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diff --git a/gcc-4.4.3/libstdc++-v3/doc/html/ext/pb_ds/pq_design.html b/gcc-4.4.3/libstdc++-v3/doc/html/ext/pb_ds/pq_design.html new file mode 100644 index 000000000..959560045 --- /dev/null +++ b/gcc-4.4.3/libstdc++-v3/doc/html/ext/pb_ds/pq_design.html @@ -0,0 +1,381 @@ +<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Transitional//EN" + "http://www.w3.org/TR/xhtml1/DTD/xhtml1-transitional.dtd"> + +<html xmlns="http://www.w3.org/1999/xhtml" xml:lang="en" lang="en"> +<head> + <meta name="generator" content= + "HTML Tidy for Linux/x86 (vers 12 April 2005), see www.w3.org" /> + + <title>Priority-Queues</title> + <meta http-equiv="Content-Type" content= + "text/html; charset=us-ascii" /> + </head> + +<body> + <div id="page"> + <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> |