1. Topological Sort

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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