1. ROUND ROBIN SCHEDULING BY NAGA SAI HANUMAN.POTTI

2. OUTLINE What is Round Robin. Complete graph. Real time Example. Relation to graph problem. Depicting graph solution. Graph colouring.

3. WHAT IS ROUND ROBIN? WHAT IS ROUND ROBIN? A round-robin story is one that is started by one person and then continued successively by others.It is an arrangement of choosing all elements in a group equally in some rational order in turn. A round robin format is problematic when the number of entries is high. For example, a tournament with 32 entries would take 496 games to complete using a round robin.

4. ROUND ROBIN SINGLE DOUBLE

5. A round-robin tournament is one in which every player plays against everyone else once. For example, with 3 players, we will have 3 matches: A-B, B-C, C-A. How many matches are needed for 4 players? 5 players? N players? If we can schedule two matches in the same time slot (round), how many rounds will it take for a 3- player round-robin tournament? 4- player tournament? 5-player tournament?

6. Complete graph: complete graph is a graph in which every pair of distinct vertices connected by an edge. K7::COMPLETE GRAPH WITH 7 VERTICES 1 2 3 45 6 7

7. Real time example: Lets take an indian premier league(IPL)- cricket tournament and let it consists of 4 teams. Team 0-----Deccan chargers. Team 1-----chennai super kings. Team 2-----kolkata knight riders. Team 3-----Mumbai indians.

8. Problem relating to graph: let us take a tournament with 4 teams (0,1,2,3).By using round robin we will depict the number of matches,number of rounds and the teams involved in each round. 1 2 3 0 0,1 0,2 2,3 1,3 1,2 0,3 Round Robin GraphMatches between teams

9. NUMBER OF ROUNDS We have n teams, and all teams play all others m times in m(n – 1) rounds. For single round robin: m=1 if n=4,i.e. 4 teams 3 rounds. For double round robin: m=2 if n=4,i.e. 4 teams 6 rounds.

10. 0,1 0,2 2,3 1,3 1,2 0,3 Matches between teamsCalculate number of matches Case 1:Single Round Robin For n teams =n(n-1)/2 Case 2:Double Round Robin For n teams=2*(n(n-1)/2)=n(n-1)

11. graph colouring problem or vertex colouring problem involves assigning colours to each vertex v Ɛ V such that no pair of adjacent vertices is assigned the same colour and the number of colours used is minimal. The minimum number of colours required to colour a particular graph is called the chromatic number". Graph colouring:

12. 0,1 0,2 2,3 1,3 1,2 0,3 Assigning matches in each round : 0,1 0,2 2,3 1,3 1,2 0,3

13. ROUND 1 st match 2 nd match 1 (0,1) (2,3) 0,1 0,2 2,3 1,3 1,2 0,3

14. ROUND 1 st match 2 nd match 2 (0,2) (1,3)

15. ROUND 1 st match 2 nd match 3 (0,3) (1,2)

16. ROUND 1 st match 2 nd match 1 (0,1) (2,3) 2 (0,2) (1,3) 3 (0,3) (1,2)

17. Round robin schedule: For 10 teams:

18. References  http://en.wikipedia.org/wiki/Round- robin_tournament. http://en.wikipedia.org/wiki/Round- robin_tournament  www.rhydlewis.eu/talks/sportsRRTalk.pdf. www.rhydlewis.eu/talks/sportsRRTalk.pdf  www.splendidcity.net www.splendidcity.net  www.tutorialspoint.com/...system/os_process _scheduling_algorithms.htm. www.tutorialspoint.com/...system/os_process _scheduling_algorithms.htm

21. Sports Scheduling and Round Robin Tournaments In a round robin tournament, a given collection of teams play a competition such that every two teams play each other a fixed number of times. A tournament is a directed graph which results from assigning unique directions to the edges of a complete graph.

22. Representations of Graphs as they relate to Round Robin Tournaments We can represent every tournament by a tournament T where the vertices of T correspond to the individual teams. The teams are represented by points and for each pair of points an arc is drawn from the visiting team to the home team. 12 4 3

23. Representations of Graphs as they relate to Round Robin Tournaments If a game i and j is played in the home-city of team i, it is a home game for i and an away game for j. Which can be represented by an arc (j,i). j i Likewise, if the game is played in the home city of team j, the game can be represented by an arc (i,j). j i

24. Graph Theory and Tournaments The in-degree of a tournament would refer to the number of home games a team would play. The out-degree of a tournament would refer to the number of away games a team would play.

25. Representations of Graphs as they relate to Round Robin Tournaments An oriented coloring in tournaments is obtained by partitioning the edges into n color classes such that no two adjacent edges have the same color. Such a coloring defines a schedule as the following: if arc (i,j) has color p, it means that team i and team j play against each other in the home city of team j on day p. i j Team i plays team j on the day assigned to the blue coloring. TR F B

26. Putting Everything Together: This graph represents a tournament T with four vertices. Each vertex of the graph represents an individual team, – In this graph we have four teams: Team 1, Team 2, Team 3, and Team 4 Each edge represents a competition between each team that it connects – In this graph their consists 6 edges and therefore there are 6 competitions in this tournament. For each pair of vertices an arc is drawn from the visiting team to the home team Each coloring corresponds to a specific day. 12 4 3 12 4 3