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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>
+ </div>
+</body>
+</html>
projects / msp430-gcc.git / blobdiff