Towards a Latin-Square Search Engine

Abstract

Latin squares are combinatorial matrices that are widely used in diverse areas of research such as codes and cryptography, software testing, mathematical research, and experimental designs. All of these fields would benefit from a search engine for Latin squares. One major obstacle to developing a Latin-square search engine is that any Latin square has a large number of equivalent Latin squares, which are contained in multiple equivalence classes, and thus we need an efficient online method for canonical labelling Latin squares. Canonical labelling usually proceeds via the Nauty graph isomorphism software, but this incurs conversion costs. Moreover, the canonical labels are practically random members of their equivalence classes. A second obstacle is how large amounts of searchable Latin-square data may be stored efficiently. In this paper, we design data structures and algorithms suitable for a Latin-square search engine. We use a tree-based data structure for storing large numbers of Latin squares that also enables efficient search capabilities. We design an efficient canonical labelling algorithm (via partial Latin squares, PLSs) which does not require graph conversion, facilitates compression, and the labels are more humanly meaningful. We implement and experiment with a skeletal prototype of the Latin-square search engine. Experimental results confirm that the PLS method is faster than Nauty, and has reduced space requirements.