Efficient way of generating latin squares (or randomly permute numbers in matrix uniquely on both axes - using NumPy)

Question

For example, if there are 5 numbers 1, 2, 3, 4, 5

I want a random result like

[[ 2, 3, 1, 4, 5]
 [ 5, 1, 2, 3, 4]
 [ 3, 2, 4, 5, 1]
 [ 1, 4, 5, 2, 3]         


        

Answer 1

This may seem odd, but you have basically described generating a random n-dimension Sudoku puzzle. From a blog post by Daniel Beer:

The basic approach to solving a Sudoku puzzle is by a backtracking search of candidate values for each cell. The general procedure is as follows:

1. Generate, for each cell, a list of candidate values by starting with the set of all possible values and eliminating those which appear in the same row, column and box as the cell being examined.

2. Choose one empty cell. If none are available, the puzzle is solved.

3. If the cell has no candidate values, the puzzle is unsolvable.

4. For each candidate value in that cell, place the value in the cell and try to recursively solve the puzzle.

There are two optimizations which greatly improve the performance of this algorithm:

1. When choosing a cell, always pick the one with the fewest candidate values. This reduces the branching factor. As values are added to the grid, the number of candidates for other cells reduces too.

2. When analysing the candidate values for empty cells, it's much quicker to start with the analysis of the previous step and modify it by removing values along the row, column and box of the last-modified cell. This is O(N) in the size of the puzzle, whereas analysing from scratch is O(N3).

In your case an "unsolvable puzzle" is an invalid matrix. Every element in the matrix will be unique on both axis in a solvable puzzle.