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<?xml version="1.0" encoding="UTF-8" standalone="no"?> <!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.1//EN" "http://www.w3.org/TR/xhtml11/DTD/xhtml11.dtd"> <html xmlns="http://www.w3.org/1999/xhtml"><head><title>Design</title><meta name="generator" content="DocBook XSL-NS Stylesheets V1.76.1"/><meta name="keywords" content=" 	ISO C++ , 	policy , 	container , 	data , 	structure , 	associated , 	tree , 	trie , 	hash , 	metaprogramming "/><meta name="keywords" content=" ISO C++ , library "/><meta name="keywords" content=" ISO C++ , runtime , library "/><link rel="home" href="../index.html" title="The GNU C++ Library"/><link rel="up" href="policy_data_structures.html" title="Chapter 22. Policy-Based Data Structures"/><link rel="prev" href="policy_data_structures_using.html" title="Using"/><link rel="next" href="policy_based_data_structures_test.html" title="Testing"/></head><body><div class="navheader"><table width="100%" summary="Navigation header"><tr><th colspan="3" align="center">Design</th></tr><tr><td align="left"><a accesskey="p" href="policy_data_structures_using.html">Prev</a> </td><th width="60%" align="center">Chapter 22. Policy-Based Data Structures</th><td align="right"> <a accesskey="n" href="policy_based_data_structures_test.html">Next</a></td></tr></table><hr/></div><div class="section" title="Design"><div class="titlepage"><div><div><h2 class="title"><a id="containers.pbds.design"/>Design</h2></div></div></div><p/><div class="section" title="Concepts"><div class="titlepage"><div><div><h3 class="title"><a id="pbds.design.concepts"/>Concepts</h3></div></div></div><div class="section" title="Null Policy Classes"><div class="titlepage"><div><div><h4 class="title"><a id="pbds.design.concepts.null_type"/>Null Policy Classes</h4></div></div></div><p> 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. </p><p> 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. </p><p> 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. </p><p> When a policy is either redundant or irrelevant, it can be replaced by <code class="classname">null_type</code>. </p><p> For example, a <span class="emphasis"><em>set</em></span> is an associative container with one of its template parameters (the one for the mapped type) replaced with <code class="classname">null_type</code>. 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. </p></div><div class="section" title="Map and Set Semantics"><div class="titlepage"><div><div><h4 class="title"><a id="pbds.design.concepts.associative_semantics"/>Map and Set Semantics</h4></div></div></div><div class="section" title="Distinguishing Between Maps and Sets"><div class="titlepage"><div><div><h5 class="title"><a id="concepts.associative_semantics.set_vs_map"/> Distinguishing Between Maps and Sets </h5></div></div></div><p> 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. </p><p> 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., <code class="classname">std::map<int, char></code> maps each <code class="classname">int</code> to a <code class="classname">char</code>, but <code class="classname">std::set<int, char></code> simply stores <code class="classname">int</code>s. In this library, however, there are no distinct classes for maps and sets. Instead, an associative container's <code class="classname">Mapped</code> template parameter is a policy: if it is instantiated by <code class="classname">null_type</code>, then it is a "set"; otherwise, it is a "map". E.g., </p><pre class="programlisting"> cc_hash_table<int, char> </pre><p> is a "map" mapping each <span class="type">int</span> value to a <span class="type"> char</span>, but </p><pre class="programlisting"> cc_hash_table<int, null_type> </pre><p> is a type that uniquely stores <span class="type">int</span> values. </p><p>Once the <code class="classname">Mapped</code> template parameter is instantiated by <code class="classname">null_type</code>, then the "set" acts very similarly to the standard's sets - it does not map each key to a distinct <code class="classname">null_type</code> object. Also, , the container's <span class="type">value_type</span> is essentially its <span class="type">key_type</span> - just as with the standard's sets .</p><p> 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 <code class="classname">std::multiset<int></code> might be used to store multiple instances of integers, but using this library's containers, one might use </p><pre class="programlisting"> tree<int, size_t> </pre><p> i.e., a <code class="classname">map</code> of <span class="type">int</span>s to <span class="type">size_t</span>s. </p><p> These "multimaps" and "multisets" might be confusing to anyone familiar with the standard's <code class="classname">std::multimap</code> and <code class="classname">std::multiset</code>, because there is no clear correspondence between the two. For example, in some cases where one uses <code class="classname">std::multiset</code> 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. </p><p> When one uses a "multimap," one should choose with care the type of container used for secondary keys. </p></div><div class="section" title="Alternatives to std::multiset and std::multimap"><div class="titlepage"><div><div><h5 class="title"><a id="concepts.associative_semantics.multi"/>Alternatives to <code class="classname">std::multiset</code> and <code class="classname">std::multimap</code></h5></div></div></div><p> Brace onself: this library does not contain containers like <code class="classname">std::multimap</code> or <code class="classname">std::multiset</code>. Instead, these data structures can be synthesized via manipulation of the <code class="classname">Mapped</code> template parameter. </p><p> 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. </p><p> Stepping back a bit, and starting in from the beginning. </p><p> 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: <code class="classname">std::multimap</code>, <code class="classname">std::multiset</code>, <code class="classname">std::tr1::unordered_multimap</code>, and <code class="classname">unordered_multiset</code>. We first discuss how these might be used, then why we think it is best to avoid them. </p><p> Suppose one builds a simple bank-account application that records for each client (identified by an <code class="classname">std::string</code>) and account-id (marked by an <span class="type">unsigned long</span>) - the balance in the account (described by a <span class="type">float</span>). 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 </p><pre class="programlisting"> std::tr1::unordered_map<std::pair<std::string, unsigned long>, float, ...> </pre><p> 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 </p><pre class="programlisting"> std::tr1::unordered_multimap<std::pair<std::string, unsigned long>, float, ...> </pre><p> 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. </p><p> Also, suppose one wants an integers' priority queue (a container that supports <code class="function">push</code>, <code class="function">pop</code>, and <code class="function">top</code> operations, the last of which returns the largest <span class="type">int</span>) that also supports operations such as <code class="function">find</code> and <code class="function">lower_bound</code>. A reasonable solution is to build an adapter over <code class="classname">std::set<int></code>. In this adapter, <code class="function">push</code> will just call the tree-based associative container's <code class="function">insert</code> method; <code class="function">pop</code> will call its <code class="function">end</code> 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 (<code class="function">push(4)</code>, will be a no-op if <code class="constant">4</code> is already in the container object). If multiple keys are necessary, then one might build the adapter over an <code class="classname">std::multiset<int></code>. </p><p> 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). </p><p> 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. </p><div class="figure"><a id="id519449"/><p class="title"><strong>Figure 22.8. Non-unique Mapping Standard Containers</strong></p><div class="figure-contents"><div class="mediaobject" style="text-align: center"><img src="../images/pbds_embedded_lists_1.png" style="text-align: middle" alt="Non-unique Mapping Standard Containers"/></div></div></div><br class="figure-break"/><p> 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. </p><div class="figure"><a id="fig.pbds_embedded_lists_2"/><p class="title"><strong>Figure 22.9. Effect of embedded lists in <code class="classname">std::multimap</code> </strong></p><div class="figure-contents"><div class="mediaobject" style="text-align: center"><img src="../images/pbds_embedded_lists_2.png" style="text-align: middle" alt="Effect of embedded lists in std::multimap"/></div></div></div><br class="figure-break"/><p> These embedded linked lists have several disadvantages. </p><div class="orderedlist"><ol class="orderedlist"><li class="listitem"><p> 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. </p></li><li class="listitem"><p> 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 <code class="function">equal_range</code>) can be expensive. </p></li><li class="listitem"><p> The primary key is stored multiply; this uses more memory. </p></li><li class="listitem"><p> 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. </p></li></ol></div><p> 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: </p><div class="orderedlist"><ol class="orderedlist"><li class="listitem"><p> 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. </p></li><li class="listitem"><p> 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. </p></li><li class="listitem"><p> While the interface "understands" that all equivalent-key values constitute a distinct list (through <code class="function">equal_range</code>), 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. </p></li></ol></div><p> 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. </p><p> 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. </p><div class="figure"><a id="id519645"/><p class="title"><strong>Figure 22.10. Non-unique Mapping Containers</strong></p><div class="figure-contents"><div class="mediaobject" style="text-align: center"><img src="../images/pbds_embedded_lists_3.png" style="text-align: middle" alt="Non-unique Mapping Containers"/></div></div></div><br class="figure-break"/><p> 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 <code class="filename">example/basic_multimap.cc</code>); in the second example, one would use an associative container mapping each <code class="classname">int</code> to some size-type indicating the number of times it logically occurs (see <code class="filename">example/basic_multiset.cc</code>. </p><p> See the discussion in list-based container types for containers especially suited as secondary associative-containers. </p></div></div><div class="section" title="Iterator Semantics"><div class="titlepage"><div><div><h4 class="title"><a id="pbds.design.concepts.iterator_semantics"/>Iterator Semantics</h4></div></div></div><div class="section" title="Point and Range Iterators"><div class="titlepage"><div><div><h5 class="title"><a id="concepts.iterator_semantics.point_and_range"/>Point and Range Iterators</h5></div></div></div><p> Iterator concepts are bifurcated in this design, and are comprised of point-type and range-type iteration. </p><p> A point-type iterator is an iterator that refers to a specific element as returned through an associative-container's <code class="function">find</code> method. </p><p> A range-type iterator is an iterator that is used to go over a sequence of elements, as returned by a container's <code class="function">find</code> method. </p><p> 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. </p><p>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. </p></div><div class="section" title="Distinguishing Point and Range Iterators"><div class="titlepage"><div><div><h5 class="title"><a id="concepts.iterator_semantics.both"/>Distinguishing Point and Range Iterators</h5></div></div></div><p>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:</p><pre class="programlisting"> point_const_iterator find(const_key_reference r_key) const; point_iterator find(const_key_reference r_key); std::pair<point_iterator,bool> insert(const_reference r_val); </pre><p>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 <span class="emphasis"><em>A</em></span> <code class="literal">iterator</code>, can always be converted to <code class="literal">point_iterator</code>. In <span class="emphasis"><em>B</em></span> shows invariants for order-preserving containers: point-type iterators are synonymous with range-type iterators. Orthogonally, <span class="emphasis"><em>C</em></span>shows invariants for "set" containers: iterators are synonymous with const iterators.</p><div class="figure"><a id="id519810"/><p class="title"><strong>Figure 22.11. Point Iterator Hierarchy</strong></p><div class="figure-contents"><div class="mediaobject" style="text-align: center"><img src="../images/pbds_point_iterator_hierarchy.png" style="text-align: middle" alt="Point Iterator Hierarchy"/></div></div></div><br class="figure-break"/><p>Note that point-type iterators in self-organizing containers (hash-based associative containers) lack movement operators, such as <code class="literal">operator++</code> - in fact, this is the reason why this library differentiates from the standard C++ librarys design on this point.</p><p>Typically, one can determine an iterator's movement capabilities using <code class="literal">std::iterator_traits<It>iterator_category</code>, which is a <code class="literal">struct</code> indicating the iterator's movement capabilities. Unfortunately, none of the standard predefined categories reflect a pointer's <span class="emphasis"><em>not</em></span> having any movement capabilities whatsoever. Consequently, <code class="literal">pb_ds</code> adds a type <code class="literal">trivial_iterator_tag</code> (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 <code class="literal">forward_iterator_tag</code> retain their common use.</p></div><div class="section" title="Invalidation Guarantees"><div class="titlepage"><div><div><h5 class="title"><a id="pbds.design.concepts.invalidation"/>Invalidation Guarantees</h5></div></div></div><p> 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. </p><p> Given three different types of associative containers, a modifying operation (in that example, <code class="function">erase</code>) 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. </p><p> 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. </p><div class="figure"><a id="id519922"/><p class="title"><strong>Figure 22.12. Invalidation Guarantee Tags Hierarchy</strong></p><div class="figure-contents"><div class="mediaobject" style="text-align: center"><img src="../images/pbds_invalidation_tag_hierarchy.png" style="text-align: middle" alt="Invalidation Guarantee Tags Hierarchy"/></div></div></div><br class="figure-break"/><div class="itemizedlist"><ul class="itemizedlist"><li class="listitem"><p> <code class="classname">basic_invalidation_guarantee</code> 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. </p></li><li class="listitem"><p> <code class="classname">point_invalidation_guarantee</code> 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. </p></li><li class="listitem"><p> <code class="classname">range_invalidation_guarantee</code> corresponds to a guarantee that a range-type iterator remains valid even if the container object is modified. </p></li></ul></div><p>To find the invalidation guarantee of a container, one can use</p><pre class="programlisting"> typename container_traits<Cntnr>::invalidation_guarantee </pre><p>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 <code class="classname">range_invalidation_guarantee</code>) can be cast to its base class (in this case <code class="classname">point_invalidation_guarantee</code>). This means that this this hierarchy can be used easily using standard metaprogramming techniques, by specializing on the type of <code class="literal">invalidation_guarantee</code>.</p><p> These types of problems were addressed, in a more general setting, in <a class="xref" href="policy_data_structures.html#biblio.meyers96more" title="More Effective C++: 35 New Ways to Improve Your Programs and Designs">[biblio.meyers96more]</a> - Item 2. In our opinion, an invalidation-guarantee hierarchy would solve these problems in all container types - not just associative containers. </p></div></div><div class="section" title="Genericity"><div class="titlepage"><div><div><h4 class="title"><a id="pbds.design.concepts.genericity"/>Genericity</h4></div></div></div><p> 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 </p><pre class="programlisting"> template<typename Cntnr> void some_op_sequence(Cntnr &r_container) { ... } </pre><p> then one needs to address the following questions in the body of <code class="function">some_op_sequence</code>: </p><div class="itemizedlist"><ul class="itemizedlist"><li class="listitem"><p> Which types and methods does <code class="literal">Cntnr</code> 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. </p></li><li class="listitem"><p> What are the exception and invalidation guarantees of <code class="literal">Cntnr</code>? 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. </p></li><li class="listitem"><p> 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. </p></li><li class="listitem"><p> How does one query a container about characteristics and capabilities? What is the relationship between two different data structures, if anything? </p></li></ul></div><p>The remainder of this section explains these issues in detail.</p><div class="section" title="Tag"><div class="titlepage"><div><div><h5 class="title"><a id="concepts.genericity.tag"/>Tag</h5></div></div></div><p> Tags are very useful for manipulating generic types. For example, if <code class="literal">It</code> is an iterator class, then <code class="literal">typename It::iterator_category</code> or <code class="literal">typename std::iterator_traits<It>::iterator_category</code> will yield its category, and <code class="literal">typename std::iterator_traits<It>::value_type</code> will yield its value type. </p><p> This library contains a container tag hierarchy corresponding to the diagram below. </p><div class="figure"><a id="id520174"/><p class="title"><strong>Figure 22.13. Container Tag Hierarchy</strong></p><div class="figure-contents"><div class="mediaobject" style="text-align: center"><img src="../images/pbds_container_tag_hierarchy.png" style="text-align: middle" alt="Container Tag Hierarchy"/></div></div></div><br class="figure-break"/><p> Given any container <span class="type">Cntnr</span>, the tag of the underlying data structure can be found via <code class="literal">typename Cntnr::container_category</code>. </p></div><div class="section" title="Traits"><div class="titlepage"><div><div><h5 class="title"><a id="concepts.genericity.traits"/>Traits</h5></div></div></div><p/><p>Additionally, a traits mechanism can be used to query a container type for its attributes. Given any container <code class="literal">Cntnr</code>, then <code class="literal"><Cntnr></code> is a traits class identifying the properties of the container.</p><p>To find if a container can throw when a key is erased (which is true for vector-based trees, for example), one can use </p><pre class="programlisting">container_traits<Cntnr>::erase_can_throw</pre><p> Some of the definitions in <code class="classname">container_traits</code> are dependent on other definitions. If <code class="classname">container_traits<Cntnr>::order_preserving</code> is <code class="constant">true</code> (which is the case for containers based on trees and tries), then the container can be split or joined; in this case, <code class="classname">container_traits<Cntnr>::split_join_can_throw</code> indicates whether splits or joins can throw exceptions (which is true for vector-based trees); otherwise <code class="classname">container_traits<Cntnr>::split_join_can_throw</code> will yield a compilation error. (This is somewhat similar to a compile-time version of the COM model). </p></div></div></div><div class="section" title="By Container"><div class="titlepage"><div><div><h3 class="title"><a id="pbds.design.container"/>By Container</h3></div></div></div><div class="section" title="hash"><div class="titlepage"><div><div><h4 class="title"><a id="pbds.design.container.hash"/>hash</h4></div></div></div><div class="section" title="Interface"><div class="titlepage"><div><div><h5 class="title"><a id="container.hash.interface"/>Interface</h5></div></div></div><p> The collision-chaining hash-based container has the following declaration.</p><pre class="programlisting"> template< typename Key, typename Mapped, typename Hash_Fn = std::hash<Key>, typename Eq_Fn = std::equal_to<Key>, typename Comb_Hash_Fn = direct_mask_range_hashing<> typename Resize_Policy = default explained below. bool Store_Hash = false, typename Allocator = std::allocator<char> > class cc_hash_table; </pre><p>The parameters have the following meaning:</p><div class="orderedlist"><ol class="orderedlist"><li class="listitem"><p><code class="classname">Key</code> is the key type.</p></li><li class="listitem"><p><code class="classname">Mapped</code> is the mapped-policy.</p></li><li class="listitem"><p><code class="classname">Hash_Fn</code> is a key hashing functor.</p></li><li class="listitem"><p><code class="classname">Eq_Fn</code> is a key equivalence functor.</p></li><li class="listitem"><p><code class="classname">Comb_Hash_Fn</code> is a range-hashing_functor; it describes how to translate hash values into positions within the table. </p></li><li class="listitem"><p><code class="classname">Resize_Policy</code> describes how a container object should change its internal size. </p></li><li class="listitem"><p><code class="classname">Store_Hash</code> indicates whether the hash value should be stored with each entry. </p></li><li class="listitem"><p><code class="classname">Allocator</code> is an allocator type.</p></li></ol></div><p>The probing hash-based container has the following declaration.</p><pre class="programlisting"> template< typename Key, typename Mapped, typename Hash_Fn = std::hash<Key>, typename Eq_Fn = std::equal_to<Key>, typename Comb_Probe_Fn = direct_mask_range_hashing<> typename Probe_Fn = default explained below. typename Resize_Policy = default explained below. bool Store_Hash = false, typename Allocator = std::allocator<char> > class gp_hash_table; </pre><p>The parameters are identical to those of the collision-chaining container, except for the following.</p><div class="orderedlist"><ol class="orderedlist"><li class="listitem"><p><code class="classname">Comb_Probe_Fn</code> describes how to transform a probe sequence into a sequence of positions within the table.</p></li><li class="listitem"><p><code class="classname">Probe_Fn</code> describes a probe sequence policy.</p></li></ol></div><p>Some of the default template values depend on the values of other parameters, and are explained below.</p></div><div class="section" title="Details"><div class="titlepage"><div><div><h5 class="title"><a id="container.hash.details"/>Details</h5></div></div></div><div class="section" title="Hash Policies"><div class="titlepage"><div><div><h6 class="title"><a id="container.hash.details.hash_policies"/>Hash Policies</h6></div></div></div><div class="section" title="General"><div class="titlepage"><div><div><h6 class="title"><a id="details.hash_policies.general"/>General</h6></div></div></div><p>Following is an explanation of some functions which hashing involves. The graphic below illustrates the discussion.</p><div class="figure"><a id="id520506"/><p class="title"><strong>Figure 22.14. Hash functions, ranged-hash functions, and range-hashing functions</strong></p><div class="figure-contents"><div class="mediaobject" style="text-align: center"><img src="../images/pbds_hash_ranged_hash_range_hashing_fns.png" style="text-align: middle" alt="Hash functions, ranged-hash functions, and range-hashing functions"/></div></div></div><br class="figure-break"/><p>Let U be a domain (e.g., the integers, or the 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</p><p> f : U × Z<sub>+</sub> → Z<sub>+</sub> </p><p>such that for any u in U ,</p><p>0 ≤ f(u, m) ≤ m - 1</p><p>and which has "good uniformity" properties (say <a class="xref" href="policy_data_structures.html#biblio.knuth98sorting" title="The Art of Computer Programming - Sorting and Searching">[biblio.knuth98sorting]</a>.) One common solution is to use the composition of the hash function</p><p>h : U → Z<sub>+</sub> ,</p><p>which maps elements of U into the non-negative integrals, and</p><p>g : Z<sub>+</sub> × Z<sub>+</sub> → Z<sub>+</sub>,</p><p>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<sub>+</sub>,</p><p>0 ≤ g(r, m) ≤ m - 1</p><p>The resulting ranged-hash function, is</p><div class="equation"><a id="id520621"/><p class="title"><strong>Equation 22.1. Ranged Hash Function</strong></p><div class="equation-contents"><span class="mathphrase"> f(u , m) = g(h(u), m) </span></div></div><br class="equation-break"/><p>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.</p><p>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.</p></div><div class="section" title="Range Hashing"><div class="titlepage"><div><div><h6 class="title"><a id="details.hash_policies.range"/>Range Hashing</h6></div></div></div><p>Some common choices for range-hashing functions are the division, multiplication, and middle-square methods (<a class="xref" href="policy_data_structures.html#biblio.knuth98sorting" title="The Art of Computer Programming - Sorting and Searching">[biblio.knuth98sorting]</a>), defined as</p><div class="equation"><a id="id520670"/><p class="title"><strong>Equation 22.2. Range-Hashing, Division Method</strong></p><div class="equation-contents"><span class="mathphrase"> g(r, m) = r mod m </span></div></div><br class="equation-break"/><p>g(r, m) = ⌈ u/v ( a r mod v ) ⌉</p><p>and</p><p>g(r, m) = ⌈ u/v ( r<sup>2</sup> mod v ) ⌉</p><p>respectively, for some positive integrals u and v (typically powers of 2), and some a. Each of these range-hashing functions works best for some different setting.</p><p>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 & (bit-mask) operation (for the case where m is a power of 2), i.e.,</p><div class="equation"><a id="id520708"/><p class="title"><strong>Equation 22.3. Division via Prime Modulo</strong></p><div class="equation-contents"><span class="mathphrase"> g(r, m) = r % m </span></div></div><br class="equation-break"/><p>and</p><div class="equation"><a id="id520723"/><p class="title"><strong>Equation 22.4. Division via Bit Mask</strong></p><div class="equation-contents"><span class="mathphrase"> g(r, m) = r & m - 1, (with m = 2<sup>k</sup> for some k) </span></div></div><br class="equation-break"/><p>respectively.</p><p>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 .</p><p>The & (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.</p></div><div class="section" title="Ranged Hash"><div class="titlepage"><div><div><h6 class="title"><a id="details.hash_policies.ranged"/>Ranged Hash</h6></div></div></div><p>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 <a class="xref" href="policy_data_structures.html#biblio.knuth98sorting" title="The Art of Computer Programming - Sorting and Searching">[biblio.knuth98sorting]</a>), since the values of m are determined by a resize policy with possibly orthogonal considerations.</p><p>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.</p><p>Let</p><p> s = [ s<sub>0</sub>,..., s<sub>t - 1</sub>] </p><p>be a string of t characters, each of which is from domain S. Consider the following ranged-hash function:</p><div class="equation"><a id="id520803"/><p class="title"><strong>Equation 22.5. A Standard String Hash Function </strong></p><div class="equation-contents"><span class="mathphrase"> f<sub>1</sub>(s, m) = ∑ <sub>i = 0</sub><sup>t - 1</sup> s<sub>i</sub> a<sup>i</sup> mod m </span></div></div><br class="equation-break"/><p>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.</p><p>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</p><p>|S|<sup>k</sup> ≥ m ,</p><p>i.e., using the hash function</p><div class="equation"><a id="id520854"/><p class="title"><strong>Equation 22.6. Only k String DNA Hash </strong></p><div class="equation-contents"><span class="mathphrase"> f<sub>2</sub>(s, m) = ∑ <sub>i = 0</sub><sup>k - 1</sup> s<sub>i</sub> a<sup>i</sup> mod m </span></div></div><br class="equation-break"/><p>requiring scanning over only</p><p>k = log<sub>4</sub>( m )</p><p>characters.</p><p>Other more elaborate hash-functions might scan k characters starting at a random position (determined at each resize), or scanning k random positions (determined at each resize), i.e., using</p><p>f<sub>3</sub>(s, m) = ∑ <sub>i = r</sub>0<sup>r<sub>0</sub> + k - 1</sup> s<sub>i</sub> a<sup>i</sup> mod m ,</p><p>or</p><p>f<sub>4</sub>(s, m) = ∑ <sub>i = 0</sub><sup>k - 1</sup> s<sub>r</sub>i a<sup>r<sub>i</sub></sup> mod m ,</p><p>respectively, for r<sub>0</sub>,..., r<sub>k-1</sub> each in the (inclusive) range [0,...,t-1].</p><p>It should be noted that the above functions cannot be decomposed as per a ranged hash composed of hash and range hashing.</p></div><div class="section" title="Implementation"><div class="titlepage"><div><div><h6 class="title"><a id="details.hash_policies.implementation"/>Implementation</h6></div></div></div><p>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.</p><div class="section" title="Range-Hashing and Ranged-Hashes in Collision-Chaining Tables"><div class="titlepage"><div><div><h6 class="title"><a id="hash_policies.implementation.collision-chaining"/> Range-Hashing and Ranged-Hashes in Collision-Chaining Tables </h6></div></div></div><p><code class="classname">cc_hash_table</code> is parametrized by <code class="classname">Hash_Fn</code> and <code class="classname">Comb_Hash_Fn</code>, a hash functor and a combining hash functor, respectively.</p><p>In general, <code class="classname">Comb_Hash_Fn</code> is considered a range-hashing functor. <code class="classname">cc_hash_table</code> synthesizes a ranged-hash function from <code class="classname">Hash_Fn</code> and <code class="classname">Comb_Hash_Fn</code>. The figure below shows an <code class="classname">insert</code> 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).</p><div class="figure"><a id="id521043"/><p class="title"><strong>Figure 22.15. Insert hash sequence diagram</strong></p><div class="figure-contents"><div class="mediaobject" style="text-align: center"><img src="../images/pbds_hash_range_hashing_seq_diagram.png" style="text-align: middle" alt="Insert hash sequence diagram"/></div></div></div><br class="figure-break"/><p>If <code class="classname">cc_hash_table</code>'s hash-functor, <code class="classname">Hash_Fn</code> is instantiated by <code class="classname">null_type</code> , then <code class="classname">Comb_Hash_Fn</code> is taken to be a ranged-hash function. The graphic below shows an <code class="function">insert</code> 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).</p><div class="figure"><a id="id521102"/><p class="title"><strong>Figure 22.16. Insert hash sequence diagram with a null policy</strong></p><div class="figure-contents"><div class="mediaobject" style="text-align: center"><img src="../images/pbds_hash_range_hashing_seq_diagram2.png" style="text-align: middle" alt="Insert hash sequence diagram with a null policy"/></div></div></div><br class="figure-break"/></div><div class="section" title="Probing tables"><div class="titlepage"><div><div><h6 class="title"><a id="hash_policies.implementation.probe"/> Probing tables </h6></div></div></div><p><code class="classname">gp_hash_table</code> is parametrized by <code class="classname">Hash_Fn</code>, <code class="classname">Probe_Fn</code>, and <code class="classname">Comb_Probe_Fn</code>. As before, if <code class="classname">Hash_Fn</code> and <code class="classname">Probe_Fn</code> are both <code class="classname">null_type</code>, then <code class="classname">Comb_Probe_Fn</code> is a ranged-probe functor. Otherwise, <code class="classname">Hash_Fn</code> is a hash functor, <code class="classname">Probe_Fn</code> is a functor for offsets from a hash value, and <code class="classname">Comb_Probe_Fn</code> transforms a probe sequence into a sequence of positions within the table.</p></div><div class="section" title="Pre-Defined Policies"><div class="titlepage"><div><div><h6 class="title"><a id="hash_policies.implementation.predefined"/> Pre-Defined Policies </h6></div></div></div><p>This library contains some pre-defined classes implementing range-hashing and probing functions:</p><div class="orderedlist"><ol class="orderedlist"><li class="listitem"><p><code class="classname">direct_mask_range_hashing</code> and <code class="classname">direct_mod_range_hashing</code> are range-hashing functions based on a bit-mask and a modulo operation, respectively.</p></li><li class="listitem"><p><code class="classname">linear_probe_fn</code>, and <code class="classname">quadratic_probe_fn</code> are a linear probe and a quadratic probe function, respectively.</p></li></ol></div><p> The graphic below shows the relationships. </p><div class="figure"><a id="id521241"/><p class="title"><strong>Figure 22.17. Hash policy class diagram</strong></p><div class="figure-contents"><div class="mediaobject" style="text-align: center"><img src="../images/pbds_hash_policy_cd.png" style="text-align: middle" alt="Hash policy class diagram"/></div></div></div><br class="figure-break"/></div></div></div><div class="section" title="Resize Policies"><div class="titlepage"><div><div><h6 class="title"><a id="container.hash.details.resize_policies"/>Resize Policies</h6></div></div></div><div class="section" title="General"><div class="titlepage"><div><div><h6 class="title"><a id="resize_policies.general"/>General</h6></div></div></div><p>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:</p><div class="orderedlist"><ol class="orderedlist"><li class="listitem"><p>A size policy indicating how a hash table should grow (e.g., it should multiply by powers of 2).</p></li><li class="listitem"><p>A trigger policy indicating when a hash table should grow (e.g., a load factor is exceeded).</p></li></ol></div></div><div class="section" title="Size Policies"><div class="titlepage"><div><div><h6 class="title"><a id="resize_policies.size"/>Size Policies</h6></div></div></div><p>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.</p></div><div class="section" title="Trigger Policies"><div class="titlepage"><div><div><h6 class="title"><a id="resize_policies.trigger"/>Trigger Policies</h6></div></div></div><p>Trigger policies determine when a hash table changes size. Following is a description of two policies: load-check policies, and collision-check policies.</p><p>Load-check policies are straightforward. The user specifies two factors, Α<sub>min</sub> and Α<sub>max</sub>, and the hash table maintains the invariant that</p><p>Α<sub>min</sub> ≤ (number of stored elements) / (hash-table size) ≤ Α<sub>max</sub><em><span class="remark">load factor min max</span></em></p><p>Collision-check policies work in the opposite direction of 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.</p><p>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.</p><div class="figure"><a id="id521400"/><p class="title"><strong>Figure 22.18. Balls and bins</strong></p><div class="figure-contents"><div class="mediaobject" style="text-align: center"><img src="../images/pbds_balls_and_bins.png" style="text-align: middle" alt="Balls and bins"/></div></div></div><br class="figure-break"/><p>Denote the probability that a probe sequence of length k appears in bin i by p<sub>i</sub>, the length of the probe sequence of bin i by l<sub>i</sub>, and assume uniform distribution. Then</p><div class="equation"><a id="id521446"/><p class="title"><strong>Equation 22.7. Probability of Probe Sequence of Length k </strong></p><div class="equation-contents"><span class="mathphrase"> p<sub>1</sub> = </span></div></div><br class="equation-break"/><p>P(l<sub>1</sub> ≥ k) =</p><p> P(l<sub>1</sub> ≥ α ( 1 + k / α - 1) ≤ (a) </p><p> e ^ ( - ( α ( k / α - 1 )<sup>2</sup> ) /2) </p><p>where (a) follows from the Chernoff bound (<a class="xref" href="policy_data_structures.html#biblio.motwani95random" title="Randomized Algorithms">[biblio.motwani95random]</a>). To calculate the probability that some bin contains a probe sequence greater than k, we note that the l<sub>i</sub> are negatively-dependent (<a class="xref" href="policy_data_structures.html#biblio.dubhashi98neg" title="Balls and bins: A study in negative dependence">[biblio.dubhashi98neg]</a>) . Let I(.) denote the indicator function. Then</p><div class="equation"><a id="id521502"/><p class="title"><strong>Equation 22.8. Probability Probe Sequence in Some Bin </strong></p><div class="equation-contents"><span class="mathphrase"> P( exists<sub>i</sub> l<sub>i</sub> ≥ k ) = </span></div></div><br class="equation-break"/><p>P ( ∑ <sub>i = 1</sub><sup>m</sup> I(l<sub>i</sub> ≥ k) ≥ 1 ) =</p><p>P ( ∑ <sub>i = 1</sub><sup>m</sup> I ( l<sub>i</sub> ≥ k ) ≥ m p<sub>1</sub> ( 1 + 1 / (m p<sub>1</sub>) - 1 ) ) ≤ (a)</p><p>e ^ ( ( - m p<sub>1</sub> ( 1 / (m p<sub>1</sub>) - 1 ) <sup>2</sup> ) / 2 ) ,</p><p>where (a) follows from the fact that the Chernoff bound can be applied to negatively-dependent variables (<a class="xref" href="policy_data_structures.html#biblio.dubhashi98neg" title="Balls and bins: A study in negative dependence">[biblio.dubhashi98neg]</a>). Inserting the first probability equation into the second one, and equating with 1/m, we obtain</p><p>k ~ √ ( 2 α ln 2 m ln(m) ) ) .</p></div><div class="section" title="Implementation"><div class="titlepage"><div><div><h6 class="title"><a id="resize_policies.impl"/>Implementation</h6></div></div></div><p>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.</p><div class="section" title="Decomposition"><div class="titlepage"><div><div><h6 class="title"><a id="resize_policies.impl.decomposition"/>Decomposition</h6></div></div></div><p>Each hash-based container is parametrized by a <code class="classname">Resize_Policy</code> parameter; the container derives <code class="classname">public</code>ly from <code class="classname">Resize_Policy</code>. For example:</p><pre class="programlisting"> cc_hash_table<typename Key, typename Mapped, ... typename Resize_Policy ...> : public Resize_Policy </pre><p>As a container object is modified, it continuously notifies its <code class="classname">Resize_Policy</code> base of internal changes (e.g., collisions encountered and elements being inserted). It queries its <code class="classname">Resize_Policy</code> base whether it needs to be resized, and if so, to what size.</p><p>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).</p><div class="figure"><a id="id521656"/><p class="title"><strong>Figure 22.19. Insert resize sequence diagram</strong></p><div class="figure-contents"><div class="mediaobject" style="text-align: center"><img src="../images/pbds_insert_resize_sequence_diagram1.png" style="text-align: middle" alt="Insert resize sequence diagram"/></div></div></div><br class="figure-break"/><p>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: <code class="classname">hash_standard_resize_policy</code> is parametrized by <code class="classname">Size_Policy</code> and <code class="classname">Trigger_Policy</code>, derives <code class="classname">public</code>ly from both, and acts as a standard delegate (<a class="xref" href="policy_data_structures.html#biblio.gof" title="Design Patterns - Elements of Reusable Object-Oriented Software">[biblio.gof]</a>) to these policies.</p><p>The two graphics immediately below show sequence diagrams illustrating the interaction between the standard resize policy and its trigger and size policies, respectively.</p><div class="figure"><a id="id521721"/><p class="title"><strong>Figure 22.20. Standard resize policy trigger sequence diagram</strong></p><div class="figure-contents"><div class="mediaobject" style="text-align: center"><img src="../images/pbds_insert_resize_sequence_diagram2.png" style="text-align: middle" alt="Standard resize policy trigger sequence diagram"/></div></div></div><br class="figure-break"/><div class="figure"><a id="id521756"/><p class="title"><strong>Figure 22.21. Standard resize policy size sequence diagram</strong></p><div class="figure-contents"><div class="mediaobject" style="text-align: center"><img src="../images/pbds_insert_resize_sequence_diagram3.png" style="text-align: middle" alt="Standard resize policy size sequence diagram"/></div></div></div><br class="figure-break"/></div><div class="section" title="Predefined Policies"><div class="titlepage"><div><div><h6 class="title"><a id="resize_policies.impl.predefined"/>Predefined Policies</h6></div></div></div><p>The library includes the following instantiations of size and trigger policies:</p><div class="orderedlist"><ol class="orderedlist"><li class="listitem"><p><code class="classname">hash_load_check_resize_trigger</code> implements a load check trigger policy.</p></li><li class="listitem"><p><code class="classname">cc_hash_max_collision_check_resize_trigger</code> implements a collision check trigger policy.</p></li><li class="listitem"><p><code class="classname">hash_exponential_size_policy</code> implements an exponential-size policy (which should be used with mask range hashing).</p></li><li class="listitem"><p><code class="classname">hash_prime_size_policy</code> implementing a size policy based on a sequence of primes (which should be used with mod range hashing</p></li></ol></div><p>The graphic below gives an overall picture of the resize-related classes. <code class="classname">basic_hash_table</code> is parametrized by <code class="classname">Resize_Policy</code>, which it subclasses publicly. This class is currently instantiated only by <code class="classname">hash_standard_resize_policy</code>. <code class="classname">hash_standard_resize_policy</code> itself is parametrized by <code class="classname">Trigger_Policy</code> and <code class="classname">Size_Policy</code>. Currently, <code class="classname">Trigger_Policy</code> is instantiated by <code class="classname">hash_load_check_resize_trigger</code>, or <code class="classname">cc_hash_max_collision_check_resize_trigger</code>; <code class="classname">Size_Policy</code> is instantiated by <code class="classname">hash_exponential_size_policy</code>, or <code class="classname">hash_prime_size_policy</code>.</p></div><div class="section" title="Controling Access to Internals"><div class="titlepage"><div><div><h6 class="title"><a id="resize_policies.impl.internals"/>Controling Access to Internals</h6></div></div></div><p>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 <code class="function">size()</code> - 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.</p><p>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.</p><p>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 <code class="classname">cc_hash_table</code> nor <code class="classname">gp_hash_table</code> contain methods for querying the actual size of the table; this is deferred to <code class="classname">hash_standard_resize_policy</code>.</p><p>Furthermore, the policies themselves are parametrized by template arguments that determine the methods they support ( <a class="xref" href="policy_data_structures.html#biblio.alexandrescu01modern" title="Modern C++ Design: Generic Programming and Design Patterns Applied">[biblio.alexandrescu01modern]</a> shows techniques for doing so). <code class="classname">hash_standard_resize_policy</code> is parametrized by <code class="classname">External_Size_Access</code> that determines whether it supports methods for querying the actual size of the table or resizing it. <code class="classname">hash_load_check_resize_trigger</code> is parametrized by <code class="classname">External_Load_Access</code> that determines whether it supports methods for querying or modifying the loads. <code class="classname">cc_hash_max_collision_check_resize_trigger</code> is parametrized by <code class="classname">External_Load_Access</code> that determines whether it supports methods for querying the load.</p><p>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 <code class="classname">private</code> <code class="classname">virtual</code> method:</p><pre class="programlisting"> virtual void do_resize (size_type new_size); </pre><p>which resizes the container. Implementations of <code class="classname">Resize_Policy</code> can export public methods for resizing the container externally; these methods internally call <code class="classname">do_resize</code> to resize the table.</p></div></div></div><div class="section" title="Policy Interactions"><div class="titlepage"><div><div><h6 class="title"><a id="container.hash.details.policy_interaction"/>Policy Interactions</h6></div></div></div><p> </p><p>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.</p><div class="section" title="probe/size/trigger"><div class="titlepage"><div><div><h6 class="title"><a id="policy_interaction.probesizetrigger"/>probe/size/trigger</h6></div></div></div><p>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.</p><p>Unfortunately, this library cannot detect such problems at compilation (they are halting reducible). It therefore defines an exception class <code class="classname">insert_error</code> to throw an exception in this case.</p></div><div class="section" title="hash/trigger"><div class="titlepage"><div><div><h6 class="title"><a id="policy_interaction.hashtrigger"/>hash/trigger</h6></div></div></div><p>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.</p><p>The library, therefore, by design, limits each operation to one resize. For each <code class="classname">insert</code>, for example, it queries only once whether a resize is needed.</p></div><div class="section" title="equivalence functors/storing hash values/hash"><div class="titlepage"><div><div><h6 class="title"><a id="policy_interaction.eqstorehash"/>equivalence functors/storing hash values/hash</h6></div></div></div><p><code class="classname">cc_hash_table</code> and <code class="classname">gp_hash_table</code> are parametrized by an equivalence functor and by a <code class="classname">Store_Hash</code> parameter. If the latter parameter is <code class="classname">true</code>, 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.</p><p>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.</p></div><div class="section" title="size/load-check trigger"><div class="titlepage"><div><div><h6 class="title"><a id="policy_interaction.sizeloadtrigger"/>size/load-check trigger</h6></div></div></div><p>Assume a size policy issues an increasing sequence of sizes a, a q, a q<sup>1</sup>, a q<sup>2</sup>, ... For example, an exponential size policy might issue the sequence of sizes 8, 16, 32, 64, ...</p><p>If a load-check trigger policy is used, with loads α<sub>min</sub> and α<sub>max</sub>, respectively, then it is a good idea to have:</p><div class="orderedlist"><ol class="orderedlist"><li class="listitem"><p>α<sub>max</sub> ~ 1 / q</p></li><li class="listitem"><p>α<sub>min</sub> < 1 / (2 q)</p></li></ol></div><p>This will ensure that the amortized hash cost of each modifying operation is at most approximately 3.</p><p>α<sub>min</sub> ~ α<sub>max</sub> is, in any case, a bad choice, and α<sub>min</sub> > α <sub>max</sub> is horrendous.</p></div></div></div></div><div class="section" title="tree"><div class="titlepage"><div><div><h4 class="title"><a id="pbds.design.container.tree"/>tree</h4></div></div></div><div class="section" title="Interface"><div class="titlepage"><div><div><h5 class="title"><a id="container.tree.interface"/>Interface</h5></div></div></div><p>The tree-based container has the following declaration:</p><pre class="programlisting"> template< typename Key, typename Mapped, typename Cmp_Fn = std::less<Key>, typename Tag = rb_tree_tag, template< typename Const_Node_Iterator, typename Node_Iterator, typename Cmp_Fn_, typename Allocator_> class Node_Update = null_node_update, typename Allocator = std::allocator<char> > class tree; </pre><p>The parameters have the following meaning:</p><div class="orderedlist"><ol class="orderedlist"><li class="listitem"><p><code class="classname">Key</code> is the key type.</p></li><li class="listitem"><p><code class="classname">Mapped</code> is the mapped-policy.</p></li><li class="listitem"><p><code class="classname">Cmp_Fn</code> is a key comparison functor</p></li><li class="listitem"><p><code class="classname">Tag</code> specifies which underlying data structure to use.</p></li><li class="listitem"><p><code class="classname">Node_Update</code> is a policy for updating node invariants.</p></li><li class="listitem"><p><code class="classname">Allocator</code> is an allocator type.</p></li></ol></div><p>The <code class="classname">Tag</code> parameter specifies which underlying data structure to use. Instantiating it by <code class="classname">rb_tree_tag</code>, <code class="classname">splay_tree_tag</code>, or <code class="classname">ov_tree_tag</code>, 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., <code class="classname">reverse_iterator</code> and <code class="classname">rbegin</code>), and different exception and invalidation guarantees.</p></div><div class="section" title="Details"><div class="titlepage"><div><div><h5 class="title"><a id="container.tree.details"/>Details</h5></div></div></div><div class="section" title="Node Invariants"><div class="titlepage"><div><div><h6 class="title"><a id="container.tree.node"/>Node Invariants</h6></div></div></div><p>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 <code class="classname">0.4</code>; the second can easily search for <code class="classname">std::make_pair(10, 41)</code>.</p><p>Each of these trees can efficiently support other queries. The first can efficiently determine that the 2rd key in the tree is <code class="constant">0.3</code>; the second can efficiently determine whether any of its intervals overlaps </p><pre class="programlisting">std::make_pair(29,42)</pre><p> (useful in geometric applications or distributed file systems with leases, for example). It should be noted that an <code class="classname">std::set</code> can only solve these types of problems with linear complexity.</p><p>In order to do so, each tree stores some metadata in each node, and maintains node invariants (see <a class="xref" href="policy_data_structures.html#biblio.clrs2001" title="Introduction to Algorithms, 2nd edition">[biblio.clrs2001]</a>.) The first stores in 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.</p><div class="figure"><a id="id522406"/><p class="title"><strong>Figure 22.22. Tree node invariants</strong></p><div class="figure-contents"><div class="mediaobject" style="text-align: center"><img src="../images/pbds_tree_node_invariants.png" style="text-align: middle" alt="Tree node invariants"/></div></div></div><br class="figure-break"/><p>Supporting such trees is difficult for a number of reasons:</p><div class="orderedlist"><ol class="orderedlist"><li class="listitem"><p>There must be a way to specify what a node's metadata should be (if any).</p></li><li class="listitem"><p>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.</p></li><li class="listitem"><p>The search paths of standard associative containers are defined by comparisons between keys, and not through metadata.</p></li><li class="listitem"><p>It is not feasible to know in advance which methods trees can support. Besides the usual <code class="classname">find</code> method, the first tree can support a <code class="classname">find_by_order</code> method, while the second can support an <code class="classname">overlaps</code> method.</p></li></ol></div><div class="figure"><a id="id522484"/><p class="title"><strong>Figure 22.23. Tree node invalidation</strong></p><div class="figure-contents"><div class="mediaobject" style="text-align: center"><img src="../images/pbds_tree_node_invalidations.png" style="text-align: middle" alt="Tree node invalidation"/></div></div></div><br class="figure-break"/><p>These problems are solved by a combination of two means: node iterators, and template-template node updater parameters.</p><div class="section" title="Node Iterators"><div class="titlepage"><div><div><h6 class="title"><a id="container.tree.node.iterators"/>Node Iterators</h6></div></div></div><p>Each tree-based container defines two additional iterator types, <code class="classname">const_node_iterator</code> and <code class="classname">node_iterator</code>. 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 <code class="classname">tree</code> supports the methods:</p><pre class="programlisting"> const_node_iterator node_begin() const; node_iterator node_begin(); const_node_iterator node_end() const; node_iterator node_end(); </pre><p>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.</p></div><div class="section" title="Node Updator"><div class="titlepage"><div><div><h6 class="title"><a id="container.tree.node.updator"/>Node Updator</h6></div></div></div><p>The tree-based containers are parametrized by a <code class="classname">Node_Update</code> template-template parameter. A tree-based container instantiates <code class="classname">Node_Update</code> to some <code class="classname">node_update</code> class, and publicly subclasses <code class="classname">node_update</code>. The graphic below shows this scheme, as well as some predefined policies (which are explained below).</p><div class="figure"><a id="id522594"/><p class="title"><strong>Figure 22.24. A tree and its update policy</strong></p><div class="figure-contents"><div class="mediaobject" style="text-align: center"><img src="../images/pbds_tree_node_updator_policy_cd.png" style="text-align: middle" alt="A tree and its update policy"/></div></div></div><br class="figure-break"/><p><code class="classname">node_update</code> (an instantiation of <code class="classname">Node_Update</code>) must define <code class="classname">metadata_type</code> as the type of metadata it requires. For order statistics, e.g., <code class="classname">metadata_type</code> might be <code class="classname">size_t</code>. The tree defines within each node a <code class="classname">metadata_type</code> object.</p><p><code class="classname">node_update</code> must also define the following method for restoring node invariants:</p><pre class="programlisting"> void operator()(node_iterator nd_it, const_node_iterator end_nd_it) </pre><p>In this method, <code class="varname">nd_it</code> is a <code class="classname">node_iterator</code> corresponding to a node whose A) all descendants have valid invariants, and B) its own invariants might be violated; <code class="classname">end_nd_it</code> is a <code class="classname">const_node_iterator</code> corresponding to a just-after-leaf node. This method should correct the node invariants of the node pointed to by <code class="classname">nd_it</code>. 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.</p><div class="figure"><a id="id522691"/><p class="title"><strong>Figure 22.25. Restoring node invariants</strong></p><div class="figure-contents"><div class="mediaobject" style="text-align: center"><img src="../images/pbds_restoring_node_invariants.png" style="text-align: middle" alt="Restoring node invariants"/></div></div></div><br class="figure-break"/><p>When a tree operation might invalidate some node invariant, it invokes this method in its <code class="classname">node_update</code> base to restore the invariant. For example, the graphic below shows an <code class="function">insert</code> 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 <code class="function">insert</code>, <code class="function">erase</code>, <code class="function">split</code> or <code class="function">join</code>, can restore all node invariants by a small number of node invariant updates (<a class="xref" href="policy_data_structures.html#biblio.clrs2001" title="Introduction to Algorithms, 2nd edition">[biblio.clrs2001]</a>) .</p><div class="figure"><a id="id522759"/><p class="title"><strong>Figure 22.26. Insert update sequence</strong></p><div class="figure-contents"><div class="mediaobject" style="text-align: center"><img src="../images/pbds_update_seq_diagram.png" style="text-align: middle" alt="Insert update sequence"/></div></div></div><br class="figure-break"/><p>To complete the description of the scheme, three questions need to be answered:</p><div class="orderedlist"><ol class="orderedlist"><li class="listitem"><p>How can a tree which supports order statistics define a method such as <code class="classname">find_by_order</code>?</p></li><li class="listitem"><p>How can the node updater base access methods of the tree?</p></li><li class="listitem"><p>How can the following cyclic dependency be resolved? <code class="classname">node_update</code> is a base class of the tree, yet it uses node iterators defined in the tree (its child).</p></li></ol></div><p>The first two questions are answered by the fact that <code class="classname">node_update</code> (an instantiation of <code class="classname">Node_Update</code>) is a <span class="emphasis"><em>public</em></span> base class of the tree. Consequently:</p><div class="orderedlist"><ol class="orderedlist"><li class="listitem"><p>Any public methods of <code class="classname">node_update</code> are automatically methods of the tree (<a class="xref" href="policy_data_structures.html#biblio.alexandrescu01modern" title="Modern C++ Design: Generic Programming and Design Patterns Applied">[biblio.alexandrescu01modern]</a>). Thus an order-statistics node updater, <code class="classname">tree_order_statistics_node_update</code> defines the <code class="function">find_by_order</code> method; any tree instantiated by this policy consequently supports this method as well.</p></li><li class="listitem"><p>In C++, if a base class declares a method as <code class="literal">virtual</code>, it is <code class="literal">virtual</code> in its subclasses. If <code class="classname">node_update</code> needs to access one of the tree's methods, say the member function <code class="function">end</code>, it simply declares that method as <code class="literal">virtual</code> abstract.</p></li></ol></div><p>The cyclic dependency is solved through template-template parameters. <code class="classname">Node_Update</code> is parametrized by the tree's node iterators, its comparison functor, and its allocator type. Thus, instantiations of <code class="classname">Node_Update</code> have all information required.</p><p>This library assumes that constructing a metadata object and modifying it are exception free. Suppose that during some method, say <code class="classname">insert</code>, a metadata-related operation (e.g., changing the value of a metadata) throws an exception. Ack! Rolling back the method is unusually complex.</p><p>Previously, a distinction was made between redundant policies and null policies. Node invariants show a case where null policies are required.</p><p>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:</p><div class="orderedlist"><ol class="orderedlist"><li class="listitem"><p>Each node would carry a useless metadata object, wasting space.</p></li><li class="listitem"><p>The tree cannot know if its <code class="classname">Node_Update</code> 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.</p></li></ol></div><div class="figure"><a id="id522945"/><p class="title"><strong>Figure 22.27. Useless update path</strong></p><div class="figure-contents"><div class="mediaobject" style="text-align: center"><img src="../images/pbds_rationale_null_node_updator.png" style="text-align: middle" alt="Useless update path"/></div></div></div><br class="figure-break"/><p>A null policy class, <code class="classname">null_node_update</code> solves both these problems. The tree detects that node invariants are irrelevant, and defines all accordingly.</p></div></div><div class="section" title="Split and Join"><div class="titlepage"><div><div><h6 class="title"><a id="container.tree.details.split"/>Split and Join</h6></div></div></div><p>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:</p><div class="orderedlist"><ol class="orderedlist"><li class="listitem"><p>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.</p></li><li class="listitem"><p>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. </p></li></ol></div></div></div></div><div class="section" title="Trie"><div class="titlepage"><div><div><h4 class="title"><a id="pbds.design.container.trie"/>Trie</h4></div></div></div><div class="section" title="Interface"><div class="titlepage"><div><div><h5 class="title"><a id="container.trie.interface"/>Interface</h5></div></div></div><p>The trie-based container has the following declaration:</p><pre class="programlisting"> template<typename Key, typename Mapped, typename Cmp_Fn = std::less<Key>, typename Tag = pat_trie_tag, template<typename Const_Node_Iterator, typename Node_Iterator, typename E_Access_Traits_, typename Allocator_> class Node_Update = null_node_update, typename Allocator = std::allocator<char> > class trie; </pre><p>The parameters have the following meaning:</p><div class="orderedlist"><ol class="orderedlist"><li class="listitem"><p><code class="classname">Key</code> is the key type.</p></li><li class="listitem"><p><code class="classname">Mapped</code> is the mapped-policy.</p></li><li class="listitem"><p><code class="classname">E_Access_Traits</code> is described in below.</p></li><li class="listitem"><p><code class="classname">Tag</code> specifies which underlying data structure to use, and is described shortly.</p></li><li class="listitem"><p><code class="classname">Node_Update</code> is a policy for updating node invariants. This is described below.</p></li><li class="listitem"><p><code class="classname">Allocator</code> is an allocator type.</p></li></ol></div><p>The <code class="classname">Tag</code> parameter specifies which underlying data structure to use. Instantiating it by <code class="classname">pat_trie_tag</code>, specifies an underlying PATRICIA trie (explained shortly); any other tag is currently illegal.</p><p>Following is a description of a (PATRICIA) trie (this implementation follows <a class="xref" href="policy_data_structures.html#biblio.okasaki98mereable" title="Fast mergeable integer maps">[biblio.okasaki98mereable]</a> and <a class="xref" href="policy_data_structures.html#biblio.filliatre2000ptset" title="Ptset: Sets of integers implemented as Patricia trees">[biblio.filliatre2000ptset]</a>). </p><p>A (PATRICIA) trie is similar to a tree, but with the following differences:</p><div class="orderedlist"><ol class="orderedlist"><li class="listitem"><p>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.</p></li><li class="listitem"><p>It is not (necessarily) binary. Each node has fan-out n + 1, where n is the number of distinct elements.</p></li><li class="listitem"><p>It stores values only at leaf nodes.</p></li><li class="listitem"><p>Internal nodes have the properties that A) each has at least two children, and B) each shares the same prefix with any of its descendant.</p></li></ol></div><p>A (PATRICIA) trie has some useful properties:</p><div class="orderedlist"><ol class="orderedlist"><li class="listitem"><p>It can be configured to use large node fan-out, giving it very efficient find performance (albeit at insertion complexity and size).</p></li><li class="listitem"><p>It works well for common-prefix keys.</p></li><li class="listitem"><p>It can support efficiently queries such as which 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.</p></li></ol></div></div><div class="section" title="Details"><div class="titlepage"><div><div><h5 class="title"><a id="container.trie.details"/>Details</h5></div></div></div><div class="section" title="Element Access Traits"><div class="titlepage"><div><div><h6 class="title"><a id="container.trie.details.etraits"/>Element Access Traits</h6></div></div></div><p>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 <code class="varname">c</code> to </p><pre class="programlisting">static_cast<size_t>(c)</pre><p>.</p><p>Seemingly, then, a trie can assume that its keys support (const) iterators, and that the <code class="classname">value_type</code> of this iterator can be cast to a <code class="classname">size_t</code>. There are several reasons, though, to decouple the mechanism by which the trie accesses its keys' elements from the trie:</p><div class="orderedlist"><ol class="orderedlist"><li class="listitem"><p>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.</p></li><li class="listitem"><p>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' <code class="classname">reverse_iterator</code>. 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.</p></li></ol></div><p>trie is, consequently, parametrized by <code class="classname">E_Access_Traits</code> - traits which instruct how to access sequences' elements. <code class="classname">string_trie_e_access_traits</code> is a traits class for strings. Each such traits define some types, like:</p><pre class="programlisting"> typename E_Access_Traits::const_iterator </pre><p>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.</p><p>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).</p><p>The leaf nodes contain values; each internal node contains two <code class="classname">typename E_Access_Traits::const_iterator</code> 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 <code class="varname">'a'</code>, and the other to <code class="varname">'s'</code>.</p><div class="figure"><a id="id523317"/><p class="title"><strong>Figure 22.28. A PATRICIA trie</strong></p><div class="figure-contents"><div class="mediaobject" style="text-align: center"><img src="../images/pbds_pat_trie.png" style="text-align: middle" alt="A PATRICIA trie"/></div></div></div><br class="figure-break"/></div><div class="section" title="Node Invariants"><div class="titlepage"><div><div><h6 class="title"><a id="container.trie.details.node"/>Node Invariants</h6></div></div></div><p>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:</p><div class="orderedlist"><ol class="orderedlist"><li class="listitem"><p>A trie's <code class="classname">Node_Update</code> template-template parameter is parametrized by <code class="classname">E_Access_Traits</code>, while a tree's <code class="classname">Node_Update</code> template-template parameter is parametrized by <code class="classname">Cmp_Fn</code>.</p></li><li class="listitem"><p>Tree-based containers store values in all nodes, while trie-based containers (at least in this implementation) store values in leafs.</p></li></ol></div><p>The graphic below shows the scheme, as well as some predefined policies (which are explained below).</p><div class="figure"><a id="id523405"/><p class="title"><strong>Figure 22.29. A trie and its update policy</strong></p><div class="figure-contents"><div class="mediaobject" style="text-align: center"><img src="../images/pbds_trie_node_updator_policy_cd.png" style="text-align: middle" alt="A trie and its update policy"/></div></div></div><br class="figure-break"/><p>This library offers the following pre-defined trie node updating policies:</p><div class="orderedlist"><ol class="orderedlist"><li class="listitem"><p> <code class="classname">trie_order_statistics_node_update</code> supports order statistics. </p></li><li class="listitem"><p><code class="classname">trie_prefix_search_node_update</code> supports searching for ranges that match a given prefix.</p></li><li class="listitem"><p><code class="classname">null_node_update</code> is the null node updater.</p></li></ol></div></div><div class="section" title="Split and Join"><div class="titlepage"><div><div><h6 class="title"><a id="container.trie.details.split"/>Split and Join</h6></div></div></div><p>Trie-based containers support split and join methods; the rationale is equal to that of tree-based containers supporting these methods.</p></div></div></div><div class="section" title="List"><div class="titlepage"><div><div><h4 class="title"><a id="pbds.design.container.list"/>List</h4></div></div></div><div class="section" title="Interface"><div class="titlepage"><div><div><h5 class="title"><a id="container.list.interface"/>Interface</h5></div></div></div><p>The list-based container has the following declaration:</p><pre class="programlisting"> template<typename Key, typename Mapped, typename Eq_Fn = std::equal_to<Key>, typename Update_Policy = move_to_front_lu_policy<>, typename Allocator = std::allocator<char> > class list_update; </pre><p>The parameters have the following meaning:</p><div class="orderedlist"><ol class="orderedlist"><li class="listitem"><p> <code class="classname">Key</code> is the key type. </p></li><li class="listitem"><p> <code class="classname">Mapped</code> is the mapped-policy. </p></li><li class="listitem"><p> <code class="classname">Eq_Fn</code> is a key equivalence functor. </p></li><li class="listitem"><p> <code class="classname">Update_Policy</code> is a policy updating positions in the list based on access patterns. It is described in the following subsection. </p></li><li class="listitem"><p> <code class="classname">Allocator</code> is an allocator type. </p></li></ol></div><p>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.</p><p>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.</p><p>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 <a class="xref" href="policy_data_structures.html#biblio.motwani95random" title="Randomized Algorithms">[biblio.motwani95random]</a> and <a class="xref" href="policy_data_structures.html#biblio.andrew04mtf" title="MTF, Bit, and COMB: A Guide to Deterministic and Randomized Algorithms for the List Update Problem">[biblio.andrew04mtf]</a>). </p></div><div class="section" title="Details"><div class="titlepage"><div><div><h5 class="title"><a id="container.list.details"/>Details</h5></div></div></div><p> </p><div class="section" title="Underlying Data Structure"><div class="titlepage"><div><div><h6 class="title"><a id="container.list.details.ds"/>Underlying Data Structure</h6></div></div></div><p>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.</p><div class="figure"><a id="id523660"/><p class="title"><strong>Figure 22.30. A simple list</strong></p><div class="figure-contents"><div class="mediaobject" style="text-align: center"><img src="../images/pbds_simple_list.png" style="text-align: middle" alt="A simple list"/></div></div></div><br class="figure-break"/><p>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.</p><p>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. </p><div class="figure"><a id="id523706"/><p class="title"><strong>Figure 22.31. The counter algorithm</strong></p><div class="figure-contents"><div class="mediaobject" style="text-align: center"><img src="../images/pbds_list_update.png" style="text-align: middle" alt="The counter algorithm"/></div></div></div><br class="figure-break"/></div><div class="section" title="Policies"><div class="titlepage"><div><div><h6 class="title"><a id="container.list.details.policies"/>Policies</h6></div></div></div><p>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).</p><p>Associative containers based on lists are parametrized by a <code class="classname">Update_Policy</code> 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. </p><p>An instantiation of <code class="classname">Update_Policy</code> must define internally <code class="classname">update_metadata</code> as the metadata it requires. Internally, each node of the list contains, besides the usual key and data, an instance of <code class="classname">typename Update_Policy::update_metadata</code>.</p><p>An instantiation of <code class="classname">Update_Policy</code> must define internally two operators:</p><pre class="programlisting"> update_metadata operator()(); bool operator()(update_metadata &); </pre><p>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. </p><p>The library contains two predefined implementations of list-update policies. The first is <code class="classname">lu_counter_policy</code>, which implements the counter algorithm described above. The second is <code class="classname">lu_move_to_front_policy</code>, 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 <a class="xref" href="policy_data_structures.html#biblio.andrew04mtf" title="MTF, Bit, and COMB: A Guide to Deterministic and Randomized Algorithms for the List Update Problem">[biblio.andrew04mtf]</a> </p></div><div class="section" title="Use in Multimaps"><div class="titlepage"><div><div><h6 class="title"><a id="container.list.details.mapped"/>Use in Multimaps</h6></div></div></div><p>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.</p><p>List-based containers are especially useful as associative containers for secondary keys. In fact, they are implemented here expressly for this purpose.</p><p>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).</p><p>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).</p><p>In order to reduce the per-container memory overhead as much as possible, they are implemented as closely as possible to singly-linked lists.</p><div class="orderedlist"><ol class="orderedlist"><li class="listitem"><p> List-based containers do not store internally the number of values that they hold. This means that their <code class="function">size</code> method has linear complexity (just like <code class="classname">std::list</code>). Note that finding the number of equivalent-key values in a standard multimap also has linear complexity (because it must be done, via <code class="function">std::distance</code> of the multimap's <code class="function">equal_range</code> method), but usually with higher constants. </p></li><li class="listitem"><p> 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. </p></li></ol></div></div></div></div><div class="section" title="Priority Queue"><div class="titlepage"><div><div><h4 class="title"><a id="pbds.design.container.priority_queue"/>Priority Queue</h4></div></div></div><div class="section" title="Interface"><div class="titlepage"><div><div><h5 class="title"><a id="container.priority_queue.interface"/>Interface</h5></div></div></div><p>The priority queue container has the following declaration: </p><pre class="programlisting"> template<typename Value_Type, typename Cmp_Fn = std::less<Value_Type>, typename Tag = pairing_heap_tag, typename Allocator = std::allocator<char > > class priority_queue; </pre><p>The parameters have the following meaning:</p><div class="orderedlist"><ol class="orderedlist"><li class="listitem"><p><code class="classname">Value_Type</code> is the value type.</p></li><li class="listitem"><p><code class="classname">Cmp_Fn</code> is a value comparison functor</p></li><li class="listitem"><p><code class="classname">Tag</code> specifies which underlying data structure to use.</p></li><li class="listitem"><p><code class="classname">Allocator</code> is an allocator type.</p></li></ol></div><p>The <code class="classname">Tag</code> parameter specifies which underlying data structure to use. Instantiating it by<code class="classname">pairing_heap_tag</code>,<code class="classname">binary_heap_tag</code>, <code class="classname">binomial_heap_tag</code>, <code class="classname">rc_binomial_heap_tag</code>, or <code class="classname">thin_heap_tag</code>, specifies, respectively, an underlying pairing heap (<a class="xref" href="policy_data_structures.html#biblio.fredman86pairing" title="The pairing heap: a new form of self-adjusting heap">[biblio.fredman86pairing]</a>), binary heap (<a class="xref" href="policy_data_structures.html#biblio.clrs2001" title="Introduction to Algorithms, 2nd edition">[biblio.clrs2001]</a>), binomial heap (<a class="xref" href="policy_data_structures.html#biblio.clrs2001" title="Introduction to Algorithms, 2nd edition">[biblio.clrs2001]</a>), a binomial heap with a redundant binary counter (<a class="xref" href="policy_data_structures.html#biblio.maverik_lowerbounds" title="Deamortization - Part 2: Binomial Heaps">[biblio.maverik_lowerbounds]</a>), or a thin heap (<a class="xref" href="policy_data_structures.html#biblio.kt99fat_heaps" title="New Heap Data Structures">[biblio.kt99fat_heaps]</a>). </p><p> As mentioned in the tutorial, <code class="classname">__gnu_pbds::priority_queue</code> shares most of the same interface with <code class="classname">std::priority_queue</code>. E.g. if <code class="varname">q</code> is a priority queue of type <code class="classname">Q</code>, then <code class="function">q.top()</code> will return the "largest" value in the container (according to <code class="classname">typename Q::cmp_fn</code>). <code class="classname">__gnu_pbds::priority_queue</code> has a larger (and very slightly different) interface than <code class="classname">std::priority_queue</code>, however, since typically <code class="classname">push</code> and <code class="classname">pop</code> are deemed insufficient for manipulating priority-queues. </p><p>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.</p></div><div class="section" title="Details"><div class="titlepage"><div><div><h5 class="title"><a id="container.priority_queue.details"/>Details</h5></div></div></div><div class="section" title="Iterators"><div class="titlepage"><div><div><h6 class="title"><a id="container.priority_queue.details.iterators"/>Iterators</h6></div></div></div><p>There are many different underlying-data structures for implementing priority queues. Unfortunately, most such structures are oriented towards making <code class="function">push</code> and <code class="function">top</code> efficient, and consequently don't allow efficient access of other elements: for instance, they cannot support an efficient <code class="function">find</code> method. In the use case where it is important to both access and "do something with" an arbitrary value, one would be out of luck. For example, many graph algorithms require modifying a value (typically increasing it in the sense of the priority queue's comparison functor).</p><p>In order to access and manipulate an arbitrary value in a priority queue, one needs to reference the internals of the priority queue from some form of an associative container - this is unavoidable. Of course, in order to maintain the encapsulation of the priority queue, this needs to be done in a way that minimizes exposure to implementation internals.</p><p>In this library the priority queue's <code class="function">insert</code> method returns an iterator, which if valid can be used for subsequent <code class="function">modify</code> and <code class="function">erase</code> operations. This both preserves the priority queue's encapsulation, and allows accessing arbitrary values (since the returned iterators from the <code class="function">push</code> operation can be stored in some form of associative container).</p><p>Priority queues' iterators present a problem regarding their invalidation guarantees. One assumes that calling <code class="function">operator++</code> 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 <code class="function">push</code> 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 <code class="function">push</code> 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 <code class="function">modify</code> and <code class="function">erase</code> operations.</p><p>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 <code class="classname">iterator</code> can always be converted to a <code class="classname">point_iterator</code>, and a <code class="classname">const_iterator</code> can always be converted to a <code class="classname">point_const_iterator</code>.</p><p>The following snippet demonstrates manipulating an arbitrary value:</p><pre class="programlisting"> // A priority queue of integers. priority_queue<int > p; // Insert some values into the priority queue. priority_queue<int >::point_iterator it = p.push(0); p.push(1); p.push(2); // Now modify a value. p.modify(it, 3); assert(p.top() == 3); </pre><p>It should be noted that an alternative design could embed an associative container in a priority queue. Could, but most probably should not. To begin with, it should be noted that one could always encapsulate a priority queue and an associative container mapping values to priority queue iterators with no performance loss. One cannot, however, "un-encapsulate" a priority queue embedding an associative container, which might lead to performance loss. Assume, that one needs to associate each value with some data unrelated to priority queues. Then using 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.</p></div><div class="section" title="Underlying Data Structure"><div class="titlepage"><div><div><h6 class="title"><a id="container.priority_queue.details.d"/>Underlying Data Structure</h6></div></div></div><p>There are three main implementations of priority queues: the first employs a binary heap, typically one which uses a sequence; the second uses a tree (or forest of trees), which is typically less structured than an associative container's tree; the third simply uses an associative container. These are shown in the graphic below, in labels A1 and A2, label B, and label C.</p><div class="figure"><a id="id524238"/><p class="title"><strong>Figure 22.32. Underlying Priority-Queue Data-Structures.</strong></p><div class="figure-contents"><div class="mediaobject" style="text-align: center"><img src="../images/pbds_priority_queue_different_underlying_dss.png" style="text-align: middle" alt="Underlying Priority-Queue Data-Structures."/></div></div></div><br class="figure-break"/><p>Roughly speaking, any value that is both pushed and popped 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 <a class="xref" href="policy_data_structures.html#biblio.clrs2001" title="Introduction to Algorithms, 2nd edition">[biblio.clrs2001]</a> and <a class="xref" href="policy_data_structures.html#biblio.brodal96priority" title="Worst-case efficient priority queues">[biblio.brodal96priority]</a>). </p><p>Most implementations do not differ in the asymptotic amortized complexity of <code class="function">push</code> and <code class="function">pop</code> operations, but they differ in the constants involved, in the complexity of other operations (e.g., <code class="function">modify</code>), 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 <code class="function">push</code> and <code class="function">pop</code> operations.</p><p>This library implements different algorithms using a single class: <code class="classname">priority_queue</code>. Instantiating the <code class="classname">Tag</code> template parameter, "selects" the implementation:</p><div class="orderedlist"><ol class="orderedlist"><li class="listitem"><p> Instantiating <code class="classname">Tag = binary_heap_tag</code> 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 <code class="classname">Value_Type</code> is instantiated by a primitive type (e.g., an <span class="type">int</span>); the latter is internally selected for all other types (e.g., <code class="classname">std::string</code>). This implementations is relatively unstructured, and so has good <code class="classname">push</code> and <code class="classname">pop</code> performance; it is the "best-in-kind" for primitive types, e.g., <span class="type">int</span>s. Conversely, it has high worst-case performance, and can support only linear-time <code class="function">modify</code> and <code class="function">erase</code> operations.</p></li><li class="listitem"><p>Instantiating <code class="classname">Tag = pairing_heap_tag</code> 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 <code class="function">push</code> and <code class="function">pop</code> performance; it is the "best-in-kind" for non-primitive types, e.g., <code class="classname">std:string</code>s. It also has very good worst-case <code class="function">push</code> and <code class="function">join</code> performance (O(1)), but has high worst-case <code class="function">pop</code> complexity.</p></li><li class="listitem"><p>Instantiating <code class="classname">Tag = binomial_heap_tag</code> 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 <code class="function">push</code> and <code class="function">pop</code> performance. Conversely, it has sub-linear worst-case bounds for <code class="function">pop</code>, e.g., and so it might be preferred in cases where responsiveness is important.</p></li><li class="listitem"><p>Instantiating <code class="classname">Tag = rc_binomial_heap_tag</code> 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 <code class="function">push</code> and <code class="function">pop</code> performance. Conversely, it guarantees O(1) <code class="function">push</code> complexity, and so it might be preferred in cases where the responsiveness of a binomial heap is insufficient.</p></li><li class="listitem"><p>Instantiating <code class="classname">Tag = thin_heap_tag</code> 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 <code class="function">push</code> and <code class="function">pop</code> 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.</p></li></ol></div><p>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 <code class="classname">std::priority_queue</code> can adapt <code class="classname">std::vector</code>). 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 <code class="function">push</code> and <code class="function">pop</code> performance.</p></div><div class="section" title="Traits"><div class="titlepage"><div><div><h6 class="title"><a id="container.priority_queue.details.traits"/>Traits</h6></div></div></div><p>It would be nice if all priority queues could share exactly the same behavior regardless of implementation. Sadly, this is not possible. Just one for instance is in join operations: joining two binary heaps might throw an exception (not corrupt any of the heaps on which it operates), but joining two pairing heaps is exception free.</p><p>Tags and traits are very useful for manipulating generic types. <code class="classname">__gnu_pbds::priority_queue</code> publicly defines <code class="classname">container_category</code> as one of the tags. Given any container <code class="classname">Cntnr</code>, the tag of the underlying data structure can be found via <code class="classname">typename Cntnr::container_category</code>; this is one of the possible tags shown in the graphic below. </p><div class="figure"><a id="id524529"/><p class="title"><strong>Figure 22.33. Priority-Queue Data-Structure Tags.</strong></p><div class="figure-contents"><div class="mediaobject" style="text-align: center"><img src="../images/pbds_priority_queue_tag_hierarchy.png" style="text-align: middle" alt="Priority-Queue Data-Structure Tags."/></div></div></div><br class="figure-break"/><p>Additionally, a traits mechanism can be used to query a container type for its attributes. Given any container <code class="classname">Cntnr</code>, then </p><pre class="programlisting">__gnu_pbds::container_traits<Cntnr></pre><p> is a traits class identifying the properties of the container.</p><p>To find if a container might throw if two of its objects are joined, one can use </p><pre class="programlisting"> container_traits<Cntnr>::split_join_can_throw </pre><p> </p><p> 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 </p><pre class="programlisting"> container_traits<Cntnr>::invalidation_guarantee </pre><p> to get the invalidation guarantee type of a priority queue.</p><p>It is easy to understand from the graphic above, what <code class="classname">container_traits<Cntnr>::invalidation_guarantee</code> will be for different implementations. All implementations of type represented by label B have <code class="classname">point_invalidation_guarantee</code>: 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 <code class="classname">basic_invalidation_guarantee</code>: the container can freely internally reallocate the array - both point-type and range-type iterators might be invalidated.</p><p> This has major implications, and constitutes a good reason to avoid using binary heaps. A binary heap can perform <code class="function">modify</code> or <code class="function">erase</code> 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 <code class="function">modify</code> or <code class="function">erase</code> operations is non-negligible (say super-logarithmic in the total sequence of operations) - binary heaps will perform badly. </p></div></div></div></div></div><div class="navfooter"><hr/><table width="100%" summary="Navigation footer"><tr><td align="left"><a accesskey="p" href="policy_data_structures_using.html">Prev</a> </td><td align="center"><a accesskey="u" href="policy_data_structures.html">Up</a></td><td align="right"> <a accesskey="n" href="policy_based_data_structures_test.html">Next</a></td></tr><tr><td align="left" valign="top">Using </td><td align="center"><a accesskey="h" href="../index.html">Home</a></td><td align="right" valign="top"> Testing</td></tr></table></div></body></html>