1. Topological Sort

2. Sorting technique over DAGs (Directed Acyclic Graphs) It creates a linear sequence (ordering) for the nodes such that: –If u has an outgoing edge to v  then u must finish before v starts Very common in ordering jobs or tasks

3. Topological Sort Example A job consists of 10 tasks with the following precedence rules: Must start with 7, 5, 4 or 9. Task 1 must follow 7. Tasks 3 & 6 must follow both 7 & 5. 8 must follow 6 & 4. 2 must follow 4. 10 must follow 2. Make a directed graph and then do DFS.

4. 9 1 7 3 5 4 6 8 2 10 Tasks shown as a directed graph.

5. Topological Sort using DFS To create a topological sort from a DAG 1- Final linked list is empty 2- Run DFS 3- When a node becomes black (finishes) insert it to the top of a linked list

6. 9 1 7 3 5 4 6 8 2 10 Example

7. 9 1 7 3 5 4 6 8 2 10 Example

8. 9 1 7 3 5 4 6 8 2 10 Example

9. 9 1 7 3 5 4 6 8 2 10 Example

10. 9 1 7 3 5 4 6 8 2 10 Example 10 head

11. 9 1 7 3 5 4 6 8 2 10 Example 2, 10 head

12. 9 1 7 3 5 4 6 8 2 10 Example 2, 10 head

13. 9 1 7 3 5 4 6 8 2 10 Example 8, 2, 10 head

14. 9 1 7 3 5 4 6 8 2 10 Example 4, 8, 2, 10 head

15. 9 1 7 3 5 4 6 8 2 10 Example 4, 8, 2, 10 head

16. 9 1 7 3 5 4 6 8 2 10 Example 4, 8, 2, 10 head

17. 9 1 7 3 5 4 6 8 2 10 Example 1, 4, 8, 2, 10 head

18. 9 1 7 3 5 4 6 8 2 10 Example 1, 4, 8, 2, 10 head

19. 9 1 7 3 5 4 6 8 2 10 Example 3, 1, 4, 8, 2, 10 head

20. 9 1 7 3 5 4 6 8 2 10 Example 3, 1, 4, 8, 2, 10 head

21. 9 1 7 3 5 4 6 8 2 10 Example 6, 3, 1, 4, 8, 2, 10 head

22. 9 1 7 3 5 4 6 8 2 10 Example 7, 6, 3, 1, 4, 8, 2, 10 head

23. 9 1 7 3 5 4 6 8 2 10 Example 7, 6, 3, 1, 4, 8, 2, 10 head

24. 9 1 7 3 5 4 6 8 2 10 Example 9, 7, 6, 3, 1, 4, 8, 2, 10 head

25. 9 1 7 3 5 4 6 8 2 10 Example 9, 7, 6, 3, 1, 4, 8, 2, 10 head

26. 9 1 7 3 5 4 6 8 2 10 Example 5, 9, 7, 6, 3, 1, 4, 8, 2, 10 head

27. 9 1 7 3 5 4 6 8 2 10 Topological Sort: Summary 5, 9, 7, 6, 3, 1, 4, 8, 2, 10 head The final order or jobs Time complexity = DFS complexity O(V + E) There can be many orders that meet the requirements