1. "Learning how to learn is life's most important skill. " - Tony Buzan
Topological Sort
"Learning how to learn is life's most important skill. "
- Tony Buzan

2. Class Prerequisites Consider part-time student planning schedule
Must take the following 10 required courses with the given prerequisites:
Can take classes in any order, if prerequisite met.
Can only take one class a semester.
What order should the classes be taken?
142
143
321
322
326
341
370
378
401
421
none
CS Data Structures

3. Graph Modelling Model problem as a directed, acyclic graph (DAG):
nodes represent courses.
contains an edge (u, v) if course u is a prerequisite of course v.
Use topological sort to determine order to take classes.
CS Data Structures

4. Topological Sort Topological sorting problem:
Given directed acyclic graph (DAG) G = (V, E) ,
find a linear ordering of vertices such that for any edge (u, v) in E, u precedes v in the ordering
may be more than one ordering
Topological Sort
CS Data Structures

5. Simple Topological Sort Algorithm
Identify vertices with no incoming edges.
The in-degree of these vertices is zero.
If no vertices with no incoming edges,
graph is not acyclic.
doesn’t have a topological ordering.
Remove one of these vertices and its edges from the graph.
Return this vertex.
Repeat until graph is empty.
Order vertices are returned is topological ordering.
CS Data Structures

6. Example: Simple Topological Sort
322
143
321
326
142
341
370
401
378
421
Identify vertices with no incoming edges.
Only vertex with no incoming edges
CS Data Structures

7. Example: Simple Topological Sort
322
143
321
326
341
370
401
378
421
Remove vertex and its edges.
Output:
142
CS Data Structures

8. Example: Simple Topological Sort
322
143
321
326
341
370
401
378
421
Identify vertices with no incoming edges.
Only vertex with no incoming edges
Output:
142
CS Data Structures

9. Example: Simple Topological Sort
322
321
326
341
370
401
378
421
Remove vertex and its edges.
Output:
CS Data Structures

10. Example: Simple Topological Sort
322
321
326
341
370
401
378
421
Identify vertices with no incoming edges.
Four vertices with no incoming edges.
Select one.
Output:
CS Data Structures

11. Example: Simple Topological Sort
322
326
341
370
401
378
421
Remove vertex and its edges.
Output:
CS Data Structures

12. Example: Simple Topological Sort
322
326
341
370
401
378
421
Identify vertices with no incoming edges.
Five vertices with no incoming edges.
Select one.
Output:
CS Data Structures

13. Example: Simple Topological Sort
322
326
370
401
378
421
Remove vertex and its edges.
Output:
CS Data Structures

14. Example: Simple Topological Sort
322
326
370
401
378
421
Identify vertices with no incoming edges.
Four vertices with no incoming edges.
Select one.
Output:
CS Data Structures

15. Example: Simple Topological Sort
322
326
401
378
421
Remove vertex and its edges.
Output:
CS Data Structures

16. Example: Simple Topological Sort
322
326
401
378
421
Identify vertices with no incoming edges.
Three vertices with no incoming edges.
Select one.
Output:
CS Data Structures

17. Example: Simple Topological Sort
322
326
401
421
Remove vertex and its edges.
Output:
CS Data Structures

18. Example: Simple Topological Sort
322
326
401
421
Identify vertices with no incoming edges.
Two vertices with no incoming edges.
Select one.
Output:
CS Data Structures

19. Example: Simple Topological Sort
Remove vertex and its edges.
326
401
421
Output:
CS Data Structures

20. Example: Simple Topological Sort
Identify vertices with no incoming edges.
326
401
421
Only vertex with no incoming edge.
Output:
CS Data Structures

21. Example: Simple Topological Sort
Remove vertex and its edges.
401
421
Output:
CS Data Structures

22. Example: Simple Topological Sort
Identify vertices with no incoming edges.
401
421
Two vertices with no incoming edges.
Select one.
Output:
CS Data Structures

23. Example: Simple Topological Sort
Remove vertex and its edges.
401
Output:
CS Data Structures

24. Example: Simple Topological Sort
Identify vertices with no incoming edges.
Only vertex with no incoming edges.
401
Output:
CS Data Structures

25. Example: Simple Topological Sort
Remove vertex and its edges.
Output:
CS Data Structures

26. Example: Simple Topological Sort
Therefore, the student can take the classes in this order:
There are many other possible orderings, depending how choose which vertex to remove if there’s more than one with no incoming edges.
CS Data Structures

27. Analysis of Simple Topological Sort
Runtime of each step:
Initialize in-degree array:
Find vertex with in-degree zero:
𝑉 vertices, takes 𝑂( 𝑉 ) to search in-degree array.
Reduce degree of vertices adjacent to removed vertex:
Output and mark removed vertex:
Total time:
CS Data Structures

28. Analysis of Simple Topological Sort
Runtime of each step:
Initialize in-degree array:
Find vertex with in-degree zero:
𝑉 vertices, takes 𝑂( 𝑉 ) to search in-degree array.
Reduce degree of vertices adjacent to removed vertex:
Output and mark removed vertex:
Total time:
𝑂( 𝐸 )
CS Data Structures

29. Analysis of Simple Topological Sort
Runtime of each step:
Initialize in-degree array:
Find vertex with in-degree zero:
𝑉 vertices, takes 𝑂( 𝑉 ) to search in-degree array.
Reduce degree of vertices adjacent to removed vertex:
Output and mark removed vertex:
Total time:
𝑂( 𝐸 )
𝑂( 𝑉 2 )
CS Data Structures

30. Analysis of Simple Topological Sort
Runtime of each step:
Initialize in-degree array:
Find vertex with in-degree zero:
𝑉 vertices, takes 𝑂( 𝑉 ) to search in-degree array.
Reduce degree of vertices adjacent to removed vertex:
Output and mark removed vertex:
Total time:
𝑂( 𝐸 )
𝑂( 𝑉 2 )
CS Data Structures

31. Analysis of Simple Topological Sort
Runtime of each step:
Initialize in-degree array:
Find vertex with in-degree zero:
𝑉 vertices, takes 𝑂( 𝑉 ) to search in-degree array.
Reduce degree of vertices adjacent to removed vertex:
Output and mark removed vertex:
Total time:
𝑂( 𝐸 )
𝑂( 𝑉 2 )
𝑂( 𝑉 )
CS Data Structures

32. Analysis of Simple Topological Sort
Runtime of each step:
Initialize in-degree array:
Find vertex with in-degree zero:
𝑉 vertices, takes 𝑂( 𝑉 ) to search in-degree array.
Reduce degree of vertices adjacent to removed vertex:
Output and mark removed vertex:
Total time:
𝑂( 𝐸 )
𝑂( 𝑉 2 )
𝑂( 𝑉 )
𝑂( 𝑉 2 + 𝐸 )
CS Data Structures

33. Topological Sort with DFS
The topological sort of a DAG G can by computed by leveraging the DFS algorithm
CS Data Structures

34. Example: TS with DFS Modelling the order a person can dress:
CS Data Structures

35. Example: TS with DFS
The v.d and v.f times for the items after complete DFS.
CS Data Structures

36. Example: TS with DFS Add items to list as finish each one.
Topological Sort
CS Data Structures

37. Analysis of Topological Sort with DFS
Runtime of each step:
DFS(G):
Add vertex to front of list:
Return list:
Total time:
CS Data Structures

38. Analysis of Topological Sort with DFS
Runtime of each step:
DFS(G):
Add vertex to front of list:
Return list:
Total time:
𝑂( 𝑉 + 𝐸 )
CS Data Structures

39. Analysis of Topological Sort with DFS
Runtime of each step:
DFS(G):
Add vertex to front of list:
Return list:
Total time:
𝑂( 𝑉 + 𝐸 )
𝑂(1)
CS Data Structures

40. Analysis of Topological Sort with DFS
Runtime of each step:
DFS(G):
Add vertex to front of list:
Return list:
Total time:
𝑂( 𝑉 + 𝐸 )
𝑂(1)
CS Data Structures

41. Analysis of Topological Sort with DFS
Runtime of each step:
DFS(G):
Add vertex to front of list:
Return list:
Total time:
𝑂( 𝑉 + 𝐸 )
𝑂(1)
CS Data Structures

42. CS Data Structures