TL;DR

A heap-based priority queue, in its basic form. Almost.

While coding cglib-perl, I was also studying for a couple of Coursera courses about #algorithms. In particular, course Algorithms, Part I deals with priority queues; you can also find a high level explanation in the relevant page about the algorithm.

Hence, it seemed just natural to leverage what I learned and implement it in some tight code that I could use in CodinGame - yes, in a copy-and-paste mode, but this is life.

What Is a Priority Queue?

A priority queue is a data structure where you can put elements in and get them out. Differently from a regular queue - where you get elements out in the same order as you put them in, in a priority queue the elements come out according to their priority: the higher (or the lower, depending on your metric) is the priority, the sooner they come out.

Implementing a Priority Queue through Heaps

In a very naive implementation, you might just keep the elements in a list. When you need to get something out, you just look for it. Alternatively, you can keep the list always ordered, so that you will always know what’s the next item to get from the queue.

These implementations would be naive because their complexity would be linear in the number of elements. We can do better.

Long story short, it’s possible to use a heap to implement a priority queue in an efficient way. You can find details in this page and, of course, in the Algorithms, Part I course.

A Perl Compact Implementation

An implementation of the heap-based priority queue can be found at BasicPriorityQueue.pm. Here it is, as of today, adapted from the page in algs4:

 1 package BasicPriorityQueue;
 2 use strict;     # Adapted from https://algs4.cs.princeton.edu/24pq/
 3 sub dequeue;    # see below
 4 sub enqueue;    # see below
 5 sub is_empty { return !$#{$_[0]{items}} }
 6 sub max { return $#{$_[0]{items}} ? $_[0][1] : () }
 7 sub new;        # see below
 8 sub size { return $#{$_[0]{items}} }
 9 
10 sub dequeue {    # includes "sink"
11    my ($is, $before, $k) = (@{$_[0]}{qw< items before >}, 1);
12    return unless $#$is;
13    my $r = ($#$is > 1) ? (splice @$is, 1, 1, pop @$is) : pop @$is;
14    while ((my $j = $k * 2) <= $#$is) {
15       ++$j if ($j < $#$is) && $before->($is->[$j + 1], $is->[$j]);
16       last if $before->($is->[$k], $is->[$j]);
17       (@{$is}[$j, $k], $k) = (@{$is}[$k, $j], $j);
18    }
19    return $r;
20 } ## end sub dequeue
21 
22 sub enqueue {    # includes "swim"
23    my ($is, $before) = (@{$_[0]}{qw< items before >});
24    push @$is, $_[1];
25    my $k = $#$is;
26    (@{$is}[$k / 2, $k], $k) = (@{$is}[$k, $k / 2], int($k / 2))
27      while ($k > 1) && $before->($is->[$k], $is->[$k / 2]);
28 } ## end sub enqueue
29 
30 sub new {
31    my $package = shift;
32    my $self = bless {((@_ && ref($_[0])) ? %{$_[0]} : @_)}, $package;
33    $self->{before} ||= sub { $_[0] < $_[1] };
34    (my $is, $self->{items}) = ($self->{items} || [], ['-']);
35    $self->enqueue($_) for @$is;
36    return $self;
37 } ## end sub new

The object is a hash reference (line 32) with two keys inside: items is the array where items are kept, and before is a reference to a function that allows us to specify what “higher priority” means. It defaults to just comparing two items numerically (line 33).

The representation is pretty much the one described in the algs4 page: an array whose first element is ignored. This is why $self->{items} is initialized with an anonymous array containing the placeholder string - as its 0-th element.

The constructor allows passing the items as an input, with key items. In case it is not empty, these items are put in the queue in the “right” way, i.e. through the enqueue function (line 35).

Lines 3 to 8 provide the main interface of the module:

  • dequeue and enqueue are function to respectively get one element from the queue or add one to it;
  • is_empty tests whether the queue is empty or not;
  • max gets the highest priority element, while it leaves the element in the queue;
  • new is a constructor for the queue;
  • size tells how many elements are in the queue.

Both is_empty and size return a value based on $#{...}, which is one less than the number of elements in the array. This is a consequence of the fact that the 0-th element is ignored.

Getting the max element (i.e. the one with the highest priority) is also straighforward, as it sits at the root of the heap, i.e. in position 1 inside the array.

The real core of the algorithm is in the enqueue and dequeue functions. This observations in this page is interesting to this regard:

Insert. We add the new item at the end of the array, increment the size of the heap, and then swim up through the heap with that item to restore the heap condition.

Remove the maximum. We take the largest item off the top, put the item from the end of the heap at the top, decrement the size of the heap, and then sink down through the heap with that item to restore the heap condition.

In other terms, the generic functions swim and sink are only used, respectively, in enqueue and dequeue. This means that we can embed them in these functions and spare a few lines. Just in the spirit of readability 🙄

The enqueue function puts the new element at the end of the array (line 24), then makes it swim up in the heap (assignments in line 26) while the element’s position is incompatible with the heap rules (line 27).

The dequeue function, too, implements the suggestion in the algorithm’s page. The highest priority element is at the root, from where it is extracted. This happens in line 13, which needs a bit of unboxing:

  • if there are 2 or more elements, then the algorithm tells us to get the root element at index 1 in the array (splice @$is, 1, 1 removes this item and returns it) and to put the last element in its place (adding a fourth argument to splice puts an element in the places left by the splicing operators, and pop @$is takes the last element);

  • if there’s only one element left in the queue, then it’s the last element and we can just pop @$is (this is the : alternative in the ternary operator).

So, line 13 gets the return value $r out of the queue, and prepares the queue itself for the sink operation that is performed in lines 14 to 18.

In both cases (enqueue and dequeue) the function in before is used to tell different priorities apart, allowing to use the same implementation for both the max and min priorities.

Summing up

Priority queues are… fun! Heaps too, and the two play well together. Are you looking for something less… basic? Take a look at PriorityQueue.pm!