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Topological Sorting

Topological Sorting. Brief overview of topological sorting in graphs. Mihail Petrov. mihail-petrov.cloudvps.com. Trainer Candidate. Table of Contents. TS Why ? TS Examples and Warnings TS Algorithm explanation Source Removal Algorithm for TS TS with DFS. TS Why ?.

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Topological Sorting

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  1. Topological Sorting Brief overview of topological sorting in graphs Mihail Petrov mihail-petrov.cloudvps.com TrainerCandidate

  2. Table of Contents • TS Why ? • TS Examples and Warnings • TS Algorithm explanation • Source Removal Algorithm for TS • TS with DFS

  3. TS Why ? Why we need to sort a graph topologically

  4. University course graph Math Programing OOP Data Structures C# Java Spring JS Java EE Find the order in which all the courses can be taken !

  5. 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.

  6. TS Examples and Warnings

  7. Sample Graph #1 A C B E D F

  8. Sample Sorting #1 • Actualy we can’t sort it: • What is the order, this: A B C D E • Or this: B A E D C

  9. Sample #1 Conclusion ! We CAN NOT topologicallysort an unordered Graph

  10. Sample Graph#2 A B C# Java Script D C WPF Front-End E Production

  11. Sample Sort Rite Sort – Left Direction A B C D E Wrong Sort – Right Direction A B C E D

  12. Sample #2 Conclusion ! Any linear ordering in which an arrow goes to the left is NOT a vald solution

  13. Sample Graph#3 A B C# Java Script D C WPF Front-End E Production

  14. Sample Sort#3 A B C D E

  15. Sample #3 Conclusion ! A directed graph with a cycle CAN NOT be topologically sorted

  16. TS Algorithm General Explanation of TS

  17. Example: • We have the fouling graph, and we have to apply the TS for it. A B C E D

  18. Step #1 • Identify and Select Node with no incoming edges. A B C E D

  19. Step #1 Warning • If, no such node than: A B TPS Not Posible D C

  20. Step #2 • Remove Selectde Node A B C E D Elements:

  21. Step #2 • Repeat step#2 B C E D A Elements:

  22. Step #2 • Repeat step #2 C E D A B Elements:

  23. Step #2 • Repeat step #2 E D A B C Elements:

  24. Result: • TS Result A B C E D

  25. TS Program Implementation

  26. Graph Representation using Adjacency List A B E B D C C D D E F F E F -

  27. Graph Representation using Adjacency Matrix The Node A is the only Node without Outgoing Edges

  28. TS using Source Removal

  29. Source Removal Implementation • Implementation Steps: • Create an Empty List • Find a Noad withoutIncoming Edges • Add this Node to the End of the List • Remove the Edge from the Graph • Repeat Step 1 throw 4 until the Graph is empty

  30. Step #1-2 The Node A is the only Node without Incoming Edges B C A D E F

  31. Step #3-4 B C D E F L A

  32. Step #2-3 B C D E F L A

  33. Step #2-3 C D E F L A B

  34. Step #2-3 C D E F L A B

  35. Step #2-3 D E F L A B C

  36. Step #2-3 D E F L A B C

  37. Step #2-3 E F L A B C D

  38. Step #2-3 E F L A B C D

  39. Result:TS A B C D E F

  40. TS using Source Removal Live Demo

  41. TS using DFS Pretty much the same

  42. DFS Implementation. • This algoritum is the oposite of SR algoritum • Realisation Steps: • Create ab empty List • Find a Noad without Outgoing Edges • 3.Mark the Node as visited • Reverse the List and get the TS, of the Elements

  43. TS using DFS Live Demo

  44. Combinatorics http://algoacademy.telerik.com

  45. 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

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