Topological SortingBrief overview of topological sorting in graphs

Mihail Petrov

mihail-petrov.cloudvps.com

Trainer Candidate

Table of Contents

1. TS Why ?

2. TS Examples and Warnings

3. TS Algorithm explanation

4. Source Removal Algorithm for TS

5. TS with DFS

2

TS Why ?Why we need to sort a graph

topologically

University course graph

4

Math

Programing OOP

C#

Data Structures

Java

JS

Spring

Java EE

Find the order in which all the courses can be taken !

Explanation Using TP Sorting we could find the right sequence of element, represented as a gaph.

Given a digraph G = (V, E), we could find a linear ordering of its vertices such that for any edge (v, w) in E, v precedes w in the ordering.

5

TS Examples and Warnings

Sample Graph #1

7

A

E

B

C

D

F

Sample Sorting #1 Actualy we can’t sort it: What is the order, this:

8

A B C D E

Or this:

B A E D C

Sample #1 Conclusion

9

We CAN NOT topologically sort an unordered Graph

!

Sample Graph#2

C#

WPF

Production

Java Script

Front-End

A B

CD

E

Sample Sort

11

A B C D E

A B C DE

Rite Sort – Left Direction

Wrong Sort – Right Direction

Sample #2 Conclusion

12

Any linear ordering in which an arrow goes to the left is NOT a vald solution

!

Sample Graph#3

C#

WPF

Production

Java Script

Front-End

A B

CD

E

Sample Sort#3

14

A B C D E

Sample #3 Conclusion

15

A directed graph with a cycle CAN NOT be topologically sorted

!

TS AlgorithmGeneral Explanation of TS

Example:

A

E

C

B

D

We have the fouling graph, and we have to apply the TS for it.

Step #1

A

E

C

B

D

Identify and Select Node with no incoming edges.

Step #1 Warning If, no such node than:

19

A

D

B

C

TPS Not Posible

Step #2

A

E

C

B

D

Remove Selectde Node

Elements:

Step #2

A

E

C

B

D

Repeat step#2

Elements:

Step #2

A

E

C

B

D

Repeat step #2

Elements:

Step #2

A

E

CB

D

Repeat step #2

Elements:

Result:

A ECB D

TS Result

TS Program Implementation

Graph Representation using Adjacency List

26

A

B

C

D

E

B E

D C

D

E F

F

F -

Graph Representation using Adjacency Matrix

27

A B C D E FABCDEF

The Node A is the only Node without Outgoing Edges

TS using Source Removal

Source Removal Implementation

29

Implementation Steps:1. Create an Empty List 2. Find a Noad without Incoming Edges3. Add this Node to the End of the List4. Remove the Edge from the Graph5. Repeat Step 1 throw 4 until the

Graph is empty

Step #1-2

30

A

E

D

B

F

C

The Node A is the only Node without Incoming Edges

Step #3-4

31

A

E

D

B

F

C

L

Step #2-3

32

A

E

D

B

F

C

L

Step #2-3

33

A

E

D

B

F

C

L

Step #2-3

34

A

E

D

B

F

C

L

Step #2-3

35

A

E

D

B

F

CL

Step #2-3

36

A

E

D

B

F

CL

Step #2-3

37

A

E

DB

F

CL

Step #2-3

38

A

E

DB

F

CL

Result:TS

39

A EDB FC

TS using Source Removal

Live Demo

TS using DFSPretty much the same

42

DFS Implementation. This algoritum is the oposite of SR

algoritum Realisation Steps:1. Create ab empty List2. Find a Noad without Outgoing Edges3. 3.Mark the Node as visited4. Reverse the List and get the TS, of

the Elements

TS using DFSLive Demo

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Questions?

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Combinatorics

http://algoacademy.telerik.com

Useful Links

Topological sorting sudo code http://

en.wikipedia.org/wiki/Topological_sorting

Graph teory http://cs.maycamp.com/?page_id=772

Source removal and DFS explanation http://

www.youtube.com/watch?v=SwN4LgOw6mo

TopCoder forum discusion about number of ways to topological sort:

http://apps.topcoder.com/forums/?module=Thread&threadID=667031&start=0&mc=10#1208539