as a new path available

53. DFS DFS – Backtrack when stuck – Chain again as soon as a new path available

54. DFS DFS – Backtrack when stuck – Chain again as soon as a new path available

55. DFS DFS – Backtrack when stuck – Chain again as soon as a new path available

56. DFS DFS – Backtrack when stuck – Chain again as soon as a new path available

57. DFS DFS – Backtrack when stuck – Chain again as soon as a new path available

58. DFS DFS – Backtrack when stuck – Chain again as soon as a new path available

59. DFS DFS – Backtrack when stuck – Chain again as soon as a new path available

60. DFS DFS – Backtrack when stuck – Chain again as soon as a new path available

61. DFS DFS – Backtrack when stuck – Chain again as soon as a new path available

62. DFS DFS – Backtrack when stuck – Chain again as soon as a new path available

63. DFS DFS – Backtrack when stuck – Chain again as soon as a new path available

64. DFS DFS – Backtrack when stuck – Chain again as soon as a new path available