<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_ack.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 IBM's Haifa Research Labs.
      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>

<xi:include xmlns:xi="http://www.w3.org/2001/XInclude" parse="xml"
	    href="policy_data_structures_biblio.xml">
</xi:include>

</chapter>
