Data Structure & Algorithm

1. Data Structure & Algorithm 10 – Graph & BFS & DFS JJCAO

2. Graph Are Not

3. Graphs • G = (V,E) • V[G] = {1,2,3,4,5,6} |V| = 6 • E[G] = {{1,2},{1,5},{2,5},{3,6}} • Note: {u,v} = (u,v) = (v,u) • (u,v): u↔v

4. Can be very complex. Excellent Tool!

5. Terminology

6. Adjacency ListRepresentation

7. Adjacency MatrixRepresentation

8. Object-OrientedRepresentation • Node: some structure, with all relevant information • Edge: name & pointers to two endpoints

9. Sub-Graphs

10. More Terminology

11. Connectivity

12. Connected Components • Every node v is connected to itself • if u and v are in the same connected component then v is connected to u and u is connected to v • Connected components form a partition of the nodes and so are disjoint:

13. Forests & Trees

14. Properties

15. Directed Graph

16. Adding Direction • An undirected graph can be transformed into a directed one

17. Terminology • endpoints of an edge = the vertices it connects • e = (u,v) is incident from u = leaves u incident to v = enters v • u is the tail • v is the head • degreeof v = indegree + outdegree • indegree= # incoming edges • outdegree= # outgoing edges • Paths and cycles are now directed

18. Simple Graphs Are graphs with no parallel edges, no self loops Properties:

19. More Properties

20. Strong Connectivity • Vertices strongly connected ↔ there is a path from each to the other • Graph Strongly connected↔every2 vertices are strongly connected • Strongly Connected Components = maximal strongly connected subgraphs

21. Strong Connectivity The strongly connected components of the above graph

22. Graph Traversals Given: a graph G a source vertex s in V[G] Goal: “visit” all the vertices of G to determine some property: – Is G connected? – Does G contain a cycle? – is there a v u path in G? – What is the shortest path connecting v and u? – Will G disconnect if we remove a single edge? A vertex? – G is the WWW - follow links to retrieve information….

23. Data Structure for BFS • First In First Out (FIFO) Queue

24. Breadth First Search

25. Example

28. BFS - Running Time Adjacency list representation:

29. Shortest Paths Goal: Find a shortest path from s to every other v Idea: - start at s - find all vertices at distance 1 - find all vertices at distance 2

30. BFS for Shortest Paths

31. Shortest-Paths Claim

32. BFS for Connectivity

33. Depth First Search Goal: Visit all vertices of the graph Idea: - start at s - keep going “deeper” into the graph whenever possible

34. Depth First Search

35. Data Structure for DFS • Last In First Out (LIFO) Stack Rewrite the procedure DFS, using a stack to eliminate recursion

36. Example

37. Time Stamping

38. Example

39. BFS vs. DFS

40. Classification of Edges in DFS Forest Relative to the same DFS tree

41. Classification of Edges

42. DFS - Running Time Adjacency list representation

43. Applications of DFS In (V+E) time, we can: • Find connected components of G • determine if G has a cycle • determine if removing an edge / vertex disconnects G • determine if G is planar • …

44. Cycles in Directed Graphs Theorem: DiGraphG has a cycle DFS forest has a back edge Proof: • back edge => cycle: obvious • cycle => back edge: Lemma: A directed graph G is acyclicif and only if a depth-first search of G yields no back edges.

45. Topological Sort an ordering "<" of V[G] such that if (u, v) E[G] => u < v

46. Application of Topological Sort Many applications use directed acyclic graphs to indicate precedence among events, such as Professor Bumsteadgets dressed in the morning.

47. Topological Sort Lemma: G can be topologically sorted <=> G is a DAG (directed acyclic graph) Proof: cycle => G can't be sorted - obvious: no cycle => G can be sorted: we will show an algorithm

48. A Simple TopSort Algorithm Source = a vertex with indegree= 0

49. A Simple TopSort Algorithm Source = a vertex with indegree= 0

50. Adjacency MatrixImplementation