Topological Sort

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