LatinSquares
This module creates Latin squares and pairs of orthogonal Latin squares. Where possible, simple number-theoretic constructions are used. Otherwise, integer programming.
Usage
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
Self orthogonal Latin squares
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
No pair of orthogonal Latin squares of order 6
There does not exist a pair of 6-by-6 orthogonal Latin squares, and this verifies that fact:
julia> A,B = ortho_latin(6);
However, the run time with the Cbc solver is very long. Changing the code to use the Gurobi solver makes this calculation feasible.
Command Line
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 &
Example
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
Other Solvers
Use the ChooseOptimizer module to select an alternative solver.
We use the Cbc 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
Academic license - for non-commercial use only
Optimize a model with 222 rows, 1296 columns and 7782 nonzeros
Variable types: 0 continuous, 1296 integer (1296 binary)
Coefficient statistics:
Matrix range [1e+00, 1e+00]
Objective range [0e+00, 0e+00]
Bounds range [0e+00, 0e+00]
RHS range [1e+00, 1e+00]
Presolve removed 42 rows and 696 columns
Presolve time: 0.01s
Presolved: 180 rows, 600 columns, 3600 nonzeros
Variable types: 0 continuous, 600 integer (600 binary)
Root relaxation: objective 0.000000e+00, 245 iterations, 0.01 seconds
Nodes | Current Node | Objective Bounds | Work
Expl Unexpl | Obj Depth IntInf | Incumbent BestBd Gap | It/Node Time
0 0 0.00000 0 135 - 0.00000 - - 0s
0 0 0.00000 0 155 - 0.00000 - - 0s
0 0 0.00000 0 122 - 0.00000 - - 0s
0 0 0.00000 0 131 - 0.00000 - - 0s
0 0 0.00000 0 26 - 0.00000 - - 0s
0 2 0.00000 0 26 - 0.00000 - - 0s
Explored 1536 nodes (52753 simplex iterations) in 3.18 seconds
Thread count was 4 (of 4 available processors)
Solution count 0
Model is infeasible
Best objective -, best bound -, gap -
ERROR: No pair of orthogonal Latin squares of order 6 can be found.