RFC 91 (v4) Arrays: part and flatten

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=head1 TITLE

Arrays: part and flatten

=head1 VERSION

  Maintainer: Jeremy Howard <j@howard.fm>
  Date: 10 Aug 2000
  Last Modified: 21 Sep 2000
  Mailing List: perl6-language-data@perl.org
  Number: 91
  Version: 4
  Status: Frozen

=head1 DISCUSSION

The discussion points for this RFC were the same as those for RFC 90.

=head1 ABSTRACT

It is proposed that two new functions, C<part> and C<flatten>, be added to
Perl. C<part($part_size, @list, $skip)> would return @list broken into
references to sub-lists, each one $list_size in size, offset from the end
of the previous sub-list by $skip. C<flatten(@list_of_lists, $nest_level)>
would take a list of lists and dereference the elements recursively, up to
$nest_level levels deep. Both functions would return an alias into the
original list, not a copy of the elements.

=head1 CHANGES

=head2 Since v2

=over 4

=item *

Described aliasing behaviour of flatten and part

=item *

Added C<flatten> builtin

=back

=head2 Since v1

=over 4

=item *

Moved list to perl6-language-data@perl.org

=item *

Changed name from C<partition>

=item *

Add optional third argument to allow skipping elements

=item *

Make difference between C<unmerge> and C<part> explicit

=back

=head1 DESCRIPTION

In order to work with lists of arbitary size, it is often necessary to
split a list into equal sized sub-lists. A C<part> function is proposed
that achieves this:

  @list = (1,2,3,4,5,6);
  @parted_list = part(2, @list);   # ([1,2],[3,4],[5,6])

This is useful to provide tuples to functions that can operate on lists of
lists, for instance:

  @sum_pairs = map {$_->[0] + $_->[1]} @parted_list;   # (3,7,11)

The optional third argument can be used to skip over elements of the list:

  @list = (1,2,3,4,5,6,7,8);
  @parted_list = part(2, @list, 1);   # ([1,2],[4,5],[7,8])

If the list to be parted is not an exact multiple of the part size, the
final list reference is not padded. For example:

  @list2 = (1..7);
  @parted_list2 = part(3, @list2);   # ([1,2,3], [4,5,6], [7])

A list that has been parted can be flattened again using the
C<flatten> function:

  @parted_list2 = ([1,2,3], [4,5,6], [7]);
  @list2 = flatten(@parted_list2);   # (1,2,3,4,5,6,7)

C<flatten> can work on arrays of greater than two dimensions:

  my @some_LOL = ([[1,2], [3,4]],
                  [[5,6], [7,8]]);
  my @flat_LOL = flatten(@some_LOL);   # ([1,2,3,4], [5,6,7,8])

The innermost list of lists is flattened first. To flatten more levels of
nesting, use the optional second argument:

  my @some_LOL = ([[1,2], [3,4]],
                  [[5,6], [7,8]]);
  my @flat_LOL = flatten(@some_LOL,2);   # (1,2,3,4,5,6,7,8)

Both C<part> and <flatten> do not make a copy of the elements of their
arguments; they simply create an alias to them:

  @parted_list2 = ([1,2,3], [4,5,6], [7]);
  @list2 = flatten(@parted_list2);   # (1,2,3,4,5,6,7)
  $list2[1] = 0;
  @parted_list2 == ([1,0,3], [4,5,6], [7]);   # True

C<merge> (see RFC 90) and C<part> can work together to allow manipulation
of arbitary sized lists. For instance, we can extend the $sum_xy function
used as an example in the C<merge> RFC, which takes two lists and returns
the sum of them multiplied together component-wise:

  $sum_xy = sub {reduce ^last+^x*^y, merge($_[0], $_[1])};

to a function $sum_mult that does the same with an arbitary number of
lists:

  # Multiply all the elements of a list together, returning the result
  $apply_times = reduce (^total * ^element, @^multiplicands);
  # Swap the rows and columns of a list of lists
  $transpose = part(
    # Find the size of each column
    scalar @^list_of_lists,
    # Interleave the rows
    merge(@^lists_of_lists);
  )

  # Take a list of references to lists, multiply them component-wise,
  #   and return their sum
  $sum_mult = reduce (
    ^total + $apply_times->( @^next_list ),
    $transpose->(^list_of_lists),
  );

  # Example usage of $sum_mult
  @a = (1,3,5);
  @b = (2,4,6);
  @c = (-1,1,-1);
  $answer = $sum_mult->(\@a, \@b, \@c);   # 1*2*-1+3*4*1+5*6*-1 = -20

A common usage of C<part> and C<unmerge> is to access the rows and columns
of a matrix stored as a flat list:

  @array = ( a1, a2, a3,
             b1, b2, b3,
             c1, c2, c3 );
  @columns = unmerge(3,@array) # Return the columns
  @rows = part(3,@array)       # Return the rows

=head1 IMPLEMENTATION

The C<part> functions should be evaluated lazily. Because it is used in
common operations such as the transposition of a matrix, its efficiency is
particularly important.

C<part> and C<flatten> return an alias into the original list, not a copy
of the elements.

C<part> and C<flatten> are just special cases of C<reshape> (from RFC
148).

=head1 REFERENCES

RFC 23: Higher order functions

RFC 76: Builtin: reduce

RFC 90: Builtins: merge() and unmerge()

RFC 148: Add reshape() for multi-dimensional array reshaping

=head1 ACKNOWLEDGEMENTS

Damian Conway: Numerous comments on first draft

• RFC 91 (v4) Arrays: part and flatten by Perl6 RFC Librarian