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ROUND ROBIN SCHEDULING BY NAGA SAI HANUMAN.POTTI

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

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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.

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ROUND ROBIN SINGLE DOUBLE

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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?

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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

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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.

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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

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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.

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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)

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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:

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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

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

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

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

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

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Round robin schedule: For 10 teams:

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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

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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.

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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

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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

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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.

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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

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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

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