1. A Batch Algorithm for Maintaining a Topological Order David J. Pearce and Paul H.J. Kelly Victoria University of Wellington, New Zealand & Imperial College, London

2. Topological Orders 1 2 3 4 76 8 5 9

3. 23 45678 1 9 1 2 3 4 76 8 5 9

4. Updating Topological Orders 1 2 3 4 76 8 5 9 23 45678 1 9

5. 23 45678 1 9 Affected Region 1 2 3 4 76 8 5 9

6. Updating Topological Orders 23 45678 1 9 Affected Region

7. Updating Topological Orders Unit Change Problem 23 45678 1 9 2004: Pearce, et al. 1990: Alpern, et al. 1996: Marchetti-Spaccamela, et al. 2006: Ajwani, et al. 2009: Bender, et al. 2008: Haeupler, et al. 2005: Katriel, et al. 2007: Liu, et al.

8. 123476859 12 3 456789

9. 123476859 123456789

10. 123476859 12 345 6789 123456789

11. 123476859 12 345 6789 123456789 12 34 5 678 9

12. Batch Problem Definition: Given a batch of edge insertions, update the topological order Q) Can we do better than by processing them one at a time?

13. Batch Problem Definition: Given a batch of edge insertions, update the topological order Q) Can we do better than by processing them one at a time? A) Yes!

14. Previous MNR Algorithm 2 3 4 5 678 1 9 Discovery – DFS to identify nodes reachable from tail in affected region

15. Previous MNR Algorithm 23 4 5 67819 Discovery – DFS to identify nodes reachable from tail in affected region

16. Previous MNR Algorithm 23 4 5 6 7819 Discovery – DFS to identify nodes reachable from tail in affected region

17. Previous MNR Algorithm 23 4 5 6 7819 Discovery – DFS to identify nodes reachable from tail in affected region

18. Previous MNR Algorithm 23 4 5 6 7819 Discovery – DFS to identify nodes reachable from tail in affected region

19. Previous MNR Algorithm 23 4 5 6 781 9 Shift – marked nodes up the order, unmarked nodes down the order (in affected region)

20. Previous MNR Algorithm 23 4 5 6 781 9 Shift – marked nodes up the order, unmarked nodes down the order (in affected region)

21. Previous MNR Algorithm 23 4 5 6 781 9 Shift – marked nodes up the order, unmarked nodes down the order (in affected region)

22. Previous MNR Algorithm 23 4 5 6 781 9 Shift – marked nodes up the order, unmarked nodes down the order (in affected region)

23. Previous MNR Algorithm 23 4 5 6 781 9 Shift – marked nodes up the order, unmarked nodes down the order (in affected region)

24. Previous MNR Algorithm 23 4 5 6 781 9 Shift – marked nodes up the order, unmarked nodes down the order (in affected region)

25. MNR Algorithm 23 4 5 6 781 9 Shift – marked nodes up the order, unmarked nodes down the order (in affected region)

26. Our Novel Batch Algorithm Observation – edge insertions which overlap can be processed independently of others

27. Our Novel Batch Algorithm Observation – edge insertions which overlap can be processed independently of others Affected Region

28. Our Novel Batch Algorithm Discovery – identify “target node” for each node reachable from an insertion in affected region 4567213

29. Our Novel Batch Algorithm Discovery – identify “root node” for each node reachable from an insertion in affected region 4567213

30. Our Novel Batch Algorithm Discovery – identify “root node” for each node reachable from an insertion in affected region 4 567213

31. Our Novel Batch Algorithm Discovery – identify “root node” for each node reachable from an insertion in affected region 4567213 444

32. Our Novel Batch Algorithm Discovery – identify “root node” for each node reachable from an insertion in affected region 4567 2 1 3 44411

33. Our Novel Batch Algorithm Shift – marked nodes up order and unmarked nodes down order; marked nodes moved just past their target 4567 2 1 3 44411

34. Our Novel Batch Algorithm Shift – marked nodes up order and unmarked nodes down order; marked nodes moved just past their target 456 7 2 1 3 44 4 11

35. Our Novel Batch Algorithm Shift – marked nodes up order and unmarked nodes down order; marked nodes moved just past their target 456 7 2 1 3 44 4 11

36. Our Novel Batch Algorithm Shift – marked nodes up order and unmarked nodes down order; marked nodes moved just past their target 456 7 2 1 3 44 41 1

37. Our Novel Batch Algorithm Shift – marked nodes up order and unmarked nodes down order; marked nodes moved just past their target 456 7 2 1 3 44 41 1

38. Our Novel Batch Algorithm Shift – marked nodes up order and unmarked nodes down order; marked nodes moved just past their target 456 7 2 1 3 44 41 1

39. Our Novel Batch Algorithm Shift – marked nodes up order and unmarked nodes down order; marked nodes moved just past their target 456 7 2 1 3 44 41 1

40. Our Novel Batch Algorithm Shift – marked nodes up order and unmarked nodes down order; marked nodes moved just past their target 456 7 2 1 3 44 41

41. Our Novel Batch Algorithm Shift – marked nodes up order and unmarked nodes down order; marked nodes moved just past their target 456 7 2 1 3 44 4

42. Our Novel Batch Algorithm Shift – marked nodes up order and unmarked nodes down order; marked nodes moved just past their target 4567 2 1 3

43. Results on Sparse Random Graphs

45. Results on Denser Random Graphs

47. Applications Extensions: –Incremental Cycle Detection Applications: –Pointer Analysis –Software Model Checking –Constraint-based Local Search Algorithms –Dead-lock Detection –Machine Learning –Multiple Sequence Alignment

48. Conclusion That’s all folks!