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  31     "><link rel="home" href="../index.html" title="The GNU C++ Library"><link rel="up" href="policy_data_structures.html" title="Chapter 22. Policy-Based Data Structures"><link rel="prev" href="policy_data_structures_using.html" title="Using"><link rel="next" href="policy_based_data_structures_test.html" title="Testing"></head><body bgcolor="white" text="black" link="#0000FF" vlink="#840084" alink="#0000FF"><div class="navheader"><table width="100%" summary="Navigation header"><tr><th colspan="3" align="center">Design</th></tr><tr><td width="20%" align="left"><a accesskey="p" href="policy_data_structures_using.html">Prev</a> </td><th width="60%" align="center">Chapter 22. Policy-Based Data Structures</th><td width="20%" align="right"> <a accesskey="n" href="policy_based_data_structures_test.html">Next</a></td></tr></table><hr></div><div class="section" title="Design"><div class="titlepage"><div><div><h2 class="title" style="clear: both"><a name="containers.pbds.design"></a>Design</h2></div></div></div><p></p><div class="section" title="Concepts"><div class="titlepage"><div><div><h3 class="title"><a name="pbds.design.concepts"></a>Concepts</h3></div></div></div><div class="section" title="Null Policy Classes"><div class="titlepage"><div><div><h4 class="title"><a name="pbds.design.concepts.null_type"></a>Null Policy Classes</h4></div></div></div><p>
  32         Associative containers are typically parametrized by various
  33         policies. For example, a hash-based associative container is
  34         parametrized by a hash-functor, transforming each key into an
  35         non-negative numerical type. Each such value is then further mapped
  36         into a position within the table. The mapping of a key into a
  37         position within the table is therefore a two-step process.
  38       </p><p>
  39         In some cases, instantiations are redundant. For example, when the
  40         keys are integers, it is possible to use a redundant hash policy,
  41         which transforms each key into its value.
  42       </p><p>
  43         In some other cases, these policies are irrelevant.  For example, a
  44         hash-based associative container might transform keys into positions
  45         within a table by a different method than the two-step method
  46         described above. In such a case, the hash functor is simply
  47         irrelevant.
  48       </p><p>
  49         When a policy is either redundant or irrelevant, it can be replaced
  50         by <code class="classname">null_type</code>.
  51       </p><p>
  52         For example, a <span class="emphasis"><em>set</em></span> is an associative
  53         container with one of its template parameters (the one for the
  54         mapped type) replaced with <code class="classname">null_type</code>. Other
  55         places simplifications are made possible with this technique
  56         include node updates in tree and trie data structures, and hash
  57         and probe functions for hash data structures.
  58       </p></div><div class="section" title="Map and Set Semantics"><div class="titlepage"><div><div><h4 class="title"><a name="pbds.design.concepts.associative_semantics"></a>Map and Set Semantics</h4></div></div></div><div class="section" title="Distinguishing Between Maps and Sets"><div class="titlepage"><div><div><h5 class="title"><a name="concepts.associative_semantics.set_vs_map"></a>
  59             Distinguishing Between Maps and Sets
  60           </h5></div></div></div><p>
  61           Anyone familiar with the standard knows that there are four kinds
  62           of associative containers: maps, sets, multimaps, and
  63           multisets. The map datatype associates each key to
  64           some data.
  65         </p><p>
  66           Sets are associative containers that simply store keys -
  67           they do not map them to anything. In the standard, each map class
  68           has a corresponding set class. E.g.,
  69           <code class="classname">std::map&lt;int, char&gt;</code> maps each
  70           <code class="classname">int</code> to a <code class="classname">char</code>, but
  71           <code class="classname">std::set&lt;int, char&gt;</code> simply stores
  72           <code class="classname">int</code>s. In this library, however, there are no
  73           distinct classes for maps and sets. Instead, an associative
  74           container's <code class="classname">Mapped</code> template parameter is a policy: if
  75           it is instantiated by <code class="classname">null_type</code>, then it
  76           is a "set"; otherwise, it is a "map". E.g.,
  77         </p><pre class="programlisting">
  78           cc_hash_table&lt;int, char&gt;
  79         </pre><p>
  80           is a "map" mapping each <span class="type">int</span> value to a <span class="type">
  81           char</span>, but
  82         </p><pre class="programlisting">
  83           cc_hash_table&lt;int, null_type&gt;
  84         </pre><p>
  85           is a type that uniquely stores <span class="type">int</span> values.
  86         </p><p>Once the <code class="classname">Mapped</code> template parameter is instantiated
  87         by <code class="classname">null_type</code>, then
  88         the "set" acts very similarly to the standard's sets - it does not
  89         map each key to a distinct <code class="classname">null_type</code> object. Also,
  90         , the container's <span class="type">value_type</span> is essentially
  91         its <span class="type">key_type</span> - just as with the standard's sets
  92         .</p><p>
  93           The standard's multimaps and multisets allow, respectively,
  94           non-uniquely mapping keys and non-uniquely storing keys. As
  95           discussed, the
  96           reasons why this might be necessary are 1) that a key might be
  97           decomposed into a primary key and a secondary key, 2) that a
  98           key might appear more than once, or 3) any arbitrary
  99           combination of 1)s and 2)s. Correspondingly,
 100           one should use 1) "maps" mapping primary keys to secondary
 101           keys, 2) "maps" mapping keys to size types, or 3) any arbitrary
 102           combination of 1)s and 2)s. Thus, for example, an
 103           <code class="classname">std::multiset&lt;int&gt;</code> might be used to store
 104           multiple instances of integers, but using this library's
 105           containers, one might use
 106         </p><pre class="programlisting">
 107           tree&lt;int, size_t&gt;
 108         </pre><p>
 109           i.e., a <code class="classname">map</code> of <span class="type">int</span>s to
 110           <span class="type">size_t</span>s.
 111         </p><p>
 112           These "multimaps" and "multisets" might be confusing to
 113           anyone familiar with the standard's <code class="classname">std::multimap</code> and
 114           <code class="classname">std::multiset</code>, because there is no clear
 115           correspondence between the two. For example, in some cases
 116           where one uses <code class="classname">std::multiset</code> in the standard, one might use
 117           in this library a "multimap" of "multisets" - i.e., a
 118           container that maps primary keys each to an associative
 119           container that maps each secondary key to the number of times
 120           it occurs.
 121         </p><p>
 122           When one uses a "multimap," one should choose with care the
 123           type of container used for secondary keys.
 124         </p></div><div class="section" title="Alternatives to std::multiset and std::multimap"><div class="titlepage"><div><div><h5 class="title"><a name="concepts.associative_semantics.multi"></a>Alternatives to <code class="classname">std::multiset</code> and <code class="classname">std::multimap</code></h5></div></div></div><p>
 125           Brace onself: this library does not contain containers like
 126           <code class="classname">std::multimap</code> or
 127           <code class="classname">std::multiset</code>. Instead, these data
 128           structures can be synthesized via manipulation of the
 129           <code class="classname">Mapped</code> template parameter.
 130         </p><p>
 131           One maps the unique part of a key - the primary key, into an
 132           associative-container of the (originally) non-unique parts of
 133           the key - the secondary key. A primary associative-container
 134           is an associative container of primary keys; a secondary
 135           associative-container is an associative container of
 136           secondary keys.
 137         </p><p>
 138           Stepping back a bit, and starting in from the beginning.
 139         </p><p>
 140           Maps (or sets) allow mapping (or storing) unique-key values.
 141           The standard library also supplies associative containers which
 142           map (or store) multiple values with equivalent keys:
 143           <code class="classname">std::multimap</code>, <code class="classname">std::multiset</code>,
 144           <code class="classname">std::tr1::unordered_multimap</code>, and
 145           <code class="classname">unordered_multiset</code>. We first discuss how these might
 146           be used, then why we think it is best to avoid them.
 147         </p><p>
 148           Suppose one builds a simple bank-account application that
 149           records for each client (identified by an <code class="classname">std::string</code>)
 150           and account-id (marked by an <span class="type">unsigned long</span>) -
 151           the balance in the account (described by a
 152           <span class="type">float</span>). Suppose further that ordering this
 153           information is not useful, so a hash-based container is
 154           preferable to a tree based container. Then one can use
 155         </p><pre class="programlisting">
 156           std::tr1::unordered_map&lt;std::pair&lt;std::string, unsigned long&gt;, float, ...&gt;
 157         </pre><p>
 158           which hashes every combination of client and account-id. This
 159           might work well, except for the fact that it is now impossible
 160           to efficiently list all of the accounts of a specific client
 161           (this would practically require iterating over all
 162           entries). Instead, one can use
 163         </p><pre class="programlisting">
 164           std::tr1::unordered_multimap&lt;std::pair&lt;std::string, unsigned long&gt;, float, ...&gt;
 165         </pre><p>
 166           which hashes every client, and decides equivalence based on
 167           client only. This will ensure that all accounts belonging to a
 168           specific user are stored consecutively.
 169         </p><p>
 170           Also, suppose one wants an integers' priority queue
 171           (a container that supports <code class="function">push</code>,
 172           <code class="function">pop</code>, and <code class="function">top</code> operations, the last of which
 173           returns the largest <span class="type">int</span>) that also supports
 174           operations such as <code class="function">find</code> and <code class="function">lower_bound</code>. A
 175           reasonable solution is to build an adapter over
 176           <code class="classname">std::set&lt;int&gt;</code>. In this adapter,
 177           <code class="function">push</code> will just call the tree-based
 178           associative container's <code class="function">insert</code> method; <code class="function">pop</code>
 179           will call its <code class="function">end</code> method, and use it to return the
 180           preceding element (which must be the largest). Then this might
 181           work well, except that the container object cannot hold
 182           multiple instances of the same integer (<code class="function">push(4)</code>,
 183           will be a no-op if <code class="constant">4</code> is already in the
 184           container object). If multiple keys are necessary, then one
 185           might build the adapter over an
 186           <code class="classname">std::multiset&lt;int&gt;</code>.
 187         </p><p>
 188           The standard library's non-unique-mapping containers are useful
 189           when (1) a key can be decomposed in to a primary key and a
 190           secondary key, (2) a key is needed multiple times, or (3) any
 191           combination of (1) and (2).
 192         </p><p>
 193           The graphic below shows how the standard library's container
 194           design works internally; in this figure nodes shaded equally
 195           represent equivalent-key values. Equivalent keys are stored
 196           consecutively using the properties of the underlying data
 197           structure: binary search trees (label A) store equivalent-key
 198           values consecutively (in the sense of an in-order walk)
 199           naturally; collision-chaining hash tables (label B) store
 200           equivalent-key values in the same bucket, the bucket can be
 201           arranged so that equivalent-key values are consecutive.
 202         </p><div class="figure"><a name="id643593"></a><p class="title"><b>Figure 22.8. Non-unique Mapping Standard Containers</b></p><div class="figure-contents"><div class="mediaobject" align="center"><img src="../images/pbds_embedded_lists_1.png" align="middle" alt="Non-unique Mapping Standard Containers"></div></div></div><br class="figure-break"><p>
 203           Put differently, the standards' non-unique mapping
 204           associative-containers are associative containers that map
 205           primary keys to linked lists that are embedded into the
 206           container. The graphic below shows again the two
 207           containers from the first graphic above, this time with
 208           the embedded linked lists of the grayed nodes marked
 209           explicitly.
 210         </p><div class="figure"><a name="fig.pbds_embedded_lists_2"></a><p class="title"><b>Figure 22.9.
 211             Effect of embedded lists in
 212             <code class="classname">std::multimap</code>
 213           </b></p><div class="figure-contents"><div class="mediaobject" align="center"><img src="../images/pbds_embedded_lists_2.png" align="middle" alt="Effect of embedded lists in std::multimap"></div></div></div><br class="figure-break"><p>
 214           These embedded linked lists have several disadvantages.
 215         </p><div class="orderedlist"><ol class="orderedlist" type="1"><li class="listitem"><p>
 216               The underlying data structure embeds the linked lists
 217               according to its own consideration, which means that the
 218               search path for a value might include several different
 219               equivalent-key values. For example, the search path for the
 220               the black node in either of the first graphic, labels A or B,
 221               includes more than a single gray node.
 222             </p></li><li class="listitem"><p>
 223               The links of the linked lists are the underlying data
 224               structures' nodes, which typically are quite structured.  In
 225               the case of tree-based containers (the grapic above, label
 226               B), each "link" is actually a node with three pointers (one
 227               to a parent and two to children), and a
 228               relatively-complicated iteration algorithm. The linked
 229               lists, therefore, can take up quite a lot of memory, and
 230               iterating over all values equal to a given key (through the
 231               return value of the standard
 232               library's <code class="function">equal_range</code>) can be
 233               expensive.
 234             </p></li><li class="listitem"><p>
 235               The primary key is stored multiply; this uses more memory.
 236             </p></li><li class="listitem"><p>
 237               Finally, the interface of this design excludes several
 238               useful underlying data structures. Of all the unordered
 239               self-organizing data structures, practically only
 240               collision-chaining hash tables can (efficiently) guarantee
 241               that equivalent-key values are stored consecutively.
 242             </p></li></ol></div><p>
 243           The above reasons hold even when the ratio of secondary keys to
 244           primary keys (or average number of identical keys) is small, but
 245           when it is large, there are more severe problems:
 246         </p><div class="orderedlist"><ol class="orderedlist" type="1"><li class="listitem"><p>
 247               The underlying data structures order the links inside each
 248               embedded linked-lists according to their internal
 249               considerations, which effectively means that each of the
 250               links is unordered. Irrespective of the underlying data
 251               structure, searching for a specific value can degrade to
 252               linear complexity.
 253             </p></li><li class="listitem"><p>
 254               Similarly to the above point, it is impossible to apply
 255               to the secondary keys considerations that apply to primary
 256               keys. For example, it is not possible to maintain secondary
 257               keys by sorted order.
 258             </p></li><li class="listitem"><p>
 259               While the interface "understands" that all equivalent-key
 260               values constitute a distinct list (through
 261               <code class="function">equal_range</code>), the underlying data
 262               structure typically does not. This means that operations such
 263               as erasing from a tree-based container all values whose keys
 264               are equivalent to a a given key can be super-linear in the
 265               size of the tree; this is also true also for several other
 266               operations that target a specific list.
 267             </p></li></ol></div><p>
 268           In this library, all associative containers map
 269           (or store) unique-key values. One can (1) map primary keys to
 270           secondary associative-containers (containers of
 271           secondary keys) or non-associative containers (2) map identical
 272           keys to a size-type representing the number of times they
 273           occur, or (3) any combination of (1) and (2). Instead of
 274           allowing multiple equivalent-key values, this library
 275           supplies associative containers based on underlying
 276           data structures that are suitable as secondary
 277           associative-containers.
 278         </p><p>
 279           In the figure below, labels A and B show the equivalent
 280           underlying data structures in this library, as mapped to the
 281           first graphic above. Labels A and B, respectively. Each shaded
 282           box represents some size-type or secondary
 283           associative-container.
 284         </p><div class="figure"><a name="id643789"></a><p class="title"><b>Figure 22.10. Non-unique Mapping Containers</b></p><div class="figure-contents"><div class="mediaobject" align="center"><img src="../images/pbds_embedded_lists_3.png" align="middle" alt="Non-unique Mapping Containers"></div></div></div><br class="figure-break"><p>
 285           In the first example above, then, one would use an associative
 286           container mapping each user to an associative container which
 287           maps each application id to a start time (see
 288           <code class="filename">example/basic_multimap.cc</code>); in the second
 289           example, one would use an associative container mapping
 290           each <code class="classname">int</code> to some size-type indicating the
 291           number of times it logically occurs
 292           (see <code class="filename">example/basic_multiset.cc</code>.
 293         </p><p>
 294           See the discussion in list-based container types for containers
 295           especially suited as secondary associative-containers.
 296         </p></div></div><div class="section" title="Iterator Semantics"><div class="titlepage"><div><div><h4 class="title"><a name="pbds.design.concepts.iterator_semantics"></a>Iterator Semantics</h4></div></div></div><div class="section" title="Point and Range Iterators"><div class="titlepage"><div><div><h5 class="title"><a name="concepts.iterator_semantics.point_and_range"></a>Point and Range Iterators</h5></div></div></div><p>
 297           Iterator concepts are bifurcated in this design, and are
 298           comprised of point-type and range-type iteration.
 299         </p><p>
 300           A point-type iterator is an iterator that refers to a specific
 301           element as returned through an
 302           associative-container's <code class="function">find</code> method.
 303         </p><p>
 304           A range-type iterator is an iterator that is used to go over a
 305           sequence of elements, as returned by a container's
 306           <code class="function">find</code> method.
 307         </p><p>
 308           A point-type method is a method that
 309           returns a point-type iterator; a range-type method is a method
 310           that returns a range-type iterator.
 311         </p><p>For most containers, these types are synonymous; for
 312         self-organizing containers, such as hash-based containers or
 313         priority queues, these are inherently different (in any
 314         implementation, including that of C++ standard library
 315         components), but in this design, it is made explicit. They are
 316         distinct types.
 317         </p></div><div class="section" title="Distinguishing Point and Range Iterators"><div class="titlepage"><div><div><h5 class="title"><a name="concepts.iterator_semantics.both"></a>Distinguishing Point and Range Iterators</h5></div></div></div><p>When using this library, is necessary to differentiate
 318         between two types of methods and iterators: point-type methods and
 319         iterators, and range-type methods and iterators. Each associative
 320         container's interface includes the methods:</p><pre class="programlisting">
 321           point_const_iterator
 322           find(const_key_reference r_key) const;
 323
 324           point_iterator
 325           find(const_key_reference r_key);
 326
 327           std::pair&lt;point_iterator,bool&gt;
 328           insert(const_reference r_val);
 329         </pre><p>The relationship between these iterator types varies between
 330         container types. The figure below
 331         shows the most general invariant between point-type and
 332         range-type iterators: In <span class="emphasis"><em>A</em></span> <code class="literal">iterator</code>, can
 333         always be converted to <code class="literal">point_iterator</code>. In <span class="emphasis"><em>B</em></span>
 334         shows invariants for order-preserving containers: point-type
 335         iterators are synonymous with range-type iterators.
 336         Orthogonally,  <span class="emphasis"><em>C</em></span>shows invariants for "set"
 337         containers: iterators are synonymous with const iterators.</p><div class="figure"><a name="id643954"></a><p class="title"><b>Figure 22.11. Point Iterator Hierarchy</b></p><div class="figure-contents"><div class="mediaobject" align="center"><img src="../images/pbds_point_iterator_hierarchy.png" align="middle" alt="Point Iterator Hierarchy"></div></div></div><br class="figure-break"><p>Note that point-type iterators in self-organizing containers
 338         (hash-based associative containers) lack movement
 339         operators, such as <code class="literal">operator++</code> - in fact, this
 340         is the reason why this library differentiates from the standard C++ librarys
 341         design on this point.</p><p>Typically, one can determine an iterator's movement
 342         capabilities using
 343         <code class="literal">std::iterator_traits&lt;It&gt;iterator_category</code>,
 344         which is a <code class="literal">struct</code> indicating the iterator's
 345         movement capabilities. Unfortunately, none of the standard predefined
 346         categories reflect a pointer's <span class="emphasis"><em>not</em></span> having any
 347         movement capabilities whatsoever. Consequently,
 348         <code class="literal">pb_ds</code> adds a type
 349         <code class="literal">trivial_iterator_tag</code> (whose name is taken from
 350         a concept in C++ standardese, which is the category of iterators
 351         with no movement capabilities.) All other standard C++ library
 352         tags, such as <code class="literal">forward_iterator_tag</code> retain their
 353         common use.</p></div><div class="section" title="Invalidation Guarantees"><div class="titlepage"><div><div><h5 class="title"><a name="pbds.design.concepts.invalidation"></a>Invalidation Guarantees</h5></div></div></div><p>
 354           If one manipulates a container object, then iterators previously
 355           obtained from it can be invalidated. In some cases a
 356           previously-obtained iterator cannot be de-referenced; in other cases,
 357           the iterator's next or previous element might have changed
 358           unpredictably. This corresponds exactly to the question whether a
 359           point-type or range-type iterator (see previous concept) is valid or
 360           not. In this design, one can query a container (in compile time) about
 361           its invalidation guarantees.
 362         </p><p>
 363           Given three different types of associative containers, a modifying
 364           operation (in that example, <code class="function">erase</code>) invalidated
 365           iterators in three different ways: the iterator of one container
 366           remained completely valid - it could be de-referenced and
 367           incremented; the iterator of a different container could not even be
 368           de-referenced; the iterator of the third container could be
 369           de-referenced, but its "next" iterator changed unpredictably.
 370         </p><p>
 371           Distinguishing between find and range types allows fine-grained
 372           invalidation guarantees, because these questions correspond exactly
 373           to the question of whether point-type iterators and range-type
 374           iterators are valid. The graphic below shows tags corresponding to
 375           different types of invalidation guarantees.
 376         </p><div class="figure"><a name="id644065"></a><p class="title"><b>Figure 22.12. Invalidation Guarantee Tags Hierarchy</b></p><div class="figure-contents"><div class="mediaobject" align="center"><img src="../images/pbds_invalidation_tag_hierarchy.png" align="middle" alt="Invalidation Guarantee Tags Hierarchy"></div></div></div><br class="figure-break"><div class="itemizedlist"><ul class="itemizedlist" type="disc"><li class="listitem"><p>
 377               <code class="classname">basic_invalidation_guarantee</code>
 378               corresponds to a basic guarantee that a point-type iterator,
 379               a found pointer, or a found reference, remains valid as long
 380               as the container object is not modified.
 381             </p></li><li class="listitem"><p>
 382               <code class="classname">point_invalidation_guarantee</code>
 383               corresponds to a guarantee that a point-type iterator, a
 384               found pointer, or a found reference, remains valid even if
 385               the container object is modified.
 386             </p></li><li class="listitem"><p>
 387               <code class="classname">range_invalidation_guarantee</code>
 388               corresponds to a guarantee that a range-type iterator remains
 389               valid even if the container object is modified.
 390             </p></li></ul></div><p>To find the invalidation guarantee of a
 391         container, one can use</p><pre class="programlisting">
 392           typename container_traits&lt;Cntnr&gt;::invalidation_guarantee
 393         </pre><p>Note that this hierarchy corresponds to the logic it
 394         represents: if a container has range-invalidation guarantees,
 395         then it must also have find invalidation guarantees;
 396         correspondingly, its invalidation guarantee (in this case
 397         <code class="classname">range_invalidation_guarantee</code>)
 398         can be cast to its base class (in this case <code class="classname">point_invalidation_guarantee</code>).
 399         This means that this this hierarchy can be used easily using
 400         standard metaprogramming techniques, by specializing on the
 401         type of <code class="literal">invalidation_guarantee</code>.</p><p>
 402           These types of problems were addressed, in a more general
 403           setting, in <a class="xref" href="policy_data_structures.html#biblio.meyers96more" title="More Effective C++: 35 New Ways to Improve Your Programs and Designs">[biblio.meyers96more]</a> - Item 2. In
 404           our opinion, an invalidation-guarantee hierarchy would solve
 405           these problems in all container types - not just associative
 406           containers.
 407         </p></div></div><div class="section" title="Genericity"><div class="titlepage"><div><div><h4 class="title"><a name="pbds.design.concepts.genericity"></a>Genericity</h4></div></div></div><p>
 408         The design attempts to address the following problem of
 409         data-structure genericity. When writing a function manipulating
 410         a generic container object, what is the behavior of the object?
 411         Suppose one writes
 412       </p><pre class="programlisting">
 413         template&lt;typename Cntnr&gt;
 414         void
 415         some_op_sequence(Cntnr &amp;r_container)
 416         {
 417         ...
 418         }
 419       </pre><p>
 420         then one needs to address the following questions in the body
 421         of <code class="function">some_op_sequence</code>:
 422       </p><div class="itemizedlist"><ul class="itemizedlist" type="disc"><li class="listitem"><p>
 423             Which types and methods does <code class="literal">Cntnr</code> support?
 424             Containers based on hash tables can be queries for the
 425             hash-functor type and object; this is meaningless for tree-based
 426             containers. Containers based on trees can be split, joined, or
 427             can erase iterators and return the following iterator; this
 428             cannot be done by hash-based containers.
 429           </p></li><li class="listitem"><p>
 430             What are the exception and invalidation guarantees
 431             of <code class="literal">Cntnr</code>? A container based on a probing
 432             hash-table invalidates all iterators when it is modified; this
 433             is not the case for containers based on node-based
 434             trees. Containers based on a node-based tree can be split or
 435             joined without exceptions; this is not the case for containers
 436             based on vector-based trees.
 437           </p></li><li class="listitem"><p>
 438             How does the container maintain its elements? Tree-based and
 439             Trie-based containers store elements by key order; others,
 440             typically, do not. A container based on a splay trees or lists
 441             with update policies "cache" "frequently accessed" elements;
 442             containers based on most other underlying data structures do
 443             not.
 444           </p></li><li class="listitem"><p>
 445             How does one query a container about characteristics and
 446             capabilities? What is the relationship between two different
 447             data structures, if anything?
 448           </p></li></ul></div><p>The remainder of this section explains these issues in
 449       detail.</p><div class="section" title="Tag"><div class="titlepage"><div><div><h5 class="title"><a name="concepts.genericity.tag"></a>Tag</h5></div></div></div><p>
 450           Tags are very useful for manipulating generic types. For example, if
 451           <code class="literal">It</code> is an iterator class, then <code class="literal">typename
 452           It::iterator_category</code> or <code class="literal">typename
 453           std::iterator_traits&lt;It&gt;::iterator_category</code> will
 454           yield its category, and <code class="literal">typename
 455           std::iterator_traits&lt;It&gt;::value_type</code> will yield its
 456           value type.
 457         </p><p>
 458           This library contains a container tag hierarchy corresponding to the
 459           diagram below.
 460         </p><div class="figure"><a name="id644317"></a><p class="title"><b>Figure 22.13. Container Tag Hierarchy</b></p><div class="figure-contents"><div class="mediaobject" align="center"><img src="../images/pbds_container_tag_hierarchy.png" align="middle" alt="Container Tag Hierarchy"></div></div></div><br class="figure-break"><p>
 461           Given any container <span class="type">Cntnr</span>, the tag of
 462           the underlying data structure can be found via <code class="literal">typename
 463           Cntnr::container_category</code>.
 464         </p></div><div class="section" title="Traits"><div class="titlepage"><div><div><h5 class="title"><a name="concepts.genericity.traits"></a>Traits</h5></div></div></div><p></p><p>Additionally, a traits mechanism can be used to query a
 465         container type for its attributes. Given any container
 466         <code class="literal">Cntnr</code>, then <code class="literal">&lt;Cntnr&gt;</code>
 467         is a traits class identifying the properties of the
 468         container.</p><p>To find if a container can throw when a key is erased (which
 469         is true for vector-based trees, for example), one can
 470         use
 471         </p><pre class="programlisting">container_traits&lt;Cntnr&gt;::erase_can_throw</pre><p>
 472           Some of the definitions in <code class="classname">container_traits</code>
 473           are dependent on other
 474           definitions. If <code class="classname">container_traits&lt;Cntnr&gt;::order_preserving</code>
 475           is <code class="constant">true</code> (which is the case for containers
 476           based on trees and tries), then the container can be split or
 477           joined; in this
 478           case, <code class="classname">container_traits&lt;Cntnr&gt;::split_join_can_throw</code>
 479           indicates whether splits or joins can throw exceptions (which is
 480           true for vector-based trees);
 481           otherwise <code class="classname">container_traits&lt;Cntnr&gt;::split_join_can_throw</code>
 482           will yield a compilation error. (This is somewhat similar to a
 483           compile-time version of the COM model).
 484         </p></div></div></div><div class="section" title="By Container"><div class="titlepage"><div><div><h3 class="title"><a name="pbds.design.container"></a>By Container</h3></div></div></div><div class="section" title="hash"><div class="titlepage"><div><div><h4 class="title"><a name="pbds.design.container.hash"></a>hash</h4></div></div></div><div class="section" title="Interface"><div class="titlepage"><div><div><h5 class="title"><a name="container.hash.interface"></a>Interface</h5></div></div></div><p>
 485           The collision-chaining hash-based container has the
 486         following declaration.</p><pre class="programlisting">
 487           template&lt;
 488           typename Key,
 489           typename Mapped,
 490           typename Hash_Fn = std::hash&lt;Key&gt;,
 491           typename Eq_Fn = std::equal_to&lt;Key&gt;,
 492           typename Comb_Hash_Fn =  direct_mask_range_hashing&lt;&gt;
 493           typename Resize_Policy = default explained below.
 494           bool Store_Hash = false,
 495           typename Allocator = std::allocator&lt;char&gt; &gt;
 496           class cc_hash_table;
 497         </pre><p>The parameters have the following meaning:</p><div class="orderedlist"><ol class="orderedlist" type="1"><li class="listitem"><p><code class="classname">Key</code> is the key type.</p></li><li class="listitem"><p><code class="classname">Mapped</code> is the mapped-policy.</p></li><li class="listitem"><p><code class="classname">Hash_Fn</code> is a key hashing functor.</p></li><li class="listitem"><p><code class="classname">Eq_Fn</code> is a key equivalence functor.</p></li><li class="listitem"><p><code class="classname">Comb_Hash_Fn</code> is a range-hashing_functor;
 498           it describes how to translate hash values into positions
 499           within the table. </p></li><li class="listitem"><p><code class="classname">Resize_Policy</code> describes how a container object
 500           should change its internal size. </p></li><li class="listitem"><p><code class="classname">Store_Hash</code> indicates whether the hash value
 501           should be stored with each entry. </p></li><li class="listitem"><p><code class="classname">Allocator</code> is an allocator
 502           type.</p></li></ol></div><p>The probing hash-based container has the following
 503         declaration.</p><pre class="programlisting">
 504           template&lt;
 505           typename Key,
 506           typename Mapped,
 507           typename Hash_Fn = std::hash&lt;Key&gt;,
 508           typename Eq_Fn = std::equal_to&lt;Key&gt;,
 509           typename Comb_Probe_Fn = direct_mask_range_hashing&lt;&gt;
 510           typename Probe_Fn = default explained below.
 511           typename Resize_Policy = default explained below.
 512           bool Store_Hash = false,
 513           typename Allocator =  std::allocator&lt;char&gt; &gt;
 514           class gp_hash_table;
 515         </pre><p>The parameters are identical to those of the
 516         collision-chaining container, except for the following.</p><div class="orderedlist"><ol class="orderedlist" type="1"><li class="listitem"><p><code class="classname">Comb_Probe_Fn</code> describes how to transform a probe
 517           sequence into a sequence of positions within the table.</p></li><li class="listitem"><p><code class="classname">Probe_Fn</code> describes a probe sequence policy.</p></li></ol></div><p>Some of the default template values depend on the values of
 518         other parameters, and are explained below.</p></div><div class="section" title="Details"><div class="titlepage"><div><div><h5 class="title"><a name="container.hash.details"></a>Details</h5></div></div></div><div class="section" title="Hash Policies"><div class="titlepage"><div><div><h6 class="title"><a name="container.hash.details.hash_policies"></a>Hash Policies</h6></div></div></div><div class="section" title="General"><div class="titlepage"><div><div><h6 class="title"><a name="details.hash_policies.general"></a>General</h6></div></div></div><p>Following is an explanation of some functions which hashing
 519             involves. The graphic below illustrates the discussion.</p><div class="figure"><a name="id644650"></a><p class="title"><b>Figure 22.14. Hash functions, ranged-hash functions, and
 520               range-hashing functions</b></p><div class="figure-contents"><div class="mediaobject" align="center"><img src="../images/pbds_hash_ranged_hash_range_hashing_fns.png" align="middle" alt="Hash functions, ranged-hash functions, and range-hashing functions"></div></div></div><br class="figure-break"><p>Let U be a domain (e.g., the integers, or the
 521             strings of 3 characters). A hash-table algorithm needs to map
 522             elements of U "uniformly" into the range [0,..., m -
 523             1] (where m is a non-negative integral value, and
 524             is, in general, time varying). I.e., the algorithm needs
 525             a ranged-hash function</p><p>
 526               f : U × Z<sub>+</sub> → Z<sub>+</sub>
 527             </p><p>such that for any u in U ,</p><p>0 ≤ f(u, m) ≤ m - 1</p><p>and which has "good uniformity" properties (say
 528             <a class="xref" href="policy_data_structures.html#biblio.knuth98sorting" title="The Art of Computer Programming - Sorting and Searching">[biblio.knuth98sorting]</a>.)
 529             One
 530             common solution is to use the composition of the hash
 531             function</p><p>h : U → Z<sub>+</sub> ,</p><p>which maps elements of U into the non-negative
 532             integrals, and</p><p>g : Z<sub>+</sub> × Z<sub>+</sub> →
 533             Z<sub>+</sub>,</p><p>which maps a non-negative hash value, and a non-negative
 534             range upper-bound into a non-negative integral in the range
 535             between 0 (inclusive) and the range upper bound (exclusive),
 536             i.e., for any r in Z<sub>+</sub>,</p><p>0 ≤ g(r, m) ≤ m - 1</p><p>The resulting ranged-hash function, is</p><div class="equation"><a name="id644765"></a><p class="title"><b>Equation 22.1. Ranged Hash Function</b></p><div class="equation-contents"><span class="mathphrase">
 537                 f(u , m) = g(h(u), m)
 538               </span></div></div><br class="equation-break"><p>From the above, it is obvious that given g and
 539             h, f can always be composed (however the converse
 540             is not true). The standard's hash-based containers allow specifying
 541             a hash function, and use a hard-wired range-hashing function;
 542             the ranged-hash function is implicitly composed.</p><p>The above describes the case where a key is to be mapped
 543             into a single position within a hash table, e.g.,
 544             in a collision-chaining table. In other cases, a key is to be
 545             mapped into a sequence of positions within a table,
 546             e.g., in a probing table. Similar terms apply in this
 547             case: the table requires a ranged probe function,
 548             mapping a key into a sequence of positions withing the table.
 549             This is typically achieved by composing a hash function
 550             mapping the key into a non-negative integral type, a
 551             probe function transforming the hash value into a
 552             sequence of hash values, and a range-hashing function
 553             transforming the sequence of hash values into a sequence of
 554             positions.</p></div><div class="section" title="Range Hashing"><div class="titlepage"><div><div><h6 class="title"><a name="details.hash_policies.range"></a>Range Hashing</h6></div></div></div><p>Some common choices for range-hashing functions are the
 555             division, multiplication, and middle-square methods (<a class="xref" href="policy_data_structures.html#biblio.knuth98sorting" title="The Art of Computer Programming - Sorting and Searching">[biblio.knuth98sorting]</a>), defined
 556             as</p><div class="equation"><a name="id644814"></a><p class="title"><b>Equation 22.2. Range-Hashing, Division Method</b></p><div class="equation-contents"><span class="mathphrase">
 557                 g(r, m) = r mod m
 558               </span></div></div><br class="equation-break"><p>g(r, m) = ⌈ u/v ( a r mod v ) ⌉</p><p>and</p><p>g(r, m) = ⌈ u/v ( r<sup>2</sup> mod v ) ⌉</p><p>respectively, for some positive integrals u and
 559             v (typically powers of 2), and some a. Each of
 560             these range-hashing functions works best for some different
 561             setting.</p><p>The division method (see above) is a
 562             very common choice. However, even this single method can be
 563             implemented in two very different ways. It is possible to
 564             implement using the low
 565             level % (modulo) operation (for any m), or the
 566             low level &amp; (bit-mask) operation (for the case where
 567             m is a power of 2), i.e.,</p><div class="equation"><a name="id644851"></a><p class="title"><b>Equation 22.3. Division via Prime Modulo</b></p><div class="equation-contents"><span class="mathphrase">
 568                 g(r, m) = r % m
 569               </span></div></div><br class="equation-break"><p>and</p><div class="equation"><a name="id644867"></a><p class="title"><b>Equation 22.4. Division via Bit Mask</b></p><div class="equation-contents"><span class="mathphrase">
 570                 g(r, m) = r &amp; m - 1, (with m =
 571                 2<sup>k</sup> for some k)
 572               </span></div></div><br class="equation-break"><p>respectively.</p><p>The % (modulo) implementation has the advantage that for
 573             m a prime far from a power of 2, g(r, m) is
 574             affected by all the bits of r (minimizing the chance of
 575             collision). It has the disadvantage of using the costly modulo
 576             operation. This method is hard-wired into SGI's implementation
 577             .</p><p>The &amp; (bit-mask) implementation has the advantage of
 578             relying on the fast bit-wise and operation. It has the
 579             disadvantage that for g(r, m) is affected only by the
 580             low order bits of r. This method is hard-wired into
 581             Dinkumware's implementation.</p></div><div class="section" title="Ranged Hash"><div class="titlepage"><div><div><h6 class="title"><a name="details.hash_policies.ranged"></a>Ranged Hash</h6></div></div></div><p>In cases it is beneficial to allow the
 582             client to directly specify a ranged-hash hash function. It is
 583             true, that the writer of the ranged-hash function cannot rely
 584             on the values of m having specific numerical properties
 585             suitable for hashing (in the sense used in <a class="xref" href="policy_data_structures.html#biblio.knuth98sorting" title="The Art of Computer Programming - Sorting and Searching">[biblio.knuth98sorting]</a>), since
 586             the values of m are determined by a resize policy with
 587             possibly orthogonal considerations.</p><p>There are two cases where a ranged-hash function can be
 588             superior. The firs is when using perfect hashing: the
 589             second is when the values of m can be used to estimate
 590             the "general" number of distinct values required. This is
 591             described in the following.</p><p>Let</p><p>
 592               s = [ s<sub>0</sub>,..., s<sub>t - 1</sub>]
 593             </p><p>be a string of t characters, each of which is from
 594             domain S. Consider the following ranged-hash
 595             function:</p><div class="equation"><a name="id644947"></a><p class="title"><b>Equation 22.5.
 596                 A Standard String Hash Function
 597               </b></p><div class="equation-contents"><span class="mathphrase">
 598                 f<sub>1</sub>(s, m) = ∑ <sub>i =
 599                 0</sub><sup>t - 1</sup> s<sub>i</sub> a<sup>i</sup> mod m
 600               </span></div></div><br class="equation-break"><p>where a is some non-negative integral value. This is
 601             the standard string-hashing function used in SGI's
 602             implementation (with a = 5). Its advantage is that
 603             it takes into account all of the characters of the string.</p><p>Now assume that s is the string representation of a
 604             of a long DNA sequence (and so S = {'A', 'C', 'G',
 605             'T'}). In this case, scanning the entire string might be
 606             prohibitively expensive. A possible alternative might be to use
 607             only the first k characters of the string, where</p><p>|S|<sup>k</sup> ≥ m ,</p><p>i.e., using the hash function</p><div class="equation"><a name="id644998"></a><p class="title"><b>Equation 22.6.
 608                 Only k String DNA Hash
 609               </b></p><div class="equation-contents"><span class="mathphrase">
 610                 f<sub>2</sub>(s, m) = ∑ <sub>i
 611                 = 0</sub><sup>k - 1</sup> s<sub>i</sub> a<sup>i</sup> mod m
 612               </span></div></div><br class="equation-break"><p>requiring scanning over only</p><p>k = log<sub>4</sub>( m )</p><p>characters.</p><p>Other more elaborate hash-functions might scan k
 613             characters starting at a random position (determined at each
 614             resize), or scanning k random positions (determined at
 615             each resize), i.e., using</p><p>f<sub>3</sub>(s, m) = ∑ <sub>i =
 616             r</sub>0<sup>r<sub>0</sub> + k - 1</sup> s<sub>i</sub>
 617             a<sup>i</sup> mod m ,</p><p>or</p><p>f<sub>4</sub>(s, m) = ∑ <sub>i = 0</sub><sup>k -
 618             1</sup> s<sub>r</sub>i a<sup>r<sub>i</sub></sup> mod
 619             m ,</p><p>respectively, for r<sub>0</sub>,..., r<sub>k-1</sub>
 620             each in the (inclusive) range [0,...,t-1].</p><p>It should be noted that the above functions cannot be
 621             decomposed as per a ranged hash composed of hash and range hashing.</p></div><div class="section" title="Implementation"><div class="titlepage"><div><div><h6 class="title"><a name="details.hash_policies.implementation"></a>Implementation</h6></div></div></div><p>This sub-subsection describes the implementation of
 622             the above in this library. It first explains range-hashing
 623             functions in collision-chaining tables, then ranged-hash
 624             functions in collision-chaining tables, then probing-based
 625             tables, and finally lists the relevant classes in this
 626             library.</p><div class="section" title="Range-Hashing and Ranged-Hashes in Collision-Chaining Tables"><div class="titlepage"><div><div><h6 class="title"><a name="hash_policies.implementation.collision-chaining"></a>
 627                 Range-Hashing and Ranged-Hashes in Collision-Chaining Tables
 628               </h6></div></div></div><p><code class="classname">cc_hash_table</code> is
 629               parametrized by <code class="classname">Hash_Fn</code> and <code class="classname">Comb_Hash_Fn</code>, a
 630               hash functor and a combining hash functor, respectively.</p><p>In general, <code class="classname">Comb_Hash_Fn</code> is considered a
 631               range-hashing functor. <code class="classname">cc_hash_table</code>
 632               synthesizes a ranged-hash function from <code class="classname">Hash_Fn</code> and
 633               <code class="classname">Comb_Hash_Fn</code>. The figure below shows an <code class="classname">insert</code> sequence
 634               diagram for this case. The user inserts an element (point A),
 635               the container transforms the key into a non-negative integral
 636               using the hash functor (points B and C), and transforms the
 637               result into a position using the combining functor (points D
 638               and E).</p><div class="figure"><a name="id645187"></a><p class="title"><b>Figure 22.15. Insert hash sequence diagram</b></p><div class="figure-contents"><div class="mediaobject" align="center"><img src="../images/pbds_hash_range_hashing_seq_diagram.png" align="middle" alt="Insert hash sequence diagram"></div></div></div><br class="figure-break"><p>If <code class="classname">cc_hash_table</code>'s
 639               hash-functor, <code class="classname">Hash_Fn</code> is instantiated by <code class="classname">null_type</code> , then <code class="classname">Comb_Hash_Fn</code> is taken to be
 640               a ranged-hash function. The graphic below shows an <code class="function">insert</code> sequence
 641               diagram. The user inserts an element (point A), the container
 642               transforms the key into a position using the combining functor
 643               (points B and C).</p><div class="figure"><a name="id645245"></a><p class="title"><b>Figure 22.16. Insert hash sequence diagram with a null policy</b></p><div class="figure-contents"><div class="mediaobject" align="center"><img src="../images/pbds_hash_range_hashing_seq_diagram2.png" align="middle" alt="Insert hash sequence diagram with a null policy"></div></div></div><br class="figure-break"></div><div class="section" title="Probing tables"><div class="titlepage"><div><div><h6 class="title"><a name="hash_policies.implementation.probe"></a>
 644                 Probing tables
 645               </h6></div></div></div><p><code class="classname">gp_hash_table</code> is parametrized by
 646               <code class="classname">Hash_Fn</code>, <code class="classname">Probe_Fn</code>,
 647               and <code class="classname">Comb_Probe_Fn</code>. As before, if
 648               <code class="classname">Hash_Fn</code> and <code class="classname">Probe_Fn</code>
 649               are both <code class="classname">null_type</code>, then
 650               <code class="classname">Comb_Probe_Fn</code> is a ranged-probe
 651               functor. Otherwise, <code class="classname">Hash_Fn</code> is a hash
 652               functor, <code class="classname">Probe_Fn</code> is a functor for offsets
 653               from a hash value, and <code class="classname">Comb_Probe_Fn</code>
 654               transforms a probe sequence into a sequence of positions within
 655               the table.</p></div><div class="section" title="Pre-Defined Policies"><div class="titlepage"><div><div><h6 class="title"><a name="hash_policies.implementation.predefined"></a>
 656                 Pre-Defined Policies
 657               </h6></div></div></div><p>This library contains some pre-defined classes
 658               implementing range-hashing and probing functions:</p><div class="orderedlist"><ol class="orderedlist" type="1"><li class="listitem"><p><code class="classname">direct_mask_range_hashing</code>
 659                 and <code class="classname">direct_mod_range_hashing</code>
 660                 are range-hashing functions based on a bit-mask and a modulo
 661                 operation, respectively.</p></li><li class="listitem"><p><code class="classname">linear_probe_fn</code>, and
 662                 <code class="classname">quadratic_probe_fn</code> are
 663                 a linear probe and a quadratic probe function,
 664                 respectively.</p></li></ol></div><p>
 665                 The graphic below shows the relationships.
 666               </p><div class="figure"><a name="id645385"></a><p class="title"><b>Figure 22.17. Hash policy class diagram</b></p><div class="figure-contents"><div class="mediaobject" align="center"><img src="../images/pbds_hash_policy_cd.png" align="middle" alt="Hash policy class diagram"></div></div></div><br class="figure-break"></div></div></div><div class="section" title="Resize Policies"><div class="titlepage"><div><div><h6 class="title"><a name="container.hash.details.resize_policies"></a>Resize Policies</h6></div></div></div><div class="section" title="General"><div class="titlepage"><div><div><h6 class="title"><a name="resize_policies.general"></a>General</h6></div></div></div><p>Hash-tables, as opposed to trees, do not naturally grow or
 667             shrink. It is necessary to specify policies to determine how
 668             and when a hash table should change its size. Usually, resize
 669             policies can be decomposed into orthogonal policies:</p><div class="orderedlist"><ol class="orderedlist" type="1"><li class="listitem"><p>A size policy indicating how a hash table
 670               should grow (e.g., it should multiply by powers of
 671               2).</p></li><li class="listitem"><p>A trigger policy indicating when a hash
 672               table should grow (e.g., a load factor is
 673               exceeded).</p></li></ol></div></div><div class="section" title="Size Policies"><div class="titlepage"><div><div><h6 class="title"><a name="resize_policies.size"></a>Size Policies</h6></div></div></div><p>Size policies determine how a hash table changes size. These
 674             policies are simple, and there are relatively few sensible
 675             options. An exponential-size policy (with the initial size and
 676             growth factors both powers of 2) works well with a mask-based
 677             range-hashing function, and is the
 678             hard-wired policy used by Dinkumware. A
 679             prime-list based policy works well with a modulo-prime range
 680             hashing function and is the hard-wired policy used by SGI's
 681             implementation.</p></div><div class="section" title="Trigger Policies"><div class="titlepage"><div><div><h6 class="title"><a name="resize_policies.trigger"></a>Trigger Policies</h6></div></div></div><p>Trigger policies determine when a hash table changes size.
 682             Following is a description of two policies: load-check
 683             policies, and collision-check policies.</p><p>Load-check policies are straightforward. The user specifies
 684             two factors, Α<sub>min</sub> and
 685             Α<sub>max</sub>, and the hash table maintains the
 686             invariant that</p><p>Α<sub>min</sub> ≤ (number of
 687             stored elements) / (hash-table size) ≤
 688             Α<sub>max</sub><em><span class="remark">load factor min max</span></em></p><p>Collision-check policies work in the opposite direction of
 689             load-check policies. They focus on keeping the number of
 690             collisions moderate and hoping that the size of the table will
 691             not grow very large, instead of keeping a moderate load-factor
 692             and hoping that the number of collisions will be small. A
 693             maximal collision-check policy resizes when the longest
 694             probe-sequence grows too large.</p><p>Consider the graphic below. Let the size of the hash table
 695             be denoted by m, the length of a probe sequence be denoted by k,
 696             and some load factor be denoted by Α. We would like to
 697             calculate the minimal length of k, such that if there were Α
 698             m elements in the hash table, a probe sequence of length k would
 699             be found with probability at most 1/m.</p><div class="figure"><a name="id645544"></a><p class="title"><b>Figure 22.18. Balls and bins</b></p><div class="figure-contents"><div class="mediaobject" align="center"><img src="../images/pbds_balls_and_bins.png" align="middle" alt="Balls and bins"></div></div></div><br class="figure-break"><p>Denote the probability that a probe sequence of length
 700             k appears in bin i by p<sub>i</sub>, the
 701             length of the probe sequence of bin i by
 702             l<sub>i</sub>, and assume uniform distribution. Then</p><div class="equation"><a name="id645590"></a><p class="title"><b>Equation 22.7.
 703                 Probability of Probe Sequence of Length k
 704               </b></p><div class="equation-contents"><span class="mathphrase">
 705                 p<sub>1</sub> =
 706               </span></div></div><br class="equation-break"><p>P(l<sub>1</sub> ≥ k) =</p><p>
 707               P(l<sub>1</sub> ≥ α ( 1 + k / α - 1) ≤ (a)
 708             </p><p>
 709               e ^ ( - ( α ( k / α - 1 )<sup>2</sup> ) /2)
 710             </p><p>where (a) follows from the Chernoff bound (<a class="xref" href="policy_data_structures.html#biblio.motwani95random" title="Randomized Algorithms">[biblio.motwani95random]</a>). To
 711             calculate the probability that some bin contains a probe
 712             sequence greater than k, we note that the
 713             l<sub>i</sub> are negatively-dependent
 714             (<a class="xref" href="policy_data_structures.html#biblio.dubhashi98neg" title="Balls and bins: A study in negative dependence">[biblio.dubhashi98neg]</a>)
 715             . Let
 716             I(.) denote the indicator function. Then</p><div class="equation"><a name="id645646"></a><p class="title"><b>Equation 22.8.
 717                 Probability Probe Sequence in Some Bin
 718               </b></p><div class="equation-contents"><span class="mathphrase">
 719                 P( exists<sub>i</sub> l<sub>i</sub> ≥ k ) =
 720               </span></div></div><br class="equation-break"><p>P ( ∑ <sub>i = 1</sub><sup>m</sup>
 721             I(l<sub>i</sub> ≥ k) ≥ 1 ) =</p><p>P ( ∑ <sub>i = 1</sub><sup>m</sup> I (
 722             l<sub>i</sub> ≥ k ) ≥ m p<sub>1</sub> ( 1 + 1 / (m
 723             p<sub>1</sub>) - 1 ) ) ≤ (a)</p><p>e ^ ( ( - m p<sub>1</sub> ( 1 / (m p<sub>1</sub>)
 724             - 1 ) <sup>2</sup> ) / 2 ) ,</p><p>where (a) follows from the fact that the Chernoff bound can
 725             be applied to negatively-dependent variables (<a class="xref" href="policy_data_structures.html#biblio.dubhashi98neg" title="Balls and bins: A study in negative dependence">[biblio.dubhashi98neg]</a>). Inserting the first probability
 726             equation into the second one, and equating with 1/m, we
 727             obtain</p><p>k ~ √ ( 2 α ln 2 m ln(m) )
 728             ) .</p></div><div class="section" title="Implementation"><div class="titlepage"><div><div><h6 class="title"><a name="resize_policies.impl"></a>Implementation</h6></div></div></div><p>This sub-subsection describes the implementation of the
 729             above in this library. It first describes resize policies and
 730             their decomposition into trigger and size policies, then
 731             describes pre-defined classes, and finally discusses controlled
 732             access the policies' internals.</p><div class="section" title="Decomposition"><div class="titlepage"><div><div><h6 class="title"><a name="resize_policies.impl.decomposition"></a>Decomposition</h6></div></div></div><p>Each hash-based container is parametrized by a
 733               <code class="classname">Resize_Policy</code> parameter; the container derives
 734               <code class="classname">public</code>ly from <code class="classname">Resize_Policy</code>. For
 735               example:</p><pre class="programlisting">
 736                 cc_hash_table&lt;typename Key,
 737                 typename Mapped,
 738                 ...
 739                 typename Resize_Policy
 740                 ...&gt; : public Resize_Policy
 741               </pre><p>As a container object is modified, it continuously notifies
 742               its <code class="classname">Resize_Policy</code> base of internal changes
 743               (e.g., collisions encountered and elements being
 744               inserted). It queries its <code class="classname">Resize_Policy</code> base whether
 745               it needs to be resized, and if so, to what size.</p><p>The graphic below shows a (possible) sequence diagram
 746               of an insert operation. The user inserts an element; the hash
 747               table notifies its resize policy that a search has started
 748               (point A); in this case, a single collision is encountered -
 749               the table notifies its resize policy of this (point B); the
 750               container finally notifies its resize policy that the search
 751               has ended (point C); it then queries its resize policy whether
 752               a resize is needed, and if so, what is the new size (points D
 753               to G); following the resize, it notifies the policy that a
 754               resize has completed (point H); finally, the element is
 755               inserted, and the policy notified (point I).</p><div class="figure"><a name="id645800"></a><p class="title"><b>Figure 22.19. Insert resize sequence diagram</b></p><div class="figure-contents"><div class="mediaobject" align="center"><img src="../images/pbds_insert_resize_sequence_diagram1.png" align="middle" alt="Insert resize sequence diagram"></div></div></div><br class="figure-break"><p>In practice, a resize policy can be usually orthogonally
 756               decomposed to a size policy and a trigger policy. Consequently,
 757               the library contains a single class for instantiating a resize
 758               policy: <code class="classname">hash_standard_resize_policy</code>
 759               is parametrized by <code class="classname">Size_Policy</code> and
 760               <code class="classname">Trigger_Policy</code>, derives <code class="classname">public</code>ly from
 761               both, and acts as a standard delegate (<a class="xref" href="policy_data_structures.html#biblio.gof" title="Design Patterns - Elements of Reusable Object-Oriented Software">[biblio.gof]</a>)
 762               to these policies.</p><p>The two graphics immediately below show sequence diagrams
 763               illustrating the interaction between the standard resize policy
 764               and its trigger and size policies, respectively.</p><div class="figure"><a name="id645865"></a><p class="title"><b>Figure 22.20. Standard resize policy trigger sequence
 765                 diagram</b></p><div class="figure-contents"><div class="mediaobject" align="center"><img src="../images/pbds_insert_resize_sequence_diagram2.png" align="middle" alt="Standard resize policy trigger sequence diagram"></div></div></div><br class="figure-break"><div class="figure"><a name="id645900"></a><p class="title"><b>Figure 22.21. Standard resize policy size sequence
 766                 diagram</b></p><div class="figure-contents"><div class="mediaobject" align="center"><img src="../images/pbds_insert_resize_sequence_diagram3.png" align="middle" alt="Standard resize policy size sequence diagram"></div></div></div><br class="figure-break"></div><div class="section" title="Predefined Policies"><div class="titlepage"><div><div><h6 class="title"><a name="resize_policies.impl.predefined"></a>Predefined Policies</h6></div></div></div><p>The library includes the following
 767               instantiations of size and trigger policies:</p><div class="orderedlist"><ol class="orderedlist" type="1"><li class="listitem"><p><code class="classname">hash_load_check_resize_trigger</code>
 768                 implements a load check trigger policy.</p></li><li class="listitem"><p><code class="classname">cc_hash_max_collision_check_resize_trigger</code>
 769                 implements a collision check trigger policy.</p></li><li class="listitem"><p><code class="classname">hash_exponential_size_policy</code>
 770                 implements an exponential-size policy (which should be used
 771                 with mask range hashing).</p></li><li class="listitem"><p><code class="classname">hash_prime_size_policy</code>
 772                 implementing a size policy based on a sequence of primes
 773                 (which should
 774                 be used with mod range hashing</p></li></ol></div><p>The graphic below gives an overall picture of the resize-related
 775               classes. <code class="classname">basic_hash_table</code>
 776               is parametrized by <code class="classname">Resize_Policy</code>, which it subclasses
 777               publicly. This class is currently instantiated only by <code class="classname">hash_standard_resize_policy</code>.
 778               <code class="classname">hash_standard_resize_policy</code>
 779               itself is parametrized by <code class="classname">Trigger_Policy</code> and
 780               <code class="classname">Size_Policy</code>. Currently, <code class="classname">Trigger_Policy</code> is
 781               instantiated by <code class="classname">hash_load_check_resize_trigger</code>,
 782               or <code class="classname">cc_hash_max_collision_check_resize_trigger</code>;
 783               <code class="classname">Size_Policy</code> is instantiated by <code class="classname">hash_exponential_size_policy</code>,
 784               or <code class="classname">hash_prime_size_policy</code>.</p></div><div class="section" title="Controling Access to Internals"><div class="titlepage"><div><div><h6 class="title"><a name="resize_policies.impl.internals"></a>Controling Access to Internals</h6></div></div></div><p>There are cases where (controlled) access to resize
 785               policies' internals is beneficial. E.g., it is sometimes
 786               useful to query a hash-table for the table's actual size (as
 787               opposed to its <code class="function">size()</code> - the number of values it
 788               currently holds); it is sometimes useful to set a table's
 789               initial size, externally resize it, or change load factors.</p><p>Clearly, supporting such methods both decreases the
 790               encapsulation of hash-based containers, and increases the
 791               diversity between different associative-containers' interfaces.
 792               Conversely, omitting such methods can decrease containers'
 793               flexibility.</p><p>In order to avoid, to the extent possible, the above
 794               conflict, the hash-based containers themselves do not address
 795               any of these questions; this is deferred to the resize policies,
 796               which are easier to change or replace. Thus, for example,
 797               neither <code class="classname">cc_hash_table</code> nor
 798               <code class="classname">gp_hash_table</code>
 799               contain methods for querying the actual size of the table; this
 800               is deferred to <code class="classname">hash_standard_resize_policy</code>.</p><p>Furthermore, the policies themselves are parametrized by
 801               template arguments that determine the methods they support
 802               (
 803               <a class="xref" href="policy_data_structures.html#biblio.alexandrescu01modern" title="Modern C++ Design: Generic Programming and Design Patterns Applied">[biblio.alexandrescu01modern]</a>
 804               shows techniques for doing so). <code class="classname">hash_standard_resize_policy</code>
 805               is parametrized by <code class="classname">External_Size_Access</code> that
 806               determines whether it supports methods for querying the actual
 807               size of the table or resizing it. <code class="classname">hash_load_check_resize_trigger</code>
 808               is parametrized by <code class="classname">External_Load_Access</code> that
 809               determines whether it supports methods for querying or
 810               modifying the loads. <code class="classname">cc_hash_max_collision_check_resize_trigger</code>
 811               is parametrized by <code class="classname">External_Load_Access</code> that
 812               determines whether it supports methods for querying the
 813               load.</p><p>Some operations, for example, resizing a container at
 814               run time, or changing the load factors of a load-check trigger
 815               policy, require the container itself to resize. As mentioned
 816               above, the hash-based containers themselves do not contain
 817               these types of methods, only their resize policies.
 818               Consequently, there must be some mechanism for a resize policy
 819               to manipulate the hash-based container. As the hash-based
 820               container is a subclass of the resize policy, this is done
 821               through virtual methods. Each hash-based container has a
 822               <code class="classname">private</code> <code class="classname">virtual</code> method:</p><pre class="programlisting">
 823                 virtual void
 824                 do_resize
 825                 (size_type new_size);
 826               </pre><p>which resizes the container. Implementations of
 827               <code class="classname">Resize_Policy</code> can export public methods for resizing
 828               the container externally; these methods internally call
 829               <code class="classname">do_resize</code> to resize the table.</p></div></div></div><div class="section" title="Policy Interactions"><div class="titlepage"><div><div><h6 class="title"><a name="container.hash.details.policy_interaction"></a>Policy Interactions</h6></div></div></div><p>
 830           </p><p>Hash-tables are unfortunately especially susceptible to
 831           choice of policies. One of the more complicated aspects of this
 832           is that poor combinations of good policies can form a poor
 833           container. Following are some considerations.</p><div class="section" title="probe/size/trigger"><div class="titlepage"><div><div><h6 class="title"><a name="policy_interaction.probesizetrigger"></a>probe/size/trigger</h6></div></div></div><p>Some combinations do not work well for probing containers.
 834             For example, combining a quadratic probe policy with an
 835             exponential size policy can yield a poor container: when an
 836             element is inserted, a trigger policy might decide that there
 837             is no need to resize, as the table still contains unused
 838             entries; the probe sequence, however, might never reach any of
 839             the unused entries.</p><p>Unfortunately, this library cannot detect such problems at
 840             compilation (they are halting reducible). It therefore defines
 841             an exception class <code class="classname">insert_error</code> to throw an
 842             exception in this case.</p></div><div class="section" title="hash/trigger"><div class="titlepage"><div><div><h6 class="title"><a name="policy_interaction.hashtrigger"></a>hash/trigger</h6></div></div></div><p>Some trigger policies are especially susceptible to poor
 843             hash functions. Suppose, as an extreme case, that the hash
 844             function transforms each key to the same hash value. After some
 845             inserts, a collision detecting policy will always indicate that
 846             the container needs to grow.</p><p>The library, therefore, by design, limits each operation to
 847             one resize. For each <code class="classname">insert</code>, for example, it queries
 848             only once whether a resize is needed.</p></div><div class="section" title="equivalence functors/storing hash values/hash"><div class="titlepage"><div><div><h6 class="title"><a name="policy_interaction.eqstorehash"></a>equivalence functors/storing hash values/hash</h6></div></div></div><p><code class="classname">cc_hash_table</code> and
 849             <code class="classname">gp_hash_table</code> are
 850             parametrized by an equivalence functor and by a
 851             <code class="classname">Store_Hash</code> parameter. If the latter parameter is
 852             <code class="classname">true</code>, then the container stores with each entry
 853             a hash value, and uses this value in case of collisions to
 854             determine whether to apply a hash value. This can lower the
 855             cost of collision for some types, but increase the cost of
 856             collisions for other types.</p><p>If a ranged-hash function or ranged probe function is
 857             directly supplied, however, then it makes no sense to store the
 858             hash value with each entry. This library's container will
 859             fail at compilation, by design, if this is attempted.</p></div><div class="section" title="size/load-check trigger"><div class="titlepage"><div><div><h6 class="title"><a name="policy_interaction.sizeloadtrigger"></a>size/load-check trigger</h6></div></div></div><p>Assume a size policy issues an increasing sequence of sizes
 860             a, a q, a q<sup>1</sup>, a q<sup>2</sup>, ... For
 861             example, an exponential size policy might issue the sequence of
 862             sizes 8, 16, 32, 64, ...</p><p>If a load-check trigger policy is used, with loads
 863             α<sub>min</sub> and α<sub>max</sub>,
 864             respectively, then it is a good idea to have:</p><div class="orderedlist"><ol class="orderedlist" type="1"><li class="listitem"><p>α<sub>max</sub> ~ 1 / q</p></li><li class="listitem"><p>α<sub>min</sub> &lt; 1 / (2 q)</p></li></ol></div><p>This will ensure that the amortized hash cost of each
 865             modifying operation is at most approximately 3.</p><p>α<sub>min</sub> ~ α<sub>max</sub> is, in
 866             any case, a bad choice, and α<sub>min</sub> &gt;
 867             α <sub>max</sub> is horrendous.</p></div></div></div></div><div class="section" title="tree"><div class="titlepage"><div><div><h4 class="title"><a name="pbds.design.container.tree"></a>tree</h4></div></div></div><div class="section" title="Interface"><div class="titlepage"><div><div><h5 class="title"><a name="container.tree.interface"></a>Interface</h5></div></div></div><p>The tree-based container has the following declaration:</p><pre class="programlisting">
 868           template&lt;
 869           typename Key,
 870           typename Mapped,
 871           typename Cmp_Fn = std::less&lt;Key&gt;,
 872           typename Tag = rb_tree_tag,
 873           template&lt;
 874           typename Const_Node_Iterator,
 875           typename Node_Iterator,
 876           typename Cmp_Fn_,
 877           typename Allocator_&gt;
 878           class Node_Update = null_node_update,
 879           typename Allocator = std::allocator&lt;char&gt; &gt;
 880           class tree;
 881         </pre><p>The parameters have the following meaning:</p><div class="orderedlist"><ol class="orderedlist" type="1"><li class="listitem"><p><code class="classname">Key</code> is the key type.</p></li><li class="listitem"><p><code class="classname">Mapped</code> is the mapped-policy.</p></li><li class="listitem"><p><code class="classname">Cmp_Fn</code> is a key comparison functor</p></li><li class="listitem"><p><code class="classname">Tag</code> specifies which underlying data structure
 882           to use.</p></li><li class="listitem"><p><code class="classname">Node_Update</code> is a policy for updating node
 883           invariants.</p></li><li class="listitem"><p><code class="classname">Allocator</code> is an allocator
 884           type.</p></li></ol></div><p>The <code class="classname">Tag</code> parameter specifies which underlying
 885         data structure to use. Instantiating it by <code class="classname">rb_tree_tag</code>, <code class="classname">splay_tree_tag</code>, or
 886         <code class="classname">ov_tree_tag</code>,
 887         specifies an underlying red-black tree, splay tree, or
 888         ordered-vector tree, respectively; any other tag is illegal.
 889         Note that containers based on the former two contain more types
 890         and methods than the latter (e.g.,
 891         <code class="classname">reverse_iterator</code> and <code class="classname">rbegin</code>), and different
 892         exception and invalidation guarantees.</p></div><div class="section" title="Details"><div class="titlepage"><div><div><h5 class="title"><a name="container.tree.details"></a>Details</h5></div></div></div><div class="section" title="Node Invariants"><div class="titlepage"><div><div><h6 class="title"><a name="container.tree.node"></a>Node Invariants</h6></div></div></div><p>Consider the two trees in the graphic below, labels A and B. The first
 893           is a tree of floats; the second is a tree of pairs, each
 894           signifying a geometric line interval. Each element in a tree is refered to as a node of the tree. Of course, each of
 895           these trees can support the usual queries: the first can easily
 896           search for <code class="classname">0.4</code>; the second can easily search for
 897           <code class="classname">std::make_pair(10, 41)</code>.</p><p>Each of these trees can efficiently support other queries.
 898           The first can efficiently determine that the 2rd key in the
 899           tree is <code class="constant">0.3</code>; the second can efficiently determine
 900           whether any of its intervals overlaps
 901           </p><pre class="programlisting">std::make_pair(29,42)</pre><p> (useful in geometric
 902           applications or distributed file systems with leases, for
 903           example).  It should be noted that an <code class="classname">std::set</code> can
 904           only solve these types of problems with linear complexity.</p><p>In order to do so, each tree stores some metadata in
 905           each node, and maintains node invariants (see <a class="xref" href="policy_data_structures.html#biblio.clrs2001" title="Introduction to Algorithms, 2nd edition">[biblio.clrs2001]</a>.) The first stores in
 906           each node the size of the sub-tree rooted at the node; the
 907           second stores at each node the maximal endpoint of the
 908           intervals at the sub-tree rooted at the node.</p><div class="figure"><a name="id646549"></a><p class="title"><b>Figure 22.22. Tree node invariants</b></p><div class="figure-contents"><div class="mediaobject" align="center"><img src="../images/pbds_tree_node_invariants.png" align="middle" alt="Tree node invariants"></div></div></div><br class="figure-break"><p>Supporting such trees is difficult for a number of
 909           reasons:</p><div class="orderedlist"><ol class="orderedlist" type="1"><li class="listitem"><p>There must be a way to specify what a node's metadata
 910             should be (if any).</p></li><li class="listitem"><p>Various operations can invalidate node
 911             invariants.  The graphic below shows how a right rotation,
 912             performed on A, results in B, with nodes x and y having
 913             corrupted invariants (the grayed nodes in C). The graphic shows
 914             how an insert, performed on D, results in E, with nodes x and y
 915             having corrupted invariants (the grayed nodes in F). It is not
 916             feasible to know outside the tree the effect of an operation on
 917             the nodes of the tree.</p></li><li class="listitem"><p>The search paths of standard associative containers are
 918             defined by comparisons between keys, and not through
 919             metadata.</p></li><li class="listitem"><p>It is not feasible to know in advance which methods trees
 920             can support. Besides the usual <code class="classname">find</code> method, the
 921             first tree can support a <code class="classname">find_by_order</code> method, while
 922             the second can support an <code class="classname">overlaps</code> method.</p></li></ol></div><div class="figure"><a name="id646628"></a><p class="title"><b>Figure 22.23. Tree node invalidation</b></p><div class="figure-contents"><div class="mediaobject" align="center"><img src="../images/pbds_tree_node_invalidations.png" align="middle" alt="Tree node invalidation"></div></div></div><br class="figure-break"><p>These problems are solved by a combination of two means:
 923           node iterators, and template-template node updater
 924           parameters.</p><div class="section" title="Node Iterators"><div class="titlepage"><div><div><h6 class="title"><a name="container.tree.node.iterators"></a>Node Iterators</h6></div></div></div><p>Each tree-based container defines two additional iterator
 925             types, <code class="classname">const_node_iterator</code>
 926             and <code class="classname">node_iterator</code>.
 927             These iterators allow descending from a node to one of its
 928             children. Node iterator allow search paths different than those
 929             determined by the comparison functor. The <code class="classname">tree</code>
 930             supports the methods:</p><pre class="programlisting">
 931               const_node_iterator
 932               node_begin() const;
 933
 934               node_iterator
 935               node_begin();
 936
 937               const_node_iterator
 938               node_end() const;
 939
 940               node_iterator
 941               node_end();
 942             </pre><p>The first pairs return node iterators corresponding to the
 943             root node of the tree; the latter pair returns node iterators
 944             corresponding to a just-after-leaf node.</p></div><div class="section" title="Node Updator"><div class="titlepage"><div><div><h6 class="title"><a name="container.tree.node.updator"></a>Node Updator</h6></div></div></div><p>The tree-based containers are parametrized by a
 945             <code class="classname">Node_Update</code> template-template parameter. A
 946             tree-based container instantiates
 947             <code class="classname">Node_Update</code> to some
 948             <code class="classname">node_update</code> class, and publicly subclasses
 949             <code class="classname">node_update</code>. The graphic below shows this
 950             scheme, as well as some predefined policies (which are explained
 951             below).</p><div class="figure"><a name="id646738"></a><p class="title"><b>Figure 22.24. A tree and its update policy</b></p><div class="figure-contents"><div class="mediaobject" align="center"><img src="../images/pbds_tree_node_updator_policy_cd.png" align="middle" alt="A tree and its update policy"></div></div></div><br class="figure-break"><p><code class="classname">node_update</code> (an instantiation of
 952             <code class="classname">Node_Update</code>) must define <code class="classname">metadata_type</code> as
 953             the type of metadata it requires. For order statistics,
 954             e.g., <code class="classname">metadata_type</code> might be <code class="classname">size_t</code>.
 955             The tree defines within each node a <code class="classname">metadata_type</code>
 956             object.</p><p><code class="classname">node_update</code> must also define the following method
 957             for restoring node invariants:</p><pre class="programlisting">
 958               void
 959               operator()(node_iterator nd_it, const_node_iterator end_nd_it)
 960             </pre><p>In this method, <code class="varname">nd_it</code> is a
 961             <code class="classname">node_iterator</code> corresponding to a node whose
 962             A) all descendants have valid invariants, and B) its own
 963             invariants might be violated; <code class="classname">end_nd_it</code> is
 964             a <code class="classname">const_node_iterator</code> corresponding to a
 965             just-after-leaf node. This method should correct the node
 966             invariants of the node pointed to by
 967             <code class="classname">nd_it</code>. For example, say node x in the
 968             graphic below label A has an invalid invariant, but its' children,
 969             y and z have valid invariants. After the invocation, all three
 970             nodes should have valid invariants, as in label B.</p><div class="figure"><a name="id646835"></a><p class="title"><b>Figure 22.25. Restoring node invariants</b></p><div class="figure-contents"><div class="mediaobject" align="center"><img src="../images/pbds_restoring_node_invariants.png" align="middle" alt="Restoring node invariants"></div></div></div><br class="figure-break"><p>When a tree operation might invalidate some node invariant,
 971             it invokes this method in its <code class="classname">node_update</code> base to
 972             restore the invariant. For example, the graphic below shows
 973             an <code class="function">insert</code> operation (point A); the tree performs some
 974             operations, and calls the update functor three times (points B,
 975             C, and D). (It is well known that any <code class="function">insert</code>,
 976             <code class="function">erase</code>, <code class="function">split</code> or <code class="function">join</code>, can restore
 977             all node invariants by a small number of node invariant updates (<a class="xref" href="policy_data_structures.html#biblio.clrs2001" title="Introduction to Algorithms, 2nd edition">[biblio.clrs2001]</a>)
 978             .</p><div class="figure"><a name="id646903"></a><p class="title"><b>Figure 22.26. Insert update sequence</b></p><div class="figure-contents"><div class="mediaobject" align="center"><img src="../images/pbds_update_seq_diagram.png" align="middle" alt="Insert update sequence"></div></div></div><br class="figure-break"><p>To complete the description of the scheme, three questions
 979             need to be answered:</p><div class="orderedlist"><ol class="orderedlist" type="1"><li class="listitem"><p>How can a tree which supports order statistics define a
 980               method such as <code class="classname">find_by_order</code>?</p></li><li class="listitem"><p>How can the node updater base access methods of the
 981               tree?</p></li><li class="listitem"><p>How can the following cyclic dependency be resolved?
 982               <code class="classname">node_update</code> is a base class of the tree, yet it
 983               uses node iterators defined in the tree (its child).</p></li></ol></div><p>The first two questions are answered by the fact that
 984             <code class="classname">node_update</code> (an instantiation of
 985             <code class="classname">Node_Update</code>) is a <span class="emphasis"><em>public</em></span> base class
 986             of the tree. Consequently:</p><div class="orderedlist"><ol class="orderedlist" type="1"><li class="listitem"><p>Any public methods of
 987               <code class="classname">node_update</code> are automatically methods of
 988               the tree (<a class="xref" href="policy_data_structures.html#biblio.alexandrescu01modern" title="Modern C++ Design: Generic Programming and Design Patterns Applied">[biblio.alexandrescu01modern]</a>).
 989               Thus an order-statistics node updater,
 990               <code class="classname">tree_order_statistics_node_update</code> defines
 991               the <code class="function">find_by_order</code> method; any tree
 992               instantiated by this policy consequently supports this method as
 993               well.</p></li><li class="listitem"><p>In C++, if a base class declares a method as
 994               <code class="literal">virtual</code>, it is
 995               <code class="literal">virtual</code> in its subclasses. If
 996               <code class="classname">node_update</code> needs to access one of the
 997               tree's methods, say the member function
 998               <code class="function">end</code>, it simply declares that method as
 999               <code class="literal">virtual</code> abstract.</p></li></ol></div><p>The cyclic dependency is solved through template-template
1000             parameters. <code class="classname">Node_Update</code> is parametrized by
1001             the tree's node iterators, its comparison functor, and its
1002             allocator type. Thus, instantiations of
1003             <code class="classname">Node_Update</code> have all information
1004             required.</p><p>This library assumes that constructing a metadata object and
1005             modifying it are exception free. Suppose that during some method,
1006             say <code class="classname">insert</code>, a metadata-related operation
1007             (e.g., changing the value of a metadata) throws an exception. Ack!
1008             Rolling back the method is unusually complex.</p><p>Previously, a distinction was made between redundant
1009             policies and null policies. Node invariants show a
1010             case where null policies are required.</p><p>Assume a regular tree is required, one which need not
1011             support order statistics or interval overlap queries.
1012             Seemingly, in this case a redundant policy - a policy which
1013             doesn't affect nodes' contents would suffice. This, would lead
1014             to the following drawbacks:</p><div class="orderedlist"><ol class="orderedlist" type="1"><li class="listitem"><p>Each node would carry a useless metadata object, wasting
1015               space.</p></li><li class="listitem"><p>The tree cannot know if its
1016               <code class="classname">Node_Update</code> policy actually modifies a
1017               node's metadata (this is halting reducible). In the graphic
1018               below, assume the shaded node is inserted. The tree would have
1019               to traverse the useless path shown to the root, applying
1020               redundant updates all the way.</p></li></ol></div><div class="figure"><a name="id647089"></a><p class="title"><b>Figure 22.27. Useless update path</b></p><div class="figure-contents"><div class="mediaobject" align="center"><img src="../images/pbds_rationale_null_node_updator.png" align="middle" alt="Useless update path"></div></div></div><br class="figure-break"><p>A null policy class, <code class="classname">null_node_update</code>
1021             solves both these problems. The tree detects that node
1022             invariants are irrelevant, and defines all accordingly.</p></div></div><div class="section" title="Split and Join"><div class="titlepage"><div><div><h6 class="title"><a name="container.tree.details.split"></a>Split and Join</h6></div></div></div><p>Tree-based containers support split and join methods.
1023           It is possible to split a tree so that it passes
1024           all nodes with keys larger than a given key to a different
1025           tree. These methods have the following advantages over the
1026           alternative of externally inserting to the destination
1027           tree and erasing from the source tree:</p><div class="orderedlist"><ol class="orderedlist" type="1"><li class="listitem"><p>These methods are efficient - red-black trees are split
1028             and joined in poly-logarithmic complexity; ordered-vector
1029             trees are split and joined at linear complexity. The
1030             alternatives have super-linear complexity.</p></li><li class="listitem"><p>Aside from orders of growth, these operations perform
1031             few allocations and de-allocations. For red-black trees, allocations are not performed,
1032             and the methods are exception-free. </p></li></ol></div></div></div></div><div class="section" title="Trie"><div class="titlepage"><div><div><h4 class="title"><a name="pbds.design.container.trie"></a>Trie</h4></div></div></div><div class="section" title="Interface"><div class="titlepage"><div><div><h5 class="title"><a name="container.trie.interface"></a>Interface</h5></div></div></div><p>The trie-based container has the following declaration:</p><pre class="programlisting">
1033           template&lt;typename Key,
1034           typename Mapped,
1035           typename Cmp_Fn = std::less&lt;Key&gt;,
1036           typename Tag = pat_trie_tag,
1037           template&lt;typename Const_Node_Iterator,
1038           typename Node_Iterator,
1039           typename E_Access_Traits_,
1040           typename Allocator_&gt;
1041           class Node_Update = null_node_update,
1042           typename Allocator = std::allocator&lt;char&gt; &gt;
1043           class trie;
1044         </pre><p>The parameters have the following meaning:</p><div class="orderedlist"><ol class="orderedlist" type="1"><li class="listitem"><p><code class="classname">Key</code> is the key type.</p></li><li class="listitem"><p><code class="classname">Mapped</code> is the mapped-policy.</p></li><li class="listitem"><p><code class="classname">E_Access_Traits</code> is described in below.</p></li><li class="listitem"><p><code class="classname">Tag</code> specifies which underlying data structure
1045           to use, and is described shortly.</p></li><li class="listitem"><p><code class="classname">Node_Update</code> is a policy for updating node
1046           invariants. This is described below.</p></li><li class="listitem"><p><code class="classname">Allocator</code> is an allocator
1047           type.</p></li></ol></div><p>The <code class="classname">Tag</code> parameter specifies which underlying
1048         data structure to use. Instantiating it by <code class="classname">pat_trie_tag</code>, specifies an
1049         underlying PATRICIA trie (explained shortly); any other tag is
1050         currently illegal.</p><p>Following is a description of a (PATRICIA) trie
1051         (this implementation follows <a class="xref" href="policy_data_structures.html#biblio.okasaki98mereable" title="Fast mergeable integer maps">[biblio.okasaki98mereable]</a> and
1052         <a class="xref" href="policy_data_structures.html#biblio.filliatre2000ptset" title="Ptset: Sets of integers implemented as Patricia trees">[biblio.filliatre2000ptset]</a>).
1053         </p><p>A (PATRICIA) trie is similar to a tree, but with the
1054         following differences:</p><div class="orderedlist"><ol class="orderedlist" type="1"><li class="listitem"><p>It explicitly views keys as a sequence of elements.
1055           E.g., a trie can view a string as a sequence of
1056           characters; a trie can view a number as a sequence of
1057           bits.</p></li><li class="listitem"><p>It is not (necessarily) binary. Each node has fan-out n
1058           + 1, where n is the number of distinct
1059           elements.</p></li><li class="listitem"><p>It stores values only at leaf nodes.</p></li><li class="listitem"><p>Internal nodes have the properties that A) each has at
1060           least two children, and B) each shares the same prefix with
1061           any of its descendant.</p></li></ol></div><p>A (PATRICIA) trie has some useful properties:</p><div class="orderedlist"><ol class="orderedlist" type="1"><li class="listitem"><p>It can be configured to use large node fan-out, giving it
1062           very efficient find performance (albeit at insertion
1063           complexity and size).</p></li><li class="listitem"><p>It works well for common-prefix keys.</p></li><li class="listitem"><p>It can support efficiently queries such as which
1064           keys match a certain prefix. This is sometimes useful in file
1065           systems and routers, and for "type-ahead" aka predictive text matching
1066           on mobile devices.</p></li></ol></div></div><div class="section" title="Details"><div class="titlepage"><div><div><h5 class="title"><a name="container.trie.details"></a>Details</h5></div></div></div><div class="section" title="Element Access Traits"><div class="titlepage"><div><div><h6 class="title"><a name="container.trie.details.etraits"></a>Element Access Traits</h6></div></div></div><p>A trie inherently views its keys as sequences of elements.
1067           For example, a trie can view a string as a sequence of
1068           characters. A trie needs to map each of n elements to a
1069           number in {0, n - 1}. For example, a trie can map a
1070           character <code class="varname">c</code> to
1071           </p><pre class="programlisting">static_cast&lt;size_t&gt;(c)</pre><p>.</p><p>Seemingly, then, a trie can assume that its keys support
1072           (const) iterators, and that the <code class="classname">value_type</code> of this
1073           iterator can be cast to a <code class="classname">size_t</code>. There are several
1074           reasons, though, to decouple the mechanism by which the trie
1075           accesses its keys' elements from the trie:</p><div class="orderedlist"><ol class="orderedlist" type="1"><li class="listitem"><p>In some cases, the numerical value of an element is
1076             inappropriate. Consider a trie storing DNA strings. It is
1077             logical to use a trie with a fan-out of 5 = 1 + |{'A', 'C',
1078             'G', 'T'}|. This requires mapping 'T' to 3, though.</p></li><li class="listitem"><p>In some cases the keys' iterators are different than what
1079             is needed. For example, a trie can be used to search for
1080             common suffixes, by using strings'
1081             <code class="classname">reverse_iterator</code>. As another example, a trie mapping
1082             UNICODE strings would have a huge fan-out if each node would
1083             branch on a UNICODE character; instead, one can define an
1084             iterator iterating over 8-bit (or less) groups.</p></li></ol></div><p>trie is,
1085           consequently, parametrized by <code class="classname">E_Access_Traits</code> -
1086           traits which instruct how to access sequences' elements.
1087           <code class="classname">string_trie_e_access_traits</code>
1088           is a traits class for strings. Each such traits define some
1089           types, like:</p><pre class="programlisting">
1090             typename E_Access_Traits::const_iterator
1091           </pre><p>is a const iterator iterating over a key's elements. The
1092           traits class must also define methods for obtaining an iterator
1093           to the first and last element of a key.</p><p>The graphic below shows a
1094           (PATRICIA) trie resulting from inserting the words: "I wish
1095           that I could ever see a poem lovely as a trie" (which,
1096           unfortunately, does not rhyme).</p><p>The leaf nodes contain values; each internal node contains
1097           two <code class="classname">typename E_Access_Traits::const_iterator</code>
1098           objects, indicating the maximal common prefix of all keys in
1099           the sub-tree. For example, the shaded internal node roots a
1100           sub-tree with leafs "a" and "as". The maximal common prefix is
1101           "a". The internal node contains, consequently, to const
1102           iterators, one pointing to <code class="varname">'a'</code>, and the other to
1103           <code class="varname">'s'</code>.</p><div class="figure"><a name="id647461"></a><p class="title"><b>Figure 22.28. A PATRICIA trie</b></p><div class="figure-contents"><div class="mediaobject" align="center"><img src="../images/pbds_pat_trie.png" align="middle" alt="A PATRICIA trie"></div></div></div><br class="figure-break"></div><div class="section" title="Node Invariants"><div class="titlepage"><div><div><h6 class="title"><a name="container.trie.details.node"></a>Node Invariants</h6></div></div></div><p>Trie-based containers support node invariants, as do
1104           tree-based containers. There are two minor
1105           differences, though, which, unfortunately, thwart sharing them
1106           sharing the same node-updating policies:</p><div class="orderedlist"><ol class="orderedlist" type="1"><li class="listitem"><p>A trie's <code class="classname">Node_Update</code> template-template
1107               parameter is parametrized by <code class="classname">E_Access_Traits</code>, while
1108               a tree's <code class="classname">Node_Update</code> template-template parameter is
1109             parametrized by <code class="classname">Cmp_Fn</code>.</p></li><li class="listitem"><p>Tree-based containers store values in all nodes, while
1110             trie-based containers (at least in this implementation) store
1111             values in leafs.</p></li></ol></div><p>The graphic below shows the scheme, as well as some predefined
1112           policies (which are explained below).</p><div class="figure"><a name="id647548"></a><p class="title"><b>Figure 22.29. A trie and its update policy</b></p><div class="figure-contents"><div class="mediaobject" align="center"><img src="../images/pbds_trie_node_updator_policy_cd.png" align="middle" alt="A trie and its update policy"></div></div></div><br class="figure-break"><p>This library offers the following pre-defined trie node
1113           updating policies:</p><div class="orderedlist"><ol class="orderedlist" type="1"><li class="listitem"><p>
1114                 <code class="classname">trie_order_statistics_node_update</code>
1115                 supports order statistics.
1116               </p></li><li class="listitem"><p><code class="classname">trie_prefix_search_node_update</code>
1117             supports searching for ranges that match a given prefix.</p></li><li class="listitem"><p><code class="classname">null_node_update</code>
1118             is the null node updater.</p></li></ol></div></div><div class="section" title="Split and Join"><div class="titlepage"><div><div><h6 class="title"><a name="container.trie.details.split"></a>Split and Join</h6></div></div></div><p>Trie-based containers support split and join methods; the
1119           rationale is equal to that of tree-based containers supporting
1120           these methods.</p></div></div></div><div class="section" title="List"><div class="titlepage"><div><div><h4 class="title"><a name="pbds.design.container.list"></a>List</h4></div></div></div><div class="section" title="Interface"><div class="titlepage"><div><div><h5 class="title"><a name="container.list.interface"></a>Interface</h5></div></div></div><p>The list-based container has the following declaration:</p><pre class="programlisting">
1121           template&lt;typename Key,
1122           typename Mapped,
1123           typename Eq_Fn = std::equal_to&lt;Key&gt;,
1124           typename Update_Policy = move_to_front_lu_policy&lt;&gt;,
1125           typename Allocator = std::allocator&lt;char&gt; &gt;
1126           class list_update;
1127         </pre><p>The parameters have the following meaning:</p><div class="orderedlist"><ol class="orderedlist" type="1"><li class="listitem"><p>
1128               <code class="classname">Key</code> is the key type.
1129             </p></li><li class="listitem"><p>
1130               <code class="classname">Mapped</code> is the mapped-policy.
1131             </p></li><li class="listitem"><p>
1132               <code class="classname">Eq_Fn</code> is a key equivalence functor.
1133             </p></li><li class="listitem"><p>
1134               <code class="classname">Update_Policy</code> is a policy updating positions in
1135               the list based on access patterns. It is described in the
1136               following subsection.
1137             </p></li><li class="listitem"><p>
1138               <code class="classname">Allocator</code> is an allocator type.
1139             </p></li></ol></div><p>A list-based associative container is a container that
1140         stores elements in a linked-list. It does not order the elements
1141         by any particular order related to the keys.  List-based
1142         containers are primarily useful for creating "multimaps". In fact,
1143         list-based containers are designed in this library expressly for
1144         this purpose.</p><p>List-based containers might also be useful for some rare
1145         cases, where a key is encapsulated to the extent that only
1146         key-equivalence can be tested. Hash-based containers need to know
1147         how to transform a key into a size type, and tree-based containers
1148         need to know if some key is larger than another.  List-based
1149         associative containers, conversely, only need to know if two keys
1150         are equivalent.</p><p>Since a list-based associative container does not order
1151         elements by keys, is it possible to order the list in some
1152         useful manner? Remarkably, many on-line competitive
1153         algorithms exist for reordering lists to reflect access
1154         prediction. (See <a class="xref" href="policy_data_structures.html#biblio.motwani95random" title="Randomized Algorithms">[biblio.motwani95random]</a> and <a class="xref" href="policy_data_structures.html#biblio.andrew04mtf" title="MTF, Bit, and COMB: A Guide to Deterministic and Randomized Algorithms for the List Update Problem">[biblio.andrew04mtf]</a>).
1155         </p></div><div class="section" title="Details"><div class="titlepage"><div><div><h5 class="title"><a name="container.list.details"></a>Details</h5></div></div></div><p>
1156         </p><div class="section" title="Underlying Data Structure"><div class="titlepage"><div><div><h6 class="title"><a name="container.list.details.ds"></a>Underlying Data Structure</h6></div></div></div><p>The graphic below shows a
1157           simple list of integer keys. If we search for the integer 6, we
1158           are paying an overhead: the link with key 6 is only the fifth
1159           link; if it were the first link, it could be accessed
1160           faster.</p><div class="figure"><a name="id647803"></a><p class="title"><b>Figure 22.30. A simple list</b></p><div class="figure-contents"><div class="mediaobject" align="center"><img src="../images/pbds_simple_list.png" align="middle" alt="A simple list"></div></div></div><br class="figure-break"><p>List-update algorithms reorder lists as elements are
1161           accessed. They try to determine, by the access history, which
1162           keys to move to the front of the list. Some of these algorithms
1163           require adding some metadata alongside each entry.</p><p>For example, in the graphic below label A shows the counter
1164           algorithm. Each node contains both a key and a count metadata
1165           (shown in bold). When an element is accessed (e.g. 6) its count is
1166           incremented, as shown in label B. If the count reaches some
1167           predetermined value, say 10, as shown in label C, the count is set
1168           to 0 and the node is moved to the front of the list, as in label
1169           D.
1170           </p><div class="figure"><a name="id647850"></a><p class="title"><b>Figure 22.31. The counter algorithm</b></p><div class="figure-contents"><div class="mediaobject" align="center"><img src="../images/pbds_list_update.png" align="middle" alt="The counter algorithm"></div></div></div><br class="figure-break"></div><div class="section" title="Policies"><div class="titlepage"><div><div><h6 class="title"><a name="container.list.details.policies"></a>Policies</h6></div></div></div><p>this library allows instantiating lists with policies
1171           implementing any algorithm moving nodes to the front of the
1172           list (policies implementing algorithms interchanging nodes are
1173           unsupported).</p><p>Associative containers based on lists are parametrized by a
1174           <code class="classname">Update_Policy</code> parameter. This parameter defines the
1175           type of metadata each node contains, how to create the
1176           metadata, and how to decide, using this metadata, whether to
1177           move a node to the front of the list. A list-based associative
1178           container object derives (publicly) from its update policy.
1179           </p><p>An instantiation of <code class="classname">Update_Policy</code> must define
1180           internally <code class="classname">update_metadata</code> as the metadata it
1181           requires. Internally, each node of the list contains, besides
1182           the usual key and data, an instance of <code class="classname">typename
1183           Update_Policy::update_metadata</code>.</p><p>An instantiation of <code class="classname">Update_Policy</code> must define
1184           internally two operators:</p><pre class="programlisting">
1185             update_metadata
1186             operator()();
1187
1188             bool
1189             operator()(update_metadata &amp;);
1190           </pre><p>The first is called by the container object, when creating a
1191           new node, to create the node's metadata. The second is called
1192           by the container object, when a node is accessed (
1193           when a find operation's key is equivalent to the key of the
1194           node), to determine whether to move the node to the front of
1195           the list.
1196           </p><p>The library contains two predefined implementations of
1197           list-update policies. The first
1198           is <code class="classname">lu_counter_policy</code>, which implements the
1199           counter algorithm described above. The second is
1200           <code class="classname">lu_move_to_front_policy</code>,
1201           which unconditionally move an accessed element to the front of
1202           the list. The latter type is very useful in this library,
1203           since there is no need to associate metadata with each element.
1204           (See <a class="xref" href="policy_data_structures.html#biblio.andrew04mtf" title="MTF, Bit, and COMB: A Guide to Deterministic and Randomized Algorithms for the List Update Problem">[biblio.andrew04mtf]</a>
1205           </p></div><div class="section" title="Use in Multimaps"><div class="titlepage"><div><div><h6 class="title"><a name="container.list.details.mapped"></a>Use in Multimaps</h6></div></div></div><p>In this library, there are no equivalents for the standard's
1206           multimaps and multisets; instead one uses an associative
1207           container mapping primary keys to secondary keys.</p><p>List-based containers are especially useful as associative
1208           containers for secondary keys. In fact, they are implemented
1209           here expressly for this purpose.</p><p>To begin with, these containers use very little per-entry
1210           structure memory overhead, since they can be implemented as
1211           singly-linked lists. (Arrays use even lower per-entry memory
1212           overhead, but they are less flexible in moving around entries,
1213           and have weaker invalidation guarantees).</p><p>More importantly, though, list-based containers use very
1214           little per-container memory overhead. The memory overhead of an
1215           empty list-based container is practically that of a pointer.
1216           This is important for when they are used as secondary
1217           associative-containers in situations where the average ratio of
1218           secondary keys to primary keys is low (or even 1).</p><p>In order to reduce the per-container memory overhead as much
1219           as possible, they are implemented as closely as possible to
1220           singly-linked lists.</p><div class="orderedlist"><ol class="orderedlist" type="1"><li class="listitem"><p>
1221                 List-based containers do not store internally the number
1222                 of values that they hold. This means that their <code class="function">size</code>
1223                 method has linear complexity (just like <code class="classname">std::list</code>).
1224                 Note that finding the number of equivalent-key values in a
1225                 standard multimap also has linear complexity (because it must be
1226                 done,  via <code class="function">std::distance</code> of the
1227                 multimap's <code class="function">equal_range</code> method), but usually with
1228                 higher constants.
1229               </p></li><li class="listitem"><p>
1230                 Most associative-container objects each hold a policy
1231                 object (a hash-based container object holds a
1232                 hash functor). List-based containers, conversely, only have
1233                 class-wide policy objects.
1234               </p></li></ol></div></div></div></div><div class="section" title="Priority Queue"><div class="titlepage"><div><div><h4 class="title"><a name="pbds.design.container.priority_queue"></a>Priority Queue</h4></div></div></div><div class="section" title="Interface"><div class="titlepage"><div><div><h5 class="title"><a name="container.priority_queue.interface"></a>Interface</h5></div></div></div><p>The priority queue container has the following
1235         declaration:
1236         </p><pre class="programlisting">
1237           template&lt;typename  Value_Type,
1238           typename  Cmp_Fn = std::less&lt;Value_Type&gt;,
1239           typename  Tag = pairing_heap_tag,
1240           typename  Allocator = std::allocator&lt;char &gt; &gt;
1241           class priority_queue;
1242         </pre><p>The parameters have the following meaning:</p><div class="orderedlist"><ol class="orderedlist" type="1"><li class="listitem"><p><code class="classname">Value_Type</code> is the value type.</p></li><li class="listitem"><p><code class="classname">Cmp_Fn</code> is a value comparison functor</p></li><li class="listitem"><p><code class="classname">Tag</code> specifies which underlying data structure
1243           to use.</p></li><li class="listitem"><p><code class="classname">Allocator</code> is an allocator
1244           type.</p></li></ol></div><p>The <code class="classname">Tag</code> parameter specifies which underlying
1245         data structure to use. Instantiating it by<code class="classname">pairing_heap_tag</code>,<code class="classname">binary_heap_tag</code>,
1246         <code class="classname">binomial_heap_tag</code>,
1247         <code class="classname">rc_binomial_heap_tag</code>,
1248         or <code class="classname">thin_heap_tag</code>,
1249         specifies, respectively,
1250         an underlying pairing heap (<a class="xref" href="policy_data_structures.html#biblio.fredman86pairing" title="The pairing heap: a new form of self-adjusting heap">[biblio.fredman86pairing]</a>),
1251         binary heap (<a class="xref" href="policy_data_structures.html#biblio.clrs2001" title="Introduction to Algorithms, 2nd edition">[biblio.clrs2001]</a>),
1252         binomial heap (<a class="xref" href="policy_data_structures.html#biblio.clrs2001" title="Introduction to Algorithms, 2nd edition">[biblio.clrs2001]</a>),
1253         a binomial heap with a redundant binary counter (<a class="xref" href="policy_data_structures.html#biblio.maverik_lowerbounds" title="Deamortization - Part 2: Binomial Heaps">[biblio.maverik_lowerbounds]</a>),
1254         or a thin heap (<a class="xref" href="policy_data_structures.html#biblio.kt99fat_heaps" title="New Heap Data Structures">[biblio.kt99fat_heaps]</a>).
1255         </p><p>
1256           As mentioned in the tutorial,
1257           <code class="classname">__gnu_pbds::priority_queue</code> shares most of the
1258           same interface with <code class="classname">std::priority_queue</code>.
1259           E.g. if <code class="varname">q</code> is a priority queue of type
1260           <code class="classname">Q</code>, then <code class="function">q.top()</code> will
1261           return the "largest" value in the container (according to
1262           <code class="classname">typename
1263           Q::cmp_fn</code>). <code class="classname">__gnu_pbds::priority_queue</code>
1264           has a larger (and very slightly different) interface than
1265           <code class="classname">std::priority_queue</code>, however, since typically
1266           <code class="classname">push</code> and <code class="classname">pop</code> are deemed
1267         insufficient for manipulating priority-queues. </p><p>Different settings require different priority-queue
1268         implementations which are described in later; see traits
1269         discusses ways to differentiate between the different traits of
1270         different implementations.</p></div><div class="section" title="Details"><div class="titlepage"><div><div><h5 class="title"><a name="container.priority_queue.details"></a>Details</h5></div></div></div><div class="section" title="Iterators"><div class="titlepage"><div><div><h6 class="title"><a name="container.priority_queue.details.iterators"></a>Iterators</h6></div></div></div><p>There are many different underlying-data structures for
1271           implementing priority queues. Unfortunately, most such
1272           structures are oriented towards making <code class="function">push</code> and
1273           <code class="function">top</code> efficient, and consequently don't allow efficient
1274           access of other elements: for instance, they cannot support an efficient
1275           <code class="function">find</code> method. In the use case where it
1276           is important to both access and "do something with" an
1277           arbitrary value, one would be out of luck. For example, many graph algorithms require
1278           modifying a value (typically increasing it in the sense of the
1279           priority queue's comparison functor).</p><p>In order to access and manipulate an arbitrary value in a
1280           priority queue, one needs to reference the internals of the
1281           priority queue from some form of an associative container -
1282           this is unavoidable. Of course, in order to maintain the
1283           encapsulation of the priority queue, this needs to be done in a
1284           way that minimizes exposure to implementation internals.</p><p>In this library the priority queue's <code class="function">insert</code>
1285           method returns an iterator, which if valid can be used for subsequent <code class="function">modify</code> and
1286           <code class="function">erase</code> operations. This both preserves the priority
1287           queue's encapsulation, and allows accessing arbitrary values (since the
1288           returned iterators from the <code class="function">push</code> operation can be
1289           stored in some form of associative container).</p><p>Priority queues' iterators present a problem regarding their
1290           invalidation guarantees. One assumes that calling
1291           <code class="function">operator++</code> on an iterator will associate it
1292           with the "next" value. Priority-queues are
1293           self-organizing: each operation changes what the "next" value
1294           means. Consequently, it does not make sense that <code class="function">push</code>
1295           will return an iterator that can be incremented - this can have
1296           no possible use. Also, as in the case of hash-based containers,
1297           it is awkward to define if a subsequent <code class="function">push</code> operation
1298           invalidates a prior returned iterator: it invalidates it in the
1299           sense that its "next" value is not related to what it
1300           previously considered to be its "next" value. However, it might not
1301           invalidate it, in the sense that it can be
1302           de-referenced and used for <code class="function">modify</code> and <code class="function">erase</code>
1303           operations.</p><p>Similarly to the case of the other unordered associative
1304           containers, this library uses a distinction between
1305           point-type and range type iterators. A priority queue's <code class="classname">iterator</code> can always be
1306           converted to a <code class="classname">point_iterator</code>, and a
1307           <code class="classname">const_iterator</code> can always be converted to a
1308           <code class="classname">point_const_iterator</code>.</p><p>The following snippet demonstrates manipulating an arbitrary
1309           value:</p><pre class="programlisting">
1310             // A priority queue of integers.
1311             priority_queue&lt;int &gt; p;
1312
1313             // Insert some values into the priority queue.
1314             priority_queue&lt;int &gt;::point_iterator it = p.push(0);
1315
1316             p.push(1);
1317             p.push(2);
1318
1319             // Now modify a value.
1320             p.modify(it, 3);
1321
1322             assert(p.top() == 3);
1323           </pre><p>It should be noted that an alternative design could embed an
1324           associative container in a priority queue. Could, but most
1325           probably should not. To begin with, it should be noted that one
1326           could always encapsulate a priority queue and an associative
1327           container mapping values to priority queue iterators with no
1328           performance loss. One cannot, however, "un-encapsulate" a priority
1329           queue embedding an associative container, which might lead to
1330           performance loss. Assume, that one needs to associate each value
1331           with some data unrelated to priority queues. Then using
1332           this library's design, one could use an
1333           associative container mapping each value to a pair consisting of
1334           this data and a priority queue's iterator. Using the embedded
1335           method would need to use two associative containers. Similar
1336           problems might arise in cases where a value can reside
1337           simultaneously in many priority queues.</p></div><div class="section" title="Underlying Data Structure"><div class="titlepage"><div><div><h6 class="title"><a name="container.priority_queue.details.d"></a>Underlying Data Structure</h6></div></div></div><p>There are three main implementations of priority queues: the
1338           first employs a binary heap, typically one which uses a
1339           sequence; the second uses a tree (or forest of trees), which is
1340           typically less structured than an associative container's tree;
1341           the third simply uses an associative container. These are
1342           shown in the graphic below, in labels A1 and A2, label B, and label C.</p><div class="figure"><a name="id648381"></a><p class="title"><b>Figure 22.32. Underlying Priority-Queue Data-Structures.</b></p><div class="figure-contents"><div class="mediaobject" align="center"><img src="../images/pbds_priority_queue_different_underlying_dss.png" align="middle" alt="Underlying Priority-Queue Data-Structures."></div></div></div><br class="figure-break"><p>Roughly speaking, any value that is both pushed and popped
1343           from a priority queue must incur a logarithmic expense (in the
1344           amortized sense). Any priority queue implementation that would
1345           avoid this, would violate known bounds on comparison-based
1346           sorting (see <a class="xref" href="policy_data_structures.html#biblio.clrs2001" title="Introduction to Algorithms, 2nd edition">[biblio.clrs2001]</a> and <a class="xref" href="policy_data_structures.html#biblio.brodal96priority" title="Worst-case efficient priority queues">[biblio.brodal96priority]</a>).
1347           </p><p>Most implementations do
1348           not differ in the asymptotic amortized complexity of
1349           <code class="function">push</code> and <code class="function">pop</code> operations, but they differ in
1350           the constants involved, in the complexity of other operations
1351           (e.g., <code class="function">modify</code>), and in the worst-case
1352           complexity of single operations. In general, the more
1353           "structured" an implementation (i.e., the more internal
1354           invariants it possesses) - the higher its amortized complexity
1355           of <code class="function">push</code> and <code class="function">pop</code> operations.</p><p>This library implements different algorithms using a
1356           single class: <code class="classname">priority_queue</code>.
1357           Instantiating the <code class="classname">Tag</code> template parameter, "selects"
1358           the implementation:</p><div class="orderedlist"><ol class="orderedlist" type="1"><li class="listitem"><p>
1359               Instantiating <code class="classname">Tag = binary_heap_tag</code> creates
1360               a binary heap of the form in represented in the graphic with labels A1 or A2. The former is internally
1361               selected by priority_queue
1362               if <code class="classname">Value_Type</code> is instantiated by a primitive type
1363               (e.g., an <span class="type">int</span>); the latter is
1364               internally selected for all other types (e.g.,
1365               <code class="classname">std::string</code>). This implementations is relatively
1366               unstructured, and so has good <code class="classname">push</code> and <code class="classname">pop</code>
1367               performance; it is the "best-in-kind" for primitive
1368               types, e.g., <span class="type">int</span>s. Conversely, it has
1369               high worst-case performance, and can support only linear-time
1370             <code class="function">modify</code> and <code class="function">erase</code> operations.</p></li><li class="listitem"><p>Instantiating <code class="classname">Tag =
1371             pairing_heap_tag</code> creates a pairing heap of the form
1372             in represented by label B in the graphic above. This
1373             implementations too is relatively unstructured, and so has good
1374             <code class="function">push</code> and <code class="function">pop</code>
1375             performance; it is the "best-in-kind" for non-primitive types,
1376             e.g., <code class="classname">std:string</code>s. It also has very good
1377             worst-case <code class="function">push</code> and
1378             <code class="function">join</code> performance (O(1)), but has high
1379             worst-case <code class="function">pop</code>
1380             complexity.</p></li><li class="listitem"><p>Instantiating <code class="classname">Tag =
1381             binomial_heap_tag</code> creates a binomial heap of the
1382             form repsented by label B in the graphic above. This
1383             implementations is more structured than a pairing heap, and so
1384             has worse <code class="function">push</code> and <code class="function">pop</code>
1385             performance. Conversely, it has sub-linear worst-case bounds for
1386             <code class="function">pop</code>, e.g., and so it might be preferred in
1387             cases where responsiveness is important.</p></li><li class="listitem"><p>Instantiating <code class="classname">Tag =
1388             rc_binomial_heap_tag</code> creates a binomial heap of the
1389             form represented in label B above, accompanied by a redundant
1390             counter which governs the trees. This implementations is
1391             therefore more structured than a binomial heap, and so has worse
1392             <code class="function">push</code> and <code class="function">pop</code>
1393             performance. Conversely, it guarantees O(1)
1394             <code class="function">push</code> complexity, and so it might be
1395             preferred in cases where the responsiveness of a binomial heap
1396             is insufficient.</p></li><li class="listitem"><p>Instantiating <code class="classname">Tag =
1397             thin_heap_tag</code> creates a thin heap of the form
1398             represented by the label B in the graphic above. This
1399             implementations too is more structured than a pairing heap, and
1400             so has worse <code class="function">push</code> and
1401             <code class="function">pop</code> performance. Conversely, it has better
1402             worst-case and identical amortized complexities than a Fibonacci
1403             heap, and so might be more appropriate for some graph
1404             algorithms.</p></li></ol></div><p>Of course, one can use any order-preserving associative
1405           container as a priority queue, as in the graphic above label C, possibly by creating an adapter class
1406           over the associative container (much as
1407           <code class="classname">std::priority_queue</code> can adapt <code class="classname">std::vector</code>).
1408           This has the advantage that no cross-referencing is necessary
1409           at all; the priority queue itself is an associative container.
1410           Most associative containers are too structured to compete with
1411           priority queues in terms of <code class="function">push</code> and <code class="function">pop</code>
1412           performance.</p></div><div class="section" title="Traits"><div class="titlepage"><div><div><h6 class="title"><a name="container.priority_queue.details.traits"></a>Traits</h6></div></div></div><p>It would be nice if all priority queues could
1413           share exactly the same behavior regardless of implementation. Sadly, this is not possible. Just one for instance is in join operations: joining
1414           two binary heaps might throw an exception (not corrupt
1415           any of the heaps on which it operates), but joining two pairing
1416           heaps is exception free.</p><p>Tags and traits are very useful for manipulating generic
1417           types. <code class="classname">__gnu_pbds::priority_queue</code>
1418           publicly defines <code class="classname">container_category</code> as one of the tags. Given any
1419           container <code class="classname">Cntnr</code>, the tag of the underlying
1420           data structure can be found via <code class="classname">typename
1421           Cntnr::container_category</code>; this is one of the possible tags shown in the graphic below.
1422           </p><div class="figure"><a name="id648673"></a><p class="title"><b>Figure 22.33. Priority-Queue Data-Structure Tags.</b></p><div class="figure-contents"><div class="mediaobject" align="center"><img src="../images/pbds_priority_queue_tag_hierarchy.png" align="middle" alt="Priority-Queue Data-Structure Tags."></div></div></div><br class="figure-break"><p>Additionally, a traits mechanism can be used to query a
1423           container type for its attributes. Given any container
1424           <code class="classname">Cntnr</code>, then </p><pre class="programlisting">__gnu_pbds::container_traits&lt;Cntnr&gt;</pre><p>
1425           is a traits class identifying the properties of the
1426           container.</p><p>To find if a container might throw if two of its objects are
1427           joined, one can use
1428           </p><pre class="programlisting">
1429             container_traits&lt;Cntnr&gt;::split_join_can_throw
1430           </pre><p>
1431           </p><p>
1432             Different priority-queue implementations have different invalidation guarantees. This is
1433             especially important, since there is no way to access an arbitrary
1434             value of priority queues except for iterators. Similarly to
1435             associative containers, one can use
1436             </p><pre class="programlisting">
1437               container_traits&lt;Cntnr&gt;::invalidation_guarantee
1438             </pre><p>
1439           to get the invalidation guarantee type of a priority queue.</p><p>It is easy to understand from the graphic above, what <code class="classname">container_traits&lt;Cntnr&gt;::invalidation_guarantee</code>
1440           will be for different implementations. All implementations of
1441           type represented by label B have <code class="classname">point_invalidation_guarantee</code>:
1442           the container can freely internally reorganize the nodes -
1443           range-type iterators are invalidated, but point-type iterators
1444           are always valid. Implementations of type represented by labels A1 and A2 have <code class="classname">basic_invalidation_guarantee</code>:
1445           the container can freely internally reallocate the array - both
1446           point-type and range-type iterators might be invalidated.</p><p>
1447             This has major implications, and constitutes a good reason to avoid
1448             using binary heaps. A binary heap can perform <code class="function">modify</code>
1449             or <code class="function">erase</code> efficiently given a valid point-type
1450             iterator. However, in order to supply it with a valid point-type
1451             iterator, one needs to iterate (linearly) over all
1452             values, then supply the relevant iterator (recall that a
1453             range-type iterator can always be converted to a point-type
1454             iterator). This means that if the number of <code class="function">modify</code> or
1455             <code class="function">erase</code> operations is non-negligible (say
1456             super-logarithmic in the total sequence of operations) - binary
1457             heaps will perform badly.
1458           </p></div></div></div></div></div><div class="navfooter"><hr><table width="100%" summary="Navigation footer"><tr><td width="40%" align="left"><a accesskey="p" href="policy_data_structures_using.html">Prev</a> </td><td width="20%" align="center"><a accesskey="u" href="policy_data_structures.html">Up</a></td><td width="40%" align="right"> <a accesskey="n" href="policy_based_data_structures_test.html">Next</a></td></tr><tr><td width="40%" align="left" valign="top">Using </td><td width="20%" align="center"><a accesskey="h" href="../index.html">Home</a></td><td width="40%" align="right" valign="top"> Testing</td></tr></table></div></body></html>