Generating uniformly distributed random latin squares

Abstract

By simulating an ergodic Markov chain whose stationary distribution is uniform over the space of n × n Latin squares, we can obtain squares that are (approximately) uniformly distributed; we offer two such chains. The central issue is the construction of “moves” that connect the squares. Our first approach uses the fact that an n × n Latin square is equivalent to an n × n × n contingency table in which each line sum equals 1. We relax the nonnegativity condition on the table's cells, allowing “improper” tables that have a single—1-cell. A simple set of moves connects this expanded space of tables [the diameter of the associated graph is bounded by 2(n − 1)3], and suggests a Markov chain whose subchain of proper tables has the desired uniform stationary distribution (with an average of approximately n steps between proper tables). By grouping these moves appropriately, we derive a class of moves that stay within the space of proper Latin squares [with graph diameter bounded by 4(n − 1)2]; these may also be used to form a suitable Markov chain. © 1996 John Wiley & Sons, Inc.