# ETOOBUSY ðŸš€ minimal blogging for the impatient

# Sorting an incidence matrix

**TL;DR**

I needed to sort an incidence matrix lexicographically.

Working on a program to find a design for Steiner design S(2, 4, 28), I needed to work on the input incidence matrices to apply a constraint programming technique and find a resolution (i.e. partitioning of matches in rounds).

To ease things, my algorithm assumed that the incidence matrix was sorted lexicographically. Henceâ€¦ I needed to do this sorting.

The heart is the following function:

```
sub lexi_pass ($M) {
$M = [
map { $_->[1] }
reverse sort { $a->[0] cmp $b->[0] }
map { [join('', $_->@*), $_] } transposed($M)->@*
]
for 1 .. 2;
return $M;
} ## end sub lexi_pass ($M)
```

It does a sorting pass both by column and by row, in this order. The
sorting is actually always applied by row, but the `transposed`

function
switches the roles to work on columns too:

```
sub transposed ($M) {
my $J = $M->[0]->@* - 1;
my @T;
for my $i (0 .. $#$M) {
for my $j (0 .. $J) {
$T[$j][$i] = $M->[$i][$j];
}
}
return \@T;
} ## end sub transposed ($M)
```

For reasons that I didnâ€™t investigate a single pass is not sufficient, hence I embedded it in a loop to repeat one more pass until we reach a stable situation. The same function also takes care to read and parse the input incidence matrix, provided as text:

```
sub lexi_parse ($incidence_text) {
$incidence_text =~ s{\s+\z}{}mxs;
my $incidence = [map { [split m{}mxs] } split m{\s+}, $incidence_text];
while ('necessary') {
$incidence = lexi_pass($incidence);
my $new = join "\n", map { join '', $_->@* } $incidence->@*;
return $incidence if $new eq $incidence_text;
$incidence_text = $new;
} ## end while ('necessary')
} ## end sub lexi_parse ($incidence_text)
```

After each pass, the incidence matrix text is reconstructed and compared to the previous iteration. When there is no change, we call it a day and return.

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