1. Nattee Niparnan

2. Graph  A pair G = (V,E)  V = set of vertices (node)  E = set of edges (pairs of vertices)  V = (1,2,3,4,5,6,7)  E = ((1,2),(2,3),(3,5),(1,4),(4, 5),(6,7)) 1 2 3 4 5 6 7

3. Term you should already know  directed, undirected graph  Weighted graph  Bipartite graph  Tree  Spanning tree  Path, simple path  Circuit, simple circuit  Degree

4. Representing a Graph  Adjacency Matrix  A = |V|x|V| matrix  a xy = 1when there is an edge connecting node x and node y  a xy = 0otherwise 1 2 3 4 5 01010 10110 01001 11001 00110 12345 1 2 3 4 5

5. Representing a Graph  Adjacency List  Use a list instead of a matrix  For each vertex, we have a linked list of their neighbor 1 2 3 4 5 124 2134...

6. Representing a Graph  Incidences Matrix  Row represent edge  Column represent node 1 2 3 4 5 11000 10010 01010 01100 00101 00011 12345

7. Exploring a Maze

8. Exploring Problem

9. Depth-First-Search procedure explore(G; v) // Input: G = (V;E) is a graph; v  V // Output: visited(u) is set to true for all nodes u reachable from v { visited(v) = true previsit(v) for each edge (v,u)  E if not visited(u) explore(u) postvisit(v) }

10. Example Explore(A)

11. Extend to Graph Traversal  Traversal is walking in the graph  We might need to visit each component in the graph  Can be done using explore  Do “explore” on all non-visited node  The result is that we will visit every node  What is the difference between just looking into V (the set of vertices?)

12. Graph Traversal using DFS procedure dfs(G) { for all v  V visited(v) = false for all v  V if not visited(v) explore(v) }

13. Complexity Analysis  Each node is visited once  Each edge is visited twice  Why?  O( |V| + |E|)

14. Another Example

16. Connectivity in Undirected Graph

17. Connected Component Problem  Input:  A graph  Output:  Marking in every vertices identify the connected component  Let it be an array ccnum, indexed by vertices

18. Solution  Define global variable cc  In previsit()  ccnum[v] = cc  Before calling each explore  cc++

19. Ordering in Visit procedure previsit(v) pre[v] = clock clock = clock + 1 procedure postvisit(v) post[v] = clock clock = clock + 1

20. Ordering in Visit The interval for node u is [pre(u),post(u)] The inverval for u,v is either Contained disjointed Never intersect

21. DFS in Directed Graph

22. Type of Edge in Directed Graph

23. Directed Acyclic Graph (DAG)  A directed Graph without a cycle  Has “source”  A node having only “out” edge  Has “sink”  A node having only “in” edge  How can we detect that a graph is a DAG  What should be the property of “source” and “sink” ?

24. Solution  A directed graph is acyclic if and only if it has no back edge  Sink  Having lowest post number  Source  Having highest post number

25. Linearization of Graph

26. Linearization One possible linearization B,A,D,C,E,F Order of work that can be done w/o violating the causality constraints

27. Topological Sorting Problem

28. Topological Sorting  Do DFS  List node by post number (descending) 1,12 2,9 3,84,5 6,7 10,11