This module creates Latin squares and pairs of orthogonal Latin squares. Where possible, simple number-theoretic constructions are used. Otherwise, integer programming.
New in version 0.4: The default solver is now
HiGHSwith non-verbose output.
To create a simple n-by-n Latin square, use latin(n):
julia> using LatinSquares
julia> latin(5)
5×5 Array{Int64,2}:
1 2 3 4 5
2 3 4 5 1
3 4 5 1 2
4 5 1 2 3
5 1 2 3 4
To create a pair of n-by-n orthogonal Latin squares, use ortho_latin(n).
julia> A,B = ortho_latin(5);
julia> 10A+B
5×5 Array{Int64,2}:
11 22 33 44 55
23 34 45 51 12
35 41 52 13 24
42 53 14 25 31
54 15 21 32 43
Or to see the two in Latin and Greek letters:
julia> print_latin(A,B)
Aα Bβ Cγ Dδ Eε
Bγ Cδ Dε Eα Aβ
Cε Dα Eβ Aγ Bδ
Dβ Eγ Aδ Bε Cα
Eδ Aε Bα Cβ Dγ
By default, we use a simple number-theoretic construction. When that fails, we switch to integer programming.
julia> A,B = ortho_latin(4);
No quick solution. Using integer programming.
julia> 10A+B
4×4 Array{Int64,2}:
43 11 34 22
14 42 23 31
32 24 41 13
21 33 12 44
A Latin square is self orthogonal provided it is orthogonal to
its transpose. Use ortho_latin(n,true) to create such a self
orthogonal Latin square.
julia> A,B = ortho_latin(5,true);
julia> 10A+B
5×5 Array{Int64,2}:
11 54 43 32 25
45 33 51 24 12
34 15 22 41 53
23 42 14 55 31
52 21 35 13 44
julia> A==B'
true
There does not exist a pair of 6-by-6 orthogonal Latin squares, and this verifies that fact:
julia> A,B = ortho_latin(6);
No quick solution. Using integer programming.
ERROR: No pair of orthogonal Latin squares of order 6 can be found.
In the src directory, the file run_latin.jl allows the user to find
orthogonal Latin squares from the command line. The synatx is
julia run_julia.jl n.
Long-running jobs can be conveniently sent to a file like this:
$ nohup julia run_latin.jl 8 > output.txt &
Using the Gurobi solver, we can find a pair of 10-by-10 orthogonal Latin square in a matter of hours. Here's the result:
Aα Bβ Cγ Dδ Eε Fζ Gη Hθ Iι Jκ
Bγ Iδ Hζ Eθ Aη Jα Dι Cκ Fε Gβ
Gι Cε Iα Fκ Hδ Eβ Bθ Jζ Dη Aγ
Hκ Dα Fη Aβ Gγ Cθ Iε Bι Jδ Eζ
Iβ Fγ Aε Jη Dθ Gδ Cζ Eα Bκ Hι
Jε Aζ Gθ Hγ Fι Dκ Eδ Iη Cβ Bα
Dζ Eι Bδ Gα Iκ Hε Jγ Fβ Aθ Cη
Cδ Hη Eκ Bε Jβ Aι Fα Dγ Gζ Iθ
Eη Jθ Dβ Cι Bζ Iγ Aκ Gε Hα Fδ
Fθ Gκ Jι Iζ Cα Bη Hβ Aδ Eγ Dε
Use the ChooseOptimizer module to select an alternative solver.
We use the HiGHS solver. If you have Gurobi on your system, that solver will run much faster. In that case, do this to switch solver.
julia> using Gurobi, LatinSquares, ChooseOptimizer
julia> set_solver(Gurobi)
GurobiSolver
julia> A,B = ortho_latin(6)
No quick solution. Using integer programming.
Academic license - for non-commercial use only
ERROR: No pair of orthogonal Latin squares of order 6 can be found.