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<chapter xmlns="http://docbook.org/ns/docbook" version="5.0"
         xml:id="manual.ext.containers.pbds" xreflabel="pbds">
  <info>
    <title>Policy-Based Data Structures</title>
    <keywordset>
      <keyword>
        ISO C++
      </keyword>
      <keyword>
        policy
      </keyword>
      <keyword>
        container
      </keyword>
      <keyword>
        data
      </keyword>
      <keyword>
        structure
      </keyword>
      <keyword>
        associated
      </keyword>
      <keyword>
        tree
      </keyword>
      <keyword>
        trie
      </keyword>
      <keyword>
        hash
      </keyword>
      <keyword>
        metaprogramming
      </keyword>
    </keywordset>
  </info>
  <?dbhtml filename="policy_data_structures.html"?>

  <!-- 2006-04-01 Ami Tavory -->
  <!-- 2011-05-25 Benjamin Kosnik -->

  <!-- S01: intro -->
  <section xml:id="pbds.intro">
    <info><title>Intro</title></info>

    <para>
      This is a library of policy-based elementary data structures:
      associative containers and priority queues. It is designed for
      high-performance, flexibility, semantic safety, and conformance to
      the corresponding containers in <literal>std</literal> and
      <literal>std::tr1</literal> (except for some points where it differs
      by design).
    </para>
    <para>
    </para>

    <section xml:id="pbds.intro.issues">
      <info><title>Performance Issues</title></info>
      <para>
      </para>

      <para>
        An attempt is made to categorize the wide variety of possible
        container designs in terms of performance-impacting factors. These
        performance factors are translated into design policies and
        incorporated into container design.
      </para>

      <para>
        There is tension between unravelling factors into a coherent set of
        policies. Every attempt is made to make a minimal set of
        factors. However, in many cases multiple factors make for long
        template names. Every attempt is made to alias and use typedefs in
        the source files, but the generated names for external symbols can
        be large for binary files or debuggers.
      </para>

      <para>
        In many cases, the longer names allow capabilities and behaviours
        controlled by macros to also be unamibiguously emitted as distinct
        generated names.
      </para>

      <para>
        Specific issues found while unraveling performance factors in the
        design of associative containers and priority queues follow.
      </para>

      <section xml:id="pbds.intro.issues.associative">
        <info><title>Associative</title></info>

        <para>
          Associative containers depend on their composite policies to a very
          large extent. Implicitly hard-wiring policies can hamper their
          performance and limit their functionality. An efficient hash-based
          container, for example, requires policies for testing key
          equivalence, hashing keys, translating hash values into positions
          within the hash table, and determining when and how to resize the
          table internally. A tree-based container can efficiently support
          order statistics, i.e. the ability to query what is the order of
          each key within the sequence of keys in the container, but only if
          the container is supplied with a policy to internally update
          meta-data. There are many other such examples.
        </para>

        <para>
          Ideally, all associative containers would share the same
          interface. Unfortunately, underlying data structures and mapping
          semantics differentiate between different containers. For example,
          suppose one writes a generic function manipulating an associative
          container.
        </para>

        <programlisting>
          template&lt;typename Cntnr&gt;
          void
          some_op_sequence(Cntnr&amp; r_cnt)
          {
          ...
          }
        </programlisting>

        <para>
          Given this, then what can one assume about the instantiating
          container? The answer varies according to its underlying data
          structure. If the underlying data structure of
          <literal>Cntnr</literal> is based on a tree or trie, then the order
          of elements is well defined; otherwise, it is not, in general. If
          the underlying data structure of <literal>Cntnr</literal> is based
          on a collision-chaining hash table, then modifying
          r_<literal>Cntnr</literal> will not invalidate its iterators' order;
          if the underlying data structure is a probing hash table, then this
          is not the case. If the underlying data structure is based on a tree
          or trie, then a reference to the container can efficiently be split;
          otherwise, it cannot, in general. If the underlying data structure
          is a red-black tree, then splitting a reference to the container is
          exception-free; if it is an ordered-vector tree, exceptions can be
          thrown.
        </para>

      </section>

      <section xml:id="pbds.intro.issues.priority_queue">
        <info><title>Priority Que</title></info>

        <para>
          Priority queues are useful when one needs to efficiently access a
          minimum (or maximum) value as the set of values changes.
        </para>

        <para>
          Most useful data structures for priority queues have a relatively
          simple structure, as they are geared toward relatively simple
          requirements. Unfortunately, these structures do not support access
          to an arbitrary value, which turns out to be necessary in many
          algorithms. Say, decreasing an arbitrary value in a graph
          algorithm. Therefore, some extra mechanism is necessary and must be
          invented for accessing arbitrary values. There are at least two
          alternatives: embedding an associative container in a priority
          queue, or allowing cross-referencing through iterators. The first
          solution adds significant overhead; the second solution requires a
          precise definition of iterator invalidation. Which is the next
          point...
        </para>

        <para>
          Priority queues, like hash-based containers, store values in an
          order that is meaningless and undefined externally. For example, a
          <code>push</code> operation can internally reorganize the
          values. Because of this characteristic, describing a priority
          queues' iterator is difficult: on one hand, the values to which
          iterators point can remain valid, but on the other, the logical
          order of iterators can change unpredictably.
        </para>

        <para>
          Roughly speaking, any element that is both inserted to a priority
          queue (e.g. through <code>push</code>) and removed
          from it (e.g., through <code>pop</code>), incurs a
          logarithmic overhead (in the amortized sense). Different underlying
          data structures place the actual cost differently: some are
          optimized for amortized complexity, whereas others guarantee that
          specific operations only have a constant cost. One underlying data
          structure might be chosen if modifying a value is frequent
          (Dijkstra's shortest-path algorithm), whereas a different one might
          be chosen otherwise. Unfortunately, an array-based binary heap - an
          underlying data structure that optimizes (in the amortized sense)
          <code>push</code> and <code>pop</code> operations, differs from the
          others in terms of its invalidation guarantees. Other design
          decisions also impact the cost and placement of the overhead, at the
          expense of more difference in the the kinds of operations that the
          underlying data structure can support. These differences pose a
          challenge when creating a uniform interface for priority queues.
        </para>
      </section>
    </section>

    <section xml:id="pbds.intro.motivation">
      <info><title>Goals</title></info>

      <para>
        Many fine associative-container libraries were already written,
        most notably, the C++ standard's associative containers. Why
        then write another library? This section shows some possible
        advantages of this library, when considering the challenges in
        the introduction. Many of these points stem from the fact that
        the ISO C++ process introduced associative-containers in a
        two-step process (first standardizing tree-based containers,
        only then adding hash-based containers, which are fundamentally
        different), did not standardize priority queues as containers,
        and (in our opinion) overloads the iterator concept.
      </para>

      <section xml:id="pbds.intro.motivation.associative">
        <info><title>Associative</title></info>
        <para>
        </para>

        <section xml:id="motivation.associative.policy">
          <info><title>Policy Choices</title></info>
          <para>
            Associative containers require a relatively large number of
            policies to function efficiently in various settings. In some
            cases this is needed for making their common operations more
            efficient, and in other cases this allows them to support a
            larger set of operations
          </para>

          <orderedlist>
            <listitem>
              <para>
                Hash-based containers, for example, support look-up and
                insertion methods (<function>find</function> and
                <function>insert</function>). In order to locate elements
                quickly, they are supplied a hash functor, which instruct
                how to transform a key object into some size type; a hash
                functor might transform <constant>"hello"</constant>
                into <constant>1123002298</constant>. A hash table, though,
                requires transforming each key object into some size-type
                type in some specific domain; a hash table with a 128-long
                table might transform <constant>"hello"</constant> into
                position <constant>63</constant>. The policy by which the
                hash value is transformed into a position within the table
                can dramatically affect performance.  Hash-based containers
                also do not resize naturally (as opposed to tree-based
                containers, for example). The appropriate resize policy is
                unfortunately intertwined with the policy that transforms
                hash value into a position within the table.
              </para>
            </listitem>

            <listitem>
              <para>
                Tree-based containers, for example, also support look-up and
                insertion methods, and are primarily useful when maintaining
                order between elements is important. In some cases, though,
                one can utilize their balancing algorithms for completely
                different purposes.
              </para>

              <para>
                Figure A shows a tree whose each node contains two entries:
                a floating-point key, and some size-type
                <emphasis>metadata</emphasis> (in bold beneath it) that is
                the number of nodes in the sub-tree. (The root has key 0.99,
                and has 5 nodes (including itself) in its sub-tree.) A
                container based on this data structure can obviously answer
                efficiently whether 0.3 is in the container object, but it
                can also answer what is the order of 0.3 among all those in
                the container object: see <xref linkend="biblio.clrs2001"/>.

              </para>

              <para>
                As another example, Figure B shows a tree whose each node
                contains two entries: a half-open geometric line interval,
                and a number <emphasis>metadata</emphasis> (in bold beneath
                it) that is the largest endpoint of all intervals in its
                sub-tree.  (The root describes the interval <constant>[20,
                36)</constant>, and the largest endpoint in its sub-tree is
                99.) A container based on this data structure can obviously
                answer efficiently whether <constant>[3, 41)</constant> is
                in the container object, but it can also answer efficiently
                whether the container object has intervals that intersect
                <constant>[3, 41)</constant>. These types of queries are
                very useful in geometric algorithms and lease-management
                algorithms.
              </para>

              <para>
                It is important to note, however, that as the trees are
                modified, their internal structure changes. To maintain
                these invariants, one must supply some policy that is aware
                of these changes.  Without this, it would be better to use a
                linked list (in itself very efficient for these purposes).
              </para>

            </listitem>
          </orderedlist>

          <figure>
            <title>Node Invariants</title>
            <mediaobject>
              <imageobject>
                <imagedata align="center" format="PNG" scale="100"
                           fileref="../images/pbds_node_invariants.png"/>
              </imageobject>
              <textobject>
                <phrase>Node Invariants</phrase>
              </textobject>
            </mediaobject>
          </figure>

        </section>

        <section xml:id="motivation.associative.underlying">
          <info><title>Underlying Data Structures</title></info>
          <para>
            The standard C++ library contains associative containers based on
            red-black trees and collision-chaining hash tables. These are
            very useful, but they are not ideal for all types of
            settings.
          </para>

          <para>
            The figure below shows the different underlying data structures
            currently supported in this library.
          </para>

          <figure>
            <title>Underlying Associative Data Structures</title>
            <mediaobject>
              <imageobject>
                <imagedata align="center" format="PNG" scale="100"
                           fileref="../images/pbds_different_underlying_dss_1.png"/>
              </imageobject>
              <textobject>
                <phrase>Underlying Associative Data Structures</phrase>
              </textobject>
            </mediaobject>
          </figure>

          <para>
            A shows a collision-chaining hash-table, B shows a probing
            hash-table, C shows a red-black tree, D shows a splay tree, E shows
            a tree based on an ordered vector(implicit in the order of the
            elements), F shows a PATRICIA trie, and G shows a list-based
            container with update policies.
          </para>

          <para>
            Each of these data structures has some performance benefits, in
            terms of speed, size or both. For now, note that vector-based trees
            and probing hash tables manipulate memory more efficiently than
            red-black trees and collision-chaining hash tables, and that
            list-based associative containers are very useful for constructing
            "multimaps".
          </para>

          <para>
            Now consider a function manipulating a generic associative
            container,
          </para>
          <programlisting>
            template&lt;class Cntnr&gt;
            int
            some_op_sequence(Cntnr &amp;r_cnt)
            {
            ...
            }
          </programlisting>

          <para>
            Ideally, the underlying data structure
            of <classname>Cntnr</classname> would not affect what can be
            done with <varname>r_cnt</varname>.  Unfortunately, this is not
            the case.
          </para>

          <para>
            For example, if <classname>Cntnr</classname>
            is <classname>std::map</classname>, then the function can
            use
          </para>
          <programlisting>
            std::for_each(r_cnt.find(foo), r_cnt.find(bar), foobar)
          </programlisting>
          <para>
            in order to apply <classname>foobar</classname> to all
            elements between <classname>foo</classname> and
            <classname>bar</classname>. If
            <classname>Cntnr</classname> is a hash-based container,
            then this call's results are undefined.
          </para>

          <para>
            Also, if <classname>Cntnr</classname> is tree-based, the type
            and object of the comparison functor can be
            accessed. If <classname>Cntnr</classname> is hash based, these
            queries are nonsensical.
          </para>

          <para>
            There are various other differences based on the container's
            underlying data structure. For one, they can be constructed by,
            and queried for, different policies. Furthermore:
          </para>

          <orderedlist>
            <listitem>
              <para>
                Containers based on C, D, E and F store elements in a
                meaningful order; the others store elements in a meaningless
                (and probably time-varying) order. By implication, only
                containers based on C, D, E and F can
                support <function>erase</function> operations taking an
                iterator and returning an iterator to the following element
                without performance loss.
              </para>
            </listitem>

            <listitem>
              <para>
                Containers based on C, D, E, and F can be split and joined
                efficiently, while the others cannot. Containers based on C
                and D, furthermore, can guarantee that this is exception-free;
                containers based on E cannot guarantee this.
              </para>
            </listitem>

            <listitem>
              <para>
                Containers based on all but E can guarantee that
                erasing an element is exception free; containers based on E
                cannot guarantee this. Containers based on all but B and E
                can guarantee that modifying an object of their type does
                not invalidate iterators or references to their elements,
                while containers based on B and E cannot. Containers based
                on C, D, and E can furthermore make a stronger guarantee,
                namely that modifying an object of their type does not
                affect the order of iterators.
              </para>
            </listitem>
          </orderedlist>

          <para>
            A unified tag and traits system (as used for the C++ standard
            library iterators, for example) can ease generic manipulation of
            associative containers based on different underlying data
            structures.
          </para>

        </section>

        <section xml:id="motivation.associative.iterators">
          <info><title>Iterators</title></info>
          <para>
            Iterators are centric to the design of the standard library
            containers, because of the container/algorithm/iterator
            decomposition that allows an algorithm to operate on a range
            through iterators of some sequence.  Iterators, then, are useful
            because they allow going over a
            specific <emphasis>sequence</emphasis>.  The standard library
            also uses iterators for accessing a
            specific <emphasis>element</emphasis>: when an associative
            container returns one through <function>find</function>. The
            standard library consistently uses the same types of iterators
            for both purposes: going over a range, and accessing a specific
            found element. Before the introduction of hash-based containers
            to the standard library, this made sense (with the exception of
            priority queues, which are discussed later).
          </para>

          <para>
            Using the standard associative containers together with
            non-order-preserving associative containers (and also because of
            priority-queues container), there is a possible need for
            different types of iterators for self-organizing containers:
            the iterator concept seems overloaded to mean two different
            things (in some cases). <remark> XXX
            "ds_gen.html#find_range">Design::Associative
            Containers::Data-Structure Genericity::Point-Type and Range-Type
            Methods</remark>.
          </para>

          <section xml:id="associative.iterators.using">
            <info>
              <title>Using Point Iterators for Range Operations</title>
            </info>
            <para>
              Suppose <classname>cntnr</classname> is some associative
              container, and say <varname>c</varname> is an object of
              type <classname>cntnr</classname>. Then what will be the outcome
              of
            </para>

            <programlisting>
              std::for_each(c.find(1), c.find(5), foo);
            </programlisting>

            <para>
              If <classname>cntnr</classname> is a tree-based container
              object, then an in-order walk will
              apply <classname>foo</classname> to the relevant elements,
              as in the graphic below, label A. If <varname>c</varname> is
              a hash-based container, then the order of elements between any
              two elements is undefined (and probably time-varying); there is
              no guarantee that the elements traversed will coincide with the
              <emphasis>logical</emphasis> elements between 1 and 5, as in
              label B.
            </para>

            <figure>
              <title>Range Iteration in Different Data Structures</title>
              <mediaobject>
                <imageobject>
                  <imagedata align="center" format="PNG" scale="100"
                             fileref="../images/pbds_point_iterators_range_ops_1.png"/>
                </imageobject>
                <textobject>
                  <phrase>Node Invariants</phrase>
                </textobject>
              </mediaobject>
            </figure>

            <para>
              In our opinion, this problem is not caused just because
              red-black trees are order preserving while
              collision-chaining hash tables are (generally) not - it
              is more fundamental. Most of the standard's containers
              order sequences in a well-defined manner that is
              determined by their <emphasis>interface</emphasis>:
              calling <function>insert</function> on a tree-based
              container modifies its sequence in a predictable way, as
              does calling <function>push_back</function> on a list or
              a vector. Conversely, collision-chaining hash tables,
              probing hash tables, priority queues, and list-based
              containers (which are very useful for "multimaps") are
              self-organizing data structures; the effect of each
              operation modifies their sequences in a manner that is
              (practically) determined by their
              <emphasis>implementation</emphasis>.
            </para>

            <para>
              Consequently, applying an algorithm to a sequence obtained from most
              containers may or may not make sense, but applying it to a
              sub-sequence of a self-organizing container does not.
            </para>
          </section>

          <section xml:id="associative.iterators.cost">
            <info>
              <title>Cost to Point Iterators to Enable Range Operations</title>
            </info>
            <para>
              Suppose <varname>c</varname> is some collision-chaining
              hash-based container object, and one calls
            </para>
            <programlisting>c.find(3)</programlisting>
            <para>
              Then what composes the returned iterator?
            </para>

            <para>
              In the graphic below, label A shows the simplest (and
              most efficient) implementation of a collision-chaining
              hash table.  The little box marked
              <classname>point_iterator</classname> shows an object
              that contains a pointer to the element's node. Note that
              this "iterator" has no way to move to the next element (
              it cannot support
              <function>operator++</function>). Conversely, the little
              box marked <classname>iterator</classname> stores both a
              pointer to the element, as well as some other
              information (the bucket number of the element). the
              second iterator, then, is "heavier" than the first one-
              it requires more time and space. If we were to use a
              different container to cross-reference into this
              hash-table using these iterators - it would take much
              more space. As noted above, nothing much can be done by
              incrementing these iterators, so why is this extra
              information needed?
            </para>

            <para>
              Alternatively, one might create a collision-chaining hash-table
              where the lists might be linked, forming a monolithic total-element
              list, as in the graphic below, label B.  Here the iterators are as
              light as can be, but the hash-table's operations are more
              complicated.
            </para>

            <figure>
              <title>Point Iteration in Hash Data Structures</title>
              <mediaobject>
                <imageobject>
                  <imagedata align="center" format="PNG" scale="100"
                             fileref="../images/pbds_point_iterators_range_ops_2.png"/>
                </imageobject>
                <textobject>
                  <phrase>Point Iteration in Hash Data Structures</phrase>
                </textobject>
              </mediaobject>
            </figure>

            <para>
              It should be noted that containers based on collision-chaining
              hash-tables are not the only ones with this type of behavior;
              many other self-organizing data structures display it as well.
            </para>
          </section>

          <section xml:id="associative.iterators.invalidation">
            <info><title>Invalidation Guarantees</title></info>
            <para>Consider the following snippet:</para>
            <programlisting>
              it = c.find(3);
              c.erase(5);
            </programlisting>

            <para>
              Following the call to <classname>erase</classname>, what is the
              validity of <classname>it</classname>: can it be de-referenced?
              can it be incremented?
            </para>

            <para>
              The answer depends on the underlying data structure of the
              container. The graphic below shows three cases: A1 and A2 show
              a red-black tree; B1 and B2 show a probing hash-table; C1 and C2
              show a collision-chaining hash table.
            </para>

            <figure>
              <title>Effect of erase in different underlying data structures</title>
              <mediaobject>
                <imageobject>
                  <imagedata align="center" format="PNG" scale="100"
                             fileref="../images/pbds_invalidation_guarantee_erase.png"/>
                </imageobject>
                <textobject>
                  <phrase>Effect of erase in different underlying data structures</phrase>
                </textobject>
              </mediaobject>
            </figure>

            <orderedlist>
              <listitem>
                <para>
                  Erasing 5 from A1 yields A2. Clearly, an iterator to 3 can
                  be de-referenced and incremented. The sequence of iterators
                  changed, but in a way that is well-defined by the interface.
                </para>
              </listitem>

              <listitem>
                <para>
                  Erasing 5 from B1 yields B2. Clearly, an iterator to 3 is
                  not valid at all - it cannot be de-referenced or
                  incremented; the order of iterators changed in a way that is
                  (practically) determined by the implementation and not by
                  the interface.
                </para>
              </listitem>

              <listitem>
                <para>
                  Erasing 5 from C1 yields C2. Here the situation is more
                  complicated. On the one hand, there is no problem in
                  de-referencing <classname>it</classname>. On the other hand,
                  the order of iterators changed in a way that is
                  (practically) determined by the implementation and not by
                  the interface.
                </para>
              </listitem>
            </orderedlist>

            <para>
              So in the standard library containers, it is not always possible
              to express whether <varname>it</varname> is valid or not. This
              is true also for <function>insert</function>. Again, the
              iterator concept seems overloaded.
            </para>
          </section>
        </section> <!--iterators-->


        <section xml:id="motivation.associative.functions">
          <info><title>Functional</title></info>
          <para>
          </para>

          <para>
            The design of the functional overlay to the underlying data
            structures differs slightly from some of the conventions used in
            the C++ standard.  A strict public interface of methods that
            comprise only operations which depend on the class's internal
            structure; other operations are best designed as external
            functions. (See <xref linkend="biblio.meyers02both"/>).With this
            rubric, the standard associative containers lack some useful
            methods, and provide other methods which would be better
            removed.
          </para>

          <section xml:id="motivation.associative.functions.erase">
            <info><title><function>erase</function></title></info>

            <orderedlist>
              <listitem>
                <para>
                  Order-preserving standard associative containers provide the
                  method
                </para>
                <programlisting>
                  iterator
                  erase(iterator it)
                </programlisting>

                <para>
                  which takes an iterator, erases the corresponding
                  element, and returns an iterator to the following
                  element. Also standardd hash-based associative
                  containers provide this method. This seemingly
                  increasesgenericity between associative containers,
                  since it is possible to use
                </para>
                <programlisting>
                  typename C::iterator it = c.begin();
                  typename C::iterator e_it = c.end();

                  while(it != e_it)
                  it = pred(*it)? c.erase(it) : ++it;
                </programlisting>

                <para>
                  in order to erase from a container object <varname>
                  c</varname> all element which match a
                  predicate <classname>pred</classname>. However, in a
                  different sense this actually decreases genericity: an
                  integral implication of this method is that tree-based
                  associative containers' memory use is linear in the total
                  number of elements they store, while hash-based
                  containers' memory use is unbounded in the total number of
                  elements they store. Assume a hash-based container is
                  allowed to decrease its size when an element is
                  erased. Then the elements might be rehashed, which means
                  that there is no "next" element - it is simply
                  undefined. Consequently, it is possible to infer from the
                  fact that the standard library's hash-based containers
                  provide this method that they cannot downsize when
                  elements are erased. As a consequence, different code is
                  needed to manipulate different containers, assuming that
                  memory should be conserved. Therefor, this library's
                  non-order preserving associative containers omit this
                  method.
                </para>
              </listitem>

              <listitem>
                <para>
                  All associative containers include a conditional-erase method
                </para>
                <programlisting>
                  template&lt;
                  class Pred&gt;
                  size_type
                  erase_if
                  (Pred pred)
                </programlisting>
                <para>
                  which erases all elements matching a predicate. This is probably the
                  only way to ensure linear-time multiple-item erase which can
                  actually downsize a container.
                </para>
              </listitem>

              <listitem>
                <para>
                  The standard associative containers provide methods for
                  multiple-item erase of the form
                </para>
                <programlisting>
                  size_type
                  erase(It b, It e)
                </programlisting>
                <para>
                  erasing a range of elements given by a pair of
                  iterators. For tree-based or trie-based containers, this can
                  implemented more efficiently as a (small) sequence of split
                  and join operations. For other, unordered, containers, this
                  method isn't much better than an external loop. Moreover,
                  if <varname>c</varname> is a hash-based container,
                  then
                </para>
                <programlisting>
                  c.erase(c.find(2), c.find(5))
                </programlisting>
                <para>
                  is almost certain to do something
                  different than erasing all elements whose keys are between 2
                  and 5, and is likely to produce other undefined behavior.
                </para>
              </listitem>
            </orderedlist>
          </section> <!-- erase -->

          <section xml:id="motivation.associative.functions.split">
            <info>
              <title>
                <function>split</function> and <function>join</function>
              </title>
            </info>
            <para>
              It is well-known that tree-based and trie-based container
              objects can be efficiently split or joined (See
              <xref linkend="biblio.clrs2001"/>). Externally splitting or
              joining trees is super-linear, and, furthermore, can throw
              exceptions. Split and join methods, consequently, seem good
              choices for tree-based container methods, especially, since as
              noted just before, they are efficient replacements for erasing
              sub-sequences.
            </para>

          </section> <!-- split -->

          <section xml:id="motivation.associative.functions.insert">
            <info>
              <title>
                <function>insert</function>
              </title>
            </info>
            <para>
              The standard associative containers provide methods of the form
            </para>
            <programlisting>
              template&lt;class It&gt;
              size_type
              insert(It b, It e);
            </programlisting>

            <para>
              for inserting a range of elements given by a pair of
              iterators. At best, this can be implemented as an external loop,
              or, even more efficiently, as a join operation (for the case of
              tree-based or trie-based containers). Moreover, these methods seem
              similar to constructors taking a range given by a pair of
              iterators; the constructors, however, are transactional, whereas
              the insert methods are not; this is possibly confusing.
            </para>

          </section> <!-- insert -->

          <section xml:id="motivation.associative.functions.compare">
            <info>
              <title>
                <function>operator==</function> and <function>operator&lt;=</function>
              </title>
            </info>

            <para>
              Associative containers are parametrized by policies allowing to
              test key equivalence: a hash-based container can do this through
              its equivalence functor, and a tree-based container can do this
              through its comparison functor. In addition, some standard
              associative containers have global function operators, like
              <function>operator==</function> and <function>operator&lt;=</function>,
              that allow comparing entire associative containers.
            </para>

            <para>
              In our opinion, these functions are better left out. To begin
              with, they do not significantly improve over an external
              loop. More importantly, however, they are possibly misleading -
              <function>operator==</function>, for example, usually checks for
              equivalence, or interchangeability, but the associative
              container cannot check for values' equivalence, only keys'
              equivalence; also, are two containers considered equivalent if
              they store the same values in different order? this is an
              arbitrary decision.
            </para>
          </section> <!-- compare -->

        </section>  <!-- functional -->

      </section> <!--associative-->

      <section xml:id="pbds.intro.motivation.priority_queue">
        <info><title>Priority Queues</title></info>

        <section xml:id="motivation.priority_queue.policy">
          <info><title>Policy Choices</title></info>

          <para>
            Priority queues are containers that allow efficiently inserting
            values and accessing the maximal value (in the sense of the
            container's comparison functor). Their interface
            supports <function>push</function>
            and <function>pop</function>. The standard
            container <classname>std::priorityqueue</classname> indeed support
            these methods, but little else. For algorithmic and
            software-engineering purposes, other methods are needed:
          </para>

          <orderedlist>
            <listitem>
              <para>
                Many graph algorithms (see
                <xref linkend="biblio.clrs2001"/>) require increasing a
                value in a priority queue (again, in the sense of the
                container's comparison functor), or joining two
                priority-queue objects.
              </para>
            </listitem>

            <listitem>
              <para>The return type of <classname>priority_queue</classname>'s
              <function>push</function> method is a point-type iterator, which can
              be used for modifying or erasing arbitrary values. For
              example:</para>
              <programlisting>
                priority_queue&lt;int&gt; p;
                priority_queue&lt;int&gt;::point_iterator it = p.push(3);
                p.modify(it, 4);
              </programlisting>

              <para>These types of cross-referencing operations are necessary
              for making priority queues useful for different applications,
              especially graph applications.</para>

            </listitem>
            <listitem>
              <para>
                It is sometimes necessary to erase an arbitrary value in a
                priority queue. For example, consider
                the <function>select</function> function for monitoring
                file descriptors:
              </para>

              <programlisting>
                int
                select(int nfds, fd_set *readfds, fd_set *writefds, fd_set *errorfds,
                struct timeval *timeout);
              </programlisting>
              <para>
                then, as the select documentation states:
              </para>
              <para>
                <quote>
                  The nfds argument specifies the range of file
                  descriptors to be tested. The select() function tests file
                descriptors in the range of 0 to nfds-1.</quote>
              </para>

              <para>
                It stands to reason, therefore, that we might wish to
                maintain a minimal value for <varname>nfds</varname>, and
                priority queues immediately come to mind. Note, though, that
                when a socket is closed, the minimal file description might
                change; in the absence of an efficient means to erase an
                arbitrary value from a priority queue, we might as well
                avoid its use altogether.
              </para>

              <para>
                The standard containers typically support iterators. It is
                somewhat unusual
                for <classname>std::priority_queue</classname> to omit them
                (See <xref linkend="biblio.meyers01stl"/>). One might
                ask why do priority queues need to support iterators, since
                they are self-organizing containers with a different purpose
                than abstracting sequences. There are several reasons:
              </para>
              <orderedlist>
                <listitem>
                  <para>
                    Iterators (even in self-organizing containers) are
                    useful for many purposes: cross-referencing
                    containers, serialization, and debugging code that uses
                    these containers.
                  </para>
                </listitem>

                <listitem>
                  <para>
                    The standard library's hash-based containers support
                    iterators, even though they too are self-organizing
                    containers with a different purpose than abstracting
                    sequences.
                  </para>
                </listitem>

                <listitem>
                  <para>
                    In standard-library-like containers, it is natural to specify the
                    interface of operations for modifying a value or erasing
                    a value (discussed previously) in terms of a iterators.
                    It should be noted that the standard
                    containers also use iterators for accessing and
                    manipulating a specific value. In hash-based
                    containers, one checks the existence of a key by
                    comparing the iterator returned by <function>find</function> to the
                    iterator returned by <function>end</function>, and not by comparing a
                    pointer returned by <function>find</function> to <type>NULL</type>.
                  </para>
                </listitem>
              </orderedlist>
            </listitem>
          </orderedlist>

        </section>

        <section xml:id="motivation.priority_queue.underlying">
          <info><title>Underlying Data Structures</title></info>

          <para>
            There are three main implementations of priority queues: the
            first employs a binary heap, typically one which uses a
            sequence; the second uses a tree (or forest of trees), which is
            typically less structured than an associative container's tree;
            the third simply uses an associative container. These are
            shown in the figure below with labels A1 and A2, B, and C.
          </para>

          <figure>
            <title>Underlying Priority Queue Data Structures</title>
            <mediaobject>
              <imageobject>
                <imagedata align="center" format="PNG" scale="100"
                           fileref="../images/pbds_different_underlying_dss_2.png"/>
              </imageobject>
              <textobject>
                <phrase>Underlying Priority Queue Data Structures</phrase>
              </textobject>
            </mediaobject>
          </figure>

          <para>
            No single implementation can completely replace any of the
            others. Some have better <function>push</function>
            and <function>pop</function> amortized performance, some have
            better bounded (worst case) response time than others, some
            optimize a single method at the expense of others, etc. In
            general the "best" implementation is dictated by the specific
            problem.
          </para>

          <para>
            As with associative containers, the more implementations
            co-exist, the more necessary a traits mechanism is for handling
            generic containers safely and efficiently. This is especially
            important for priority queues, since the invalidation guarantees
            of one of the most useful data structures - binary heaps - is
            markedly different than those of most of the others.
          </para>

        </section>

        <section xml:id="motivation.priority_queue.binary_heap">
          <info><title>Binary Heaps</title></info>


          <para>
            Binary heaps are one of the most useful underlying
            data structures for priority queues. They are very efficient in
            terms of memory (since they don't require per-value structure
            metadata), and have the best amortized <function>push</function> and
            <function>pop</function> performance for primitive types like
            <type>int</type>.
          </para>

          <para>
            The standard library's <classname>priority_queue</classname>
            implements this data structure as an adapter over a sequence,
            typically
            <classname>std::vector</classname>
            or <classname>std::deque</classname>, which correspond to labels
            A1 and A2 respectively in the graphic above.
          </para>

          <para>
            This is indeed an elegant example of the adapter concept and
            the algorithm/container/iterator decomposition. (See <xref linkend="biblio.nelson96stlpq"/>). There are
            several reasons why a binary-heap priority queue
            may be better implemented as a container instead of a
            sequence adapter:
          </para>

          <orderedlist>
            <listitem>
              <para>
                <classname>std::priority_queue</classname> cannot erase values
                from its adapted sequence (irrespective of the sequence
                type). This means that the memory use of
                an <classname>std::priority_queue</classname> object is always
                proportional to the maximal number of values it ever contained,
                and not to the number of values that it currently
                contains. (See <filename>performance/priority_queue_text_pop_mem_usage.cc</filename>.)
                This implementation of binary heaps acts very differently than
                other underlying data structures (See also pairing heaps).
              </para>
            </listitem>

            <listitem>
              <para>
                Some combinations of adapted sequences and value types
                are very inefficient or just don't make sense. If one uses
                <classname>std::priority_queue&lt;std::vector&lt;std::string&gt;
                &gt; &gt;</classname>, for example, then not only will each
                operation perform a logarithmic number of
                <classname>std::string</classname> assignments, but, furthermore, any
                operation (including <function>pop</function>) can render the container
                useless due to exceptions. Conversely, if one uses
                <classname>std::priority_queue&lt;std::deque&lt;int&gt; &gt;
                &gt;</classname>, then each operation uses incurs a logarithmic
                number of indirect accesses (through pointers) unnecessarily.
                It might be better to let the container make a conservative
                deduction whether to use the structure in the graphic above, labels A1 or A2.
              </para>
            </listitem>

            <listitem>
              <para>
                There does not seem to be a systematic way to determine
                what exactly can be done with the priority queue.
              </para>
              <orderedlist>
                <listitem>
                  <para>
                    If <classname>p</classname> is a priority queue adapting an
                    <classname>std::vector</classname>, then it is possible to iterate over
                    all values by using <function>&amp;p.top()</function> and
                    <function>&amp;p.top() + p.size()</function>, but this will not work
                    if <varname>p</varname> is adapting an <classname>std::deque</classname>; in any
                    case, one cannot use <classname>p.begin()</classname> and
                    <classname>p.end()</classname>. If a different sequence is adapted, it
                    is even more difficult to determine what can be
                    done.
                  </para>
                </listitem>

                <listitem>
                  <para>
                    If <varname>p</varname> is a priority queue adapting an
                    <classname>std::deque</classname>, then the reference return by
                  </para>
                  <programlisting>
                    p.top()
                  </programlisting>
                  <para>
                    will remain valid until it is popped,
                    but if <varname>p</varname> adapts an <classname>std::vector</classname>, the
                    next <function>push</function> will invalidate it. If a different
                    sequence is adapted, it is even more difficult to
                    determine what can be done.
                  </para>
                </listitem>
              </orderedlist>
            </listitem>

            <listitem>
              <para>
                Sequence-based binary heaps can still implement
                linear-time <function>erase</function> and <function>modify</function> operations.
                This means that if one needs to erase a small
                (say logarithmic) number of values, then one might still
                choose this underlying data structure. Using
                <classname>std::priority_queue</classname>, however, this will generally
                change the order of growth of the entire sequence of
                operations.
              </para>
            </listitem>
          </orderedlist>

        </section>
      </section>
    </section> <!-- goals/motivation -->
  </section> <!-- intro -->

  <!-- S02: Using -->
  <section xml:id="containers.pbds.using">
    <info><title>Using</title></info>
    <?dbhtml filename="policy_data_structures_using.html"?>

    <section xml:id="pbds.using.prereq">
      <info><title>Prerequisites</title></info>

      <para>The library contains only header files, and does not require any
      other libraries except the standard C++ library . All classes are
      defined in namespace <code>__gnu_pbds</code>. The library internally
      uses macros beginning with <code>PB_DS</code>, but
      <code>#undef</code>s anything it <code>#define</code>s (except for
      header guards). Compiling the library in an environment where macros
      beginning in <code>PB_DS</code> are defined, may yield unpredictable
      results in compilation, execution, or both.</para>

      <para>
        Further dependencies are necessary to create the visual output
        for the performance tests. To create these graphs, an
        additional package is needed: <command>pychart</command>.
      </para>
    </section>

    <section xml:id="pbds.using.organization">
      <info><title>Organization</title></info>

      <para>
        The various data structures are organized as follows.
      </para>

      <itemizedlist>
        <listitem>
          <para>
            Branch-Based
          </para>

          <itemizedlist>
            <listitem>
              <para>
                <classname>basic_branch</classname>
                is an abstract base class for branched-based
                associative-containers
              </para>
            </listitem>

            <listitem>
              <para>
                <classname>tree</classname>
                is a concrete base class for tree-based
                associative-containers
              </para>
            </listitem>

            <listitem>
              <para>
                <classname>trie</classname>
                is a concrete base class trie-based
                associative-containers
              </para>
            </listitem>
          </itemizedlist>
        </listitem>

        <listitem>
          <para>
            Hash-Based
          </para>
          <itemizedlist>
            <listitem>
              <para>
                <classname>basic_hash_table</classname>
                is an abstract base class for hash-based
                associative-containers
              </para>
            </listitem>

            <listitem>
              <para>
                <classname>cc_hash_table</classname>
                is a concrete collision-chaining hash-based
                associative-containers
              </para>
            </listitem>

            <listitem>
              <para>
                <classname>gp_hash_table</classname>
                is a concrete (general) probing hash-based
                associative-containers
              </para>
            </listitem>
          </itemizedlist>
        </listitem>

        <listitem>
          <para>
            List-Based
          </para>
          <itemizedlist>
            <listitem>
              <para>
                <classname>list_update</classname>
                list-based update-policy associative container
              </para>
            </listitem>
          </itemizedlist>
        </listitem>
        <listitem>
          <para>
            Heap-Based
          </para>
          <itemizedlist>
            <listitem>
              <para>
                <classname>priority_queue</classname>
                A priority queue.
              </para>
            </listitem>
          </itemizedlist>
        </listitem>
      </itemizedlist>

      <para>
        The hierarchy is composed naturally so that commonality is
        captured by base classes. Thus <function>operator[]</function>
        is defined at the base of any hierarchy, since all derived
        containers support it. Conversely <function>split</function> is
        defined in <classname>basic_branch</classname>, since only
        tree-like containers support it.
      </para>

      <para>
        In addition, there are the following diagnostics classes,
        used to report errors specific to this library's data
        structures.
      </para>

      <figure>
        <title>Exception Hierarchy</title>
        <mediaobject>
          <imageobject>
            <imagedata align="center" format="PDF" scale="75"
                       fileref="../images/pbds_exception_hierarchy.pdf"/>
          </imageobject>
          <imageobject>
            <imagedata align="center" format="PNG" scale="100"
                       fileref="../images/pbds_exception_hierarchy.png"/>
          </imageobject>
          <textobject>
            <phrase>Exception Hierarchy</phrase>
          </textobject>
        </mediaobject>
      </figure>

    </section>

    <section xml:id="pbds.using.tutorial">
      <info><title>Tutorial</title></info>

      <section xml:id="pbds.using.tutorial.basic">
        <info><title>Basic Use</title></info>

        <para>
          For the most part, the policy-based containers containers in
          namespace <literal>__gnu_pbds</literal> have the same interface as
          the equivalent containers in the standard C++ library, except for
          the names used for the container classes themselves. For example,
          this shows basic operations on a collision-chaining hash-based
          container:
        </para>
        <programlisting>
          #include &lt;ext/pb_ds/assoc_container.h&gt;

          int main()
          {
          __gnu_pbds::cc_hash_table&lt;int, char&gt; c;
          c[2] = 'b';
          assert(c.find(1) == c.end());
          };
        </programlisting>

        <para>
          The container is called
          <classname>__gnu_pbds::cc_hash_table</classname> instead of
          <classname>std::unordered_map</classname>, since <quote>unordered
          map</quote> does not necessarily mean a hash-based map as implied by
          the C++ library (C++11 or TR1). For example, list-based associative
          containers, which are very useful for the construction of
          "multimaps," are also unordered.
        </para>

        <para>This snippet shows a red-black tree based container:</para>

        <programlisting>
          #include &lt;ext/pb_ds/assoc_container.h&gt;

          int main()
          {
          __gnu_pbds::tree&lt;int, char&gt; c;
          c[2] = 'b';
          assert(c.find(2) != c.end());
          };
        </programlisting>

        <para>The container is called <classname>tree</classname> instead of
        <classname>map</classname> since the underlying data structures are
        being named with specificity.
        </para>

        <para>
          The member function naming convention is to strive to be the same as
          the equivalent member functions in other C++ standard library
          containers. The familiar methods are unchanged:
          <function>begin</function>, <function>end</function>,
          <function>size</function>, <function>empty</function>, and
          <function>clear</function>.
        </para>

        <para>
          This isn't to say that things are exactly as one would expect, given
          the container requirments and interfaces in the C++ standard.
        </para>

        <para>
          The names of containers' policies and policy accessors are
          different then the usual. For example, if <type>hash_type</type> is
        some type of hash-based container, then</para>

        <programlisting>
          hash_type::hash_fn
        </programlisting>

        <para>
          gives the type of its hash functor, and if <varname>obj</varname> is
          some hash-based container object, then
        </para>

        <programlisting>
          obj.get_hash_fn()
        </programlisting>

        <para>will return a reference to its hash-functor object.</para>


        <para>
          Similarly, if <type>tree_type</type> is some type of tree-based
          container, then
        </para>

        <programlisting>
          tree_type::cmp_fn
        </programlisting>

        <para>
          gives the type of its comparison functor, and if
          <varname>obj</varname> is some tree-based container object,
          then
        </para>

        <programlisting>
          obj.get_cmp_fn()
        </programlisting>

        <para>will return a reference to its comparison-functor object.</para>

        <para>
          It would be nice to give names consistent with those in the existing
          C++ standard (inclusive of TR1). Unfortunately, these standard
          containers don't consistently name types and methods. For example,
          <classname>std::tr1::unordered_map</classname> uses
          <type>hasher</type> for the hash functor, but
          <classname>std::map</classname> uses <type>key_compare</type> for
          the comparison functor. Also, we could not find an accessor for
          <classname>std::tr1::unordered_map</classname>'s hash functor, but
          <classname>std::map</classname> uses <classname>compare</classname>
          for accessing the comparison functor.
        </para>

        <para>
          Instead, <literal>__gnu_pbds</literal> attempts to be internally
          consistent, and uses standard-derived terminology if possible.
        </para>

        <para>
          Another source of difference is in scope:
          <literal>__gnu_pbds</literal> contains more types of associative
          containers than the standard C++ library, and more opportunities
          to configure these new containers, since different types of
          associative containers are useful in different settings.
        </para>

        <para>
          Namespace <literal>__gnu_pbds</literal> contains different classes for
          hash-based containers, tree-based containers, trie-based containers,
          and list-based containers.
        </para>

        <para>
          Since associative containers share parts of their interface, they
          are organized as a class hierarchy.
        </para>

        <para>Each type or method is defined in the most-common ancestor
        in which it makes sense.
        </para>

        <para>For example, all associative containers support iteration
        expressed in the following form:
        </para>

        <programlisting>
          const_iterator
          begin() const;

          iterator
          begin();

          const_iterator
          end() const;

          iterator
          end();
        </programlisting>

        <para>
          But not all containers contain or use hash functors. Yet, both
          collision-chaining and (general) probing hash-based associative
          containers have a hash functor, so
          <classname>basic_hash_table</classname> contains the interface:
        </para>

        <programlisting>
          const hash_fn&amp;
          get_hash_fn() const;

          hash_fn&amp;
          get_hash_fn();
        </programlisting>

        <para>
          so all hash-based associative containers inherit the same
          hash-functor accessor methods.
        </para>

      </section> <!--basic use -->

      <section xml:id="pbds.using.tutorial.configuring">
        <info>
          <title>
            Configuring via Template Parameters
          </title>
        </info>

        <para>
          In general, each of this library's containers is
          parametrized by more policies than those of the standard library. For
          example, the standard hash-based container is parametrized as
          follows:
        </para>
        <programlisting>
          template&lt;typename Key, typename Mapped, typename Hash,
          typename Pred, typename Allocator, bool Cache_Hashe_Code&gt;
          class unordered_map;
        </programlisting>

        <para>
          and so can be configured by key type, mapped type, a functor
          that translates keys to unsigned integral types, an equivalence
          predicate, an allocator, and an indicator whether to store hash
          values with each entry. this library's collision-chaining
          hash-based container is parametrized as
        </para>
        <programlisting>
          template&lt;typename Key, typename Mapped, typename Hash_Fn,
          typename Eq_Fn, typename Comb_Hash_Fn,
          typename Resize_Policy, bool Store_Hash
          typename Allocator&gt;
          class cc_hash_table;
        </programlisting>

        <para>
          and so can be configured by the first four types of
          <classname>std::tr1::unordered_map</classname>, then a
          policy for translating the key-hash result into a position
          within the table, then a policy by which the table resizes,
          an indicator whether to store hash values with each entry,
          and an allocator (which is typically the last template
          parameter in standard containers).
        </para>

        <para>
          Nearly all policy parameters have default values, so this
          need not be considered for casual use. It is important to
          note, however, that hash-based containers' policies can
          dramatically alter their performance in different settings,
          and that tree-based containers' policies can make them
          useful for other purposes than just look-up.
        </para>


        <para>As opposed to associative containers, priority queues have
        relatively few configuration options. The priority queue is
        parametrized as follows:</para>
        <programlisting>
          template&lt;typename Value_Type, typename Cmp_Fn,typename Tag,
          typename Allocator&gt;
          class priority_queue;
        </programlisting>

        <para>The <classname>Value_Type</classname>, <classname>Cmp_Fn</classname>, and
        <classname>Allocator</classname> parameters are the container's value type,
        comparison-functor type, and allocator type, respectively;
        these are very similar to the standard's priority queue. The
        <classname>Tag</classname> parameter is different: there are a number of
        pre-defined tag types corresponding to binary heaps, binomial
        heaps, etc., and <classname>Tag</classname> should be instantiated
        by one of them.</para>

        <para>Note that as opposed to the
        <classname>std::priority_queue</classname>,
        <classname>__gnu_pbds::priority_queue</classname> is not a
        sequence-adapter; it is a regular container.</para>

      </section>

      <section xml:id="pbds.using.tutorial.traits">
        <info>
          <title>
            Querying Container Attributes
          </title>
        </info>
        <para></para>

        <para>A containers underlying data structure
        affect their performance; Unfortunately, they can also affect
        their interface. When manipulating generically associative
        containers, it is often useful to be able to statically
        determine what they can support and what the cannot.
        </para>

        <para>Happily, the standard provides a good solution to a similar
        problem - that of the different behavior of iterators. If
        <classname>It</classname> is an iterator, then
        </para>
        <programlisting>
          typename std::iterator_traits&lt;It&gt;::iterator_category
        </programlisting>

        <para>is one of a small number of pre-defined tag classes, and
        </para>
        <programlisting>
          typename std::iterator_traits&lt;It&gt;::value_type
        </programlisting>

        <para>is the value type to which the iterator "points".</para>

        <para>
          Similarly, in this library, if <type>C</type> is a
          container, then <classname>container_traits</classname> is a
          trait class that stores information about the kind of
          container that is implemented.
        </para>
        <programlisting>
          typename container_traits&lt;C&gt;::container_category
        </programlisting>
        <para>
          is one of a small number of predefined tag structures that
          uniquely identifies the type of underlying data structure.
        </para>

        <para>In most cases, however, the exact underlying data
        structure is not really important, but what is important is
        one of its other attributes: whether it guarantees storing
        elements by key order, for example. For this one can
        use</para>
        <programlisting>
          typename container_traits&lt;C&gt;::order_preserving
        </programlisting>
        <para>
          Also,
        </para>
        <programlisting>
          typename container_traits&lt;C&gt;::invalidation_guarantee
        </programlisting>

        <para>is the container's invalidation guarantee. Invalidation
        guarantees are especially important regarding priority queues,
        since in this library's design, iterators are practically the
        only way to manipulate them.</para>
      </section>

      <section xml:id="pbds.using.tutorial.point_range_iteration">
        <info>
          <title>
            Point and Range Iteration
          </title>
        </info>
        <para></para>

        <para>This library differentiates between two types of methods
        and iterators: point-type, and range-type. For example,
        <function>find</function> and <function>insert</function> are point-type methods, since
        they each deal with a specific element; their returned
        iterators are point-type iterators. <function>begin</function> and
        <function>end</function> are range-type methods, since they are not used to
        find a specific element, but rather to go over all elements in
        a container object; their returned iterators are range-type
        iterators.
        </para>

        <para>Most containers store elements in an order that is
        determined by their interface. Correspondingly, it is fine that
        their point-type iterators are synonymous with their range-type
        iterators. For example, in the following snippet
        </para>
        <programlisting>
          std::for_each(c.find(1), c.find(5), foo);
        </programlisting>
        <para>
          two point-type iterators (returned by <function>find</function>) are used
          for a range-type purpose - going over all elements whose key is
          between 1 and 5.
        </para>

        <para>
          Conversely, the above snippet makes no sense for
          self-organizing containers - ones that order (and reorder)
          their elements by implementation. It would be nice to have a
          uniform iterator system that would allow the above snippet to
          compile only if it made sense.
        </para>

        <para>
          This could trivially be done by specializing
          <function>std::for_each</function> for the case of iterators returned by
          <classname>std::tr1::unordered_map</classname>, but this would only solve the
          problem for one algorithm and one container. Fundamentally, the
          problem is that one can loop using a self-organizing
          container's point-type iterators.
        </para>

        <para>
          This library's containers define two families of
          iterators: <type>point_const_iterator</type> and
          <type>point_iterator</type> are the iterator types returned by
          point-type methods; <type>const_iterator</type> and
          <type>iterator</type> are the iterator types returned by range-type
          methods.
        </para>
        <programlisting>
          class &lt;- some container -&gt;
          {
          public:
          ...

          typedef &lt;- something -&gt; const_iterator;

          typedef &lt;- something -&gt; iterator;

          typedef &lt;- something -&gt; point_const_iterator;

          typedef &lt;- something -&gt; point_iterator;

          ...

          public:
          ...

          const_iterator begin () const;

          iterator begin();

          point_const_iterator find(...) const;

          point_iterator find(...);
          };
        </programlisting>

        <para>For
        containers whose interface defines sequence order , it
        is very simple: point-type and range-type iterators are exactly
        the same, which means that the above snippet will compile if it
        is used for an order-preserving associative container.
        </para>

        <para>
          For self-organizing containers, however, (hash-based
          containers as a special example), the preceding snippet will
          not compile, because their point-type iterators do not support
          <function>operator++</function>.
        </para>

        <para>In any case, both for order-preserving and self-organizing
        containers, the following snippet will compile:
        </para>
        <programlisting>
          typename Cntnr::point_iterator it = c.find(2);
        </programlisting>

        <para>
          because a range-type iterator can always be converted to a
          point-type iterator.
        </para>

        <para>Distingushing between iterator types also
        raises the point that a container's iterators might have
        different invalidation rules concerning their de-referencing
        abilities and movement abilities. This now corresponds exactly
        to the question of whether point-type and range-type iterators
        are valid. As explained above, <classname>container_traits</classname> allows
        querying a container for its data structure attributes. The
        iterator-invalidation guarantees are certainly a property of
        the underlying data structure, and so
        </para>
        <programlisting>
          container_traits&lt;C&gt;::invalidation_guarantee
        </programlisting>

        <para>
          gives one of three pre-determined types that answer this
          query.
        </para>

      </section>
    </section> <!-- tutorial -->

    <section xml:id="pbds.using.examples">
      <info><title>Examples</title></info>
      <para>
        Additional code examples are provided in the source
        distribution, as part of the regression and performance
        testsuite.
      </para>

      <section xml:id="pbds.using.examples.basic">
        <info><title>Intermediate Use</title></info>

        <itemizedlist>
          <listitem>
            <para>
              Basic use of maps:
              <filename>basic_map.cc</filename>
            </para>
          </listitem>

          <listitem>
            <para>
              Basic use of sets:
              <filename>basic_set.cc</filename>
            </para>
          </listitem>

          <listitem>
            <para>
              Conditionally erasing values from an associative container object:
              <filename>erase_if.cc</filename>
            </para>
          </listitem>

          <listitem>
            <para>
              Basic use of multimaps:
              <filename>basic_multimap.cc</filename>
            </para>
          </listitem>

          <listitem>
            <para>
              Basic use of multisets:
              <filename>basic_multiset.cc</filename>
            </para>
          </listitem>

          <listitem>
            <para>
              Basic use of priority queues:
              <filename>basic_priority_queue.cc</filename>
            </para>
          </listitem>

          <listitem>
            <para>
              Splitting and joining priority queues:
              <filename>priority_queue_split_join.cc</filename>
            </para>
          </listitem>

          <listitem>
            <para>
              Conditionally erasing values from a priority queue:
              <filename>priority_queue_erase_if.cc</filename>
            </para>
          </listitem>
        </itemizedlist>

      </section>

      <section xml:id="pbds.using.examples.query">
        <info><title>Querying with <classname>container_traits</classname> </title></info>
        <itemizedlist>
          <listitem>
            <para>
              Using <classname>container_traits</classname> to query
              about underlying data structure behavior:
              <filename>assoc_container_traits.cc</filename>
            </para>
          </listitem>

          <listitem>
            <para>
              A non-compiling example showing wrong use of finding keys in
              hash-based containers: <filename>hash_find_neg.cc</filename>
            </para>
          </listitem>
          <listitem>
            <para>
              Using <classname>container_traits</classname>
              to query about underlying data structure behavior:
              <filename>priority_queue_container_traits.cc</filename>
            </para>
          </listitem>

        </itemizedlist>

      </section>

      <section xml:id="pbds.using.examples.container">
        <info><title>By Container Method</title></info>
        <para></para>

        <section xml:id="pbds.using.examples.container.hash">
          <info><title>Hash-Based</title></info>

          <section xml:id="pbds.using.examples.container.hash.resize">
            <info><title>size Related</title></info>

            <itemizedlist>
              <listitem>
                <para>
                  Setting the initial size of a hash-based container
                  object:
                  <filename>hash_initial_size.cc</filename>
                </para>
              </listitem>

              <listitem>
                <para>
                  A non-compiling example showing how not to resize a
                  hash-based container object:
                  <filename>hash_resize_neg.cc</filename>
                </para>
              </listitem>

              <listitem>
                <para>
                  Resizing the size of a hash-based container object:
                  <filename>hash_resize.cc</filename>
                </para>
              </listitem>

              <listitem>
                <para>
                  Showing an illegal resize of a hash-based container
                  object:
                  <filename>hash_illegal_resize.cc</filename>
                </para>
              </listitem>

              <listitem>
                <para>
                  Changing the load factors of a hash-based container
                  object: <filename>hash_load_set_change.cc</filename>
                </para>
              </listitem>
            </itemizedlist>
          </section>

          <section xml:id="pbds.using.examples.container.hash.hashor">
            <info><title>Hashing Function Related</title></info>
            <para></para>

            <itemizedlist>
              <listitem>
                <para>
                  Using a modulo range-hashing function for the case of an
                  unknown skewed key distribution:
                  <filename>hash_mod.cc</filename>
                </para>
              </listitem>

              <listitem>
                <para>
                  Writing a range-hashing functor for the case of a known
                  skewed key distribution:
                  <filename>shift_mask.cc</filename>
                </para>
              </listitem>

              <listitem>
                <para>
                  Storing the hash value along with each key:
                  <filename>store_hash.cc</filename>
                </para>
              </listitem>

              <listitem>
                <para>
                  Writing a ranged-hash functor:
                  <filename>ranged_hash.cc</filename>
                </para>
              </listitem>
            </itemizedlist>

          </section>

        </section>

        <section xml:id="pbds.using.examples.container.branch">
          <info><title>Branch-Based</title></info>


          <section xml:id="pbds.using.examples.container.branch.split">
            <info><title>split or join Related</title></info>

            <itemizedlist>
              <listitem>
                <para>
                  Joining two tree-based container objects:
                  <filename>tree_join.cc</filename>
                </para>
              </listitem>

              <listitem>
                <para>
                  Splitting a PATRICIA trie container object:
                  <filename>trie_split.cc</filename>
                </para>
              </listitem>

              <listitem>
                <para>
                  Order statistics while joining two tree-based container
                  objects:
                  <filename>tree_order_statistics_join.cc</filename>
                </para>
              </listitem>
            </itemizedlist>

          </section>

          <section xml:id="pbds.using.examples.container.branch.invariants">
            <info><title>Node Invariants</title></info>

            <itemizedlist>
              <listitem>
                <para>
                  Using trees for order statistics:
                  <filename>tree_order_statistics.cc</filename>
                </para>
              </listitem>

              <listitem>
                <para>
                  Augmenting trees to support operations on line
                  intervals:
                  <filename>tree_intervals.cc</filename>
                </para>
              </listitem>
            </itemizedlist>

          </section>

          <section xml:id="pbds.using.examples.container.branch.trie">
            <info><title>trie</title></info>
            <itemizedlist>
              <listitem>
                <para>
                  Using a PATRICIA trie for DNA strings:
                  <filename>trie_dna.cc</filename>
                </para>
              </listitem>

              <listitem>
                <para>
                  Using a PATRICIA
                  trie for finding all entries whose key matches a given prefix:
                  <filename>trie_prefix_search.cc</filename>
                </para>
              </listitem>
            </itemizedlist>

          </section>

        </section>

        <section xml:id="pbds.using.examples.container.priority_queue">
          <info><title>Priority Queues</title></info>
          <itemizedlist>
            <listitem>
              <para>
                Cross referencing an associative container and a priority
                queue: <filename>priority_queue_xref.cc</filename>
              </para>
            </listitem>

            <listitem>
              <para>
                Cross referencing a vector and a priority queue using a
                very simple version of Dijkstra's shortest path
                algorithm:
                <filename>priority_queue_dijkstra.cc</filename>
              </para>
            </listitem>
          </itemizedlist>

        </section>


      </section>

    </section>

  </section> <!-- using -->

  <!-- S03: Design -->


<section xml:id="containers.pbds.design">
  <info><title>Design</title></info>
  <?dbhtml filename="policy_data_structures_design.html"?>
  <para></para>

  <section xml:id="pbds.design.concepts">
    <info><title>Concepts</title></info>

    <section xml:id="pbds.design.concepts.null_type">
      <info><title>Null Policy Classes</title></info>

      <para>
        Associative containers are typically parametrized by various
        policies. For example, a hash-based associative container is
        parametrized by a hash-functor, transforming each key into an
        non-negative numerical type. Each such value is then further mapped
        into a position within the table. The mapping of a key into a
        position within the table is therefore a two-step process.
      </para>

      <para>
        In some cases, instantiations are redundant. For example, when the
        keys are integers, it is possible to use a redundant hash policy,
        which transforms each key into its value.
      </para>

      <para>
        In some other cases, these policies are irrelevant.  For example, a
        hash-based associative container might transform keys into positions
        within a table by a different method than the two-step method
        described above. In such a case, the hash functor is simply
        irrelevant.
      </para>

      <para>
        When a policy is either redundant or irrelevant, it can be replaced
        by <classname>null_type</classname>.
      </para>

      <para>
        For example, a <emphasis>set</emphasis> is an associative
        container with one of its template parameters (the one for the
        mapped type) replaced with <classname>null_type</classname>. Other
        places simplifications are made possible with this technique
        include node updates in tree and trie data structures, and hash
        and probe functions for hash data structures.
      </para>
    </section>

    <section xml:id="pbds.design.concepts.associative_semantics">
      <info><title>Map and Set Semantics</title></info>

      <section xml:id="concepts.associative_semantics.set_vs_map">
        <info>
          <title>
            Distinguishing Between Maps and Sets
          </title>
        </info>

        <para>
          Anyone familiar with the standard knows that there are four kinds
          of associative containers: maps, sets, multimaps, and
          multisets. The map datatype associates each key to
          some data.
        </para>

        <para>
          Sets are associative containers that simply store keys -
          they do not map them to anything. In the standard, each map class
          has a corresponding set class. E.g.,
          <classname>std::map&lt;int, char&gt;</classname> maps each
          <classname>int</classname> to a <classname>char</classname>, but
          <classname>std::set&lt;int, char&gt;</classname> simply stores
          <classname>int</classname>s. In this library, however, there are no
          distinct classes for maps and sets. Instead, an associative
          container's <classname>Mapped</classname> template parameter is a policy: if
          it is instantiated by <classname>null_type</classname>, then it
          is a "set"; otherwise, it is a "map". E.g.,
        </para>
        <programlisting>
          cc_hash_table&lt;int, char&gt;
        </programlisting>
        <para>
          is a "map" mapping each <type>int</type> value to a <type>
          char</type>, but
        </para>
        <programlisting>
          cc_hash_table&lt;int, null_type&gt;
        </programlisting>
        <para>
          is a type that uniquely stores <type>int</type> values.
        </para>
        <para>Once the <classname>Mapped</classname> template parameter is instantiated
        by <classname>null_type</classname>, then
        the "set" acts very similarly to the standard's sets - it does not
        map each key to a distinct <classname>null_type</classname> object. Also,
        , the container's <type>value_type</type> is essentially
        its <type>key_type</type> - just as with the standard's sets
        .</para>

        <para>
          The standard's multimaps and multisets allow, respectively,
          non-uniquely mapping keys and non-uniquely storing keys. As
          discussed, the
          reasons why this might be necessary are 1) that a key might be
          decomposed into a primary key and a secondary key, 2) that a
          key might appear more than once, or 3) any arbitrary
          combination of 1)s and 2)s. Correspondingly,
          one should use 1) "maps" mapping primary keys to secondary
          keys, 2) "maps" mapping keys to size types, or 3) any arbitrary
          combination of 1)s and 2)s. Thus, for example, an
          <classname>std::multiset&lt;int&gt;</classname> might be used to store
          multiple instances of integers, but using this library's
          containers, one might use
        </para>
        <programlisting>
          tree&lt;int, size_t&gt;
        </programlisting>

        <para>
          i.e., a <classname>map</classname> of <type>int</type>s to
          <type>size_t</type>s.
        </para>
        <para>
          These "multimaps" and "multisets" might be confusing to
          anyone familiar with the standard's <classname>std::multimap</classname> and
          <classname>std::multiset</classname>, because there is no clear
          correspondence between the two. For example, in some cases
          where one uses <classname>std::multiset</classname> in the standard, one might use
          in this library a "multimap" of "multisets" - i.e., a
          container that maps primary keys each to an associative
          container that maps each secondary key to the number of times
          it occurs.
        </para>

        <para>
          When one uses a "multimap," one should choose with care the
          type of container used for secondary keys.
        </para>
      </section> <!-- map vs set -->


      <section xml:id="concepts.associative_semantics.multi">
        <info><title>Alternatives to <classname>std::multiset</classname> and <classname>std::multimap</classname></title></info>

        <para>
          Brace onself: this library does not contain containers like
          <classname>std::multimap</classname> or
          <classname>std::multiset</classname>. Instead, these data
          structures can be synthesized via manipulation of the
          <classname>Mapped</classname> template parameter.
        </para>
        <para>
          One maps the unique part of a key - the primary key, into an
          associative-container of the (originally) non-unique parts of
          the key - the secondary key. A primary associative-container
          is an associative container of primary keys; a secondary
          associative-container is an associative container of
          secondary keys.
        </para>

        <para>
          Stepping back a bit, and starting in from the beginning.
        </para>


        <para>
          Maps (or sets) allow mapping (or storing) unique-key values.
          The standard library also supplies associative containers which
          map (or store) multiple values with equivalent keys:
          <classname>std::multimap</classname>, <classname>std::multiset</classname>,
          <classname>std::tr1::unordered_multimap</classname>, and
          <classname>unordered_multiset</classname>. We first discuss how these might
          be used, then why we think it is best to avoid them.
        </para>

        <para>
          Suppose one builds a simple bank-account application that
          records for each client (identified by an <classname>std::string</classname>)
          and account-id (marked by an <type>unsigned long</type>) -
          the balance in the account (described by a
          <type>float</type>). Suppose further that ordering this
          information is not useful, so a hash-based container is
          preferable to a tree based container. Then one can use
        </para>

        <programlisting>
          std::tr1::unordered_map&lt;std::pair&lt;std::string, unsigned long&gt;, float, ...&gt;
        </programlisting>

        <para>
          which hashes every combination of client and account-id. This
          might work well, except for the fact that it is now impossible
          to efficiently list all of the accounts of a specific client
          (this would practically require iterating over all
          entries). Instead, one can use
        </para>

        <programlisting>
          std::tr1::unordered_multimap&lt;std::pair&lt;std::string, unsigned long&gt;, float, ...&gt;
        </programlisting>

        <para>
          which hashes every client, and decides equivalence based on
          client only. This will ensure that all accounts belonging to a
          specific user are stored consecutively.
        </para>

        <para>
          Also, suppose one wants an integers' priority queue
          (a container that supports <function>push</function>,
          <function>pop</function>, and <function>top</function> operations, the last of which
          returns the largest <type>int</type>) that also supports
          operations such as <function>find</function> and <function>lower_bound</function>. A
          reasonable solution is to build an adapter over
          <classname>std::set&lt;int&gt;</classname>. In this adapter,
          <function>push</function> will just call the tree-based
          associative container's <function>insert</function> method; <function>pop</function>
          will call its <function>end</function> method, and use it to return the
          preceding element (which must be the largest). Then this might
          work well, except that the container object cannot hold
          multiple instances of the same integer (<function>push(4)</function>,
          will be a no-op if <constant>4</constant> is already in the
          container object). If multiple keys are necessary, then one
          might build the adapter over an
          <classname>std::multiset&lt;int&gt;</classname>.
        </para>

        <para>
          The standard library's non-unique-mapping containers are useful
          when (1) a key can be decomposed in to a primary key and a
          secondary key, (2) a key is needed multiple times, or (3) any
          combination of (1) and (2).
        </para>

        <para>
          The graphic below shows how the standard library's container
          design works internally; in this figure nodes shaded equally
          represent equivalent-key values. Equivalent keys are stored
          consecutively using the properties of the underlying data
          structure: binary search trees (label A) store equivalent-key
          values consecutively (in the sense of an in-order walk)
          naturally; collision-chaining hash tables (label B) store
          equivalent-key values in the same bucket, the bucket can be
          arranged so that equivalent-key values are consecutive.
        </para>

        <figure>
          <title>Non-unique Mapping Standard Containers</title>
          <mediaobject>
            <imageobject>
              <imagedata align="center" format="PNG" scale="100"
                         fileref="../images/pbds_embedded_lists_1.png"/>
            </imageobject>
            <textobject>
              <phrase>Non-unique Mapping Standard Containers</phrase>
            </textobject>
          </mediaobject>
        </figure>

        <para>
          Put differently, the standards' non-unique mapping
          associative-containers are associative containers that map
          primary keys to linked lists that are embedded into the
          container. The graphic below shows again the two
          containers from the first graphic above, this time with
          the embedded linked lists of the grayed nodes marked
          explicitly.
        </para>

        <figure xml:id="fig.pbds_embedded_lists_2">
          <title>
            Effect of embedded lists in
            <classname>std::multimap</classname>
          </title>
          <mediaobject>
            <imageobject>
              <imagedata align="center" format="PNG" scale="100"
                         fileref="../images/pbds_embedded_lists_2.png"/>
            </imageobject>
            <textobject>
              <phrase>
                Effect of embedded lists in
                <classname>std::multimap</classname>
              </phrase>
            </textobject>
          </mediaobject>
        </figure>

        <para>
          These embedded linked lists have several disadvantages.
        </para>

        <orderedlist>
          <listitem>
            <para>
              The underlying data structure embeds the linked lists
              according to its own consideration, which means that the
              search path for a value might include several different
              equivalent-key values. For example, the search path for the
              the black node in either of the first graphic, labels A or B,
              includes more than a single gray node.
            </para>
          </listitem>

          <listitem>
            <para>
              The links of the linked lists are the underlying data
              structures' nodes, which typically are quite structured.  In
              the case of tree-based containers (the grapic above, label
              B), each "link" is actually a node with three pointers (one
              to a parent and two to children), and a
              relatively-complicated iteration algorithm. The linked
              lists, therefore, can take up quite a lot of memory, and
              iterating over all values equal to a given key (through the
              return value of the standard
              library's <function>equal_range</function>) can be
              expensive.
            </para>
          </listitem>

          <listitem>
            <para>
              The primary key is stored multiply; this uses more memory.
            </para>
          </listitem>

          <listitem>
            <para>
              Finally, the interface of this design excludes several
              useful underlying data structures. Of all the unordered
              self-organizing data structures, practically only
              collision-chaining hash tables can (efficiently) guarantee
              that equivalent-key values are stored consecutively.
            </para>
          </listitem>
        </orderedlist>

        <para>
          The above reasons hold even when the ratio of secondary keys to
          primary keys (or average number of identical keys) is small, but
          when it is large, there are more severe problems:
        </para>

        <orderedlist>
          <listitem>
            <para>
              The underlying data structures order the links inside each
              embedded linked-lists according to their internal
              considerations, which effectively means that each of the
              links is unordered. Irrespective of the underlying data
              structure, searching for a specific value can degrade to
              linear complexity.
            </para>
          </listitem>

          <listitem>
            <para>
              Similarly to the above point, it is impossible to apply
              to the secondary keys considerations that apply to primary
              keys. For example, it is not possible to maintain secondary
              keys by sorted order.
            </para>
          </listitem>

          <listitem>
            <para>
              While the interface "understands" that all equivalent-key
              values constitute a distinct list (through
              <function>equal_range</function>), the underlying data
              structure typically does not. This means that operations such
              as erasing from a tree-based container all values whose keys
              are equivalent to a a given key can be super-linear in the
              size of the tree; this is also true also for several other
              operations that target a specific list.
            </para>
          </listitem>

        </orderedlist>

        <para>
          In this library, all associative containers map
          (or store) unique-key values. One can (1) map primary keys to
          secondary associative-containers (containers of
          secondary keys) or non-associative containers (2) map identical
          keys to a size-type representing the number of times they
          occur, or (3) any combination of (1) and (2). Instead of
          allowing multiple equivalent-key values, this library
          supplies associative containers based on underlying
          data structures that are suitable as secondary
          associative-containers.
        </para>

        <para>
          In the figure below, labels A and B show the equivalent
          underlying data structures in this library, as mapped to the
          first graphic above. Labels A and B, respectively. Each shaded
          box represents some size-type or secondary
          associative-container.
        </para>

        <figure>
          <title>Non-unique Mapping Containers</title>
          <mediaobject>
            <imageobject>
              <imagedata align="center" format="PNG" scale="100"
                         fileref="../images/pbds_embedded_lists_3.png"/>
            </imageobject>
            <textobject>
              <phrase>Non-unique Mapping Containers</phrase>
            </textobject>
          </mediaobject>
        </figure>

        <para>
          In the first example above, then, one would use an associative
          container mapping each user to an associative container which
          maps each application id to a start time (see
          <filename>example/basic_multimap.cc</filename>); in the second
          example, one would use an associative container mapping
          each <classname>int</classname> to some size-type indicating the
          number of times it logically occurs
          (see <filename>example/basic_multiset.cc</filename>.
        </para>

        <para>
          See the discussion in list-based container types for containers
          especially suited as secondary associative-containers.
        </para>
      </section>

    </section> <!-- map and set semantics -->

    <section xml:id="pbds.design.concepts.iterator_semantics">
      <info><title>Iterator Semantics</title></info>

      <section xml:id="concepts.iterator_semantics.point_and_range">
        <info><title>Point and Range Iterators</title></info>

        <para>
          Iterator concepts are bifurcated in this design, and are
          comprised of point-type and range-type iteration.
        </para>

        <para>
          A point-type iterator is an iterator that refers to a specific
          element as returned through an
          associative-container's <function>find</function> method.
        </para>

        <para>
          A range-type iterator is an iterator that is used to go over a
          sequence of elements, as returned by a container's
          <function>find</function> method.
        </para>

        <para>
          A point-type method is a method that
          returns a point-type iterator; a range-type method is a method
          that returns a range-type iterator.
        </para>

        <para>For most containers, these types are synonymous; for
        self-organizing containers, such as hash-based containers or
        priority queues, these are inherently different (in any
        implementation, including that of C++ standard library
        components), but in this design, it is made explicit. They are
        distinct types.
        </para>
      </section>


      <section xml:id="concepts.iterator_semantics.both">
        <info><title>Distinguishing Point and Range Iterators</title></info>

        <para>When using this library, is necessary to differentiate
        between two types of methods and iterators: point-type methods and
        iterators, and range-type methods and iterators. Each associative
        container's interface includes the methods:</para>
        <programlisting>
          point_const_iterator
          find(const_key_reference r_key) const;

          point_iterator
          find(const_key_reference r_key);

          std::pair&lt;point_iterator,bool&gt;
          insert(const_reference r_val);
        </programlisting>

        <para>The relationship between these iterator types varies between
        container types. The figure below
        shows the most general invariant between point-type and
        range-type iterators: In <emphasis>A</emphasis> <literal>iterator</literal>, can
        always be converted to <literal>point_iterator</literal>. In <emphasis>B</emphasis>
        shows invariants for order-preserving containers: point-type
        iterators are synonymous with range-type iterators.
        Orthogonally,  <emphasis>C</emphasis>shows invariants for "set"
        containers: iterators are synonymous with const iterators.</para>

        <figure>
          <title>Point Iterator Hierarchy</title>
          <mediaobject>
            <imageobject>
              <imagedata align="center" format="PNG" scale="100"
                         fileref="../images/pbds_point_iterator_hierarchy.png"/>
            </imageobject>
            <textobject>
              <phrase>Point Iterator Hierarchy</phrase>
            </textobject>
          </mediaobject>
        </figure>


        <para>Note that point-type iterators in self-organizing containers
        (hash-based associative containers) lack movement
        operators, such as <literal>operator++</literal> - in fact, this
        is the reason why this library differentiates from the standard C++ librarys
        design on this point.</para>

        <para>Typically, one can determine an iterator's movement
        capabilities using
        <literal>std::iterator_traits&lt;It&gt;iterator_category</literal>,
        which is a <literal>struct</literal> indicating the iterator's
        movement capabilities. Unfortunately, none of the standard predefined
        categories reflect a pointer's <emphasis>not</emphasis> having any
        movement capabilities whatsoever. Consequently,
        <literal>pb_ds</literal> adds a type
        <literal>trivial_iterator_tag</literal> (whose name is taken from
        a concept in C++ standardese, which is the category of iterators
        with no movement capabilities.) All other standard C++ library
        tags, such as <literal>forward_iterator_tag</literal> retain their
        common use.</para>

      </section>

      <section xml:id="pbds.design.concepts.invalidation">
        <info><title>Invalidation Guarantees</title></info>
        <para>
          If one manipulates a container object, then iterators previously
          obtained from it can be invalidated. In some cases a
          previously-obtained iterator cannot be de-referenced; in other cases,
          the iterator's next or previous element might have changed
          unpredictably. This corresponds exactly to the question whether a
          point-type or range-type iterator (see previous concept) is valid or
          not. In this design, one can query a container (in compile time) about
          its invalidation guarantees.
        </para>


        <para>
          Given three different types of associative containers, a modifying
          operation (in that example, <function>erase</function>) invalidated
          iterators in three different ways: the iterator of one container
          remained completely valid - it could be de-referenced and
          incremented; the iterator of a different container could not even be
          de-referenced; the iterator of the third container could be
          de-referenced, but its "next" iterator changed unpredictably.
        </para>

        <para>
          Distinguishing between find and range types allows fine-grained
          invalidation guarantees, because these questions correspond exactly
          to the question of whether point-type iterators and range-type
          iterators are valid. The graphic below shows tags corresponding to
          different types of invalidation guarantees.
        </para>

        <figure>
          <title>Invalidation Guarantee Tags Hierarchy</title>
          <mediaobject>
            <imageobject>
              <imagedata align="center" format="PDF" scale="75"
                         fileref="../images/pbds_invalidation_tag_hierarchy.pdf"/>
            </imageobject>
            <imageobject>
              <imagedata align="center" format="PNG" scale="100"
                         fileref="../images/pbds_invalidation_tag_hierarchy.png"/>
            </imageobject>
            <textobject>
              <phrase>Invalidation Guarantee Tags Hierarchy</phrase>
            </textobject>
          </mediaobject>
        </figure>

        <itemizedlist>
          <listitem>
            <para>
              <classname>basic_invalidation_guarantee</classname>
              corresponds to a basic guarantee that a point-type iterator,
              a found pointer, or a found reference, remains valid as long
              as the container object is not modified.
            </para>
          </listitem>

          <listitem>
            <para>
              <classname>point_invalidation_guarantee</classname>
              corresponds to a guarantee that a point-type iterator, a
              found pointer, or a found reference, remains valid even if
              the container object is modified.
            </para>
          </listitem>

          <listitem>
            <para>
              <classname>range_invalidation_guarantee</classname>
              corresponds to a guarantee that a range-type iterator remains
              valid even if the container object is modified.
            </para>
          </listitem>
        </itemizedlist>

        <para>To find the invalidation guarantee of a
        container, one can use</para>
        <programlisting>
          typename container_traits&lt;Cntnr&gt;::invalidation_guarantee
        </programlisting>

        <para>Note that this hierarchy corresponds to the logic it
        represents: if a container has range-invalidation guarantees,
        then it must also have find invalidation guarantees;
        correspondingly, its invalidation guarantee (in this case
        <classname>range_invalidation_guarantee</classname>)
        can be cast to its base class (in this case <classname>point_invalidation_guarantee</classname>).
        This means that this this hierarchy can be used easily using
        standard metaprogramming techniques, by specializing on the
        type of <literal>invalidation_guarantee</literal>.</para>

        <para>
          These types of problems were addressed, in a more general
          setting, in <xref linkend="biblio.meyers96more"/> - Item 2. In
          our opinion, an invalidation-guarantee hierarchy would solve
          these problems in all container types - not just associative
          containers.
        </para>

      </section>
    </section> <!-- iterator semantics -->

    <section xml:id="pbds.design.concepts.genericity">
      <info><title>Genericity</title></info>

      <para>
        The design attempts to address the following problem of
        data-structure genericity. When writing a function manipulating
        a generic container object, what is the behavior of the object?
        Suppose one writes
      </para>
      <programlisting>
        template&lt;typename Cntnr&gt;
        void
        some_op_sequence(Cntnr &amp;r_container)
        {
        ...
        }
      </programlisting>

      <para>
        then one needs to address the following questions in the body
        of <function>some_op_sequence</function>:
      </para>

      <itemizedlist>
        <listitem>
          <para>
            Which types and methods does <literal>Cntnr</literal> support?
            Containers based on hash tables can be queries for the
            hash-functor type and object; this is meaningless for tree-based
            containers. Containers based on trees can be split, joined, or
            can erase iterators and return the following iterator; this
            cannot be done by hash-based containers.
          </para>
        </listitem>

        <listitem>
          <para>
            What are the exception and invalidation guarantees
            of <literal>Cntnr</literal>? A container based on a probing
            hash-table invalidates all iterators when it is modified; this
            is not the case for containers based on node-based
            trees. Containers based on a node-based tree can be split or
            joined without exceptions; this is not the case for containers
            based on vector-based trees.
          </para>
        </listitem>

        <listitem>
          <para>
            How does the container maintain its elements? Tree-based and
            Trie-based containers store elements by key order; others,
            typically, do not. A container based on a splay trees or lists
            with update policies "cache" "frequently accessed" elements;
            containers based on most other underlying data structures do
            not.
          </para>
        </listitem>
        <listitem>
          <para>
            How does one query a container about characteristics and
            capabilities? What is the relationship between two different
            data structures, if anything?
          </para>
        </listitem>
      </itemizedlist>

      <para>The remainder of this section explains these issues in
      detail.</para>


      <section xml:id="concepts.genericity.tag">
        <info><title>Tag</title></info>
        <para>
          Tags are very useful for manipulating generic types. For example, if
          <literal>It</literal> is an iterator class, then <literal>typename
          It::iterator_category</literal> or <literal>typename
          std::iterator_traits&lt;It&gt;::iterator_category</literal> will
          yield its category, and <literal>typename
          std::iterator_traits&lt;It&gt;::value_type</literal> will yield its
          value type.
        </para>

        <para>
          This library contains a container tag hierarchy corresponding to the
          diagram below.
        </para>

        <figure>
          <title>Container Tag Hierarchy</title>
          <mediaobject>
            <imageobject>
              <imagedata align="center" format="PDF" scale="75"
                         fileref="../images/pbds_container_tag_hierarchy.pdf"/>
            </imageobject>
            <imageobject>
              <imagedata align="center" format="PNG" scale="100"
                         fileref="../images/pbds_container_tag_hierarchy.png"/>
            </imageobject>
            <textobject>
              <phrase>Container Tag Hierarchy</phrase>
            </textobject>
          </mediaobject>
        </figure>

        <para>
          Given any container <type>Cntnr</type>, the tag of
          the underlying data structure can be found via <literal>typename
          Cntnr::container_category</literal>.
        </para>

      </section> <!-- tag -->

      <section xml:id="concepts.genericity.traits">
        <info><title>Traits</title></info>
        <para></para>

        <para>Additionally, a traits mechanism can be used to query a
        container type for its attributes. Given any container
        <literal>Cntnr</literal>, then <literal>&lt;Cntnr&gt;</literal>
        is a traits class identifying the properties of the
        container.</para>

        <para>To find if a container can throw when a key is erased (which
        is true for vector-based trees, for example), one can
        use
        </para>
        <programlisting>container_traits&lt;Cntnr&gt;::erase_can_throw</programlisting>

        <para>
          Some of the definitions in <classname>container_traits</classname>
          are dependent on other
          definitions. If <classname>container_traits&lt;Cntnr&gt;::order_preserving</classname>
          is <constant>true</constant> (which is the case for containers
          based on trees and tries), then the container can be split or
          joined; in this
          case, <classname>container_traits&lt;Cntnr&gt;::split_join_can_throw</classname>
          indicates whether splits or joins can throw exceptions (which is
          true for vector-based trees);
          otherwise <classname>container_traits&lt;Cntnr&gt;::split_join_can_throw</classname>
          will yield a compilation error. (This is somewhat similar to a
          compile-time version of the COM model).
        </para>

      </section> <!-- traits -->

    </section> <!-- genericity -->
  </section> <!-- concepts -->

  <section xml:id="pbds.design.container">
    <info><title>By Container</title></info>

    <!-- hash -->
    <section xml:id="pbds.design.container.hash">
      <info><title>hash</title></info>

      <!--

// hash policies
/// general terms / background
/// range hashing policies
/// ranged-hash policies
/// implementation

// resize policies
/// general
/// size policies
/// trigger policies
/// implementation

// policy interactions
/// probe/size/trigger
/// hash/trigger
/// eq/hash/storing hash values
/// size/load-check trigger
      -->
      <section xml:id="container.hash.interface">
        <info><title>Interface</title></info>



        <para>
          The collision-chaining hash-based container has the
        following declaration.</para>
        <programlisting>
          template&lt;
          typename Key,
          typename Mapped,
          typename Hash_Fn = std::hash&lt;Key&gt;,
          typename Eq_Fn = std::equal_to&lt;Key&gt;,
          typename Comb_Hash_Fn =  direct_mask_range_hashing&lt;&gt;
          typename Resize_Policy = default explained below.
          bool Store_Hash = false,
          typename Allocator = std::allocator&lt;char&gt; &gt;
          class cc_hash_table;
        </programlisting>

        <para>The parameters have the following meaning:</para>

        <orderedlist>
          <listitem><para><classname>Key</classname> is the key type.</para></listitem>

          <listitem><para><classname>Mapped</classname> is the mapped-policy.</para></listitem>

          <listitem><para><classname>Hash_Fn</classname> is a key hashing functor.</para></listitem>

          <listitem><para><classname>Eq_Fn</classname> is a key equivalence functor.</para></listitem>

          <listitem><para><classname>Comb_Hash_Fn</classname> is a range-hashing_functor;
          it describes how to translate hash values into positions
          within the table. </para></listitem>

          <listitem><para><classname>Resize_Policy</classname> describes how a container object
          should change its internal size. </para></listitem>

          <listitem><para><classname>Store_Hash</classname> indicates whether the hash value
          should be stored with each entry. </para></listitem>

          <listitem><para><classname>Allocator</classname> is an allocator
          type.</para></listitem>
        </orderedlist>

        <para>The probing hash-based container has the following
        declaration.</para>
        <programlisting>
          template&lt;
          typename Key,
          typename Mapped,
          typename Hash_Fn = std::hash&lt;Key&gt;,
          typename Eq_Fn = std::equal_to&lt;Key&gt;,
          typename Comb_Probe_Fn = direct_mask_range_hashing&lt;&gt;
          typename Probe_Fn = default explained below.
          typename Resize_Policy = default explained below.
          bool Store_Hash = false,
          typename Allocator =  std::allocator&lt;char&gt; &gt;
          class gp_hash_table;
        </programlisting>

        <para>The parameters are identical to those of the
        collision-chaining container, except for the following.</para>

        <orderedlist>
          <listitem><para><classname>Comb_Probe_Fn</classname> describes how to transform a probe
          sequence into a sequence of positions within the table.</para></listitem>

          <listitem><para><classname>Probe_Fn</classname> describes a probe sequence policy.</para></listitem>
        </orderedlist>

        <para>Some of the default template values depend on the values of
        other parameters, and are explained below.</para>

      </section>
      <section xml:id="container.hash.details">
        <info><title>Details</title></info>

        <section xml:id="container.hash.details.hash_policies">
          <info><title>Hash Policies</title></info>

          <section xml:id="details.hash_policies.general">
            <info><title>General</title></info>

            <para>Following is an explanation of some functions which hashing
            involves. The graphic below illustrates the discussion.</para>

            <figure>
              <title>Hash functions, ranged-hash functions, and
              range-hashing functions</title>
              <mediaobject>
                <imageobject>
                  <imagedata align="center" format="PNG" scale="100"
                             fileref="../images/pbds_hash_ranged_hash_range_hashing_fns.png"/>
                </imageobject>
                <textobject>
                  <phrase>Hash functions, ranged-hash functions, and
                  range-hashing functions</phrase>
                </textobject>
              </mediaobject>
            </figure>
            
            <para>Let U be a domain (e.g., the integers, or the
            strings of 3 characters). A hash-table algorithm needs to map
            elements of U "uniformly" into the range [0,..., m -
            1] (where m is a non-negative integral value, and
            is, in general, time varying). I.e., the algorithm needs
            a ranged-hash function</para>

            <para>
              f : U × Z<subscript>+</subscript> → Z<subscript>+</subscript>
            </para>

            <para>such that for any u in U ,</para>

            <para>0 ≤ f(u, m) ≤ m - 1</para>

            <para>and which has "good uniformity" properties (say
            <xref linkend="biblio.knuth98sorting"/>.)
            One
            common solution is to use the composition of the hash
            function</para>

            <para>h : U → Z<subscript>+</subscript> ,</para>

            <para>which maps elements of U into the non-negative
            integrals, and</para>

            <para>g : Z<subscript>+</subscript> × Z<subscript>+</subscript> →
            Z<subscript>+</subscript>,</para>

            <para>which maps a non-negative hash value, and a non-negative
            range upper-bound into a non-negative integral in the range
            between 0 (inclusive) and the range upper bound (exclusive),
            i.e., for any r in Z<subscript>+</subscript>,</para>

            <para>0 ≤ g(r, m) ≤ m - 1</para>


            <para>The resulting ranged-hash function, is</para>

            <!-- ranged_hash_composed_of_hash_and_range_hashing -->
            <equation>
              <title>Ranged Hash Function</title>
              <mathphrase>
                f(u , m) = g(h(u), m)
              </mathphrase>
            </equation>

            <para>From the above, it is obvious that given g and
            h, f can always be composed (however the converse
            is not true). The standard's hash-based containers allow specifying
            a hash function, and use a hard-wired range-hashing function;
            the ranged-hash function is implicitly composed.</para>

            <para>The above describes the case where a key is to be mapped
            into a single position within a hash table, e.g.,
            in a collision-chaining table. In other cases, a key is to be
            mapped into a sequence of positions within a table,
            e.g., in a probing table. Similar terms apply in this
            case: the table requires a ranged probe function,
            mapping a key into a sequence of positions withing the table.
            This is typically achieved by composing a hash function
            mapping the key into a non-negative integral type, a
            probe function transforming the hash value into a
            sequence of hash values, and a range-hashing function
            transforming the sequence of hash values into a sequence of
            positions.</para>

          </section>

          <section xml:id="details.hash_policies.range">
            <info><title>Range Hashing</title></info>

            <para>Some common choices for range-hashing functions are the
            division, multiplication, and middle-square methods (<xref linkend="biblio.knuth98sorting"/>), defined
            as</para>

            <equation>
              <title>Range-Hashing, Division Method</title>
              <mathphrase>
                g(r, m) = r mod m
              </mathphrase>
            </equation>



            <para>g(r, m) = ⌈ u/v ( a r mod v ) ⌉</para>

            <para>and</para>

            <para>g(r, m) = ⌈ u/v ( r<superscript>2</superscript> mod v ) ⌉</para>

            <para>respectively, for some positive integrals u and
            v (typically powers of 2), and some a. Each of
            these range-hashing functions works best for some different
            setting.</para>

            <para>The division method (see above) is a
            very common choice. However, even this single method can be
            implemented in two very different ways. It is possible to
            implement using the low
            level % (modulo) operation (for any m), or the
            low level &amp; (bit-mask) operation (for the case where
            m is a power of 2), i.e.,</para>

            <equation>
              <title>Division via Prime Modulo</title>
              <mathphrase>
                g(r, m) = r % m
              </mathphrase>
            </equation>

            <para>and</para>

            <equation>
              <title>Division via Bit Mask</title>
              <mathphrase>
                g(r, m) = r &amp; m - 1, (with m =
                2<superscript>k</superscript> for some k)
              </mathphrase>
            </equation>


            <para>respectively.</para>

            <para>The % (modulo) implementation has the advantage that for
            m a prime far from a power of 2, g(r, m) is
            affected by all the bits of r (minimizing the chance of
            collision). It has the disadvantage of using the costly modulo
            operation. This method is hard-wired into SGI's implementation
            .</para>

            <para>The &amp; (bit-mask) implementation has the advantage of
            relying on the fast bit-wise and operation. It has the
            disadvantage that for g(r, m) is affected only by the
            low order bits of r. This method is hard-wired into
            Dinkumware's implementation.</para>


          </section>

          <section xml:id="details.hash_policies.ranged">
            <info><title>Ranged Hash</title></info>

            <para>In cases it is beneficial to allow the
            client to directly specify a ranged-hash hash function. It is
            true, that the writer of the ranged-hash function cannot rely
            on the values of m having specific numerical properties
            suitable for hashing (in the sense used in <xref linkend="biblio.knuth98sorting"/>), since
            the values of m are determined by a resize policy with
            possibly orthogonal considerations.</para>

            <para>There are two cases where a ranged-hash function can be
            superior. The firs is when using perfect hashing: the
            second is when the values of m can be used to estimate
            the "general" number of distinct values required. This is
            described in the following.</para>

            <para>Let</para>

            <para>
              s = [ s<subscript>0</subscript>,..., s<subscript>t - 1</subscript>]
            </para>

            <para>be a string of t characters, each of which is from
            domain S. Consider the following ranged-hash
            function:</para>
            <equation>
              <title>
                A Standard String Hash Function
              </title>
              <mathphrase>
                f<subscript>1</subscript>(s, m) = ∑ <subscript>i =
                0</subscript><superscript>t - 1</superscript> s<subscript>i</subscript> a<superscript>i</superscript> mod m
              </mathphrase>
            </equation>
            

            <para>where a is some non-negative integral value. This is
            the standard string-hashing function used in SGI's
            implementation (with a = 5). Its advantage is that
            it takes into account all of the characters of the string.</para>

            <para>Now assume that s is the string representation of a
            of a long DNA sequence (and so S = {'A', 'C', 'G',
            'T'}). In this case, scanning the entire string might be
            prohibitively expensive. A possible alternative might be to use
            only the first k characters of the string, where</para>

            <para>|S|<superscript>k</superscript> ≥ m ,</para>

            <para>i.e., using the hash function</para>

            <equation>
              <title>
                Only k String DNA Hash
              </title>
              <mathphrase>
                f<subscript>2</subscript>(s, m) = ∑ <subscript>i
                = 0</subscript><superscript>k - 1</superscript> s<subscript>i</subscript> a<superscript>i</superscript> mod m 
              </mathphrase>
            </equation>

            <para>requiring scanning over only</para>

            <para>k = log<subscript>4</subscript>( m )</para>

            <para>characters.</para>

            <para>Other more elaborate hash-functions might scan k
            characters starting at a random position (determined at each
            resize), or scanning k random positions (determined at
            each resize), i.e., using</para>

            <para>f<subscript>3</subscript>(s, m) = ∑ <subscript>i =
            r</subscript>0<superscript>r<subscript>0</subscript> + k - 1</superscript> s<subscript>i</subscript>
            a<superscript>i</superscript> mod m ,</para>

            <para>or</para>

            <para>f<subscript>4</subscript>(s, m) = ∑ <subscript>i = 0</subscript><superscript>k -
            1</superscript> s<subscript>r</subscript>i a<superscript>r<subscript>i</subscript></superscript> mod
            m ,</para>

            <para>respectively, for r<subscript>0</subscript>,..., r<subscript>k-1</subscript>
            each in the (inclusive) range [0,...,t-1].</para>

            <para>It should be noted that the above functions cannot be
            decomposed as per a ranged hash composed of hash and range hashing.</para>


          </section>

          <section xml:id="details.hash_policies.implementation">
            <info><title>Implementation</title></info>

            <para>This sub-subsection describes the implementation of
            the above in this library. It first explains range-hashing
            functions in collision-chaining tables, then ranged-hash
            functions in collision-chaining tables, then probing-based
            tables, and finally lists the relevant classes in this
            library.</para>

            <section xml:id="hash_policies.implementation.collision-chaining">
              <info><title>
                Range-Hashing and Ranged-Hashes in Collision-Chaining Tables
              </title></info>


              <para><classname>cc_hash_table</classname> is
              parametrized by <classname>Hash_Fn</classname> and <classname>Comb_Hash_Fn</classname>, a
              hash functor and a combining hash functor, respectively.</para>

              <para>In general, <classname>Comb_Hash_Fn</classname> is considered a
              range-hashing functor. <classname>cc_hash_table</classname>
              synthesizes a ranged-hash function from <classname>Hash_Fn</classname> and
              <classname>Comb_Hash_Fn</classname>. The figure below shows an <classname>insert</classname> sequence
              diagram for this case. The user inserts an element (point A),
              the container transforms the key into a non-negative integral
              using the hash functor (points B and C), and transforms the
              result into a position using the combining functor (points D
              and E).</para>

              <figure>
                <title>Insert hash sequence diagram</title>
                <mediaobject>
                  <imageobject>
                    <imagedata align="center" format="PNG" scale="100"
                               fileref="../images/pbds_hash_range_hashing_seq_diagram.png"/>
                  </imageobject>
                  <textobject>
                    <phrase>Insert hash sequence diagram</phrase>
                  </textobject>
                </mediaobject>
              </figure>
              
              <para>If <classname>cc_hash_table</classname>'s
              hash-functor, <classname>Hash_Fn</classname> is instantiated by <classname>null_type</classname> , then <classname>Comb_Hash_Fn</classname> is taken to be
              a ranged-hash function. The graphic below shows an <function>insert</function> sequence
              diagram. The user inserts an element (point A), the container
              transforms the key into a position using the combining functor
              (points B and C).</para>

              <figure>
                <title>Insert hash sequence diagram with a null policy</title>
                <mediaobject>
                  <imageobject>
                    <imagedata align="center" format="PNG" scale="100"
                               fileref="../images/pbds_hash_range_hashing_seq_diagram2.png"/>
                  </imageobject>
                  <textobject>
                    <phrase>Insert hash sequence diagram with a null policy</phrase>
                  </textobject>
                </mediaobject>
              </figure>
              
            </section>

            <section xml:id="hash_policies.implementation.probe">
              <info><title>
                Probing tables
              </title></info>
              <para><classname>gp_hash_table</classname> is parametrized by
              <classname>Hash_Fn</classname>, <classname>Probe_Fn</classname>,
              and <classname>Comb_Probe_Fn</classname>. As before, if
              <classname>Hash_Fn</classname> and <classname>Probe_Fn</classname>
              are both <classname>null_type</classname>, then
              <classname>Comb_Probe_Fn</classname> is a ranged-probe
              functor. Otherwise, <classname>Hash_Fn</classname> is a hash
              functor, <classname>Probe_Fn</classname> is a functor for offsets
              from a hash value, and <classname>Comb_Probe_Fn</classname>
              transforms a probe sequence into a sequence of positions within
              the table.</para>

            </section>

            <section xml:id="hash_policies.implementation.predefined">
              <info><title>
                Pre-Defined Policies
              </title></info>

              <para>This library contains some pre-defined classes
              implementing range-hashing and probing functions:</para>

              <orderedlist>
                <listitem><para><classname>direct_mask_range_hashing</classname>
                and <classname>direct_mod_range_hashing</classname>
                are range-hashing functions based on a bit-mask and a modulo
                operation, respectively.</para></listitem>

                <listitem><para><classname>linear_probe_fn</classname>, and
                <classname>quadratic_probe_fn</classname> are
                a linear probe and a quadratic probe function,
                respectively.</para></listitem>
              </orderedlist>

              <para>
                The graphic below shows the relationships.
              </para>
              <figure>
                <title>Hash policy class diagram</title>
                <mediaobject>
                  <imageobject>
                    <imagedata align="center" format="PNG" scale="100"
                               fileref="../images/pbds_hash_policy_cd.png"/>
                  </imageobject>
                  <textobject>
                    <phrase>Hash policy class diagram</phrase>
                  </textobject>
                </mediaobject>
              </figure>


            </section>

          </section> <!-- impl -->

        </section>

        <section xml:id="container.hash.details.resize_policies">
          <info><title>Resize Policies</title></info>

          <section xml:id="resize_policies.general">
            <info><title>General</title></info>

            <para>Hash-tables, as opposed to trees, do not naturally grow or
            shrink. It is necessary to specify policies to determine how
            and when a hash table should change its size. Usually, resize
            policies can be decomposed into orthogonal policies:</para>

            <orderedlist>
              <listitem><para>A size policy indicating how a hash table
              should grow (e.g., it should multiply by powers of
              2).</para></listitem>

              <listitem><para>A trigger policy indicating when a hash
              table should grow (e.g., a load factor is
              exceeded).</para></listitem>
            </orderedlist>

          </section>

          <section xml:id="resize_policies.size">
            <info><title>Size Policies</title></info>


            <para>Size policies determine how a hash table changes size. These
            policies are simple, and there are relatively few sensible
            options. An exponential-size policy (with the initial size and
            growth factors both powers of 2) works well with a mask-based
            range-hashing function, and is the
            hard-wired policy used by Dinkumware. A
            prime-list based policy works well with a modulo-prime range
            hashing function and is the hard-wired policy used by SGI's
            implementation.</para>

          </section>

          <section xml:id="resize_policies.trigger">
            <info><title>Trigger Policies</title></info>

            <para>Trigger policies determine when a hash table changes size.
            Following is a description of two policies: load-check
            policies, and collision-check policies.</para>

            <para>Load-check policies are straightforward. The user specifies
            two factors, Α<subscript>min</subscript> and
            Α<subscript>max</subscript>, and the hash table maintains the
            invariant that</para>

            <para>Α<subscript>min</subscript> ≤ (number of
            stored elements) / (hash-table size) ≤
            Α<subscript>max</subscript><remark>load factor min max</remark></para>

            <para>Collision-check policies work in the opposite direction of
            load-check policies. They focus on keeping the number of
            collisions moderate and hoping that the size of the table will
            not grow very large, instead of keeping a moderate load-factor
            and hoping that the number of collisions will be small. A
            maximal collision-check policy resizes when the longest
            probe-sequence grows too large.</para>

            <para>Consider the graphic below. Let the size of the hash table
            be denoted by m, the length of a probe sequence be denoted by k,
            and some load factor be denoted by Α. We would like to
            calculate the minimal length of k, such that if there were Α
            m elements in the hash table, a probe sequence of length k would
            be found with probability at most 1/m.</para>

            <figure>
              <title>Balls and bins</title>
              <mediaobject>
                <imageobject>
                  <imagedata align="center" format="PNG" scale="100"
                             fileref="../images/pbds_balls_and_bins.png"/>
                </imageobject>
                <textobject>
                  <phrase>Balls and bins</phrase>
                </textobject>
              </mediaobject>
            </figure>

            <para>Denote the probability that a probe sequence of length
            k appears in bin i by p<subscript>i</subscript>, the
            length of the probe sequence of bin i by
            l<subscript>i</subscript>, and assume uniform distribution. Then</para>



            <equation>
              <title>
                Probability of Probe Sequence of Length k
              </title>
              <mathphrase>
                p<subscript>1</subscript> = 
              </mathphrase>
            </equation>

            <para>P(l<subscript>1</subscript> ≥ k) =</para>

            <para>
              P(l<subscript>1</subscript> ≥ α ( 1 + k / α - 1) ≤ (a)
            </para>

            <para>
              e ^ ( - ( α ( k / α - 1 )<superscript>2</superscript> ) /2)
            </para>

            <para>where (a) follows from the Chernoff bound (<xref linkend="biblio.motwani95random"/>). To
            calculate the probability that some bin contains a probe
            sequence greater than k, we note that the
            l<subscript>i</subscript> are negatively-dependent
            (<xref linkend="biblio.dubhashi98neg"/>)
            . Let
            I(.) denote the indicator function. Then</para>

            <equation>
              <title>
                Probability Probe Sequence in Some Bin
              </title>
              <mathphrase>
                P( exists<subscript>i</subscript> l<subscript>i</subscript> ≥ k ) = 
              </mathphrase>
            </equation>

            <para>P ( ∑ <subscript>i = 1</subscript><superscript>m</superscript>
            I(l<subscript>i</subscript> ≥ k) ≥ 1 ) =</para>

            <para>P ( ∑ <subscript>i = 1</subscript><superscript>m</superscript> I (
            l<subscript>i</subscript> ≥ k ) ≥ m p<subscript>1</subscript> ( 1 + 1 / (m
            p<subscript>1</subscript>) - 1 ) ) ≤ (a)</para>

            <para>e ^ ( ( - m p<subscript>1</subscript> ( 1 / (m p<subscript>1</subscript>)
            - 1 ) <superscript>2</superscript> ) / 2 ) ,</para>

            <para>where (a) follows from the fact that the Chernoff bound can
            be applied to negatively-dependent variables (<xref
            linkend="biblio.dubhashi98neg"/>). Inserting the first probability
            equation into the second one, and equating with 1/m, we
            obtain</para>


            <para>k ~ √ ( 2 α ln 2 m ln(m) )
            ) .</para>

          </section>

          <section xml:id="resize_policies.impl">
            <info><title>Implementation</title></info>

            <para>This sub-subsection describes the implementation of the
            above in this library. It first describes resize policies and
            their decomposition into trigger and size policies, then
            describes pre-defined classes, and finally discusses controlled
            access the policies' internals.</para>

            <section xml:id="resize_policies.impl.decomposition">
              <info><title>Decomposition</title></info>


              <para>Each hash-based container is parametrized by a
              <classname>Resize_Policy</classname> parameter; the container derives
              <classname>public</classname>ly from <classname>Resize_Policy</classname>. For
              example:</para>
              <programlisting>
                cc_hash_table&lt;typename Key,
                typename Mapped,
                ...
                typename Resize_Policy
                ...&gt; : public Resize_Policy
              </programlisting>

              <para>As a container object is modified, it continuously notifies
              its <classname>Resize_Policy</classname> base of internal changes
              (e.g., collisions encountered and elements being
              inserted). It queries its <classname>Resize_Policy</classname> base whether
              it needs to be resized, and if so, to what size.</para>

              <para>The graphic below shows a (possible) sequence diagram
              of an insert operation. The user inserts an element; the hash
              table notifies its resize policy that a search has started
              (point A); in this case, a single collision is encountered -
              the table notifies its resize policy of this (point B); the
              container finally notifies its resize policy that the search
              has ended (point C); it then queries its resize policy whether
              a resize is needed, and if so, what is the new size (points D
              to G); following the resize, it notifies the policy that a
              resize has completed (point H); finally, the element is
              inserted, and the policy notified (point I).</para>

              <figure>
                <title>Insert resize sequence diagram</title>
                <mediaobject>
                  <imageobject>
                    <imagedata align="center" format="PNG" scale="100"
                               fileref="../images/pbds_insert_resize_sequence_diagram1.png"/>
                  </imageobject>
                  <textobject>
                    <phrase>Insert resize sequence diagram</phrase>
                  </textobject>
                </mediaobject>
              </figure>


              <para>In practice, a resize policy can be usually orthogonally
              decomposed to a size policy and a trigger policy. Consequently,
              the library contains a single class for instantiating a resize
              policy: <classname>hash_standard_resize_policy</classname>
              is parametrized by <classname>Size_Policy</classname> and
              <classname>Trigger_Policy</classname>, derives <classname>public</classname>ly from
              both, and acts as a standard delegate (<xref linkend="biblio.gof"/>)
              to these policies.</para>

              <para>The two graphics immediately below show sequence diagrams
              illustrating the interaction between the standard resize policy
              and its trigger and size policies, respectively.</para>

              <figure>
                <title>Standard resize policy trigger sequence
                diagram</title>
                <mediaobject>
                  <imageobject>
                    <imagedata align="center" format="PNG" scale="100"
                               fileref="../images/pbds_insert_resize_sequence_diagram2.png"/>
                  </imageobject>
                  <textobject>
                    <phrase>Standard resize policy trigger sequence
                    diagram</phrase>
                  </textobject>
                </mediaobject>
              </figure>

              <figure>
                <title>Standard resize policy size sequence
                diagram</title>
                <mediaobject>
                  <imageobject>
                    <imagedata align="center" format="PNG" scale="100"
                               fileref="../images/pbds_insert_resize_sequence_diagram3.png"/>
                  </imageobject>
                  <textobject>
                    <phrase>Standard resize policy size sequence
                    diagram</phrase>
                  </textobject>
                </mediaobject>
              </figure>


            </section>

            <section xml:id="resize_policies.impl.predefined">
              <info><title>Predefined Policies</title></info>
              <para>The library includes the following
              instantiations of size and trigger policies:</para>

              <orderedlist>
                <listitem><para><classname>hash_load_check_resize_trigger</classname>
                implements a load check trigger policy.</para></listitem>

                <listitem><para><classname>cc_hash_max_collision_check_resize_trigger</classname>
                implements a collision check trigger policy.</para></listitem>

                <listitem><para><classname>hash_exponential_size_policy</classname>
                implements an exponential-size policy (which should be used
                with mask range hashing).</para></listitem>

                <listitem><para><classname>hash_prime_size_policy</classname>
                implementing a size policy based on a sequence of primes
                (which should
                be used with mod range hashing</para></listitem>
              </orderedlist>

              <para>The graphic below gives an overall picture of the resize-related
              classes. <classname>basic_hash_table</classname>
              is parametrized by <classname>Resize_Policy</classname>, which it subclasses
              publicly. This class is currently instantiated only by <classname>hash_standard_resize_policy</classname>. 
              <classname>hash_standard_resize_policy</classname>
              itself is parametrized by <classname>Trigger_Policy</classname> and
              <classname>Size_Policy</classname>. Currently, <classname>Trigger_Policy</classname> is
              instantiated by <classname>hash_load_check_resize_trigger</classname>,
              or <classname>cc_hash_max_collision_check_resize_trigger</classname>;
              <classname>Size_Policy</classname> is instantiated by <classname>hash_exponential_size_policy</classname>,
              or <classname>hash_prime_size_policy</classname>.</para>

            </section>

            <section xml:id="resize_policies.impl.internals">
              <info><title>Controling Access to Internals</title></info>

              <para>There are cases where (controlled) access to resize
              policies' internals is beneficial. E.g., it is sometimes
              useful to query a hash-table for the table's actual size (as
              opposed to its <function>size()</function> - the number of values it
              currently holds); it is sometimes useful to set a table's
              initial size, externally resize it, or change load factors.</para>

              <para>Clearly, supporting such methods both decreases the
              encapsulation of hash-based containers, and increases the
              diversity between different associative-containers' interfaces.
              Conversely, omitting such methods can decrease containers'
              flexibility.</para>

              <para>In order to avoid, to the extent possible, the above
              conflict, the hash-based containers themselves do not address
              any of these questions; this is deferred to the resize policies,
              which are easier to change or replace. Thus, for example,
              neither <classname>cc_hash_table</classname> nor
              <classname>gp_hash_table</classname>
              contain methods for querying the actual size of the table; this
              is deferred to <classname>hash_standard_resize_policy</classname>.</para>

              <para>Furthermore, the policies themselves are parametrized by
              template arguments that determine the methods they support
              (
              <xref linkend="biblio.alexandrescu01modern"/>
              shows techniques for doing so). <classname>hash_standard_resize_policy</classname>
              is parametrized by <classname>External_Size_Access</classname> that
              determines whether it supports methods for querying the actual
              size of the table or resizing it. <classname>hash_load_check_resize_trigger</classname>
              is parametrized by <classname>External_Load_Access</classname> that
              determines whether it supports methods for querying or
              modifying the loads. <classname>cc_hash_max_collision_check_resize_trigger</classname>
              is parametrized by <classname>External_Load_Access</classname> that
              determines whether it supports methods for querying the
              load.</para>

              <para>Some operations, for example, resizing a container at
              run time, or changing the load factors of a load-check trigger
              policy, require the container itself to resize. As mentioned
              above, the hash-based containers themselves do not contain
              these types of methods, only their resize policies.
              Consequently, there must be some mechanism for a resize policy
              to manipulate the hash-based container. As the hash-based
              container is a subclass of the resize policy, this is done
              through virtual methods. Each hash-based container has a
              <classname>private</classname> <classname>virtual</classname> method:</para>
              <programlisting>
                virtual void
                do_resize
                (size_type new_size);
              </programlisting>

              <para>which resizes the container. Implementations of
              <classname>Resize_Policy</classname> can export public methods for resizing
              the container externally; these methods internally call
              <classname>do_resize</classname> to resize the table.</para>


            </section>

          </section>


        </section> <!-- resize policies -->

        <section xml:id="container.hash.details.policy_interaction">
          <info><title>Policy Interactions</title></info>
          <para>
          </para>
          <para>Hash-tables are unfortunately especially susceptible to
          choice of policies. One of the more complicated aspects of this
          is that poor combinations of good policies can form a poor
          container. Following are some considerations.</para>

          <section xml:id="policy_interaction.probesizetrigger">
            <info><title>probe/size/trigger</title></info>

            <para>Some combinations do not work well for probing containers.
            For example, combining a quadratic probe policy with an
            exponential size policy can yield a poor container: when an
            element is inserted, a trigger policy might decide that there
            is no need to resize, as the table still contains unused
            entries; the probe sequence, however, might never reach any of
            the unused entries.</para>

            <para>Unfortunately, this library cannot detect such problems at
            compilation (they are halting reducible). It therefore defines
            an exception class <classname>insert_error</classname> to throw an
            exception in this case.</para>

          </section>

          <section xml:id="policy_interaction.hashtrigger">
            <info><title>hash/trigger</title></info>

            <para>Some trigger policies are especially susceptible to poor
            hash functions. Suppose, as an extreme case, that the hash
            function transforms each key to the same hash value. After some
            inserts, a collision detecting policy will always indicate that
            the container needs to grow.</para>

            <para>The library, therefore, by design, limits each operation to
            one resize. For each <classname>insert</classname>, for example, it queries
            only once whether a resize is needed.</para>

          </section>

          <section xml:id="policy_interaction.eqstorehash">
            <info><title>equivalence functors/storing hash values/hash</title></info>

            <para><classname>cc_hash_table</classname> and
            <classname>gp_hash_table</classname> are
            parametrized by an equivalence functor and by a
            <classname>Store_Hash</classname> parameter. If the latter parameter is
            <classname>true</classname>, then the container stores with each entry
            a hash value, and uses this value in case of collisions to
            determine whether to apply a hash value. This can lower the
            cost of collision for some types, but increase the cost of
            collisions for other types.</para>

            <para>If a ranged-hash function or ranged probe function is
            directly supplied, however, then it makes no sense to store the
            hash value with each entry. This library's container will
            fail at compilation, by design, if this is attempted.</para>

          </section>

          <section xml:id="policy_interaction.sizeloadtrigger">
            <info><title>size/load-check trigger</title></info>

            <para>Assume a size policy issues an increasing sequence of sizes
            a, a q, a q<superscript>1</superscript>, a q<superscript>2</superscript>, ... For
            example, an exponential size policy might issue the sequence of
            sizes 8, 16, 32, 64, ...</para>

            <para>If a load-check trigger policy is used, with loads
            α<subscript>min</subscript> and α<subscript>max</subscript>,
            respectively, then it is a good idea to have:</para>

            <orderedlist>
              <listitem><para>α<subscript>max</subscript> ~ 1 / q</para></listitem>

              <listitem><para>α<subscript>min</subscript> &lt; 1 / (2 q)</para></listitem>
            </orderedlist>

            <para>This will ensure that the amortized hash cost of each
            modifying operation is at most approximately 3.</para>

            <para>α<subscript>min</subscript> ~ α<subscript>max</subscript> is, in
            any case, a bad choice, and α<subscript>min</subscript> &gt;
            α <subscript>max</subscript> is horrendous.</para>

          </section>

        </section>

      </section> <!-- details -->

    </section> <!-- hash -->

    <!-- tree -->
    <section xml:id="pbds.design.container.tree">
      <info><title>tree</title></info>

      <section xml:id="container.tree.interface">
        <info><title>Interface</title></info>

        <para>The tree-based container has the following declaration:</para>
        <programlisting>
          template&lt;
          typename Key,
          typename Mapped,
          typename Cmp_Fn = std::less&lt;Key&gt;,
          typename Tag = rb_tree_tag,
          template&lt;
          typename Const_Node_Iterator,
          typename Node_Iterator,
          typename Cmp_Fn_,
          typename Allocator_&gt;
          class Node_Update = null_node_update,
          typename Allocator = std::allocator&lt;char&gt; &gt;
          class tree;
        </programlisting>

        <para>The parameters have the following meaning:</para>

        <orderedlist>
          <listitem>
          <para><classname>Key</classname> is the key type.</para></listitem>

          <listitem>
          <para><classname>Mapped</classname> is the mapped-policy.</para></listitem>

          <listitem>
          <para><classname>Cmp_Fn</classname> is a key comparison functor</para></listitem>

          <listitem>
            <para><classname>Tag</classname> specifies which underlying data structure
          to use.</para></listitem>

          <listitem>
            <para><classname>Node_Update</classname> is a policy for updating node
          invariants.</para></listitem>

          <listitem>
            <para><classname>Allocator</classname> is an allocator
          type.</para></listitem>
        </orderedlist>

        <para>The <classname>Tag</classname> parameter specifies which underlying
        data structure to use. Instantiating it by <classname>rb_tree_tag</classname>, <classname>splay_tree_tag</classname>, or
        <classname>ov_tree_tag</classname>,
        specifies an underlying red-black tree, splay tree, or
        ordered-vector tree, respectively; any other tag is illegal.
        Note that containers based on the former two contain more types
        and methods than the latter (e.g.,
        <classname>reverse_iterator</classname> and <classname>rbegin</classname>), and different
        exception and invalidation guarantees.</para>

      </section>

      <section xml:id="container.tree.details">
        <info><title>Details</title></info>

        <section xml:id="container.tree.node">
          <info><title>Node Invariants</title></info>


          <para>Consider the two trees in the graphic below, labels A and B. The first
          is a tree of floats; the second is a tree of pairs, each
          signifying a geometric line interval. Each element in a tree is refered to as a node of the tree. Of course, each of
          these trees can support the usual queries: the first can easily
          search for <classname>0.4</classname>; the second can easily search for
          <classname>std::make_pair(10, 41)</classname>.</para>

          <para>Each of these trees can efficiently support other queries.
          The first can efficiently determine that the 2rd key in the
          tree is <constant>0.3</constant>; the second can efficiently determine
          whether any of its intervals overlaps
          <programlisting>std::make_pair(29,42)</programlisting> (useful in geometric
          applications or distributed file systems with leases, for
          example).  It should be noted that an <classname>std::set</classname> can
          only solve these types of problems with linear complexity.</para>

          <para>In order to do so, each tree stores some metadata in
          each node, and maintains node invariants (see <xref linkend="biblio.clrs2001"/>.) The first stores in
          each node the size of the sub-tree rooted at the node; the
          second stores at each node the maximal endpoint of the
          intervals at the sub-tree rooted at the node.</para>

          <figure>
            <title>Tree node invariants</title>
            <mediaobject>
              <imageobject>
                <imagedata align="center" format="PNG" scale="100"
                           fileref="../images/pbds_tree_node_invariants.png"/>
              </imageobject>
              <textobject>
                <phrase>Tree node invariants</phrase>
              </textobject>
            </mediaobject>
          </figure>
          
          <para>Supporting such trees is difficult for a number of
          reasons:</para>

          <orderedlist>
            <listitem><para>There must be a way to specify what a node's metadata
            should be (if any).</para></listitem>

            <listitem><para>Various operations can invalidate node
            invariants.  The graphic below shows how a right rotation,
            performed on A, results in B, with nodes x and y having
            corrupted invariants (the grayed nodes in C). The graphic shows
            how an insert, performed on D, results in E, with nodes x and y
            having corrupted invariants (the grayed nodes in F). It is not
            feasible to know outside the tree the effect of an operation on
            the nodes of the tree.</para></listitem>

            <listitem><para>The search paths of standard associative containers are
            defined by comparisons between keys, and not through
            metadata.</para></listitem>

            <listitem><para>It is not feasible to know in advance which methods trees
            can support. Besides the usual <classname>find</classname> method, the
            first tree can support a <classname>find_by_order</classname> method, while
            the second can support an <classname>overlaps</classname> method.</para></listitem>
          </orderedlist>

          <figure>
            <title>Tree node invalidation</title>
            <mediaobject>
              <imageobject>
                <imagedata align="center" format="PNG" scale="100"
                           fileref="../images/pbds_tree_node_invalidations.png"/>
              </imageobject>
              <textobject>
                <phrase>Tree node invalidation</phrase>
              </textobject>
            </mediaobject>
          </figure>

          <para>These problems are solved by a combination of two means:
          node iterators, and template-template node updater
          parameters.</para>

          <section xml:id="container.tree.node.iterators">
            <info><title>Node Iterators</title></info>


            <para>Each tree-based container defines two additional iterator
            types, <classname>const_node_iterator</classname>
            and <classname>node_iterator</classname>.
            These iterators allow descending from a node to one of its
            children. Node iterator allow search paths different than those
            determined by the comparison functor. The <classname>tree</classname>
            supports the methods:</para>
            <programlisting>
              const_node_iterator
              node_begin() const;

              node_iterator
              node_begin();

              const_node_iterator
              node_end() const;

              node_iterator
              node_end(); 
            </programlisting>

            <para>The first pairs return node iterators corresponding to the
            root node of the tree; the latter pair returns node iterators
            corresponding to a just-after-leaf node.</para>
          </section>

          <section xml:id="container.tree.node.updator">
            <info><title>Node Updator</title></info>

            <para>The tree-based containers are parametrized by a
            <classname>Node_Update</classname> template-template parameter. A
            tree-based container instantiates
            <classname>Node_Update</classname> to some
            <classname>node_update</classname> class, and publicly subclasses
            <classname>node_update</classname>. The graphic below shows this
            scheme, as well as some predefined policies (which are explained
            below).</para>

            <figure>
              <title>A tree and its update policy</title>
              <mediaobject>
                <imageobject>
                  <imagedata align="center" format="PNG" scale="100"
                             fileref="../images/pbds_tree_node_updator_policy_cd.png"/>
                </imageobject>
                <textobject>
                  <phrase>A tree and its update policy</phrase>
                </textobject>
              </mediaobject>
            </figure>

            <para><classname>node_update</classname> (an instantiation of
            <classname>Node_Update</classname>) must define <classname>metadata_type</classname> as
            the type of metadata it requires. For order statistics,
            e.g., <classname>metadata_type</classname> might be <classname>size_t</classname>.
            The tree defines within each node a <classname>metadata_type</classname>
            object.</para>

            <para><classname>node_update</classname> must also define the following method
            for restoring node invariants:</para>
            <programlisting>
              void 
              operator()(node_iterator nd_it, const_node_iterator end_nd_it)
            </programlisting>

            <para>In this method, <varname>nd_it</varname> is a
            <classname>node_iterator</classname> corresponding to a node whose
            A) all descendants have valid invariants, and B) its own
            invariants might be violated; <classname>end_nd_it</classname> is
            a <classname>const_node_iterator</classname> corresponding to a
            just-after-leaf node. This method should correct the node
            invariants of the node pointed to by
            <classname>nd_it</classname>. For example, say node x in the
            graphic below label A has an invalid invariant, but its' children,
            y and z have valid invariants. After the invocation, all three
            nodes should have valid invariants, as in label B.</para>


            <figure>
              <title>Restoring node invariants</title>
              <mediaobject>
                <imageobject>
                  <imagedata align="center" format="PNG" scale="100"
                             fileref="../images/pbds_restoring_node_invariants.png"/>
                </imageobject>
                <textobject>
                  <phrase>Restoring node invariants</phrase>
                </textobject>
              </mediaobject>
            </figure>

            <para>When a tree operation might invalidate some node invariant,
            it invokes this method in its <classname>node_update</classname> base to
            restore the invariant. For example, the graphic below shows
            an <function>insert</function> operation (point A); the tree performs some
            operations, and calls the update functor three times (points B,
            C, and D). (It is well known that any <function>insert</function>,
            <function>erase</function>, <function>split</function> or <function>join</function>, can restore
            all node invariants by a small number of node invariant updates (<xref linkend="biblio.clrs2001"/>)
            .</para>

            <figure>
              <title>Insert update sequence</title>
              <mediaobject>
                <imageobject>
                  <imagedata align="center" format="PNG" scale="100"
                             fileref="../images/pbds_update_seq_diagram.png"/>
                </imageobject>
                <textobject>
                  <phrase>Insert update sequence</phrase>
                </textobject>
              </mediaobject>
            </figure>

            <para>To complete the description of the scheme, three questions
            need to be answered:</para>

            <orderedlist>
              <listitem><para>How can a tree which supports order statistics define a
              method such as <classname>find_by_order</classname>?</para></listitem>

              <listitem><para>How can the node updater base access methods of the
              tree?</para></listitem>

              <listitem><para>How can the following cyclic dependency be resolved?
              <classname>node_update</classname> is a base class of the tree, yet it
              uses node iterators defined in the tree (its child).</para></listitem>
            </orderedlist>

            <para>The first two questions are answered by the fact that
            <classname>node_update</classname> (an instantiation of
            <classname>Node_Update</classname>) is a <emphasis>public</emphasis> base class
            of the tree. Consequently:</para>

            <orderedlist>
              <listitem><para>Any public methods of
              <classname>node_update</classname> are automatically methods of
              the tree (<xref linkend="biblio.alexandrescu01modern"/>).
              Thus an order-statistics node updater,
              <classname>tree_order_statistics_node_update</classname> defines
              the <function>find_by_order</function> method; any tree
              instantiated by this policy consequently supports this method as
              well.</para></listitem>

              <listitem><para>In C++, if a base class declares a method as
              <literal>virtual</literal>, it is
              <literal>virtual</literal> in its subclasses. If
              <classname>node_update</classname> needs to access one of the
              tree's methods, say the member function
              <function>end</function>, it simply declares that method as
              <literal>virtual</literal> abstract.</para></listitem>
            </orderedlist>

            <para>The cyclic dependency is solved through template-template
            parameters. <classname>Node_Update</classname> is parametrized by
            the tree's node iterators, its comparison functor, and its
            allocator type. Thus, instantiations of
            <classname>Node_Update</classname> have all information
            required.</para>

            <para>This library assumes that constructing a metadata object and
            modifying it are exception free. Suppose that during some method,
            say <classname>insert</classname>, a metadata-related operation
            (e.g., changing the value of a metadata) throws an exception. Ack!
            Rolling back the method is unusually complex.</para>

            <para>Previously, a distinction was made between redundant
            policies and null policies. Node invariants show a
            case where null policies are required.</para>

            <para>Assume a regular tree is required, one which need not
            support order statistics or interval overlap queries.
            Seemingly, in this case a redundant policy - a policy which
            doesn't affect nodes' contents would suffice. This, would lead
            to the following drawbacks:</para>

            <orderedlist>
              <listitem><para>Each node would carry a useless metadata object, wasting
              space.</para></listitem>

              <listitem><para>The tree cannot know if its
              <classname>Node_Update</classname> policy actually modifies a
              node's metadata (this is halting reducible). In the graphic
              below, assume the shaded node is inserted. The tree would have
              to traverse the useless path shown to the root, applying
              redundant updates all the way.</para></listitem>
            </orderedlist>
            <figure>
              <title>Useless update path</title>
              <mediaobject>
                <imageobject>
                  <imagedata align="center" format="PNG" scale="100"
                             fileref="../images/pbds_rationale_null_node_updator.png"/>
                </imageobject>
                <textobject>
                  <phrase>Useless update path</phrase>
                </textobject>
              </mediaobject>
            </figure>


            <para>A null policy class, <classname>null_node_update</classname>
            solves both these problems. The tree detects that node
            invariants are irrelevant, and defines all accordingly.</para>

          </section>

        </section> 

        <section xml:id="container.tree.details.split">
          <info><title>Split and Join</title></info>

          <para>Tree-based containers support split and join methods.
          It is possible to split a tree so that it passes
          all nodes with keys larger than a given key to a different
          tree. These methods have the following advantages over the
          alternative of externally inserting to the destination
          tree and erasing from the source tree:</para>

          <orderedlist>
            <listitem><para>These methods are efficient - red-black trees are split
            and joined in poly-logarithmic complexity; ordered-vector
            trees are split and joined at linear complexity. The
            alternatives have super-linear complexity.</para></listitem>

            <listitem><para>Aside from orders of growth, these operations perform
            few allocations and de-allocations. For red-black trees, allocations are not performed,
            and the methods are exception-free. </para></listitem>
          </orderedlist>
        </section>

      </section> <!-- details -->

    </section> <!-- tree -->

    <!-- trie -->
    <section xml:id="pbds.design.container.trie">
      <info><title>Trie</title></info>

      <section xml:id="container.trie.interface">
        <info><title>Interface</title></info>

        <para>The trie-based container has the following declaration:</para>
        <programlisting>
          template&lt;typename Key,
          typename Mapped,
          typename Cmp_Fn = std::less&lt;Key&gt;,
          typename Tag = pat_trie_tag,
          template&lt;typename Const_Node_Iterator,
          typename Node_Iterator,
          typename E_Access_Traits_,
          typename Allocator_&gt;
          class Node_Update = null_node_update,
          typename Allocator = std::allocator&lt;char&gt; &gt;
          class trie;
        </programlisting>

        <para>The parameters have the following meaning:</para>

        <orderedlist>
          <listitem><para><classname>Key</classname> is the key type.</para></listitem>

          <listitem><para><classname>Mapped</classname> is the mapped-policy.</para></listitem>

          <listitem><para><classname>E_Access_Traits</classname> is described in below.</para></listitem>

          <listitem><para><classname>Tag</classname> specifies which underlying data structure
          to use, and is described shortly.</para></listitem>

          <listitem><para><classname>Node_Update</classname> is a policy for updating node
          invariants. This is described below.</para></listitem>

          <listitem><para><classname>Allocator</classname> is an allocator
          type.</para></listitem>
        </orderedlist>

        <para>The <classname>Tag</classname> parameter specifies which underlying
        data structure to use. Instantiating it by <classname>pat_trie_tag</classname>, specifies an
        underlying PATRICIA trie (explained shortly); any other tag is
        currently illegal.</para>

        <para>Following is a description of a (PATRICIA) trie
        (this implementation follows <xref linkend="biblio.okasaki98mereable"/> and 
        <xref linkend="biblio.filliatre2000ptset"/>). 
        </para>

        <para>A (PATRICIA) trie is similar to a tree, but with the
        following differences:</para>

        <orderedlist>
          <listitem><para>It explicitly views keys as a sequence of elements.
          E.g., a trie can view a string as a sequence of
          characters; a trie can view a number as a sequence of
          bits.</para></listitem>

          <listitem><para>It is not (necessarily) binary. Each node has fan-out n
          + 1, where n is the number of distinct
          elements.</para></listitem>

          <listitem><para>It stores values only at leaf nodes.</para></listitem>

          <listitem><para>Internal nodes have the properties that A) each has at
          least two children, and B) each shares the same prefix with
          any of its descendant.</para></listitem>
        </orderedlist>

        <para>A (PATRICIA) trie has some useful properties:</para>

        <orderedlist>
          <listitem><para>It can be configured to use large node fan-out, giving it
          very efficient find performance (albeit at insertion
          complexity and size).</para></listitem>

          <listitem><para>It works well for common-prefix keys.</para></listitem>

          <listitem><para>It can support efficiently queries such as which
          keys match a certain prefix. This is sometimes useful in file
          systems and routers, and for "type-ahead" aka predictive text matching
          on mobile devices.</para></listitem>
        </orderedlist>


      </section>

      <section xml:id="container.trie.details">
        <info><title>Details</title></info>

        <section xml:id="container.trie.details.etraits">
          <info><title>Element Access Traits</title></info>

          <para>A trie inherently views its keys as sequences of elements.
          For example, a trie can view a string as a sequence of
          characters. A trie needs to map each of n elements to a
          number in {0, n - 1}. For example, a trie can map a
          character <varname>c</varname> to
          <programlisting>static_cast&lt;size_t&gt;(c)</programlisting>.</para>

          <para>Seemingly, then, a trie can assume that its keys support
          (const) iterators, and that the <classname>value_type</classname> of this
          iterator can be cast to a <classname>size_t</classname>. There are several
          reasons, though, to decouple the mechanism by which the trie
          accesses its keys' elements from the trie:</para>

          <orderedlist>
            <listitem><para>In some cases, the numerical value of an element is
            inappropriate. Consider a trie storing DNA strings. It is
            logical to use a trie with a fan-out of 5 = 1 + |{'A', 'C',
            'G', 'T'}|. This requires mapping 'T' to 3, though.</para></listitem>

            <listitem><para>In some cases the keys' iterators are different than what
            is needed. For example, a trie can be used to search for
            common suffixes, by using strings'
            <classname>reverse_iterator</classname>. As another example, a trie mapping
            UNICODE strings would have a huge fan-out if each node would
            branch on a UNICODE character; instead, one can define an
            iterator iterating over 8-bit (or less) groups.</para></listitem>
          </orderedlist>

          <para>trie is,
          consequently, parametrized by <classname>E_Access_Traits</classname> -
          traits which instruct how to access sequences' elements.
          <classname>string_trie_e_access_traits</classname>
          is a traits class for strings. Each such traits define some
          types, like:</para>
          <programlisting>
            typename E_Access_Traits::const_iterator
          </programlisting>

          <para>is a const iterator iterating over a key's elements. The
          traits class must also define methods for obtaining an iterator
          to the first and last element of a key.</para>

          <para>The graphic below shows a
          (PATRICIA) trie resulting from inserting the words: "I wish
          that I could ever see a poem lovely as a trie" (which,
          unfortunately, does not rhyme).</para>

          <para>The leaf nodes contain values; each internal node contains
          two <classname>typename E_Access_Traits::const_iterator</classname>
          objects, indicating the maximal common prefix of all keys in
          the sub-tree. For example, the shaded internal node roots a
          sub-tree with leafs "a" and "as". The maximal common prefix is
          "a". The internal node contains, consequently, to const
          iterators, one pointing to <varname>'a'</varname>, and the other to
          <varname>'s'</varname>.</para>

          <figure>
            <title>A PATRICIA trie</title>
            <mediaobject>
              <imageobject>
                <imagedata align="center" format="PNG" scale="100"
                           fileref="../images/pbds_pat_trie.png"/>
              </imageobject>
              <textobject>
                <phrase>A PATRICIA trie</phrase>
              </textobject>
            </mediaobject>
          </figure>

        </section>

        <section xml:id="container.trie.details.node">
          <info><title>Node Invariants</title></info>

          <para>Trie-based containers support node invariants, as do
          tree-based containers. There are two minor
          differences, though, which, unfortunately, thwart sharing them
          sharing the same node-updating policies:</para>

          <orderedlist>
            <listitem>
              <para>A trie's <classname>Node_Update</classname> template-template
              parameter is parametrized by <classname>E_Access_Traits</classname>, while
              a tree's <classname>Node_Update</classname> template-template parameter is
            parametrized by <classname>Cmp_Fn</classname>.</para></listitem>

            <listitem><para>Tree-based containers store values in all nodes, while
            trie-based containers (at least in this implementation) store
            values in leafs.</para></listitem>
          </orderedlist>

          <para>The graphic below shows the scheme, as well as some predefined
          policies (which are explained below).</para>

          <figure>
            <title>A trie and its update policy</title>
            <mediaobject>
              <imageobject>
                <imagedata align="center" format="PNG" scale="100"
                           fileref="../images/pbds_trie_node_updator_policy_cd.png"/>
              </imageobject>
              <textobject>
                <phrase>A trie and its update policy</phrase>
              </textobject>
            </mediaobject>
          </figure>


          <para>This library offers the following pre-defined trie node
          updating policies:</para>

          <orderedlist>
            <listitem>
              <para>
                <classname>trie_order_statistics_node_update</classname>
                supports order statistics.
              </para>
            </listitem>

            <listitem><para><classname>trie_prefix_search_node_update</classname>
            supports searching for ranges that match a given prefix.</para></listitem>

            <listitem><para><classname>null_node_update</classname>
            is the null node updater.</para></listitem>
          </orderedlist>

        </section>

        <section xml:id="container.trie.details.split">
          <info><title>Split and Join</title></info>
          <para>Trie-based containers support split and join methods; the
          rationale is equal to that of tree-based containers supporting
          these methods.</para>
        </section>

      </section> <!-- details -->

    </section> <!-- trie -->

    <!-- list_update -->
    <section xml:id="pbds.design.container.list">
      <info><title>List</title></info>

      <section xml:id="container.list.interface">
        <info><title>Interface</title></info>

        <para>The list-based container has the following declaration:</para>
        <programlisting>
          template&lt;typename Key,
          typename Mapped,
          typename Eq_Fn = std::equal_to&lt;Key&gt;,
          typename Update_Policy = move_to_front_lu_policy&lt;&gt;,
          typename Allocator = std::allocator&lt;char&gt; &gt;
          class list_update;
        </programlisting>

        <para>The parameters have the following meaning:</para>

        <orderedlist>
          <listitem>
            <para>
              <classname>Key</classname> is the key type.
            </para>
          </listitem>

          <listitem>
            <para>
              <classname>Mapped</classname> is the mapped-policy.
            </para>
          </listitem>

          <listitem>
            <para>
              <classname>Eq_Fn</classname> is a key equivalence functor.
            </para>
          </listitem>

          <listitem>
            <para>
              <classname>Update_Policy</classname> is a policy updating positions in
              the list based on access patterns. It is described in the
              following subsection.
            </para>
          </listitem>

          <listitem>
            <para>
              <classname>Allocator</classname> is an allocator type.
            </para>
          </listitem>
        </orderedlist>

        <para>A list-based associative container is a container that
        stores elements in a linked-list. It does not order the elements
        by any particular order related to the keys.  List-based
        containers are primarily useful for creating "multimaps". In fact,
        list-based containers are designed in this library expressly for
        this purpose.</para>

        <para>List-based containers might also be useful for some rare
        cases, where a key is encapsulated to the extent that only
        key-equivalence can be tested. Hash-based containers need to know
        how to transform a key into a size type, and tree-based containers
        need to know if some key is larger than another.  List-based
        associative containers, conversely, only need to know if two keys
        are equivalent.</para>

        <para>Since a list-based associative container does not order
        elements by keys, is it possible to order the list in some
        useful manner? Remarkably, many on-line competitive
        algorithms exist for reordering lists to reflect access
        prediction. (See <xref linkend="biblio.motwani95random"/> and <xref linkend="biblio.andrew04mtf"/>).
        </para>

      </section>

      <section xml:id="container.list.details">
        <info><title>Details</title></info>
        <para>
        </para>
        <section xml:id="container.list.details.ds">
          <info><title>Underlying Data Structure</title></info>

          <para>The graphic below shows a
          simple list of integer keys. If we search for the integer 6, we
          are paying an overhead: the link with key 6 is only the fifth
          link; if it were the first link, it could be accessed
          faster.</para>

          <figure>
            <title>A simple list</title>
            <mediaobject>
              <imageobject>
                <imagedata align="center" format="PNG" scale="100"
                           fileref="../images/pbds_simple_list.png"/>
              </imageobject>
              <textobject>
                <phrase>A simple list</phrase>
              </textobject>
            </mediaobject>
          </figure>

          <para>List-update algorithms reorder lists as elements are
          accessed. They try to determine, by the access history, which
          keys to move to the front of the list. Some of these algorithms
          require adding some metadata alongside each entry.</para>

          <para>For example, in the graphic below label A shows the counter
          algorithm. Each node contains both a key and a count metadata
          (shown in bold). When an element is accessed (e.g. 6) its count is
          incremented, as shown in label B. If the count reaches some
          predetermined value, say 10, as shown in label C, the count is set
          to 0 and the node is moved to the front of the list, as in label
          D.
          </para>

          <figure>
            <title>The counter algorithm</title>
            <mediaobject>
              <imageobject>
                <imagedata align="center" format="PNG" scale="100"
                           fileref="../images/pbds_list_update.png"/>
              </imageobject>
              <textobject>
                <phrase>The counter algorithm</phrase>
              </textobject>
            </mediaobject>
          </figure>


        </section>

        <section xml:id="container.list.details.policies">
          <info><title>Policies</title></info>

          <para>this library allows instantiating lists with policies
          implementing any algorithm moving nodes to the front of the
          list (policies implementing algorithms interchanging nodes are
          unsupported).</para>

          <para>Associative containers based on lists are parametrized by a
          <classname>Update_Policy</classname> parameter. This parameter defines the
          type of metadata each node contains, how to create the
          metadata, and how to decide, using this metadata, whether to
          move a node to the front of the list. A list-based associative
          container object derives (publicly) from its update policy.
          </para>

          <para>An instantiation of <classname>Update_Policy</classname> must define
          internally <classname>update_metadata</classname> as the metadata it
          requires. Internally, each node of the list contains, besides
          the usual key and data, an instance of <classname>typename
          Update_Policy::update_metadata</classname>.</para>

          <para>An instantiation of <classname>Update_Policy</classname> must define
          internally two operators:</para>
          <programlisting>
            update_metadata
            operator()();

            bool
            operator()(update_metadata &amp;);
          </programlisting>

          <para>The first is called by the container object, when creating a
          new node, to create the node's metadata. The second is called
          by the container object, when a node is accessed (
          when a find operation's key is equivalent to the key of the
          node), to determine whether to move the node to the front of
          the list.
          </para>

          <para>The library contains two predefined implementations of
          list-update policies. The first
          is <classname>lu_counter_policy</classname>, which implements the
          counter algorithm described above. The second is
          <classname>lu_move_to_front_policy</classname>,
          which unconditionally move an accessed element to the front of
          the list. The latter type is very useful in this library,
          since there is no need to associate metadata with each element.
          (See <xref linkend="biblio.andrew04mtf"/> 
          </para>

        </section>

        <section xml:id="container.list.details.mapped">
          <info><title>Use in Multimaps</title></info>

          <para>In this library, there are no equivalents for the standard's
          multimaps and multisets; instead one uses an associative
          container mapping primary keys to secondary keys.</para>

          <para>List-based containers are especially useful as associative
          containers for secondary keys. In fact, they are implemented
          here expressly for this purpose.</para>

          <para>To begin with, these containers use very little per-entry
          structure memory overhead, since they can be implemented as
          singly-linked lists. (Arrays use even lower per-entry memory
          overhead, but they are less flexible in moving around entries,
          and have weaker invalidation guarantees).</para>

          <para>More importantly, though, list-based containers use very
          little per-container memory overhead. The memory overhead of an
          empty list-based container is practically that of a pointer.
          This is important for when they are used as secondary
          associative-containers in situations where the average ratio of
          secondary keys to primary keys is low (or even 1).</para>

          <para>In order to reduce the per-container memory overhead as much
          as possible, they are implemented as closely as possible to
          singly-linked lists.</para>

          <orderedlist>
            <listitem>
              <para>
                List-based containers do not store internally the number
                of values that they hold. This means that their <function>size</function>
                method has linear complexity (just like <classname>std::list</classname>).
                Note that finding the number of equivalent-key values in a
                standard multimap also has linear complexity (because it must be
                done,  via <function>std::distance</function> of the
                multimap's <function>equal_range</function> method), but usually with
                higher constants.
              </para>
            </listitem>

            <listitem>
              <para>
                Most associative-container objects each hold a policy
                object (a hash-based container object holds a
                hash functor). List-based containers, conversely, only have
                class-wide policy objects.
              </para>
            </listitem>
          </orderedlist>


        </section>

      </section> <!-- details -->

    </section> <!-- list -->


    <!-- priority_queue -->
    <section xml:id="pbds.design.container.priority_queue">
      <info><title>Priority Queue</title></info>

      <section xml:id="container.priority_queue.interface">
        <info><title>Interface</title></info>

        <para>The priority queue container has the following
        declaration:
        </para>
        <programlisting>
          template&lt;typename  Value_Type,
          typename  Cmp_Fn = std::less&lt;Value_Type&gt;,
          typename  Tag = pairing_heap_tag,
          typename  Allocator = std::allocator&lt;char &gt; &gt;
          class priority_queue;
        </programlisting>

        <para>The parameters have the following meaning:</para>

        <orderedlist>
          <listitem><para><classname>Value_Type</classname> is the value type.</para></listitem>

          <listitem><para><classname>Cmp_Fn</classname> is a value comparison functor</para></listitem>

          <listitem><para><classname>Tag</classname> specifies which underlying data structure
          to use.</para></listitem>

          <listitem><para><classname>Allocator</classname> is an allocator
          type.</para></listitem>
        </orderedlist>

        <para>The <classname>Tag</classname> parameter specifies which underlying
        data structure to use. Instantiating it by<classname>pairing_heap_tag</classname>,<classname>binary_heap_tag</classname>,
        <classname>binomial_heap_tag</classname>,
        <classname>rc_binomial_heap_tag</classname>,
        or <classname>thin_heap_tag</classname>,
        specifies, respectively, 
        an underlying pairing heap (<xref linkend="biblio.fredman86pairing"/>),
        binary heap (<xref linkend="biblio.clrs2001"/>),
        binomial heap (<xref linkend="biblio.clrs2001"/>),
        a binomial heap with a redundant binary counter (<xref linkend="biblio.maverik_lowerbounds"/>),
        or a thin heap (<xref linkend="biblio.kt99fat_heaps"/>).
        </para>

        <para>
          As mentioned in the tutorial,
          <classname>__gnu_pbds::priority_queue</classname> shares most of the
          same interface with <classname>std::priority_queue</classname>.
          E.g. if <varname>q</varname> is a priority queue of type
          <classname>Q</classname>, then <function>q.top()</function> will
          return the "largest" value in the container (according to
          <classname>typename
          Q::cmp_fn</classname>). <classname>__gnu_pbds::priority_queue</classname>
          has a larger (and very slightly different) interface than
          <classname>std::priority_queue</classname>, however, since typically
          <classname>push</classname> and <classname>pop</classname> are deemed
        insufficient for manipulating priority-queues. </para>

        <para>Different settings require different priority-queue
        implementations which are described in later; see traits
        discusses ways to differentiate between the different traits of
        different implementations.</para>


      </section>

      <section xml:id="container.priority_queue.details">
        <info><title>Details</title></info>

        <section xml:id="container.priority_queue.details.iterators">
          <info><title>Iterators</title></info>

          <para>There are many different underlying-data structures for
          implementing priority queues. Unfortunately, most such
          structures are oriented towards making <function>push</function> and
          <function>top</function> efficient, and consequently don't allow efficient
          access of other elements: for instance, they cannot support an efficient
          <function>find</function> method. In the use case where it
          is important to both access and "do something with" an
          arbitrary value, one would be out of luck. For example, many graph algorithms require
          modifying a value (typically increasing it in the sense of the
          priority queue's comparison functor).</para>

          <para>In order to access and manipulate an arbitrary value in a
          priority queue, one needs to reference the internals of the
          priority queue from some form of an associative container -
          this is unavoidable. Of course, in order to maintain the
          encapsulation of the priority queue, this needs to be done in a
          way that minimizes exposure to implementation internals.</para>

          <para>In this library the priority queue's <function>insert</function>
          method returns an iterator, which if valid can be used for subsequent <function>modify</function> and
          <function>erase</function> operations. This both preserves the priority
          queue's encapsulation, and allows accessing arbitrary values (since the
          returned iterators from the <function>push</function> operation can be
          stored in some form of associative container).</para>

          <para>Priority queues' iterators present a problem regarding their
          invalidation guarantees. One assumes that calling
          <function>operator++</function> on an iterator will associate it
          with the "next" value. Priority-queues are
          self-organizing: each operation changes what the "next" value
          means. Consequently, it does not make sense that <function>push</function>
          will return an iterator that can be incremented - this can have
          no possible use. Also, as in the case of hash-based containers,
          it is awkward to define if a subsequent <function>push</function> operation
          invalidates a prior returned iterator: it invalidates it in the
          sense that its "next" value is not related to what it
          previously considered to be its "next" value. However, it might not
          invalidate it, in the sense that it can be
          de-referenced and used for <function>modify</function> and <function>erase</function>
          operations.</para>

          <para>Similarly to the case of the other unordered associative
          containers, this library uses a distinction between
          point-type and range type iterators. A priority queue's <classname>iterator</classname> can always be
          converted to a <classname>point_iterator</classname>, and a
          <classname>const_iterator</classname> can always be converted to a
          <classname>point_const_iterator</classname>.</para>

          <para>The following snippet demonstrates manipulating an arbitrary
          value:</para>
          <programlisting>
            // A priority queue of integers.
            priority_queue&lt;int &gt; p;

            // Insert some values into the priority queue.
            priority_queue&lt;int &gt;::point_iterator it = p.push(0);

            p.push(1);
            p.push(2);

            // Now modify a value.
            p.modify(it, 3);

            assert(p.top() == 3);
          </programlisting>

          
          <para>It should be noted that an alternative design could embed an
          associative container in a priority queue. Could, but most
          probably should not. To begin with, it should be noted that one
          could always encapsulate a priority queue and an associative
          container mapping values to priority queue iterators with no
          performance loss. One cannot, however, "un-encapsulate" a priority
          queue embedding an associative container, which might lead to
          performance loss. Assume, that one needs to associate each value
          with some data unrelated to priority queues. Then using
          this library's design, one could use an
          associative container mapping each value to a pair consisting of
          this data and a priority queue's iterator. Using the embedded
          method would need to use two associative containers. Similar
          problems might arise in cases where a value can reside
          simultaneously in many priority queues.</para>

        </section>


        <section xml:id="container.priority_queue.details.d">
          <info><title>Underlying Data Structure</title></info>

          <para>There are three main implementations of priority queues: the
          first employs a binary heap, typically one which uses a
          sequence; the second uses a tree (or forest of trees), which is
          typically less structured than an associative container's tree;
          the third simply uses an associative container. These are
          shown in the graphic below, in labels A1 and A2, label B, and label C.</para>

          <figure>
            <title>Underlying Priority-Queue Data-Structures.</title>
            <mediaobject>
              <imageobject>
                <imagedata align="center" format="PNG" scale="100"
                           fileref="../images/pbds_priority_queue_different_underlying_dss.png"/>
              </imageobject>
              <textobject>
                <phrase>Underlying Priority-Queue Data-Structures.</phrase>
              </textobject>
            </mediaobject>
          </figure>

          <para>Roughly speaking, any value that is both pushed and popped
          from a priority queue must incur a logarithmic expense (in the
          amortized sense). Any priority queue implementation that would
          avoid this, would violate known bounds on comparison-based
          sorting (see <xref linkend="biblio.clrs2001"/> and <xref linkend="biblio.brodal96priority"/>).
          </para>

          <para>Most implementations do
          not differ in the asymptotic amortized complexity of
          <function>push</function> and <function>pop</function> operations, but they differ in
          the constants involved, in the complexity of other operations
          (e.g., <function>modify</function>), and in the worst-case
          complexity of single operations. In general, the more
          "structured" an implementation (i.e., the more internal
          invariants it possesses) - the higher its amortized complexity
          of <function>push</function> and <function>pop</function> operations.</para>

          <para>This library implements different algorithms using a
          single class: <classname>priority_queue</classname>.
          Instantiating the <classname>Tag</classname> template parameter, "selects"
          the implementation:</para>

          <orderedlist>
            <listitem><para>
              Instantiating <classname>Tag = binary_heap_tag</classname> creates
              a binary heap of the form in represented in the graphic with labels A1 or A2. The former is internally
              selected by priority_queue
              if <classname>Value_Type</classname> is instantiated by a primitive type
              (e.g., an <type>int</type>); the latter is
              internally selected for all other types (e.g.,
              <classname>std::string</classname>). This implementations is relatively
              unstructured, and so has good <classname>push</classname> and <classname>pop</classname>
              performance; it is the "best-in-kind" for primitive
              types, e.g., <type>int</type>s. Conversely, it has
              high worst-case performance, and can support only linear-time
            <function>modify</function> and <function>erase</function> operations.</para></listitem>

            <listitem><para>Instantiating <classname>Tag =
            pairing_heap_tag</classname> creates a pairing heap of the form
            in represented by label B in the graphic above. This
            implementations too is relatively unstructured, and so has good
            <function>push</function> and <function>pop</function>
            performance; it is the "best-in-kind" for non-primitive types,
            e.g., <classname>std:string</classname>s. It also has very good
            worst-case <function>push</function> and
            <function>join</function> performance (O(1)), but has high
            worst-case <function>pop</function>
            complexity.</para></listitem>

            <listitem><para>Instantiating <classname>Tag =
            binomial_heap_tag</classname> creates a binomial heap of the
            form repsented by label B in the graphic above. This
            implementations is more structured than a pairing heap, and so
            has worse <function>push</function> and <function>pop</function>
            performance. Conversely, it has sub-linear worst-case bounds for
            <function>pop</function>, e.g., and so it might be preferred in
            cases where responsiveness is important.</para></listitem>

            <listitem><para>Instantiating <classname>Tag =
            rc_binomial_heap_tag</classname> creates a binomial heap of the
            form represented in label B above, accompanied by a redundant
            counter which governs the trees. This implementations is
            therefore more structured than a binomial heap, and so has worse
            <function>push</function> and <function>pop</function>
            performance. Conversely, it guarantees O(1)
            <function>push</function> complexity, and so it might be
            preferred in cases where the responsiveness of a binomial heap
            is insufficient.</para></listitem>

            <listitem><para>Instantiating <classname>Tag =
            thin_heap_tag</classname> creates a thin heap of the form
            represented by the label B in the graphic above. This
            implementations too is more structured than a pairing heap, and
            so has worse <function>push</function> and
            <function>pop</function> performance. Conversely, it has better
            worst-case and identical amortized complexities than a Fibonacci
            heap, and so might be more appropriate for some graph
            algorithms.</para></listitem>
          </orderedlist>

          <para>Of course, one can use any order-preserving associative
          container as a priority queue, as in the graphic above label C, possibly by creating an adapter class
          over the associative container (much as 
          <classname>std::priority_queue</classname> can adapt <classname>std::vector</classname>).
          This has the advantage that no cross-referencing is necessary
          at all; the priority queue itself is an associative container.
          Most associative containers are too structured to compete with
          priority queues in terms of <function>push</function> and <function>pop</function>
          performance.</para>



        </section>

        <section xml:id="container.priority_queue.details.traits">
          <info><title>Traits</title></info>

          <para>It would be nice if all priority queues could
          share exactly the same behavior regardless of implementation. Sadly, this is not possible. Just one for instance is in join operations: joining
          two binary heaps might throw an exception (not corrupt
          any of the heaps on which it operates), but joining two pairing
          heaps is exception free.</para>

          <para>Tags and traits are very useful for manipulating generic
          types. <classname>__gnu_pbds::priority_queue</classname>
          publicly defines <classname>container_category</classname> as one of the tags. Given any
          container <classname>Cntnr</classname>, the tag of the underlying
          data structure can be found via <classname>typename 
          Cntnr::container_category</classname>; this is one of the possible tags shown in the graphic below.
          </para>

          <figure>
            <title>Priority-Queue Data-Structure Tags.</title>
            <mediaobject>
              <imageobject>
                <imagedata align="center" format="PNG" scale="100"
                           fileref="../images/pbds_priority_queue_tag_hierarchy.png"/>
              </imageobject>
              <textobject>
                <phrase>Priority-Queue Data-Structure Tags.</phrase>
              </textobject>
            </mediaobject>
          </figure>


          <para>Additionally, a traits mechanism can be used to query a
          container type for its attributes. Given any container
          <classname>Cntnr</classname>, then <programlisting>__gnu_pbds::container_traits&lt;Cntnr&gt;</programlisting>
          is a traits class identifying the properties of the
          container.</para>

          <para>To find if a container might throw if two of its objects are
          joined, one can use 
          <programlisting>
            container_traits&lt;Cntnr&gt;::split_join_can_throw
          </programlisting>
          </para>

          <para>
            Different priority-queue implementations have different invalidation guarantees. This is
            especially important, since there is no way to access an arbitrary
            value of priority queues except for iterators. Similarly to
            associative containers, one can use
            <programlisting>
              container_traits&lt;Cntnr&gt;::invalidation_guarantee
            </programlisting>
          to get the invalidation guarantee type of a priority queue.</para>

          <para>It is easy to understand from the graphic above, what <classname>container_traits&lt;Cntnr&gt;::invalidation_guarantee</classname>
          will be for different implementations. All implementations of
          type represented by label B have <classname>point_invalidation_guarantee</classname>:
          the container can freely internally reorganize the nodes -
          range-type iterators are invalidated, but point-type iterators
          are always valid. Implementations of type represented by labels A1 and A2 have <classname>basic_invalidation_guarantee</classname>:
          the container can freely internally reallocate the array - both
          point-type and range-type iterators might be invalidated.</para>

          <para>
            This has major implications, and constitutes a good reason to avoid
            using binary heaps. A binary heap can perform <function>modify</function>
            or <function>erase</function> efficiently given a valid point-type
            iterator. However, in order to supply it with a valid point-type
            iterator, one needs to iterate (linearly) over all
            values, then supply the relevant iterator (recall that a
            range-type iterator can always be converted to a point-type
            iterator). This means that if the number of <function>modify</function> or
            <function>erase</function> operations is non-negligible (say
            super-logarithmic in the total sequence of operations) - binary
            heaps will perform badly.
          </para>

        </section>

      </section> <!-- details -->

    </section> <!-- priority_queue -->



  </section> <!-- container -->

  </section> <!-- design -->



  <!-- S04: Test -->
  <xi:include xmlns:xi="http://www.w3.org/2001/XInclude" parse="xml"
              href="test_policy_data_structures.xml">
  </xi:include>

  <!-- S05: Reference/Acknowledgments -->
  <section xml:id="pbds.ack">
    <info><title>Acknowledgments</title></info>
    <?dbhtml filename="policy_data_structures_biblio.html"?>

    <para>
      Written by Ami Tavory and Vladimir Dreizin (IBM Haifa Research
      Laboratories), and Benjamin Kosnik (Red Hat).
    </para>

    <para>
      This library was partially written at
      <link xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="http://www.haifa.il.ibm.com/">IBM's Haifa Research Labs</link>.
      It is based heavily on policy-based design and uses many useful
      techniques from Modern C++ Design: Generic Programming and Design
      Patterns Applied by Andrei Alexandrescu.
    </para>

    <para>
      Two ideas are borrowed from the SGI-STL implementation:
    </para>

    <orderedlist>
      <listitem>
        <para>
          The prime-based resize policies use a list of primes taken from
          the SGI-STL implementation.
        </para>
      </listitem>

      <listitem>
        <para>
          The red-black trees contain both a root node and a header node
          (containing metadata), connected in a way that forward and
          reverse iteration can be performed efficiently.
        </para>
      </listitem>
    </orderedlist>

    <para>
      Some test utilities borrow ideas from
      <link xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="http://www.boost.org/doc/libs/release/libs/timer/index.html">boost::timer</link>.
    </para>

    <para>
      We would like to thank Scott Meyers for useful comments (without
      attributing to him any flaws in the design or implementation of the
      library).
    </para>
    <para>We would like to thank Matt Austern for the suggestion to
    include tries.</para>
  </section>

  <!-- S06: Biblio -->
  <bibliography xml:id="pbds.biblio">
    <info>
      <title>
        Bibliography
      </title>
    </info>
    <?dbhtml filename="policy_data_structures_biblio.html"?>

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              Josuttis
            </surname>
          </personname>
        </author>
      </authorgroup>
      <publisher>
        <publishername>
          Addison-Wesley Publishing Company
        </publishername>
      </publisher>
    </biblioentry>


    <!-- 45 -->
    <biblioentry xml:id="biblio.wickland96thirty">
      <title>
        <link xmlns:xlink="http://www.w3.org/1999/xlink"
              xlink:href="http://myweb.wvnet.edu/~gsa00121/books/amongdead30.zip">
          Thirty Years Among the Dead
        </link>
      </title>
      <date>
        1996
      </date>

      <author>
        <personname>
          <firstname>
            C. A.
          </firstname>
          <surname>
            Wickland
          </surname>
        </personname>
      </author>

      <publisher>
        <publishername>
          National Psychological Institute
        </publishername>
      </publisher>
    </biblioentry>


  </bibliography>

</chapter>