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<h1>Priority-Queue Design</h1>
<h2><a name="overview" id="overview">Overview</a></h2>
<p>The priority-queue container has the following
declaration:</p>
<pre>
<b>template</b><
<b>typename</b> Value_Type,
<b>typename</b> Cmp_Fn = std::less<Value_Type>,
<b>typename</b> Tag = <a href="pairing_heap_tag.html">pairing_heap_tag</a>,
<b>typename</b> Allocator = std::allocator<<b>char</b>> >
<b>class</b> <a href="priority_queue.html">priority_queue</a>;
</pre>
<p>The parameters have the following meaning:</p>
<ol>
<li><tt>Value_Type</tt> is the value type.</li>
<li><tt>Cmp_Fn</tt> is a value comparison functor</li>
<li><tt>Tag</tt> specifies which underlying data structure
to use.</li>
<li><tt>Allocator</tt> is an allocator
type.</li>
</ol>
<p>The <tt>Tag</tt> parameter specifies which underlying
data structure to use. Instantiating it by <a href=
"pairing_heap_tag.html"><tt>pairing_heap_tag</tt></a>,
<a href=
"binary_heap_tag.html"><tt>binary_heap_tag</tt></a>,
<a href=
"binomial_heap_tag.html"><tt>binomial_heap_tag</tt></a>,
<a href=
"rc_binomial_heap_tag.html"><tt>rc_binomial_heap_tag</tt></a>,
or <a href=
"thin_heap_tag.html"><tt>thin_heap_tag</tt></a>,
specifies, respectively, an underlying pairing heap [<a href=
"references.html#fredman86pairing">fredman86pairing</a>],
binary heap [<a href="references.html#clrs2001">clrs2001</a>],
binomial heap [<a href=
"references.html#clrs2001">clrs2001</a>], a binomial heap with
a redundant binary counter [<a href=
"references.html#maverik_lowerbounds">maverik_lowerbounds</a>],
or a thin heap [<a href=
"references.html#kt99fat_heaps">kt99fat_heas</a>]. These are
explained further in <a href="#pq_imp">Implementations</a>.</p>
<p>As mentioned in <a href=
"tutorial.html#pq">Tutorial::Priority Queues</a>,
<a href=
"priority_queue.html"><tt>__gnu_pbds::priority_queue</tt></a>
shares most of the same interface with <tt>std::priority_queue</tt>.
<i>E.g.</i> if <tt>q</tt> is a priority queue of type
<tt>Q</tt>, then <tt>q.top()</tt> will return the "largest"
value in the container (according to <tt><b>typename</b>
Q::cmp_fn</tt>). <a href=
"priority_queue.html"><tt>__gnu_pbds::priority_queue</tt></a>
has a larger (and very slightly different) interface than
<tt>std::priority_queue</tt>, however, since typically
<tt>push</tt> and <tt>pop</tt> are deemed insufficient for
manipulating priority-queues. </p>
<p>Different settings require different priority-queue
implementations which are described in <a href=
"#pq_imp">Implementations</a>; <a href="#pq_traits">Traits</a>
discusses ways to differentiate between the different traits of
different implementations.</p>
<h2><a name="pq_it" id="pq_it">Iterators</a></h2>
<p>There are many different underlying-data structures for
implementing priority queues. Unfortunately, most such
structures are oriented towards making <tt>push</tt> and
<tt>top</tt> efficient, and consequently don't allow efficient
access of other elements: for instance, they cannot support an efficient
<tt>find</tt> method. In the use case where it
is important to both access and "do something with" an
arbitrary value, one would be out of luck. For example, many graph algorithms require
modifying a value (typically increasing it in the sense of the
priority queue's comparison functor).</p>
<p>In order to access and manipulate an arbitrary value in a
priority queue, one needs to reference the internals of the
priority queue from some form of an associative container -
this is unavoidable. Of course, in order to maintain the
encapsulation of the priority queue, this needs to be done in a
way that minimizes exposure to implementation internals.</p>
<p>In <tt>pb_ds</tt> the priority queue's <tt>insert</tt>
method returns an iterator, which if valid can be used for subsequent <tt>modify</tt> and
<tt>erase</tt> operations. This both preserves the priority
queue's encapsulation, and allows accessing arbitrary values (since the
returned iterators from the <tt>push</tt> operation can be
stored in some form of associative container).</p>
<p>Priority queues' iterators present a problem regarding their
invalidation guarantees. One assumes that calling
<tt><b>operator</b>++</tt> on an iterator will associate it
with the "next" value. Priority-queues are
self-organizing: each operation changes what the "next" value
means. Consequently, it does not make sense that <tt>push</tt>
will return an iterator that can be incremented - this can have
no possible use. Also, as in the case of hash-based containers,
it is awkward to define if a subsequent <tt>push</tt> operation
invalidates a prior returned iterator: it invalidates it in the
sense that its "next" value is not related to what it
previously considered to be its "next" value. However, it might not
invalidate it, in the sense that it can be
de-referenced and used for <tt>modify</tt> and <tt>erase</tt>
operations.</p>
<p>Similarly to the case of the other unordered associative
containers, <tt>pb_ds</tt> uses a distinction between
point-type and range type iterators. A priority queue's <tt>iterator</tt> can always be
converted to a <tt>point_iterator</tt>, and a
<tt>const_iterator</tt> can always be converted to a
<tt>const_point_iterator</tt>.</p>
<p>The following snippet demonstrates manipulating an arbitrary
value:</p>
<pre>
<i>// A priority queue of integers.</i>
<a href=
"priority_queue.html">priority_queue</a><<b>int</b>> p;
<i>// Insert some values into the priority queue.</i>
<a href=
"priority_queue.html">priority_queue</a><<b>int</b>>::point_iterator it = p.push(0);
p.push(1);
p.push(2);
<i>// Now modify a value.</i>
p.modify(it, 3);
assert(p.top() == 3);
</pre>
<p>(<a href="pq_examples.html#xref">Priority Queue
Examples::Cross-Referencing</a> shows a more detailed
example.)</p>
<p>It should be noted that an alternative design could embed an
associative container in a priority queue. Could, but most probably should not. To begin with, it should be noted that one
could always encapsulate a priority queue and an associative
container mapping values to priority queue iterators with no
performance loss. One cannot, however, "un-encapsulate" a
priority queue embedding an associative container, which might
lead to performance loss. Assume, that one needs to
associate each value with some data unrelated to priority
queues. Then using <tt>pb_ds</tt>'s design, one could use an
associative container mapping each value to a pair consisting
of this data and a priority queue's iterator. Using the
embedded method would need to use two associative
containers. Similar problems might arise in cases where a value
can reside simultaneously in many priority queues.</p>
<h2><a name="pq_imp" id="pq_imp">Implementations</a></h2>
<p>There are three main implementations of priority queues: the
first employs a binary heap, typically one which uses a
sequence; the second uses a tree (or forest of trees), which is
typically less structured than an associative container's tree;
the third simply uses an associative container. These are
shown, respectively, in Figures <a href=
"#pq_different_underlying_dss">Underlying Priority-Queue
Data-Structures</a> A1 and A2, Figure <a href=
"#pq_different_underlying_dss">Underlying Priority-Queue
Data-Structures</a> B, and Figures <a href=
"#pq_different_underlying_dss">Underlying Priority-Queue
Data-Structures</a> C.</p>
<h6 class="c1"><a name="pq_different_underlying_dss" id=
"pq_different_underlying_dss"><img src=
"pq_different_underlying_dss.png" alt="no image" /></a></h6>
<h6 class="c1">Underlying Priority-Queue Data-Structures.</h6>
<p>Roughly speaking, any value that is both pushed and popped
from a priority queue must incur a logarithmic expense (in the
amortized sense). Any priority queue implementation that would
avoid this, would violate known bounds on comparison-based
sorting (see, <i>e.g.</i>, [<a href=
"references.html#clrs2001">clrs2001</a>] and <a href=
"references.html#brodal96priority">brodal96priority</a>]).</p>
<p>Most implementations do
not differ in the asymptotic amortized complexity of
<tt>push</tt> and <tt>pop</tt> operations, but they differ in
the constants involved, in the complexity of other operations
(<i>e.g.</i>, <tt>modify</tt>), and in the worst-case
complexity of single operations. In general, the more
"structured" an implementation (<i>i.e.</i>, the more internal
invariants it possesses) - the higher its amortized complexity
of <tt>push</tt> and <tt>pop</tt> operations.</p>
<p><tt>pb_ds</tt> implements different algorithms using a
single class: <a href="priority_queue.html">priority_queue</a>.
Instantiating the <tt>Tag</tt> template parameter, "selects"
the implementation:</p>
<ol>
<li>Instantiating <tt>Tag = <a href=
"binary_heap_tag.html">binary_heap_tag</a></tt> creates
a binary heap of the form in Figures <a href=
"#pq_different_underlying_dss">Underlying Priority-Queue
Data-Structures</a> A1 or A2. The former is internally
selected by <a href="priority_queue.html">priority_queue</a>
if <tt>Value_Type</tt> is instantiated by a primitive type
(<i>e.g.</i>, an <tt><b>int</b></tt>); the latter is
internally selected for all other types (<i>e.g.</i>,
<tt>std::string</tt>). This implementations is relatively
unstructured, and so has good <tt>push</tt> and <tt>pop</tt>
performance; it is the "best-in-kind" for primitive
types, <i>e.g.</i>, <tt><b>int</b></tt>s. Conversely, it has
high worst-case performance, and can support only linear-time
<tt>modify</tt> and <tt>erase</tt> operations; this is
explained further in <a href="#pq_traits">Traits</a>.</li>
<li>Instantiating <tt>Tag = <a href=
"pairing_heap_tag.html">pairing_heap_tag</a></tt>
creates a pairing heap of the form in Figure <a href=
"#pq_different_underlying_dss">Underlying Priority-Queue
Data-Structures</a> B. This implementations too is relatively
unstructured, and so has good <tt>push</tt> and <tt>pop</tt>
performance; it is the "best-in-kind" for non-primitive
types, <i>e.g.</i>, <tt>std:string</tt>s. It also has very
good worst-case <tt>push</tt> and <tt>join</tt> performance
(<i>O(1)</i>), but has high worst-case <tt>pop</tt>
complexity.</li>
<li>Instantiating <tt>Tag = <a href=
"binomial_heap_tag.html">binomial_heap_tag</a></tt>
creates a binomial heap of the form in Figure <a href=
"#pq_different_underlying_dss">Underlying Priority-Queue
Data-Structures</a> B. This implementations is more
structured than a pairing heap, and so has worse
<tt>push</tt> and <tt>pop</tt> performance. Conversely, it
has sub-linear worst-case bounds for <tt>pop</tt>,
<i>e.g.</i>, and so it might be preferred in cases where
responsiveness is important.</li>
<li>Instantiating <tt>Tag = <a href=
"rc_binomial_heap_tag.html">rc_binomial_heap_tag</a></tt>
creates a binomial heap of the form in Figure <a href=
"#pq_different_underlying_dss">Underlying Priority-Queue
Data-Structures</a> B, accompanied by a redundant counter
which governs the trees. This implementations is therefore
more structured than a binomial heap, and so has worse
<tt>push</tt> and <tt>pop</tt> performance. Conversely, it
guarantees <i>O(1)</i> <tt>push</tt> complexity, and so it
might be preferred in cases where the responsiveness of a
binomial heap is insufficient.</li>
<li>Instantiating <tt>Tag = <a href=
"thin_heap_tag.html">thin_heap_tag</a></tt> creates a
thin heap of the form in Figure <a href=
"#pq_different_underlying_dss">Underlying Priority-Queue
Data-Structures</a> B. This implementations too is more
structured than a pairing heap, and so has worse
<tt>push</tt> and <tt>pop</tt> performance. Conversely, it
has better worst-case and identical amortized complexities
than a Fibonacci heap, and so might be more appropriate for
some graph algorithms.</li>
</ol>
<p><a href="pq_performance_tests.html">Priority-Queue
Performance Tests</a> shows some results for the above, and
discusses these points further.</p>
<p>Of course, one can use any order-preserving associative
container as a priority queue, as in Figure <a href=
"#pq_different_underlying_dss">Underlying Priority-Queue
Data-Structures</a> C, possibly by creating an adapter class
over the associative container (much as
<tt>std::priority_queue</tt> can adapt <tt>std::vector</tt>).
This has the advantage that no cross-referencing is necessary
at all; the priority queue itself is an associative container.
Most associative containers are too structured to compete with
priority queues in terms of <tt>push</tt> and <tt>pop</tt>
performance.</p>
<h2><a name="pq_traits" id="pq_traits">Traits</a></h2>
<p>It would be nice if all priority queues could
share exactly the same behavior regardless of implementation. Sadly, this is not possible. Just one for instance is in join operations: joining
two binary heaps might throw an exception (not corrupt
any of the heaps on which it operates), but joining two pairing
heaps is exception free.</p>
<p>Tags and traits are very useful for manipulating generic
types. <a href=
"priority_queue.html"><tt>__gnu_pbds::priority_queue</tt></a>
publicly defines <tt>container_category</tt> as one of the tags
discussed in <a href="#pq_imp">Implementations</a>. Given any
container <tt>Cntnr</tt>, the tag of the underlying
data structure can be found via <tt><b>typename</b>
Cntnr::container_category</tt>; this is one of the types shown in
Figure <a href="#pq_tag_cd">Data-structure tag class
hierarchy</a>.</p>
<h6 class="c1"><a name="pq_tag_cd" id=
"pq_tag_cd"><img src="priority_queue_tag_cd.png" alt=
"no image" /></a></h6>
<h6 class="c1">Data-structure tag class hierarchy.</h6>
<p>Additionally, a traits mechanism can be used to query a
container type for its attributes. Given any container
<tt>Cntnr</tt>, then <tt><a href=
"assoc_container_traits.html">__gnu_pbds::container_traits</a><Cntnr></tt>
is a traits class identifying the properties of the
container.</p>
<p>To find if a container might throw if two of its objects are
joined, one can use <a href=
"assoc_container_traits.html"><tt>container_traits</tt></a><tt><Cntnr>::split_join_can_throw</tt>,
for example.</p>
<p>Different priority-queue implementations have different invalidation guarantees. This is
especially important, since as explained in <a href=
"#pq_it">Iterators</a>, there is no way to access an arbitrary
value of priority queues except for iterators. Similarly to
associative containers, one can use
<a href=
"assoc_container_traits.html"><tt>container_traits</tt></a><tt><Cntnr>::invalidation_guarantee</tt>
to get the invalidation guarantee type of a priority queue.</p>
<p>It is easy to understand from Figure <a href=
"#pq_different_underlying_dss">Underlying Priority-Queue
Data-Structures</a>, what <a href=
"assoc_container_traits.html"><tt>container_traits</tt></a><tt><Cntnr>::invalidation_guarantee</tt>
will be for different implementations. All implementations of
type <a href="#pq_different_underlying_dss">Underlying
Priority-Queue Data-Structures</a> B have <a href=
"point_invalidation_guarantee.html"><tt>point_invalidation_guarantee</tt></a>:
the container can freely internally reorganize the nodes -
range-type iterators are invalidated, but point-type iterators
are always valid. Implementations of type <a href=
"#pq_different_underlying_dss">Underlying Priority-Queue
Data-Structures</a> A1 and A2 have <a href=
"basic_invalidation_guarantee.html"><tt>basic_invalidation_guarantee</tt></a>:
the container can freely internally reallocate the array - both
point-type and range-type iterators might be invalidated.</p>
<p>This has major implications, and constitutes a good reason to avoid
using binary heaps. A binary heap can perform <tt>modify</tt>
or <tt>erase</tt> efficiently <u>given a valid point-type
iterator</u>. However, inn order to supply it with a valid point-type
iterator, one needs to iterate (linearly) over all
values, then supply the relevant iterator (recall that a
range-type iterator can always be converted to a point-type
iterator). This means that if the number of <tt>modify</tt> or
<tt>erase</tt> operations is non-negligible (say
super-logarithmic in the total sequence of operations) - binary
heaps will perform badly.</p>
<pre>
</pre>
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