package Algorithm::Combinatorics;
use 5.006002;
use strict;
our $VERSION = '0.27';
use XSLoader;
XSLoader::load('Algorithm::Combinatorics', $VERSION);
use Carp;
use Scalar::Util qw(reftype);
use Exporter;
use base 'Exporter';
our @EXPORT_OK = qw(
combinations
combinations_with_repetition
variations
variations_with_repetition
tuples
tuples_with_repetition
permutations
circular_permutations
derangements
complete_permutations
partitions
subsets
);
our %EXPORT_TAGS = (all => [ @EXPORT_OK ]);
sub combinations {
my ($data, $k) = @_;
__check_params($data, $k);
return __contextualize(__null_iter()) if $k < 0;
return __contextualize(__once_iter()) if $k == 0;
if ($k > @$data) {
carp("Parameter k is greater than the size of data");
return __contextualize(__null_iter());
}
my @indices = 0..($k-1);
my $iter = Algorithm::Combinatorics::Iterator->new(sub {
__next_combination(\@indices, @$data-1) == -1 ? undef : [ @{$data}[@indices] ];
}, [ @{$data}[@indices] ]);
return __contextualize($iter);
}
sub combinations_with_repetition {
my ($data, $k) = @_;
__check_params($data, $k);
return __contextualize(__null_iter()) if $k < 0;
return __contextualize(__once_iter()) if $k == 0;
my @indices = (0) x $k;
my $iter = Algorithm::Combinatorics::Iterator->new(sub {
__next_combination_with_repetition(\@indices, @$data-1) == -1 ? undef : [ @{$data}[@indices] ];
}, [ @{$data}[@indices] ]);
return __contextualize($iter);
}
sub subsets {
my ($data, $k) = @_;
__check_params($data, $k, 1);
return combinations($data, $k) if defined $k;
my $finished = 0;
my @odometer = (1) x @$data;
my $iter = Algorithm::Combinatorics::Iterator->new(sub {
return if $finished;
my $subset = __next_subset($data, \@odometer);
$finished = 1 if @$subset == 0;
$subset;
});
return __contextualize($iter);
}
sub variations {
my ($data, $k) = @_;
__check_params($data, $k);
return __contextualize(__null_iter()) if $k < 0;
return __contextualize(__once_iter()) if $k == 0;
if ($k > @$data) {
carp("Parameter k is greater than the size of data");
return __contextualize(__null_iter());
}
# permutations() is more efficient because it knows
# all indices are always used
return permutations($data) if @$data == $k;
my @indices = 0..($k-1);
my @used = ((1) x $k, (0) x (@$data-$k));
my $iter = Algorithm::Combinatorics::Iterator->new(sub {
__next_variation(\@indices, \@used, @$data-1) == -1 ? undef : [ @{$data}[@indices] ];
}, [ @{$data}[@indices] ]);
return __contextualize($iter);
}
*tuples = \&variations;
sub variations_with_repetition {
my ($data, $k) = @_;
__check_params($data, $k);
return __contextualize(__null_iter()) if $k < 0;
return __contextualize(__once_iter()) if $k == 0;
my @indices = (0) x $k;
my $iter = Algorithm::Combinatorics::Iterator->new(sub {
__next_variation_with_repetition(\@indices, @$data-1) == -1 ? undef : [ @{$data}[@indices] ];
}, [ @{$data}[@indices] ]);
return __contextualize($iter);
}
*tuples_with_repetition = \&variations_with_repetition;
sub __variations_with_repetition_gray_code {
my ($data, $k) = @_;
__check_params($data, $k);
return __contextualize(__null_iter()) if $k < 0;
return __contextualize(__once_iter()) if $k == 0;
my @indices = (0) x $k;
my @focus_pointers = 0..$k; # yeah, length $k+1
my @directions = (1) x $k;
my $iter = Algorithm::Combinatorics::Iterator->new(sub {
__next_variation_with_repetition_gray_code(
\@indices,
\@focus_pointers,
\@directions,
@$data-1,
) == -1 ? undef : [ @{$data}[@indices] ];
}, [ @{$data}[@indices] ]);
return __contextualize($iter);
}
sub permutations {
my ($data) = @_;
__check_params($data, 0);
return __contextualize(__once_iter()) if @$data == 0;
my @indices = 0..(@$data-1);
my $iter = Algorithm::Combinatorics::Iterator->new(sub {
__next_permutation(\@indices) == -1 ? undef : [ @{$data}[@indices] ];
}, [ @{$data}[@indices] ]);
return __contextualize($iter);
}
sub circular_permutations {
my ($data) = @_;
__check_params($data, 0);
return __contextualize(__once_iter()) if @$data == 0;
return __contextualize(__once_iter([@$data])) if @$data == 1 || @$data == 2;
my @indices = 1..(@$data-1);
my $iter = Algorithm::Combinatorics::Iterator->new(sub {
__next_permutation(\@indices) == -1 ? undef : [ @{$data}[0, @indices] ];
}, [ @{$data}[0, @indices] ]);
return __contextualize($iter);
}
sub __permutations_heap {
my ($data) = @_;
__check_params($data, 0);
return __contextualize(__once_iter()) if @$data == 0;
my @a = 0..(@$data-1);
my @c = (0) x (@$data+1); # yeah, there's an spurious $c[0] to make the notation coincide
my $iter = Algorithm::Combinatorics::Iterator->new(sub {
__next_permutation_heap(\@a, \@c) == -1 ? undef : [ @{$data}[@a] ];
}, [ @{$data}[@a] ]);
return __contextualize($iter);
}
sub derangements {
my ($data) = @_;
__check_params($data, 0);
return __contextualize(__once_iter()) if @$data == 0;
return __contextualize(__null_iter()) if @$data == 1;
my @indices = 0..(@$data-1);
@indices[$_, $_+1] = @indices[$_+1, $_] for map { 2*$_ } 0..((@$data-2)/2);
@indices[-1, -2] = @indices[-2, -1] if @$data % 2;
my $iter = Algorithm::Combinatorics::Iterator->new(sub {
__next_derangement(\@indices) == -1 ? undef : [ @{$data}[@indices] ];
}, [ @{$data}[@indices] ]);
return __contextualize($iter);
}
*complete_permutations = \&derangements;
sub partitions {
my ($data, $k) = @_;
if (defined $k) {
__partitions_of_size_p($data, $k);
} else {
__partitions_of_all_sizes($data);
}
}
sub __partitions_of_all_sizes {
my ($data) = @_;
__check_params($data, 0);
return __contextualize(__once_iter()) if @$data == 0;
my @k = (0) x @$data;
my @M = (0) x @$data;
my $iter = Algorithm::Combinatorics::Iterator->new(sub {
__next_partition(\@k, \@M) == -1 ? undef : __slice_partition(\@k, \@M, $data);
}, __slice_partition(\@k, \@M, $data));
return __contextualize($iter);
}
# We use @k and $p here and sacrifice the uniform usage of $k
# to follow the notation in [3].
sub __partitions_of_size_p {
my ($data, $p) = @_;
__check_params($data, $p);
return __contextualize(__null_iter()) if $p < 0;
return __contextualize(__once_iter()) if @$data == 0 && $p == 0;
return __contextualize(__null_iter()) if $p == 0;
if ($p > @$data) {
carp("Parameter k is greater than the size of data");
return __contextualize(__null_iter());
}
my $q = @$data - $p + 1;
my @k = (0) x $q;
my @M = (0) x $q;
push @k, $_ - $q + 1 for $q..(@$data-1);
push @M, $_ - $q + 1 for $q..(@$data-1);
my $iter = Algorithm::Combinatorics::Iterator->new(sub {
__next_partition_of_size_p(\@k, \@M, $p) == -1 ? undef : __slice_partition_of_size_p(\@k, $p, $data);
}, __slice_partition_of_size_p(\@k, $p, $data));
return __contextualize($iter);
}
sub __slice_partition {
my ($k, $M, $data) = @_;
my @partition = ();
my $size = $M->[-1] + 1; # $M->[0] is always 0 in our code
push @partition, [] for 1..$size;
my $i = 0;
foreach my $x (@$data) {
push @{$partition[$k->[$i]]}, $x;
++$i;
}
return \@partition;
}
# We use @k and $p here and sacrifice the uniform usage of $k
# to follow the notation in [3].
sub __slice_partition_of_size_p {
my ($k, $p, $data) = @_;
my @partition = ();
push @partition, [] for 1..$p;
my $i = 0;
foreach my $x (@$data) {
push @{$partition[$k->[$i]]}, $x;
++$i;
}
return \@partition;
}
sub __check_params {
my ($data, $k, $k_is_not_required) = @_;
if (not defined $data) {
croak("Missing parameter data");
}
unless ($k_is_not_required || defined $k) {
croak("Missing parameter k");
}
my $type = reftype $data;
if (!defined($type) || $type ne "ARRAY") {
croak("Parameter data is not an arrayref");
}
carp("Parameter k is negative") if !$k_is_not_required && $k < 0;
}
# Given an iterator that responds to the next() method this
# subrutine returns the iterator in scalar context, loops
# over the iterator to build and return an array of results
# in list context, and does nothing but issue a warning in
# void context.
sub __contextualize {
my $iter = shift;
my $w = wantarray;
if (defined $w) {
if ($w) {
my @result = ();
while (my $c = $iter->next) {
push @result, $c;
}
return @result;
} else {
return $iter;
}
} else {
my $sub = (caller(1))[3];
carp("Useless use of $sub in void context");
}
}
sub __null_iter {
return Algorithm::Combinatorics::Iterator->new(sub { return });
}
sub __once_iter {
my $tuple = shift;
$tuple ? Algorithm::Combinatorics::Iterator->new(sub { return }, $tuple) :
Algorithm::Combinatorics::Iterator->new(sub { return }, []);
}
# This is a bit dirty by now, the objective is to be able to
# pass an initial sequence to the iterator and avoid a test
# in each iteration saying whether the sequence was already
# returned or not, since that might potentially be done a lot
# of times.
#
# The solution is to return an iterator that has a first sequence
# associated. The first time you call it that sequence is returned
# and the iterator rebless itself to become just a wrapped coderef.
#
# Note that the public contract is that responds to next(), no
# iterator class name is documented.
package Algorithm::Combinatorics::Iterator;
sub new {
my ($class, $coderef, $first_seq) = @_;
if (defined $first_seq) {
return bless [$coderef, $first_seq], $class;
} else {
return bless $coderef, 'Algorithm::Combinatorics::JustCoderef';
}
}
sub next {
my ($self) = @_;
$_[0] = $self->[0];
bless $_[0], 'Algorithm::Combinatorics::JustCoderef';
return $self->[1];
}
package Algorithm::Combinatorics::JustCoderef;
sub next {
my ($self) = @_;
return $self->();
}
1;
__END__
=head1 NAME
Algorithm::Combinatorics - Efficient generation of combinatorial sequences
=head1 SYNOPSIS
use Algorithm::Combinatorics qw(permutations);
my @data = qw(a b c);
# scalar context gives an iterator
my $iter = permutations(\@data);
while (my $p = $iter->next) {
# ...
}
# list context slurps
my @all_permutations = permutations(\@data);
=head1 VERSION
This documentation refers to Algorithm::Combinatorics version 0.26.
=head1 DESCRIPTION
Algorithm::Combinatorics is an efficient generator of combinatorial sequences. Algorithms are selected from the literature (work in progress, see L</REFERENCES>). Iterators do not use recursion, nor stacks, and are written in C.
Tuples are generated in lexicographic order, except in C<subsets()>.
=head1 SUBROUTINES
Algorithm::Combinatorics provides these subroutines:
permutations(\@data)
circular_permutations(\@data)
derangements(\@data)
complete_permutations(\@data)
variations(\@data, $k)
variations_with_repetition(\@data, $k)
tuples(\@data, $k)
tuples_with_repetition(\@data, $k)
combinations(\@data, $k)
combinations_with_repetition(\@data, $k)
partitions(\@data[, $k])
subsets(\@data[, $k])
All of them are context-sensitive:
=over 4
=item *
In scalar context subroutines return an iterator that responds to the C<next()> method. Using this object you can iterate over the sequence of tuples one by one this way:
my $iter = combinations(\@data, $k);
while (my $c = $iter->next) {
# ...
}
The C<next()> method returns an arrayref to the next tuple, if any, or C<undef> if the
sequence is exhausted.
Memory usage is minimal, no recursion and no stacks are involved.
=item *
In list context subroutines slurp the entire set of tuples. This behaviour is offered
for convenience, but take into account that the resulting array may be really huge:
my @all_combinations = combinations(\@data, $k);
=back
=head2 permutations(\@data)
The permutations of C<@data> are all its reorderings. For example, the permutations of C<@data = (1, 2, 3)> are:
(1, 2, 3)
(1, 3, 2)
(2, 1, 3)
(2, 3, 1)
(3, 1, 2)
(3, 2, 1)
The number of permutations of C<n> elements is:
n! = 1, if n = 0
n! = n*(n-1)*...*1, if n > 0
See some values at L<http://www.research.att.com/~njas/sequences/A000142>.
=head2 circular_permutations(\@data)
The circular permutations of C<@data> are its arrangements around a circle, where only relative order of elements matter, rather than their actual position. Think possible arrangements of people around a circular table for dinner according to whom they have to their right and left, no matter the actual chair they sit on.
For example the circular permutations of C<@data = (1, 2, 3, 4)> are:
(1, 2, 3, 4)
(1, 2, 4, 3)
(1, 3, 2, 4)
(1, 3, 4, 2)
(1, 4, 2, 3)
(1, 4, 3, 2)
The number of circular permutations of C<n> elements is:
n! = 1, if 0 <= n <= 1
(n-1)! = (n-1)*(n-2)*...*1, if n > 1
See a few numbers in a comment of L<http://www.research.att.com/~njas/sequences/A000142>.
=head2 derangements(\@data)
The derangements of C<@data> are those reorderings that have no element
in its original place. In jargon those are the permutations of C<@data>
with no fixed points. For example, the derangements of C<@data = (1, 2,
3)> are:
(2, 3, 1)
(3, 1, 2)
The number of derangements of C<n> elements is:
d(n) = 1, if n = 0
d(n) = n*d(n-1) + (-1)**n, if n > 0
See some values at L<http://www.research.att.com/~njas/sequences/A000166>.
=head2 complete_permutations(\@data)
This is an alias for C<derangements>, documented above.
=head2 variations(\@data, $k)
The variations of length C<$k> of C<@data> are all the tuples of length C<$k> consisting of elements of C<@data>. For example, for C<@data = (1, 2, 3)> and C<$k = 2>:
(1, 2)
(1, 3)
(2, 1)
(2, 3)
(3, 1)
(3, 2)
For this to make sense, C<$k> has to be less than or equal to the length of C<@data>.
Note that
permutations(\@data);
is equivalent to
variations(\@data, scalar @data);
The number of variations of C<n> elements taken in groups of C<k> is:
v(n, k) = 1, if k = 0
v(n, k) = n*(n-1)*...*(n-k+1), if 0 < k <= n
=head2 variations_with_repetition(\@data, $k)
The variations with repetition of length C<$k> of C<@data> are all the tuples of length C<$k> consisting of elements of C<@data>, including repetitions. For example, for C<@data = (1, 2, 3)> and C<$k = 2>:
(1, 1)
(1, 2)
(1, 3)
(2, 1)
(2, 2)
(2, 3)
(3, 1)
(3, 2)
(3, 3)
Note that C<$k> can be greater than the length of C<@data>. For example, for C<@data = (1, 2)> and C<$k = 3>:
(1, 1, 1)
(1, 1, 2)
(1, 2, 1)
(1, 2, 2)
(2, 1, 1)
(2, 1, 2)
(2, 2, 1)
(2, 2, 2)
The number of variations with repetition of C<n> elements taken in groups of C<< k >= 0 >> is:
vr(n, k) = n**k
=head2 tuples(\@data, $k)
This is an alias for C<variations>, documented above.
=head2 tuples_with_repetition(\@data, $k)
This is an alias for C<variations_with_repetition>, documented above.
=head2 combinations(\@data, $k)
The combinations of length C<$k> of C<@data> are all the sets of size C<$k> consisting of elements of C<@data>. For example, for C<@data = (1, 2, 3, 4)> and C<$k = 3>:
(1, 2, 3)
(1, 2, 4)
(1, 3, 4)
(2, 3, 4)
For this to make sense, C<$k> has to be less than or equal to the length of C<@data>.
The number of combinations of C<n> elements taken in groups of C<< 0 <= k <= n >> is:
n choose k = n!/(k!*(n-k)!)
=head2 combinations_with_repetition(\@data, $k);
The combinations of length C<$k> of an array C<@data> are all the bags of size C<$k> consisting of elements of C<@data>, with repetitions. For example, for C<@data = (1, 2, 3)> and C<$k = 2>:
(1, 1)
(1, 2)
(1, 3)
(2, 2)
(2, 3)
(3, 3)
Note that C<$k> can be greater than the length of C<@data>. For example, for C<@data = (1, 2, 3)> and C<$k = 4>:
(1, 1, 1, 1)
(1, 1, 1, 2)
(1, 1, 1, 3)
(1, 1, 2, 2)
(1, 1, 2, 3)
(1, 1, 3, 3)
(1, 2, 2, 2)
(1, 2, 2, 3)
(1, 2, 3, 3)
(1, 3, 3, 3)
(2, 2, 2, 2)
(2, 2, 2, 3)
(2, 2, 3, 3)
(2, 3, 3, 3)
(3, 3, 3, 3)
The number of combinations with repetition of C<n> elements taken in groups of C<< k >= 0 >> is:
n+k-1 over k = (n+k-1)!/(k!*(n-1)!)
=head2 partitions(\@data[, $k])
A partition of C<@data> is a division of C<@data> in separate pieces. Technically that's a set of subsets of C<@data> which are non-empty, disjoint, and whose union is C<@data>. For example, the partitions of C<@data = (1, 2, 3)> are:
((1, 2, 3))
((1, 2), (3))
((1, 3), (2))
((1), (2, 3))
((1), (2), (3))
This subroutine returns in consequence tuples of tuples. The top-level tuple (an arrayref) represents the partition itself, whose elements are tuples (arrayrefs) in turn, each one representing a subset of C<@data>.
The number of partitions of a set of C<n> elements are known as Bell numbers, and satisfy the recursion:
B(0) = 1
B(n+1) = (n over 0)B(0) + (n over 1)B(1) + ... + (n over n)B(n)
See some values at L<http://www.research.att.com/~njas/sequences/A000110>.
If you pass the optional parameter C<$k>, the subroutine generates only partitions of size C<$k>. This uses an specific algorithm for partitions of known size, which is more efficient than generating all partitions and filtering them by size.
Note that in that case the subsets themselves may have several sizes, it is the number of elements I<of the partition> which is C<$k>. For instance if C<@data> has 5 elements there are partitions of size 2 that consist of a subset of size 2 and its complement of size 3; and partitions of size 2 that consist of a subset of size 1 and its complement of size 4. In both cases the partitions have the same size, they have two elements.
The number of partitions of size C<k> of a set of C<n> elements are known as Stirling numbers of the second kind, and satisfy the recursion:
S(0, 0) = 1
S(n, 0) = 0 if n > 0
S(n, 1) = S(n, n) = 1
S(n, k) = S(n-1, k-1) + kS(n-1, k)
=head2 subsets(\@data[, $k])
This subroutine iterates over the subsets of data, which is assumed to represent a set. If you pass the optional parameter C<$k> the iteration runs over subsets of data of size C<$k>.
The number of subsets of a set of C<n> elements is
2**n
See some values at L<http://www.research.att.com/~njas/sequences/A000079>.
=head1 CORNER CASES
Since version 0.05 subroutines are more forgiving for unsual values of C<$k>:
=over 4
=item *
If C<$k> is less than zero no tuple exists. Thus, the very first call to
the iterator's C<next()> method returns C<undef>, and a call in list
context returns the empty list. (See L</DIAGNOSTICS>.)
=item *
If C<$k> is zero we have one tuple, the empty tuple. This is a different
case than the former: when C<$k> is negative there are no tuples at all,
when C<$k> is zero there is one tuple. The rationale for this behaviour
is the same rationale for n choose 0 = 1: the empty tuple is a subset of
C<@data> with C<$k = 0> elements, so it complies with the definition.
=item *
If C<$k> is greater than the size of C<@data>, and we are calling a
subroutine that does not generate tuples with repetitions, no tuple
exists. Thus, the very first call to the iterator's C<next()> method
returns C<undef>, and a call in list context returns the empty
list. (See L</DIAGNOSTICS>.)
=back
In addition, since 0.05 empty C<@data>s are supported as well.
=head1 EXPORT
Algorithm::Combinatorics exports nothing by default. Each of the subroutines can be exported on demand, as in
use Algorithm::Combinatorics qw(combinations);
and the tag C<all> exports them all:
use Algorithm::Combinatorics qw(:all);
=head1 DIAGNOSTICS
=head2 Warnings
The following warnings may be issued:
=over
=item Useless use of %s in void context
A subroutine was called in void context.
=item Parameter k is negative
A subroutine was called with a negative k.
=item Parameter k is greater than the size of data
A subroutine that does not generate tuples with repetitions was called with a k greater than the size of data.
=back
=head2 Errors
The following errors may be thrown:
=over
=item Missing parameter data
A subroutine was called with no parameters.
=item Missing parameter k
A subroutine that requires a second parameter k was called without one.
=item Parameter data is not an arrayref
The first parameter is not an arrayref (tested with "reftype()" from Scalar::Util.)
=back
=head1 DEPENDENCIES
Algorithm::Combinatorics is known to run under perl 5.6.2. The
distribution uses L<Test::More> and L<FindBin> for testing,
L<Scalar::Util> for C<reftype()>, and L<XSLoader> for XS.
=head1 BUGS
Please report any bugs or feature requests to
C<bug-algorithm-combinatorics@rt.cpan.org>, or through the web interface at
L<http://rt.cpan.org/NoAuth/ReportBug.html?Queue=Algorithm-Combinatorics>.
=head1 SEE ALSO
L<Math::Combinatorics> is a pure Perl module that offers similar features.
L<List::PowerSet> offers a fast pure-Perl generator of power sets that
Algorithm::Combinatorics copies and translates to XS.
=head1 BENCHMARKS
There are some benchmarks in the F<benchmarks> directory of the distribution.
=head1 REFERENCES
[1] Donald E. Knuth, I<The Art of Computer Programming, Volume 4, Fascicle 2: Generating All Tuples and Permutations>. Addison Wesley Professional, 2005. ISBN 0201853930.
[2] Donald E. Knuth, I<The Art of Computer Programming, Volume 4, Fascicle 3: Generating All Combinations and Partitions>. Addison Wesley Professional, 2005. ISBN 0201853949.
[3] Michael Orlov, I<Efficient Generation of Set Partitions>, L<http://www.informatik.uni-ulm.de/ni/Lehre/WS03/DMM/Software/partitions.pdf>.
=head1 AUTHOR
Xavier Noria (FXN), E<lt>fxn@cpan.orgE<gt>
=head1 COPYRIGHT & LICENSE
Copyright 2005-2012 Xavier Noria, all rights reserved.
This program is free software; you can redistribute it and/or modify it
under the same terms as Perl itself.
=cut