# Round Robin Tournament Scheduling

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by Richard A. DeVenezia, Back to Home

Just looking for tables ? Go here...

The problem of scheduling a tournament in which all participants compete against each other one-on-one in a series of rounds probably goes back to pre-history. This page presents some thoughts of the author on the problem and some software to explore various ways to look at it.

The terminology of a round robin may differ according to it's application. The pairing of two items (one-on-one) might be known as a game, match, outing. An item that is paired might be known as a player or team. A round might be known as a week or meet.

## Whist

Whist tables for 4n people.

A whist tournament is a variation of a round robin. In a whist tournament the team a player is on varies over the course of the tournament. By the end of the tournament each player has been teamed with each player one time, and opposed each player two times. It might help to think of round robins as based on pairs, and whist as based on pairs of pairs.

An examination of whist as social teams of 2 for 12 players

Find additional information and schedules (1 to 24 Bridge tables, 4n or 4n+1 players) at Durango Bill's website.

## Cyclic algorithm

Cyclic tables

While updating the first fit algorithm (see below) I did some newsgroup searches and turned up a cyclic algorithm for scheduling. One item is locked while the others rotate. At each step of the rotation the round is planned by pairing items.

• sci.math.num-analysis newsgroup thread discussing planning and proving a round robin algorithm
• Some javascript for visualizing the cyclic algorithm
• Java applet based on the Ron Shepard newsgroup post
• Visualize the cyclic algorithm for ten items:
Press 'Go' to start rotating the reddish cells clockwise.
 3 4 5 6 1 2 10 9 8 7 Pair 1 Pair 2 Pair 3 Pair 4 Pair 5 0

## Balanced schedules

Balanced tables, balancing Home and Away.

Over the course of a balanced schedule everyone will play in every match place and on both sides of each match place. No one will have unfair advantage due to repeated placement at a specific match place or side. Because there is one less round than players, each player will be in one match place where they play on just one side.

## First fit algorithm

First fit tables

My first analysis of the problem (and by no means complete, succinct, accurate or appropriate) was in 1991. The analysis looked at the problem from the standpoint of a first fitting pair algorithm that would scan lists of pairs and determine their eligibility for inclusion in a round. The products of this analysis are thus:

• Mathematical musings on the problem
• Original C source code implementing the first fit algorithm (has since been improved)
• The Scheduler, a windows application written by the author.
• Tournament tables, captured schedules from "The Scheduler" presented on the rounds page.
 Show me N= 2 4 6 8 10 12 14 16 18 20 22 24 26 28 32
• Some interesting edge graphs of the schedules.

## Social squares

Plan a schedule for M teams of M players. In each round all M teams will play. M+1 rounds will be played. In each round each player is teamed with all new people; in other words, each player is teamed with each other player only once.

## Team play

Plan a tournament for 12 players. Two person teams. Everyone gets teamed with everyone else one time. Everyone plays against a team having everyone else on it twice. In combinatoric literature this is a Whist tournament.

## Tidbits

• Eelke Spaak's Tournament Scheduler page can generate schedules using names and locations you enter. I like the ease of use and minimalist interface.
• If you have an odd number of players, invite a ghost to play. Whoever plays the ghost gets a bye.
• Random House dictionary word of the day, April 26, 2001. Link contributed by Ian Wakeling.
• Why is it called Round-Robin anyway?
I have read various sources that state the term first appeared to describe how people would sign documents of grievance in a circular fashion to preserve anonymity of status within the group. E.g. Naval officers ( see here ) or 17th-century French politicians. Back in those days of yore it seems the head griever quickly became headless.

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