Graph Algorithms: Topological Sort

1. Graph Algorithms: Topological Sort The topological sorting problem: given a directed, acyclic graph G = (V, E) , find a linear ordering of the vertices such that for all (v, w)  E, v precedes w in the ordering. B C A F D E

2. Graph Algorithms: Topological Sort The topological sorting problem: given a directed, acyclic graph G = (V, E) , find a linear ordering of the vertices such that for all (v, w)  E, v precedes w in the ordering. B C A A F D E B C F D E

3. Graph Algorithms: Topological Sort The topological sorting problem: given a directed, acyclic graph G = (V, E) , find a linear ordering of the vertices such that for all (v, w)  E, v precedes w in the ordering. Any linear ordering in which all the arrows go to the right. B C A A F D E B C F D E

4. Graph Algorithms: Topological Sort The topological sorting problem: given a directed, acyclic graph G = (V, E) , find a linear ordering of the vertices such that for all (v, w)  E, v precedes w in the ordering. Any linear ordering in which all the arrows go to the right. B C A A E D F B C F D E This is not a topological ordering.

5. Graph Algorithms: Topological Sort The topological sorting algorithm: Identify the subset of vertices that have no incoming edge. B C A F D E

6. Graph Algorithms: Topological Sort The topological sorting algorithm: Identify the subset of vertices that have no incoming edge. (In general, this subset must be nonempty—why?) B C A F D E

7. Graph Algorithms: Topological Sort The topological sorting algorithm: Identify the subset of vertices that have no incoming edge. (In general, this subset must be nonempty—because the graph is acyclic.) B C A F D E

8. Graph Algorithms: Topological Sort The topological sorting algorithm: Identify the subset of vertices that have no incoming edge. Select one of them. B C A F D E

9. Graph Algorithms: Topological Sort The topological sorting algorithm: Remove it, and its outgoing edges, and add it to the output. B C A F D E

10. Graph Algorithms: Topological Sort The topological sorting algorithm: Again, identify the subset of vertices that have no incoming edge, . . . B C A F D E

11. Graph Algorithms: Topological Sort The topological sorting algorithm: Again, identify the subset of vertices that have no incoming edge, select one of them, . . . B C A F D E

12. Graph Algorithms: Topological Sort The topological sorting algorithm: Again, identify the subset of vertices that have no incoming edge, select one of them, remove it and any outgoing edges, and put it in the output. B C A F D E

13. Graph Algorithms: Topological Sort The topological sorting algorithm: Again, identify the subset of vertices that have no incoming edge, select one of them, remove it and any outgoing edges, and put it in the output. B C A F D E

14. Graph Algorithms: Topological Sort The topological sorting algorithm: Again, identify the subset of vertices that have no incoming edge, select one of them, remove it and any outgoing edges, and put it in the output. C A F B D E

15. Graph Algorithms: Topological Sort The topological sorting algorithm: Again, identify the subset of vertices that have no incoming edge, select one of them, remove it and any outgoing edges, and put it in the output. C A F B D E

16. Graph Algorithms: Topological Sort The topological sorting algorithm: Again, identify the subset of vertices that have no incoming edge, select one of them, remove it and any outgoing edges, and put it in the output. A F B C D E

17. Graph Algorithms: Topological Sort The topological sorting algorithm: Again, identify the subset of vertices that have no incoming edge, select one of them, remove it and any outgoing edges, and put it in the output. A F B C D E

18. Graph Algorithms: Topological Sort The topological sorting algorithm: Again, identify the subset of vertices that have no incoming edge, select one of them, remove it and any outgoing edges, and put it in the output. A F B C D E

19. Graph Algorithms: Topological Sort The topological sorting algorithm: Again, identify the subset of vertices that have no incoming edge, select one of them, remove it and any outgoing edges, and put it in the output. A F B C D E

20. Graph Algorithms: Topological Sort The topological sorting algorithm: Again, identify the subset of vertices that have no incoming edge, select one of them, remove it and any outgoing edges, and put it in the output. A F B C D E

21. Graph Algorithms: Topological Sort The topological sorting algorithm: finished! B C A A F B C D E F D E

22. Graph Algorithms: Topological Sort The topological sorting algorithm: Time bound? B C A A F B C D E F D E

23. Graph Algorithms: Topological Sort The topological sorting algorithm: Time bound: Break down into total time to: Find vertices with no predecessors: ? Remove edges: ? Place vertices in output: ? B C A A F B C D E F D E

24. Graph Algorithms: Topological Sort The topological sorting algorithm: Time bound: Break down into total time to: Find vertices with no predecessors: ? Remove edges: O(|E|) Place vertices in output: ? B C A A F B C D E F D E

25. Graph Algorithms: Topological Sort The topological sorting algorithm: Time bound: Break down into total time to: Find vertices with no predecessors: ? Remove edges: O(|E|) Place vertices in output: O(|V|) B C A A F B C D E F D E

26. Graph Algorithms: Topological Sort Find vertices with no predecessors: ? Assume an adjacency list representation: A B C D E F B D B C C E A D F E D E

27. Graph Algorithms: Topological Sort The topological sorting algorithm: …and initialize and maintain for each vertex its no. of predecessors. B 0 D A B C D E F 1 1 1 B C 0 C 1 0 E A D F 2 E 2 2 2 D E 0

28. Graph Algorithms: Topological Sort Find vertices with no predecessors: ? Time for each vertex: O(|V|) B 0 D A B C D E F 1 B C C 1 E A D F 2 E 2 D E 0

29. Graph Algorithms: Topological Sort Find vertices with no predecessors: ? 2 Total time: O(|V| ) B 0 D A B C D E F 1 B C C 1 E A D F 2 E 2 D E 0

30. Graph Algorithms: Topological Sort The topological sorting algorithm: Time bound: Break down into total time to: Find vertices with no predecessors: O(|V| ) Remove edges: O(|E|) Place vertices in output: O(|V|) 2

31. Graph Algorithms: Topological Sort The topological sorting algorithm: Time bound: Break down into total time to: Find vertices with no predecessors: O(|V| ) Remove edges: O(|E|) Place vertices in output: O(|V|) 2 2 Total: O(|V| + |E|)

32. Graph Algorithms: Topological Sort The topological sorting algorithm: Time bound: Break down into total time to: Find vertices with no predecessors: O(|V| ) Remove edges: O(|E|) Place vertices in output: O(|V|) 2 2 Total: O(|V| + |E|) Too much!

33. Graph Algorithms: Topological Sort The topological sorting algorithm: We need a faster way to do this step: Find vertices with no predecessors.

34. Graph Algorithms: Topological Sort The topological sorting algorithm: Key idea: initialize and maintain a queue (or stack) holding pointers to the vertices with 0 predecessors A B C D E F B 0 D 1 1 B 1 C 0 C 0 1 A E D F 2 E 2 2 2 D E 0

35. Graph Algorithms: Topological Sort The topological sorting algorithm: As each vertex is removed, update the predecessor counts, and for any vertex whose count has become zero, put it in the queue. A B C D E F B 0 D 1 1 B 1 C 0 C 0 1 A E D F 2 E 2 2 2 D E 0

36. Graph Algorithms: Topological Sort The topological sorting algorithm: As each vertex is removed, update the predecessor counts, and for any vertex whose count has become zero, put it in the queue. No scan is required, so O(1). A B C D E F 0 0 1 B 0 C C 0 1 E D F 1 E 1 2 2 D Output: A E 0

37. Graph Algorithms: Topological Sort The topological sorting algorithm: As each vertex is removed, update the predecessor counts, and for any vertex whose count has become zero, put it in the queue. A B C D E F 0 0 1 B 0 C C 1 E D 1 E 1 2 2 D Output: A F E 0

38. Graph Algorithms: Topological Sort The topological sorting algorithm: As each vertex is removed, update the predecessor counts, and for any vertex whose count has become zero, put it in the queue. A B C D E F 0 0 0 C 0 E D 1 E 1 2 2 D Output: A F B E 0

39. Graph Algorithms: Topological Sort The topological sorting algorithm: As each vertex is removed, update the predecessor counts, and for any vertex whose count has become zero, put it in the queue. A B C D E F 0 0 0 0 E 0 1 1 D Output: A F B C E 0

40. Graph Algorithms: Topological Sort The topological sorting algorithm: As each vertex is removed, update the predecessor counts, and for any vertex whose count has become zero, put it in the queue. A B C D E F 0 0 0 0 0 0 Output: A F B C D E 0

41. Graph Algorithms: Topological Sort The topological sorting algorithm: As each vertex is removed, update the predecessor counts, and for any vertex whose count has become zero, put it in the queue. Finished! A B C D E F 0 0 0 0 0 Output: A F B C D E 0

42. Graph Algorithms: Topological Sort The topological sorting algorithm: Time bound: Now the time for each part is Find vertices with no predecessors: O(|V|) Remove edges: O(|E|) Place vertices in output: O(|V|) Total: O(|V|+|E|) Linear in |V|+|E|. Much better!