Randomize matrix in perl, keeping row and column totals the same

Question

I have a matrix that I want to randomize a couple of thousand times, while keeping the row and column totals the same:

     1 2 3 
   A 0 0 1 
   B 1 1 0 
   C 1 0 0      

An example of a valid random matrix would be:

     1 2 3
   A 1 0 0
   B 1 1 0
   C 0 0 1

My actual matrix is a lot bigger (about 600x600 items), so I really need an approach that is computationally efficient.

My initial (inefficient) approach consisted of shuffling arrays using the Perl Cookbook shuffle

I pasted my current code below. I've got extra code in place to start with a new shuffled list of numbers, if no solution is found in the while loop. The algorithm works fine for a small matrix, but as soon as I start scaling up it takes forever to find a random matrix that fits the requirements.

Is there a more efficient way to accomplish what I'm searching for? Thanks a lot!

#!/usr/bin/perl -w
use strict;

my %matrix = ( 'A' => {'3'  => 1 },
           'B' => {'1'  => 1,
               '2'  => 1 },
           'C' => {'1'  => 1 }
    );

my @letters = ();
my @numbers = ();

foreach my $letter (keys %matrix){
    foreach my $number (keys %{$matrix{$letter}}){
    push (@letters, $letter);
    push (@numbers, $number);
    }
}

my %random_matrix = ();

&shuffle(\@numbers);
foreach my $letter (@letters){
    while (exists($random_matrix{$letter}{$numbers[0]})){
    &shuffle (\@numbers);
    }
    my $chosen_number = shift (@numbers);
    $random_matrix{$letter}{$chosen_number} = 1;
}

sub shuffle {
    my $array = shift;
    my $i = scalar(@$array);
    my $j;
    foreach my $item (@$array )
    {
        --$i;
        $j = int rand ($i+1);
        next if $i == $j;
        @$array [$i,$j] = @$array[$j,$i];
    }
    return @$array;
}

Answer 1

The problem with your current algorithm is that you are trying to shuffle your way out of dead ends -- specifically, when your @letters and @numbers arrays (after the initial shuffle of @numbers) yield the same cell more than once. That approach works when the matrix is small, because it doesn't take too many tries to find a viable re-shuffle. However, it's a killer when the lists are big. Even if you could hunt for alternatives more efficiently -- for example, trying permutations rather than random shuffling -- the approach is probably doomed.

Rather than shuffling entire lists, you might tackle the problem by making small modifications to an existing matrix.

For example, let's start with your example matrix (call it M1). Randomly pick one cell to change (say, A1). At this point the matrix is in an illegal state. Our goal will be to fix it in the minimum number of edits -- specifically 3 more edits. You implement these 3 additional edits by "walking" around the matrix, with each repair of a row or column yielding another problem to be solved, until you have walked full circle (err ... full rectangle).

For example, after changing A1 from 0 to 1, there are 3 ways to walk for the next repair: A3, B1, and C1. Let's decide that the 1st edit should fix rows. So we pick A3. On the second edit, we will fix the column, so we have choices: B3 or C3 (say, C3). The final repair offers only one choice (C1), because we need to return to the column of our original edit. The end result is a new, valid matrix.

    Orig         Change A1     Change A3     Change C3     Change C1
    M1                                                     M2

    1 2 3        1 2 3         1 2 3         1 2 3         1 2 3
    -----        -----         -----         -----         -----
A | 0 0 1        1 0 1         1 0 0         1 0 0         1 0 0
B | 1 1 0        1 1 0         1 1 0         1 1 0         1 1 0
C | 1 0 0        1 0 0         1 0 0         1 0 1         0 0 1

If an editing path leads to a dead end, you backtrack. If all of the repair paths fail, the initial edit can be rejected.

This approach will generate new, valid matrixes quickly. It will not necessarily produce random outcomes: M1 and M2 will still be highly correlated with each other, a point that will become more directly evident as the size of the matrix grows.

How do you increase the randomness? You mentioned that most cells (99% or more) are zeros. One idea would be to proceed like this: for each 1 in the matrix, set its value to 0 and then repair the matrix using the 4-edit method outlined above. In effect, you would be moving all of the ones to new, random locations.

Here is an illustration. There are probably further speed optimizations in here, but this approach yielded 10 new 600x600 matrixes, at 0.5% density, in 30 seconds or so on my Windows box. Don't know if that's fast enough.

use strict;
use warnings;

# Args: N rows, N columns, density, N iterations.
main(@ARGV);

sub main {
    my $n_iter = pop;
    my $matrix = init_matrix(@_);
    print_matrix($matrix);
    for my $n (1 .. $n_iter){
        warn $n, "\n"; # Show progress.
        edit_matrix($matrix);
        print_matrix($matrix);
    }
}

sub init_matrix {
    # Generate initial matrix, given N of rows, N of cols, and density.
    my ($rows, $cols, $density) = @_;
    my @matrix;
    for my $r (1 .. $rows){
        push @matrix, [ map { rand() < $density ? 1 : 0  } 1 .. $cols ];
    }
    return \@matrix;
}

sub print_matrix {
    # Dump out a matrix for checking.
    my $matrix = shift;
    print "\n";
    for my $row (@$matrix){
        my @vals = map { $_ ? 1 : ''} @$row;
        print join("\t", @vals), "\n";
    }
}

sub edit_matrix {
    # Takes a matrix and moves all of the non-empty cells somewhere else.
    my $matrix = shift;
    my $move_these = cells_to_move($matrix);
    for my $cell (@$move_these){
        my ($i, $j) = @$cell;
        # Move the cell, provided that the cell hasn't been moved
        # already and the subsequent edits don't lead to a dead end.
        $matrix->[$i][$j] = 0
            if $matrix->[$i][$j]
            and other_edits($matrix, $cell, 0, $j);
    }
}

sub cells_to_move {
    # Returns a list of non-empty cells.
    my $matrix = shift;
    my $i = -1;
    my @cells = ();
    for my $row (@$matrix){
        $i ++;
        for my $j (0 .. @$row - 1){
            push @cells, [$i, $j] if $matrix->[$i][$j];
        }
    }
    return \@cells;
}

sub other_edits {
    my ($matrix, $cell, $step, $last_j) = @_;

    # We have succeeded if we've already made 3 edits.
    $step ++;
    return 1 if $step > 3;

    # Determine the roster of next edits to fix the row or
    # column total upset by our prior edit.
    my ($i, $j) = @$cell;
    my @fixes;
    if ($step == 1){
        @fixes = 
            map  { [$i, $_] }
            grep { $_ != $j and not $matrix->[$i][$_] }
            0 .. @{$matrix->[0]} - 1
        ;
        shuffle(\@fixes);
    }
    elsif ($step == 2) {
        @fixes = 
            map  { [$_, $j] }
            grep { $_ != $i and $matrix->[$_][$j] }
            0 .. @$matrix - 1
        ;
        shuffle(\@fixes);
    }
    else {
        # On the last edit, the column of the fix must be
        # the same as the column of the initial edit.
        @fixes = ([$i, $last_j]) unless $matrix->[$i][$last_j];
    }

    for my $f (@fixes){
        # If all subsequent fixes succeed, we are golden: make
        # the current fix and return true.
        if ( other_edits($matrix, [@$f], $step, $last_j) ){
            $matrix->[$f->[0]][$f->[1]] = $step == 2 ? 0 : 1;
            return 1;
        }
    }

    # Failure if we get here.
    return;
}

sub shuffle {
    my $array = shift;
    my $i = scalar(@$array);
    my $j;
    for (@$array ){
        $i --;
        $j = int rand($i + 1);
        @$array[$i, $j] = @$array[$j, $i] unless $i == $j;
    }
}

Answer 2

Step 1: First I would initialize the matrix to zeros and calculate the required row and column totals.

Step 2: Now pick a random row, weighted by the count of 1s that must be in that row (so a row with count 300 is more likely to be picked than a row with weight 5).

Step 3: For this row, pick a random column, weighted by the count of 1s in that column (except ignore any cells that may already contain a 1 - more on this later).

Step 4: Place a one in this cell and reduce both the row and column count for the appropriate row and column.

Step 5: Go back to step 2 until no rows have non-zero count.

The problem though is that this algorithm can fail to terminate because you may have a row where you need to place a one, and a column that needs a one, but you've already placed a one in that cell, so you get 'stuck'. I'm not sure how likely this is to happen, but I wouldn't be surprised if it happened very frequently - enough to make the algorithm unusable. If this is a problem I can think of two ways to fix it:

a) Construct the above algorithm recursively and allow backtracking on failure.

b) Allow a cell to contain a value greater than 1 if there is no other option and keep going. Then at the end you have a correct row and column count but some cells may contain numbers greater than 1. You can fix this by finding a grouping that looks like this:

2 . . . . 0
. . . . . .
. . . . . .
0 . . . . 1

and changing it to:

1 . . . . 1
. . . . . .
. . . . . .
1 . . . . 0

It should be easy to find such a grouping if you have many zeros. I think b) is likely to be faster.

I'm not sure it's the best way, but it's probably faster than shuffling arrays. I'll be tracking this question to see what other people come up with.

Answer 3

I'm not a mathematician, but I figure that if you need to keep the same column and row totals, then random versions of the matrix will have the same quantity of ones and zeros.

Correct me if I'm wrong, but that would mean that making subsequent versions of the matrix would only require you to shuffle around the rows and columns.

Randomly shuffling columns won't change your totals for rows and columns, and randomly shuffling rows won't either. So, what I would do, is first shuffle rows, and then shuffle columns.

That should be pretty fast.