CSE 2331/5331

1. CSE 2331/5331 Topic 11: Basic Graph Alg. Representations Undirected graph Directed graph Topological sort CSE 2331/5331

2. a b c d e f What Is A Graph • Graph G = (V, E) • V: set of nodes • E: set of edges • Example: • V ={ a, b, c, d, e, f } • E ={(a, b), (a, d), (a, e), (b, c), (b, d), (b, e), (c, e), (e,f)} CSE 2331/5331

3. a b c d e f • Un-directed graph • Directed graph CSE 2331/5331

4. Un-directed graphs CSE 2331/5331

5. a b c d e f Vertex Degree • deg(v) = degree of v = # edges incident on v • Lemma: CSE 2331/5331

6. Some Special Graphs • Complete graph • Path • Cycle • Planar graph • Tree CSE 2331/5331

7. Representations of Graphs • Adjacency lists • Each vertex u has a list, recording its neighbors • i.e., all v’s such that (u, v)  E • An array of V lists • V[i].degree = size of adj list for node • V[i].AdjList= adjacency list for node CSE 2331/5331

8. Adjacency Lists • For vertex v  V, its adjacency list • has size: deg(v) • decide whether (v, u)  E or not in time O(deg(v)) • Size of data structure (space complexity): • O(|V| + |E|) = O(V+E) CSE 2331/5331

9. Adjacency Matrix • matrix • if is an edge • Otherwise, CSE 2331/5331

10. Adjacency Matrix • Size of data structure: • O ( V  V) • Time to determine if (v, u)  E : • O(1) • Though larger, it is simpler compared to adjacency list. CSE 2331/5331

11. Sample Graph Algorithm • Input: Graph G represented by adjacency lists Running time: O(V + E) CSE 2331/5331

12. Connectivity • A path in a graph is a sequence of vertices such that there is an edge between any two adjacent vertices in the sequence • Two vertices are connected if there is a path in G from u to w. • We also say that w is reachable from u. • A graph G is connected if every pair of nodes are connected. CSE 2331/5331

13. Connectivity Checking • How to check if the graph is connected? • One approach: Graph traversal • BFS: breadth-first search • DFS: depth-first search CSE 2331/5331

14. BFS: Breadth-first search • Input: • Given graph G = (V, E), and a source node s  V • Output: • Will visit all nodes in V reachable from s • For each output a value • distance (smallest # of edges) from s to v. • if v is not reachable from s. CSE 2331/5331

15. Intuition • Starting from source node s, • Spread a wavefront to visit other nodes • First visit all nodes one edge away from s • Then all nodes two edges away from s • … Need a data-structure to store nodes to be explored. CSE 2331/5331

16. Intuition cont. • A node can be: • un-discovered • discovered, but not explored • explored (finished) • : • is set when node v is first discovered. • Need a data structure to store discovered but un-explored nodes • FIFO ! Queue CSE 2331/5331

17. Pseudo-code Use adjacency list representation Time complexity: O(V+E) CSE 2331/5331

18. Correctness of Algorithm • A node not reachable from s will not be visited • A node reachable from s will be visited • computed is correct: • Intuitively, if all nodes k distance away from s are in level k, and no other nodes are in level k, • Then all nodes (k+1)-distance away from s must be in level (k+1). • Rigorous proof by induction CSE 2331/5331

19. Connectivity-Checking Time complexity: O(V+E) CSE 2331/5331

20. Summary for BFS • Starting from source node s, visits remaining nodes of graph from small distance to large distance • This is one way to traverse an input graph • With some special property where nodes are visited in non-decreasing distance to the source node s. • Return distance between s to any reachable node in time O(|V| + |E|) CSE 2331/5331

21. DFS: Depth-First Search • Another graph traversal algorithm • BFS: • Go as broad as possible in the algorithm • DFS: • Go as deep as possible in the algorithm CSE 2331/5331

22. Example • Perform DFS starting from • What if we add edge CSE 2331/5331

23. DFS • Time complexity • O(V+E) CSE 2331/5331

24. Depth First Search Tree • If v is discovered when exploring u • Set • The collection of edges • form a tree, • called Depth-first search tree. CSE 2331/5331

25. DFS with DFS Tree CSE 2331/5331

26. Example • Perform DFS starting from CSE 2331/5331

27. Another Connectivity Test CSE 2331/5331

28. Traverse Entire Graph CSE 2331/5331

29. Remarks • DFS(G, k) • Another way to compute all nodes reachable to the node • Same time complexity as BFS • There are nice properties of DFS and DFS tree that we are not reviewing in this class. CSE 2331/5331

30. Directed Graphs CSE 2331/5331

31. a b c d e f • Un-directed graph • Directed graph • Each edge is directed from u to v CSE 2331/5331

32. a b c d e f Vertex Degree • indeg(v) = # edges of the form • outdeg(v) = # edges of the form • Lemma: • Lemma: CSE 2331/5331

33. Representations of Graphs • Adjacency lists • Each vertex u has a list, recording its neighbors • i.e., all v’s such that (u, v)  E • An array of V lists • V[i].degree = size of adj list for node • V[i].AdjList= adjacency list for node CSE 2331/5331

34. Adjacency Lists • For vertex v  V, its adjacency list • has size: outdeg(v) • decide whether (v, u)  E or not in time O(outdeg(v)) • Size of data structure (space complexity): • O(V+E) CSE 2331/5331

35. Adjacency Matrix • matrix • if is an edge • Otherwise, CSE 2331/5331

36. Adjacency Matrix • Size of data structure: • O ( V  V) • Time to determine if (v, u)  E : • O(1) • Though larger, it is simpler compared to adjacency list. CSE 2331/5331

37. Sample Graph Algorithm • Input: Directed graph G represented by adjacency list Running time: O(V + E) CSE 2331/5331

38. Connectivity • A path in a graph is a sequence of vertices such that there is an edge between any two adjacent vertices in the sequence • Given two vertices we say that wis reachable from u if there is a path in G from u to w. • Note: w is reachable from uDOES NOT necessarily mean that u is reachable from w. CSE 2331/5331

39. Reachability Test • How many (or which) vertices are reachable from a source node, say ? CSE 2331/5331

40. BFS and DFS • The algorithms for BFS and DFS remain the same • Each edge is now understood as a directed edge • BFS(V,E, s) : • visits all nodes reachable from s in non-decreasing order CSE 2331/5331

41. BFS • Starting from source node s, • Spread a wavefront to visit other nodes • First visit all nodes one edge away from s • Then all nodes two edges away from s • … CSE 2331/5331

42. Pseudo-code Use adjacency list representation Time complexity: O(V+E) CSE 2331/5331

43. Number of Reachable Nodes Compute # nodes reachable from Time complexity: O(V+E) CSE 2331/5331

44. DFS: Depth-First Search • Similarly, DFS remains the same • Each edge is now a directed edge • If we start with all nodes unvisited, • Then DFS(G, visits all nodes reachable to node • BFS from previous NumReachable() procedure can be replaced with DFS. CSE 2331/5331

45. More Example • Is reachable from ? • Is reachable from ? CSE 2331/5331

46. DFS above can be replaced with BFS DFS(G, k); CSE 2331/5331

47. Topological Sort CSE 2331/5331

48. Directed Acyclic Graph • A directed cycle is a sequence such that there is a directed edge between any two consecutive nodes. • DAG: directed acyclic graph • Is a directed graph with no directed cycles. CSE 2331/5331

49. undershorts socks watch pants shoes shirt belt tie jacket Topological Sort • A topological sort of a DAG G = (V, E) • A linear ordering A of all vertices from V • If edge (u,v)  E => A[u] < A[v] CSE 2331/5331

50. Another Example Why requires DAG? Is the sorting order unique? CSE 2331/5331