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3 <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>
4 Associative containers are typically parametrized by various
5 policies. For example, a hash-based associative container is
6 parametrized by a hash-functor, transforming each key into an
7 non-negative numerical type. Each such value is then further mapped
8 into a position within the table. The mapping of a key into a
9 position within the table is therefore a two-step process.
11 In some cases, instantiations are redundant. For example, when the
12 keys are integers, it is possible to use a redundant hash policy,
13 which transforms each key into its value.
15 In some other cases, these policies are irrelevant. For example, a
16 hash-based associative container might transform keys into positions
17 within a table by a different method than the two-step method
18 described above. In such a case, the hash functor is simply
21 When a policy is either redundant or irrelevant, it can be replaced
22 by <code class="classname">null_type</code>.
24 For example, a <span class="emphasis"><em>set</em></span> is an associative
25 container with one of its template parameters (the one for the
26 mapped type) replaced with <code class="classname">null_type</code>. Other
27 places simplifications are made possible with this technique
28 include node updates in tree and trie data structures, and hash
29 and probe functions for hash data structures.
30 </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"/>
31 Distinguishing Between Maps and Sets
32 </h5></div></div></div><p>
33 Anyone familiar with the standard knows that there are four kinds
34 of associative containers: maps, sets, multimaps, and
35 multisets. The map datatype associates each key to
38 Sets are associative containers that simply store keys -
39 they do not map them to anything. In the standard, each map class
40 has a corresponding set class. E.g.,
41 <code class="classname">std::map<int, char></code> maps each
42 <code class="classname">int</code> to a <code class="classname">char</code>, but
43 <code class="classname">std::set<int, char></code> simply stores
44 <code class="classname">int</code>s. In this library, however, there are no
45 distinct classes for maps and sets. Instead, an associative
46 container's <code class="classname">Mapped</code> template parameter is a policy: if
47 it is instantiated by <code class="classname">null_type</code>, then it
48 is a "set"; otherwise, it is a "map". E.g.,
49 </p><pre class="programlisting">
50 cc_hash_table<int, char>
52 is a "map" mapping each <span class="type">int</span> value to a <span class="type">
54 </p><pre class="programlisting">
55 cc_hash_table<int, null_type>
57 is a type that uniquely stores <span class="type">int</span> values.
58 </p><p>Once the <code class="classname">Mapped</code> template parameter is instantiated
59 by <code class="classname">null_type</code>, then
60 the "set" acts very similarly to the standard's sets - it does not
61 map each key to a distinct <code class="classname">null_type</code> object. Also,
62 , the container's <span class="type">value_type</span> is essentially
63 its <span class="type">key_type</span> - just as with the standard's sets
65 The standard's multimaps and multisets allow, respectively,
66 non-uniquely mapping keys and non-uniquely storing keys. As
68 reasons why this might be necessary are 1) that a key might be
69 decomposed into a primary key and a secondary key, 2) that a
70 key might appear more than once, or 3) any arbitrary
71 combination of 1)s and 2)s. Correspondingly,
72 one should use 1) "maps" mapping primary keys to secondary
73 keys, 2) "maps" mapping keys to size types, or 3) any arbitrary
74 combination of 1)s and 2)s. Thus, for example, an
75 <code class="classname">std::multiset<int></code> might be used to store
76 multiple instances of integers, but using this library's
77 containers, one might use
78 </p><pre class="programlisting">
79 tree<int, size_t>
81 i.e., a <code class="classname">map</code> of <span class="type">int</span>s to
82 <span class="type">size_t</span>s.
84 These "multimaps" and "multisets" might be confusing to
85 anyone familiar with the standard's <code class="classname">std::multimap</code> and
86 <code class="classname">std::multiset</code>, because there is no clear
87 correspondence between the two. For example, in some cases
88 where one uses <code class="classname">std::multiset</code> in the standard, one might use
89 in this library a "multimap" of "multisets" - i.e., a
90 container that maps primary keys each to an associative
91 container that maps each secondary key to the number of times
94 When one uses a "multimap," one should choose with care the
95 type of container used for secondary keys.
96 </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>
97 Brace onself: this library does not contain containers like
98 <code class="classname">std::multimap</code> or
99 <code class="classname">std::multiset</code>. Instead, these data
100 structures can be synthesized via manipulation of the
101 <code class="classname">Mapped</code> template parameter.
103 One maps the unique part of a key - the primary key, into an
104 associative-container of the (originally) non-unique parts of
105 the key - the secondary key. A primary associative-container
106 is an associative container of primary keys; a secondary
107 associative-container is an associative container of
110 Stepping back a bit, and starting in from the beginning.
112 Maps (or sets) allow mapping (or storing) unique-key values.
113 The standard library also supplies associative containers which
114 map (or store) multiple values with equivalent keys:
115 <code class="classname">std::multimap</code>, <code class="classname">std::multiset</code>,
116 <code class="classname">std::tr1::unordered_multimap</code>, and
117 <code class="classname">unordered_multiset</code>. We first discuss how these might
118 be used, then why we think it is best to avoid them.
120 Suppose one builds a simple bank-account application that
121 records for each client (identified by an <code class="classname">std::string</code>)
122 and account-id (marked by an <span class="type">unsigned long</span>) -
123 the balance in the account (described by a
124 <span class="type">float</span>). Suppose further that ordering this
125 information is not useful, so a hash-based container is
126 preferable to a tree based container. Then one can use
127 </p><pre class="programlisting">
128 std::tr1::unordered_map<std::pair<std::string, unsigned long>, float, ...>
130 which hashes every combination of client and account-id. This
131 might work well, except for the fact that it is now impossible
132 to efficiently list all of the accounts of a specific client
133 (this would practically require iterating over all
134 entries). Instead, one can use
135 </p><pre class="programlisting">
136 std::tr1::unordered_multimap<std::pair<std::string, unsigned long>, float, ...>
138 which hashes every client, and decides equivalence based on
139 client only. This will ensure that all accounts belonging to a
140 specific user are stored consecutively.
142 Also, suppose one wants an integers' priority queue
143 (a container that supports <code class="function">push</code>,
144 <code class="function">pop</code>, and <code class="function">top</code> operations, the last of which
145 returns the largest <span class="type">int</span>) that also supports
146 operations such as <code class="function">find</code> and <code class="function">lower_bound</code>. A
147 reasonable solution is to build an adapter over
148 <code class="classname">std::set<int></code>. In this adapter,
149 <code class="function">push</code> will just call the tree-based
150 associative container's <code class="function">insert</code> method; <code class="function">pop</code>
151 will call its <code class="function">end</code> method, and use it to return the
152 preceding element (which must be the largest). Then this might
153 work well, except that the container object cannot hold
154 multiple instances of the same integer (<code class="function">push(4)</code>,
155 will be a no-op if <code class="constant">4</code> is already in the
156 container object). If multiple keys are necessary, then one
157 might build the adapter over an
158 <code class="classname">std::multiset<int></code>.
160 The standard library's non-unique-mapping containers are useful
161 when (1) a key can be decomposed in to a primary key and a
162 secondary key, (2) a key is needed multiple times, or (3) any
163 combination of (1) and (2).
165 The graphic below shows how the standard library's container
166 design works internally; in this figure nodes shaded equally
167 represent equivalent-key values. Equivalent keys are stored
168 consecutively using the properties of the underlying data
169 structure: binary search trees (label A) store equivalent-key
170 values consecutively (in the sense of an in-order walk)
171 naturally; collision-chaining hash tables (label B) store
172 equivalent-key values in the same bucket, the bucket can be
173 arranged so that equivalent-key values are consecutive.
174 </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>
175 Put differently, the standards' non-unique mapping
176 associative-containers are associative containers that map
177 primary keys to linked lists that are embedded into the
178 container. The graphic below shows again the two
179 containers from the first graphic above, this time with
180 the embedded linked lists of the grayed nodes marked
182 </p><div class="figure"><a id="fig.pbds_embedded_lists_2"/><p class="title"><strong>Figure 22.9.
183 Effect of embedded lists in
184 <code class="classname">std::multimap</code>
185 </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>
186 These embedded linked lists have several disadvantages.
187 </p><div class="orderedlist"><ol class="orderedlist"><li class="listitem"><p>
188 The underlying data structure embeds the linked lists
189 according to its own consideration, which means that the
190 search path for a value might include several different
191 equivalent-key values. For example, the search path for the
192 the black node in either of the first graphic, labels A or B,
193 includes more than a single gray node.
194 </p></li><li class="listitem"><p>
195 The links of the linked lists are the underlying data
196 structures' nodes, which typically are quite structured. In
197 the case of tree-based containers (the grapic above, label
198 B), each "link" is actually a node with three pointers (one
199 to a parent and two to children), and a
200 relatively-complicated iteration algorithm. The linked
201 lists, therefore, can take up quite a lot of memory, and
202 iterating over all values equal to a given key (through the
203 return value of the standard
204 library's <code class="function">equal_range</code>) can be
206 </p></li><li class="listitem"><p>
207 The primary key is stored multiply; this uses more memory.
208 </p></li><li class="listitem"><p>
209 Finally, the interface of this design excludes several
210 useful underlying data structures. Of all the unordered
211 self-organizing data structures, practically only
212 collision-chaining hash tables can (efficiently) guarantee
213 that equivalent-key values are stored consecutively.
214 </p></li></ol></div><p>
215 The above reasons hold even when the ratio of secondary keys to
216 primary keys (or average number of identical keys) is small, but
217 when it is large, there are more severe problems:
218 </p><div class="orderedlist"><ol class="orderedlist"><li class="listitem"><p>
219 The underlying data structures order the links inside each
220 embedded linked-lists according to their internal
221 considerations, which effectively means that each of the
222 links is unordered. Irrespective of the underlying data
223 structure, searching for a specific value can degrade to
225 </p></li><li class="listitem"><p>
226 Similarly to the above point, it is impossible to apply
227 to the secondary keys considerations that apply to primary
228 keys. For example, it is not possible to maintain secondary
229 keys by sorted order.
230 </p></li><li class="listitem"><p>
231 While the interface "understands" that all equivalent-key
232 values constitute a distinct list (through
233 <code class="function">equal_range</code>), the underlying data
234 structure typically does not. This means that operations such
235 as erasing from a tree-based container all values whose keys
236 are equivalent to a a given key can be super-linear in the
237 size of the tree; this is also true also for several other
238 operations that target a specific list.
239 </p></li></ol></div><p>
240 In this library, all associative containers map
241 (or store) unique-key values. One can (1) map primary keys to
242 secondary associative-containers (containers of
243 secondary keys) or non-associative containers (2) map identical
244 keys to a size-type representing the number of times they
245 occur, or (3) any combination of (1) and (2). Instead of
246 allowing multiple equivalent-key values, this library
247 supplies associative containers based on underlying
248 data structures that are suitable as secondary
249 associative-containers.
251 In the figure below, labels A and B show the equivalent
252 underlying data structures in this library, as mapped to the
253 first graphic above. Labels A and B, respectively. Each shaded
254 box represents some size-type or secondary
255 associative-container.
256 </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>
257 In the first example above, then, one would use an associative
258 container mapping each user to an associative container which
259 maps each application id to a start time (see
260 <code class="filename">example/basic_multimap.cc</code>); in the second
261 example, one would use an associative container mapping
262 each <code class="classname">int</code> to some size-type indicating the
263 number of times it logically occurs
264 (see <code class="filename">example/basic_multiset.cc</code>.
266 See the discussion in list-based container types for containers
267 especially suited as secondary associative-containers.
268 </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>
269 Iterator concepts are bifurcated in this design, and are
270 comprised of point-type and range-type iteration.
272 A point-type iterator is an iterator that refers to a specific
273 element as returned through an
274 associative-container's <code class="function">find</code> method.
276 A range-type iterator is an iterator that is used to go over a
277 sequence of elements, as returned by a container's
278 <code class="function">find</code> method.
280 A point-type method is a method that
281 returns a point-type iterator; a range-type method is a method
282 that returns a range-type iterator.
283 </p><p>For most containers, these types are synonymous; for
284 self-organizing containers, such as hash-based containers or
285 priority queues, these are inherently different (in any
286 implementation, including that of C++ standard library
287 components), but in this design, it is made explicit. They are
289 </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
290 between two types of methods and iterators: point-type methods and
291 iterators, and range-type methods and iterators. Each associative
292 container's interface includes the methods:</p><pre class="programlisting">
294 find(const_key_reference r_key) const;
297 find(const_key_reference r_key);
299 std::pair<point_iterator,bool>
300 insert(const_reference r_val);
301 </pre><p>The relationship between these iterator types varies between
302 container types. The figure below
303 shows the most general invariant between point-type and
304 range-type iterators: In <span class="emphasis"><em>A</em></span> <code class="literal">iterator</code>, can
305 always be converted to <code class="literal">point_iterator</code>. In <span class="emphasis"><em>B</em></span>
306 shows invariants for order-preserving containers: point-type
307 iterators are synonymous with range-type iterators.
308 Orthogonally, <span class="emphasis"><em>C</em></span>shows invariants for "set"
309 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
310 (hash-based associative containers) lack movement
311 operators, such as <code class="literal">operator++</code> - in fact, this
312 is the reason why this library differentiates from the standard C++ librarys
313 design on this point.</p><p>Typically, one can determine an iterator's movement
315 <code class="literal">std::iterator_traits<It>iterator_category</code>,
316 which is a <code class="literal">struct</code> indicating the iterator's
317 movement capabilities. Unfortunately, none of the standard predefined
318 categories reflect a pointer's <span class="emphasis"><em>not</em></span> having any
319 movement capabilities whatsoever. Consequently,
320 <code class="literal">pb_ds</code> adds a type
321 <code class="literal">trivial_iterator_tag</code> (whose name is taken from
322 a concept in C++ standardese, which is the category of iterators
323 with no movement capabilities.) All other standard C++ library
324 tags, such as <code class="literal">forward_iterator_tag</code> retain their
325 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>
326 If one manipulates a container object, then iterators previously
327 obtained from it can be invalidated. In some cases a
328 previously-obtained iterator cannot be de-referenced; in other cases,
329 the iterator's next or previous element might have changed
330 unpredictably. This corresponds exactly to the question whether a
331 point-type or range-type iterator (see previous concept) is valid or
332 not. In this design, one can query a container (in compile time) about
333 its invalidation guarantees.
335 Given three different types of associative containers, a modifying
336 operation (in that example, <code class="function">erase</code>) invalidated
337 iterators in three different ways: the iterator of one container
338 remained completely valid - it could be de-referenced and
339 incremented; the iterator of a different container could not even be
340 de-referenced; the iterator of the third container could be
341 de-referenced, but its "next" iterator changed unpredictably.
343 Distinguishing between find and range types allows fine-grained
344 invalidation guarantees, because these questions correspond exactly
345 to the question of whether point-type iterators and range-type
346 iterators are valid. The graphic below shows tags corresponding to
347 different types of invalidation guarantees.
348 </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>
349 <code class="classname">basic_invalidation_guarantee</code>
350 corresponds to a basic guarantee that a point-type iterator,
351 a found pointer, or a found reference, remains valid as long
352 as the container object is not modified.
353 </p></li><li class="listitem"><p>
354 <code class="classname">point_invalidation_guarantee</code>
355 corresponds to a guarantee that a point-type iterator, a
356 found pointer, or a found reference, remains valid even if
357 the container object is modified.
358 </p></li><li class="listitem"><p>
359 <code class="classname">range_invalidation_guarantee</code>
360 corresponds to a guarantee that a range-type iterator remains
361 valid even if the container object is modified.
362 </p></li></ul></div><p>To find the invalidation guarantee of a
363 container, one can use</p><pre class="programlisting">
364 typename container_traits<Cntnr>::invalidation_guarantee
365 </pre><p>Note that this hierarchy corresponds to the logic it
366 represents: if a container has range-invalidation guarantees,
367 then it must also have find invalidation guarantees;
368 correspondingly, its invalidation guarantee (in this case
369 <code class="classname">range_invalidation_guarantee</code>)
370 can be cast to its base class (in this case <code class="classname">point_invalidation_guarantee</code>).
371 This means that this this hierarchy can be used easily using
372 standard metaprogramming techniques, by specializing on the
373 type of <code class="literal">invalidation_guarantee</code>.</p><p>
374 These types of problems were addressed, in a more general
375 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
376 our opinion, an invalidation-guarantee hierarchy would solve
377 these problems in all container types - not just associative
379 </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>
380 The design attempts to address the following problem of
381 data-structure genericity. When writing a function manipulating
382 a generic container object, what is the behavior of the object?
384 </p><pre class="programlisting">
385 template<typename Cntnr>
387 some_op_sequence(Cntnr &r_container)
392 then one needs to address the following questions in the body
393 of <code class="function">some_op_sequence</code>:
394 </p><div class="itemizedlist"><ul class="itemizedlist"><li class="listitem"><p>
395 Which types and methods does <code class="literal">Cntnr</code> support?
396 Containers based on hash tables can be queries for the
397 hash-functor type and object; this is meaningless for tree-based
398 containers. Containers based on trees can be split, joined, or
399 can erase iterators and return the following iterator; this
400 cannot be done by hash-based containers.
401 </p></li><li class="listitem"><p>
402 What are the exception and invalidation guarantees
403 of <code class="literal">Cntnr</code>? A container based on a probing
404 hash-table invalidates all iterators when it is modified; this
405 is not the case for containers based on node-based
406 trees. Containers based on a node-based tree can be split or
407 joined without exceptions; this is not the case for containers
408 based on vector-based trees.
409 </p></li><li class="listitem"><p>
410 How does the container maintain its elements? Tree-based and
411 Trie-based containers store elements by key order; others,
412 typically, do not. A container based on a splay trees or lists
413 with update policies "cache" "frequently accessed" elements;
414 containers based on most other underlying data structures do
416 </p></li><li class="listitem"><p>
417 How does one query a container about characteristics and
418 capabilities? What is the relationship between two different
419 data structures, if anything?
420 </p></li></ul></div><p>The remainder of this section explains these issues in
421 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>
422 Tags are very useful for manipulating generic types. For example, if
423 <code class="literal">It</code> is an iterator class, then <code class="literal">typename
424 It::iterator_category</code> or <code class="literal">typename
425 std::iterator_traits<It>::iterator_category</code> will
426 yield its category, and <code class="literal">typename
427 std::iterator_traits<It>::value_type</code> will yield its
430 This library contains a container tag hierarchy corresponding to the
432 </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>
433 Given any container <span class="type">Cntnr</span>, the tag of
434 the underlying data structure can be found via <code class="literal">typename
435 Cntnr::container_category</code>.
436 </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
437 container type for its attributes. Given any container
438 <code class="literal">Cntnr</code>, then <code class="literal"><Cntnr></code>
439 is a traits class identifying the properties of the
440 container.</p><p>To find if a container can throw when a key is erased (which
441 is true for vector-based trees, for example), one can
443 </p><pre class="programlisting">container_traits<Cntnr>::erase_can_throw</pre><p>
444 Some of the definitions in <code class="classname">container_traits</code>
445 are dependent on other
446 definitions. If <code class="classname">container_traits<Cntnr>::order_preserving</code>
447 is <code class="constant">true</code> (which is the case for containers
448 based on trees and tries), then the container can be split or
450 case, <code class="classname">container_traits<Cntnr>::split_join_can_throw</code>
451 indicates whether splits or joins can throw exceptions (which is
452 true for vector-based trees);
453 otherwise <code class="classname">container_traits<Cntnr>::split_join_can_throw</code>
454 will yield a compilation error. (This is somewhat similar to a
455 compile-time version of the COM model).
456 </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>
457 The collision-chaining hash-based container has the
458 following declaration.</p><pre class="programlisting">
462 typename Hash_Fn = std::hash<Key>,
463 typename Eq_Fn = std::equal_to<Key>,
464 typename Comb_Hash_Fn = direct_mask_range_hashing<>
465 typename Resize_Policy = default explained below.
466 bool Store_Hash = false,
467 typename Allocator = std::allocator<char> >
469 </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;
470 it describes how to translate hash values into positions
471 within the table. </p></li><li class="listitem"><p><code class="classname">Resize_Policy</code> describes how a container object
472 should change its internal size. </p></li><li class="listitem"><p><code class="classname">Store_Hash</code> indicates whether the hash value
473 should be stored with each entry. </p></li><li class="listitem"><p><code class="classname">Allocator</code> is an allocator
474 type.</p></li></ol></div><p>The probing hash-based container has the following
475 declaration.</p><pre class="programlisting">
479 typename Hash_Fn = std::hash<Key>,
480 typename Eq_Fn = std::equal_to<Key>,
481 typename Comb_Probe_Fn = direct_mask_range_hashing<>
482 typename Probe_Fn = default explained below.
483 typename Resize_Policy = default explained below.
484 bool Store_Hash = false,
485 typename Allocator = std::allocator<char> >
487 </pre><p>The parameters are identical to those of the
488 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
489 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
490 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
491 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
492 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
493 strings of 3 characters). A hash-table algorithm needs to map
494 elements of U "uniformly" into the range [0,..., m -
495 1] (where m is a non-negative integral value, and
496 is, in general, time varying). I.e., the algorithm needs
497 a ranged-hash function</p><p>
498 f : U × Z<sub>+</sub> → Z<sub>+</sub>
499 </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
500 <a class="xref" href="policy_data_structures.html#biblio.knuth98sorting" title="The Art of Computer Programming - Sorting and Searching">[biblio.knuth98sorting]</a>.)
502 common solution is to use the composition of the hash
503 function</p><p>h : U → Z<sub>+</sub> ,</p><p>which maps elements of U into the non-negative
504 integrals, and</p><p>g : Z<sub>+</sub> × Z<sub>+</sub> →
505 Z<sub>+</sub>,</p><p>which maps a non-negative hash value, and a non-negative
506 range upper-bound into a non-negative integral in the range
507 between 0 (inclusive) and the range upper bound (exclusive),
508 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">
509 f(u , m) = g(h(u), m)
510 </span></div></div><br class="equation-break"/><p>From the above, it is obvious that given g and
511 h, f can always be composed (however the converse
512 is not true). The standard's hash-based containers allow specifying
513 a hash function, and use a hard-wired range-hashing function;
514 the ranged-hash function is implicitly composed.</p><p>The above describes the case where a key is to be mapped
515 into a single position within a hash table, e.g.,
516 in a collision-chaining table. In other cases, a key is to be
517 mapped into a sequence of positions within a table,
518 e.g., in a probing table. Similar terms apply in this
519 case: the table requires a ranged probe function,
520 mapping a key into a sequence of positions withing the table.
521 This is typically achieved by composing a hash function
522 mapping the key into a non-negative integral type, a
523 probe function transforming the hash value into a
524 sequence of hash values, and a range-hashing function
525 transforming the sequence of hash values into a sequence of
526 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
527 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
528 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">
530 </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
531 v (typically powers of 2), and some a. Each of
532 these range-hashing functions works best for some different
533 setting.</p><p>The division method (see above) is a
534 very common choice. However, even this single method can be
535 implemented in two very different ways. It is possible to
536 implement using the low
537 level % (modulo) operation (for any m), or the
538 low level & (bit-mask) operation (for the case where
539 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">
541 </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">
542 g(r, m) = r & m - 1, (with m =
543 2<sup>k</sup> for some k)
544 </span></div></div><br class="equation-break"/><p>respectively.</p><p>The % (modulo) implementation has the advantage that for
545 m a prime far from a power of 2, g(r, m) is
546 affected by all the bits of r (minimizing the chance of
547 collision). It has the disadvantage of using the costly modulo
548 operation. This method is hard-wired into SGI's implementation
549 .</p><p>The & (bit-mask) implementation has the advantage of
550 relying on the fast bit-wise and operation. It has the
551 disadvantage that for g(r, m) is affected only by the
552 low order bits of r. This method is hard-wired into
553 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
554 client to directly specify a ranged-hash hash function. It is
555 true, that the writer of the ranged-hash function cannot rely
556 on the values of m having specific numerical properties
557 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
558 the values of m are determined by a resize policy with
559 possibly orthogonal considerations.</p><p>There are two cases where a ranged-hash function can be
560 superior. The firs is when using perfect hashing: the
561 second is when the values of m can be used to estimate
562 the "general" number of distinct values required. This is
563 described in the following.</p><p>Let</p><p>
564 s = [ s<sub>0</sub>,..., s<sub>t - 1</sub>]
565 </p><p>be a string of t characters, each of which is from
566 domain S. Consider the following ranged-hash
567 function:</p><div class="equation"><a id="id520803"/><p class="title"><strong>Equation 22.5.
568 A Standard String Hash Function
569 </strong></p><div class="equation-contents"><span class="mathphrase">
570 f<sub>1</sub>(s, m) = ∑ <sub>i =
571 0</sub><sup>t - 1</sup> s<sub>i</sub> a<sup>i</sup> mod m
572 </span></div></div><br class="equation-break"/><p>where a is some non-negative integral value. This is
573 the standard string-hashing function used in SGI's
574 implementation (with a = 5). Its advantage is that
575 it takes into account all of the characters of the string.</p><p>Now assume that s is the string representation of a
576 of a long DNA sequence (and so S = {'A', 'C', 'G',
577 'T'}). In this case, scanning the entire string might be
578 prohibitively expensive. A possible alternative might be to use
579 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.
580 Only k String DNA Hash
581 </strong></p><div class="equation-contents"><span class="mathphrase">
582 f<sub>2</sub>(s, m) = ∑ <sub>i
583 = 0</sub><sup>k - 1</sup> s<sub>i</sub> a<sup>i</sup> mod m
584 </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
585 characters starting at a random position (determined at each
586 resize), or scanning k random positions (determined at
587 each resize), i.e., using</p><p>f<sub>3</sub>(s, m) = ∑ <sub>i =
588 r</sub>0<sup>r<sub>0</sub> + k - 1</sup> s<sub>i</sub>
589 a<sup>i</sup> mod m ,</p><p>or</p><p>f<sub>4</sub>(s, m) = ∑ <sub>i = 0</sub><sup>k -
590 1</sup> s<sub>r</sub>i a<sup>r<sub>i</sub></sup> mod
591 m ,</p><p>respectively, for r<sub>0</sub>,..., r<sub>k-1</sub>
592 each in the (inclusive) range [0,...,t-1].</p><p>It should be noted that the above functions cannot be
593 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
594 the above in this library. It first explains range-hashing
595 functions in collision-chaining tables, then ranged-hash
596 functions in collision-chaining tables, then probing-based
597 tables, and finally lists the relevant classes in this
598 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"/>
599 Range-Hashing and Ranged-Hashes in Collision-Chaining Tables
600 </h6></div></div></div><p><code class="classname">cc_hash_table</code> is
601 parametrized by <code class="classname">Hash_Fn</code> and <code class="classname">Comb_Hash_Fn</code>, a
602 hash functor and a combining hash functor, respectively.</p><p>In general, <code class="classname">Comb_Hash_Fn</code> is considered a
603 range-hashing functor. <code class="classname">cc_hash_table</code>
604 synthesizes a ranged-hash function from <code class="classname">Hash_Fn</code> and
605 <code class="classname">Comb_Hash_Fn</code>. The figure below shows an <code class="classname">insert</code> sequence
606 diagram for this case. The user inserts an element (point A),
607 the container transforms the key into a non-negative integral
608 using the hash functor (points B and C), and transforms the
609 result into a position using the combining functor (points D
610 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
611 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
612 a ranged-hash function. The graphic below shows an <code class="function">insert</code> sequence
613 diagram. The user inserts an element (point A), the container
614 transforms the key into a position using the combining functor
615 (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"/>
617 </h6></div></div></div><p><code class="classname">gp_hash_table</code> is parametrized by
618 <code class="classname">Hash_Fn</code>, <code class="classname">Probe_Fn</code>,
619 and <code class="classname">Comb_Probe_Fn</code>. As before, if
620 <code class="classname">Hash_Fn</code> and <code class="classname">Probe_Fn</code>
621 are both <code class="classname">null_type</code>, then
622 <code class="classname">Comb_Probe_Fn</code> is a ranged-probe
623 functor. Otherwise, <code class="classname">Hash_Fn</code> is a hash
624 functor, <code class="classname">Probe_Fn</code> is a functor for offsets
625 from a hash value, and <code class="classname">Comb_Probe_Fn</code>
626 transforms a probe sequence into a sequence of positions within
627 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"/>
629 </h6></div></div></div><p>This library contains some pre-defined classes
630 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>
631 and <code class="classname">direct_mod_range_hashing</code>
632 are range-hashing functions based on a bit-mask and a modulo
633 operation, respectively.</p></li><li class="listitem"><p><code class="classname">linear_probe_fn</code>, and
634 <code class="classname">quadratic_probe_fn</code> are
635 a linear probe and a quadratic probe function,
636 respectively.</p></li></ol></div><p>
637 The graphic below shows the relationships.
638 </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
639 shrink. It is necessary to specify policies to determine how
640 and when a hash table should change its size. Usually, resize
641 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
642 should grow (e.g., it should multiply by powers of
643 2).</p></li><li class="listitem"><p>A trigger policy indicating when a hash
644 table should grow (e.g., a load factor is
645 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
646 policies are simple, and there are relatively few sensible
647 options. An exponential-size policy (with the initial size and
648 growth factors both powers of 2) works well with a mask-based
649 range-hashing function, and is the
650 hard-wired policy used by Dinkumware. A
651 prime-list based policy works well with a modulo-prime range
652 hashing function and is the hard-wired policy used by SGI's
653 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.
654 Following is a description of two policies: load-check
655 policies, and collision-check policies.</p><p>Load-check policies are straightforward. The user specifies
656 two factors, Α<sub>min</sub> and
657 Α<sub>max</sub>, and the hash table maintains the
658 invariant that</p><p>Α<sub>min</sub> ≤ (number of
659 stored elements) / (hash-table size) ≤
660 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
661 load-check policies. They focus on keeping the number of
662 collisions moderate and hoping that the size of the table will
663 not grow very large, instead of keeping a moderate load-factor
664 and hoping that the number of collisions will be small. A
665 maximal collision-check policy resizes when the longest
666 probe-sequence grows too large.</p><p>Consider the graphic below. Let the size of the hash table
667 be denoted by m, the length of a probe sequence be denoted by k,
668 and some load factor be denoted by Α. We would like to
669 calculate the minimal length of k, such that if there were Α
670 m elements in the hash table, a probe sequence of length k would
671 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
672 k appears in bin i by p<sub>i</sub>, the
673 length of the probe sequence of bin i by
674 l<sub>i</sub>, and assume uniform distribution. Then</p><div class="equation"><a id="id521446"/><p class="title"><strong>Equation 22.7.
675 Probability of Probe Sequence of Length k
676 </strong></p><div class="equation-contents"><span class="mathphrase">
678 </span></div></div><br class="equation-break"/><p>P(l<sub>1</sub> ≥ k) =</p><p>
679 P(l<sub>1</sub> ≥ α ( 1 + k / α - 1) ≤ (a)
681 e ^ ( - ( α ( k / α - 1 )<sup>2</sup> ) /2)
682 </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
683 calculate the probability that some bin contains a probe
684 sequence greater than k, we note that the
685 l<sub>i</sub> are negatively-dependent
686 (<a class="xref" href="policy_data_structures.html#biblio.dubhashi98neg" title="Balls and bins: A study in negative dependence">[biblio.dubhashi98neg]</a>)
688 I(.) denote the indicator function. Then</p><div class="equation"><a id="id521502"/><p class="title"><strong>Equation 22.8.
689 Probability Probe Sequence in Some Bin
690 </strong></p><div class="equation-contents"><span class="mathphrase">
691 P( exists<sub>i</sub> l<sub>i</sub> ≥ k ) =
692 </span></div></div><br class="equation-break"/><p>P ( ∑ <sub>i = 1</sub><sup>m</sup>
693 I(l<sub>i</sub> ≥ k) ≥ 1 ) =</p><p>P ( ∑ <sub>i = 1</sub><sup>m</sup> I (
694 l<sub>i</sub> ≥ k ) ≥ m p<sub>1</sub> ( 1 + 1 / (m
695 p<sub>1</sub>) - 1 ) ) ≤ (a)</p><p>e ^ ( ( - m p<sub>1</sub> ( 1 / (m p<sub>1</sub>)
696 - 1 ) <sup>2</sup> ) / 2 ) ,</p><p>where (a) follows from the fact that the Chernoff bound can
697 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
698 equation into the second one, and equating with 1/m, we
699 obtain</p><p>k ~ √ ( 2 α ln 2 m ln(m) )
700 ) .</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
701 above in this library. It first describes resize policies and
702 their decomposition into trigger and size policies, then
703 describes pre-defined classes, and finally discusses controlled
704 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
705 <code class="classname">Resize_Policy</code> parameter; the container derives
706 <code class="classname">public</code>ly from <code class="classname">Resize_Policy</code>. For
707 example:</p><pre class="programlisting">
708 cc_hash_table<typename Key,
711 typename Resize_Policy
712 ...> : public Resize_Policy
713 </pre><p>As a container object is modified, it continuously notifies
714 its <code class="classname">Resize_Policy</code> base of internal changes
715 (e.g., collisions encountered and elements being
716 inserted). It queries its <code class="classname">Resize_Policy</code> base whether
717 it needs to be resized, and if so, to what size.</p><p>The graphic below shows a (possible) sequence diagram
718 of an insert operation. The user inserts an element; the hash
719 table notifies its resize policy that a search has started
720 (point A); in this case, a single collision is encountered -
721 the table notifies its resize policy of this (point B); the
722 container finally notifies its resize policy that the search
723 has ended (point C); it then queries its resize policy whether
724 a resize is needed, and if so, what is the new size (points D
725 to G); following the resize, it notifies the policy that a
726 resize has completed (point H); finally, the element is
727 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
728 decomposed to a size policy and a trigger policy. Consequently,
729 the library contains a single class for instantiating a resize
730 policy: <code class="classname">hash_standard_resize_policy</code>
731 is parametrized by <code class="classname">Size_Policy</code> and
732 <code class="classname">Trigger_Policy</code>, derives <code class="classname">public</code>ly from
733 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>)
734 to these policies.</p><p>The two graphics immediately below show sequence diagrams
735 illustrating the interaction between the standard resize policy
736 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
737 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
738 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
739 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>
740 implements a load check trigger policy.</p></li><li class="listitem"><p><code class="classname">cc_hash_max_collision_check_resize_trigger</code>
741 implements a collision check trigger policy.</p></li><li class="listitem"><p><code class="classname">hash_exponential_size_policy</code>
742 implements an exponential-size policy (which should be used
743 with mask range hashing).</p></li><li class="listitem"><p><code class="classname">hash_prime_size_policy</code>
744 implementing a size policy based on a sequence of primes
746 be used with mod range hashing</p></li></ol></div><p>The graphic below gives an overall picture of the resize-related
747 classes. <code class="classname">basic_hash_table</code>
748 is parametrized by <code class="classname">Resize_Policy</code>, which it subclasses
749 publicly. This class is currently instantiated only by <code class="classname">hash_standard_resize_policy</code>.
750 <code class="classname">hash_standard_resize_policy</code>
751 itself is parametrized by <code class="classname">Trigger_Policy</code> and
752 <code class="classname">Size_Policy</code>. Currently, <code class="classname">Trigger_Policy</code> is
753 instantiated by <code class="classname">hash_load_check_resize_trigger</code>,
754 or <code class="classname">cc_hash_max_collision_check_resize_trigger</code>;
755 <code class="classname">Size_Policy</code> is instantiated by <code class="classname">hash_exponential_size_policy</code>,
756 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
757 policies' internals is beneficial. E.g., it is sometimes
758 useful to query a hash-table for the table's actual size (as
759 opposed to its <code class="function">size()</code> - the number of values it
760 currently holds); it is sometimes useful to set a table's
761 initial size, externally resize it, or change load factors.</p><p>Clearly, supporting such methods both decreases the
762 encapsulation of hash-based containers, and increases the
763 diversity between different associative-containers' interfaces.
764 Conversely, omitting such methods can decrease containers'
765 flexibility.</p><p>In order to avoid, to the extent possible, the above
766 conflict, the hash-based containers themselves do not address
767 any of these questions; this is deferred to the resize policies,
768 which are easier to change or replace. Thus, for example,
769 neither <code class="classname">cc_hash_table</code> nor
770 <code class="classname">gp_hash_table</code>
771 contain methods for querying the actual size of the table; this
772 is deferred to <code class="classname">hash_standard_resize_policy</code>.</p><p>Furthermore, the policies themselves are parametrized by
773 template arguments that determine the methods they support
775 <a class="xref" href="policy_data_structures.html#biblio.alexandrescu01modern" title="Modern C++ Design: Generic Programming and Design Patterns Applied">[biblio.alexandrescu01modern]</a>
776 shows techniques for doing so). <code class="classname">hash_standard_resize_policy</code>
777 is parametrized by <code class="classname">External_Size_Access</code> that
778 determines whether it supports methods for querying the actual
779 size of the table or resizing it. <code class="classname">hash_load_check_resize_trigger</code>
780 is parametrized by <code class="classname">External_Load_Access</code> that
781 determines whether it supports methods for querying or
782 modifying the loads. <code class="classname">cc_hash_max_collision_check_resize_trigger</code>
783 is parametrized by <code class="classname">External_Load_Access</code> that
784 determines whether it supports methods for querying the
785 load.</p><p>Some operations, for example, resizing a container at
786 run time, or changing the load factors of a load-check trigger
787 policy, require the container itself to resize. As mentioned
788 above, the hash-based containers themselves do not contain
789 these types of methods, only their resize policies.
790 Consequently, there must be some mechanism for a resize policy
791 to manipulate the hash-based container. As the hash-based
792 container is a subclass of the resize policy, this is done
793 through virtual methods. Each hash-based container has a
794 <code class="classname">private</code> <code class="classname">virtual</code> method:</p><pre class="programlisting">
797 (size_type new_size);
798 </pre><p>which resizes the container. Implementations of
799 <code class="classname">Resize_Policy</code> can export public methods for resizing
800 the container externally; these methods internally call
801 <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>
802 </p><p>Hash-tables are unfortunately especially susceptible to
803 choice of policies. One of the more complicated aspects of this
804 is that poor combinations of good policies can form a poor
805 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.
806 For example, combining a quadratic probe policy with an
807 exponential size policy can yield a poor container: when an
808 element is inserted, a trigger policy might decide that there
809 is no need to resize, as the table still contains unused
810 entries; the probe sequence, however, might never reach any of
811 the unused entries.</p><p>Unfortunately, this library cannot detect such problems at
812 compilation (they are halting reducible). It therefore defines
813 an exception class <code class="classname">insert_error</code> to throw an
814 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
815 hash functions. Suppose, as an extreme case, that the hash
816 function transforms each key to the same hash value. After some
817 inserts, a collision detecting policy will always indicate that
818 the container needs to grow.</p><p>The library, therefore, by design, limits each operation to
819 one resize. For each <code class="classname">insert</code>, for example, it queries
820 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
821 <code class="classname">gp_hash_table</code> are
822 parametrized by an equivalence functor and by a
823 <code class="classname">Store_Hash</code> parameter. If the latter parameter is
824 <code class="classname">true</code>, then the container stores with each entry
825 a hash value, and uses this value in case of collisions to
826 determine whether to apply a hash value. This can lower the
827 cost of collision for some types, but increase the cost of
828 collisions for other types.</p><p>If a ranged-hash function or ranged probe function is
829 directly supplied, however, then it makes no sense to store the
830 hash value with each entry. This library's container will
831 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
832 a, a q, a q<sup>1</sup>, a q<sup>2</sup>, ... For
833 example, an exponential size policy might issue the sequence of
834 sizes 8, 16, 32, 64, ...</p><p>If a load-check trigger policy is used, with loads
835 α<sub>min</sub> and α<sub>max</sub>,
836 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
837 modifying operation is at most approximately 3.</p><p>α<sub>min</sub> ~ α<sub>max</sub> is, in
838 any case, a bad choice, and α<sub>min</sub> >
839 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">
843 typename Cmp_Fn = std::less<Key>,
844 typename Tag = rb_tree_tag,
846 typename Const_Node_Iterator,
847 typename Node_Iterator,
849 typename Allocator_>
850 class Node_Update = null_node_update,
851 typename Allocator = std::allocator<char> >
853 </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
854 to use.</p></li><li class="listitem"><p><code class="classname">Node_Update</code> is a policy for updating node
855 invariants.</p></li><li class="listitem"><p><code class="classname">Allocator</code> is an allocator
856 type.</p></li></ol></div><p>The <code class="classname">Tag</code> parameter specifies which underlying
857 data structure to use. Instantiating it by <code class="classname">rb_tree_tag</code>, <code class="classname">splay_tree_tag</code>, or
858 <code class="classname">ov_tree_tag</code>,
859 specifies an underlying red-black tree, splay tree, or
860 ordered-vector tree, respectively; any other tag is illegal.
861 Note that containers based on the former two contain more types
862 and methods than the latter (e.g.,
863 <code class="classname">reverse_iterator</code> and <code class="classname">rbegin</code>), and different
864 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
865 is a tree of floats; the second is a tree of pairs, each
866 signifying a geometric line interval. Each element in a tree is refered to as a node of the tree. Of course, each of
867 these trees can support the usual queries: the first can easily
868 search for <code class="classname">0.4</code>; the second can easily search for
869 <code class="classname">std::make_pair(10, 41)</code>.</p><p>Each of these trees can efficiently support other queries.
870 The first can efficiently determine that the 2rd key in the
871 tree is <code class="constant">0.3</code>; the second can efficiently determine
872 whether any of its intervals overlaps
873 </p><pre class="programlisting">std::make_pair(29,42)</pre><p> (useful in geometric
874 applications or distributed file systems with leases, for
875 example). It should be noted that an <code class="classname">std::set</code> can
876 only solve these types of problems with linear complexity.</p><p>In order to do so, each tree stores some metadata in
877 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
878 each node the size of the sub-tree rooted at the node; the
879 second stores at each node the maximal endpoint of the
880 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
881 reasons:</p><div class="orderedlist"><ol class="orderedlist"><li class="listitem"><p>There must be a way to specify what a node's metadata
882 should be (if any).</p></li><li class="listitem"><p>Various operations can invalidate node
883 invariants. The graphic below shows how a right rotation,
884 performed on A, results in B, with nodes x and y having
885 corrupted invariants (the grayed nodes in C). The graphic shows
886 how an insert, performed on D, results in E, with nodes x and y
887 having corrupted invariants (the grayed nodes in F). It is not
888 feasible to know outside the tree the effect of an operation on
889 the nodes of the tree.</p></li><li class="listitem"><p>The search paths of standard associative containers are
890 defined by comparisons between keys, and not through
891 metadata.</p></li><li class="listitem"><p>It is not feasible to know in advance which methods trees
892 can support. Besides the usual <code class="classname">find</code> method, the
893 first tree can support a <code class="classname">find_by_order</code> method, while
894 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:
895 node iterators, and template-template node updater
896 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
897 types, <code class="classname">const_node_iterator</code>
898 and <code class="classname">node_iterator</code>.
899 These iterators allow descending from a node to one of its
900 children. Node iterator allow search paths different than those
901 determined by the comparison functor. The <code class="classname">tree</code>
902 supports the methods:</p><pre class="programlisting">
914 </pre><p>The first pairs return node iterators corresponding to the
915 root node of the tree; the latter pair returns node iterators
916 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
917 <code class="classname">Node_Update</code> template-template parameter. A
918 tree-based container instantiates
919 <code class="classname">Node_Update</code> to some
920 <code class="classname">node_update</code> class, and publicly subclasses
921 <code class="classname">node_update</code>. The graphic below shows this
922 scheme, as well as some predefined policies (which are explained
923 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
924 <code class="classname">Node_Update</code>) must define <code class="classname">metadata_type</code> as
925 the type of metadata it requires. For order statistics,
926 e.g., <code class="classname">metadata_type</code> might be <code class="classname">size_t</code>.
927 The tree defines within each node a <code class="classname">metadata_type</code>
928 object.</p><p><code class="classname">node_update</code> must also define the following method
929 for restoring node invariants:</p><pre class="programlisting">
931 operator()(node_iterator nd_it, const_node_iterator end_nd_it)
932 </pre><p>In this method, <code class="varname">nd_it</code> is a
933 <code class="classname">node_iterator</code> corresponding to a node whose
934 A) all descendants have valid invariants, and B) its own
935 invariants might be violated; <code class="classname">end_nd_it</code> is
936 a <code class="classname">const_node_iterator</code> corresponding to a
937 just-after-leaf node. This method should correct the node
938 invariants of the node pointed to by
939 <code class="classname">nd_it</code>. For example, say node x in the
940 graphic below label A has an invalid invariant, but its' children,
941 y and z have valid invariants. After the invocation, all three
942 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,
943 it invokes this method in its <code class="classname">node_update</code> base to
944 restore the invariant. For example, the graphic below shows
945 an <code class="function">insert</code> operation (point A); the tree performs some
946 operations, and calls the update functor three times (points B,
947 C, and D). (It is well known that any <code class="function">insert</code>,
948 <code class="function">erase</code>, <code class="function">split</code> or <code class="function">join</code>, can restore
949 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>)
950 .</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
951 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
952 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
953 tree?</p></li><li class="listitem"><p>How can the following cyclic dependency be resolved?
954 <code class="classname">node_update</code> is a base class of the tree, yet it
955 uses node iterators defined in the tree (its child).</p></li></ol></div><p>The first two questions are answered by the fact that
956 <code class="classname">node_update</code> (an instantiation of
957 <code class="classname">Node_Update</code>) is a <span class="emphasis"><em>public</em></span> base class
958 of the tree. Consequently:</p><div class="orderedlist"><ol class="orderedlist"><li class="listitem"><p>Any public methods of
959 <code class="classname">node_update</code> are automatically methods of
960 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>).
961 Thus an order-statistics node updater,
962 <code class="classname">tree_order_statistics_node_update</code> defines
963 the <code class="function">find_by_order</code> method; any tree
964 instantiated by this policy consequently supports this method as
965 well.</p></li><li class="listitem"><p>In C++, if a base class declares a method as
966 <code class="literal">virtual</code>, it is
967 <code class="literal">virtual</code> in its subclasses. If
968 <code class="classname">node_update</code> needs to access one of the
969 tree's methods, say the member function
970 <code class="function">end</code>, it simply declares that method as
971 <code class="literal">virtual</code> abstract.</p></li></ol></div><p>The cyclic dependency is solved through template-template
972 parameters. <code class="classname">Node_Update</code> is parametrized by
973 the tree's node iterators, its comparison functor, and its
974 allocator type. Thus, instantiations of
975 <code class="classname">Node_Update</code> have all information
976 required.</p><p>This library assumes that constructing a metadata object and
977 modifying it are exception free. Suppose that during some method,
978 say <code class="classname">insert</code>, a metadata-related operation
979 (e.g., changing the value of a metadata) throws an exception. Ack!
980 Rolling back the method is unusually complex.</p><p>Previously, a distinction was made between redundant
981 policies and null policies. Node invariants show a
982 case where null policies are required.</p><p>Assume a regular tree is required, one which need not
983 support order statistics or interval overlap queries.
984 Seemingly, in this case a redundant policy - a policy which
985 doesn't affect nodes' contents would suffice. This, would lead
986 to the following drawbacks:</p><div class="orderedlist"><ol class="orderedlist"><li class="listitem"><p>Each node would carry a useless metadata object, wasting
987 space.</p></li><li class="listitem"><p>The tree cannot know if its
988 <code class="classname">Node_Update</code> policy actually modifies a
989 node's metadata (this is halting reducible). In the graphic
990 below, assume the shaded node is inserted. The tree would have
991 to traverse the useless path shown to the root, applying
992 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>
993 solves both these problems. The tree detects that node
994 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.
995 It is possible to split a tree so that it passes
996 all nodes with keys larger than a given key to a different
997 tree. These methods have the following advantages over the
998 alternative of externally inserting to the destination
999 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
1000 and joined in poly-logarithmic complexity; ordered-vector
1001 trees are split and joined at linear complexity. The
1002 alternatives have super-linear complexity.</p></li><li class="listitem"><p>Aside from orders of growth, these operations perform
1003 few allocations and de-allocations. For red-black trees, allocations are not performed,
1004 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">
1005 template<typename Key,
1007 typename Cmp_Fn = std::less<Key>,
1008 typename Tag = pat_trie_tag,
1009 template<typename Const_Node_Iterator,
1010 typename Node_Iterator,
1011 typename E_Access_Traits_,
1012 typename Allocator_>
1013 class Node_Update = null_node_update,
1014 typename Allocator = std::allocator<char> >
1016 </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
1017 to use, and is described shortly.</p></li><li class="listitem"><p><code class="classname">Node_Update</code> is a policy for updating node
1018 invariants. This is described below.</p></li><li class="listitem"><p><code class="classname">Allocator</code> is an allocator
1019 type.</p></li></ol></div><p>The <code class="classname">Tag</code> parameter specifies which underlying
1020 data structure to use. Instantiating it by <code class="classname">pat_trie_tag</code>, specifies an
1021 underlying PATRICIA trie (explained shortly); any other tag is
1022 currently illegal.</p><p>Following is a description of a (PATRICIA) trie
1023 (this implementation follows <a class="xref" href="policy_data_structures.html#biblio.okasaki98mereable" title="Fast mergeable integer maps">[biblio.okasaki98mereable]</a> and
1024 <a class="xref" href="policy_data_structures.html#biblio.filliatre2000ptset" title="Ptset: Sets of integers implemented as Patricia trees">[biblio.filliatre2000ptset]</a>).
1025 </p><p>A (PATRICIA) trie is similar to a tree, but with the
1026 following differences:</p><div class="orderedlist"><ol class="orderedlist"><li class="listitem"><p>It explicitly views keys as a sequence of elements.
1027 E.g., a trie can view a string as a sequence of
1028 characters; a trie can view a number as a sequence of
1029 bits.</p></li><li class="listitem"><p>It is not (necessarily) binary. Each node has fan-out n
1030 + 1, where n is the number of distinct
1031 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
1032 least two children, and B) each shares the same prefix with
1033 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
1034 very efficient find performance (albeit at insertion
1035 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
1036 keys match a certain prefix. This is sometimes useful in file
1037 systems and routers, and for "type-ahead" aka predictive text matching
1038 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.
1039 For example, a trie can view a string as a sequence of
1040 characters. A trie needs to map each of n elements to a
1041 number in {0, n - 1}. For example, a trie can map a
1042 character <code class="varname">c</code> to
1043 </p><pre class="programlisting">static_cast<size_t>(c)</pre><p>.</p><p>Seemingly, then, a trie can assume that its keys support
1044 (const) iterators, and that the <code class="classname">value_type</code> of this
1045 iterator can be cast to a <code class="classname">size_t</code>. There are several
1046 reasons, though, to decouple the mechanism by which the trie
1047 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
1048 inappropriate. Consider a trie storing DNA strings. It is
1049 logical to use a trie with a fan-out of 5 = 1 + |{'A', 'C',
1050 '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
1051 is needed. For example, a trie can be used to search for
1052 common suffixes, by using strings'
1053 <code class="classname">reverse_iterator</code>. As another example, a trie mapping
1054 UNICODE strings would have a huge fan-out if each node would
1055 branch on a UNICODE character; instead, one can define an
1056 iterator iterating over 8-bit (or less) groups.</p></li></ol></div><p>trie is,
1057 consequently, parametrized by <code class="classname">E_Access_Traits</code> -
1058 traits which instruct how to access sequences' elements.
1059 <code class="classname">string_trie_e_access_traits</code>
1060 is a traits class for strings. Each such traits define some
1061 types, like:</p><pre class="programlisting">
1062 typename E_Access_Traits::const_iterator
1063 </pre><p>is a const iterator iterating over a key's elements. The
1064 traits class must also define methods for obtaining an iterator
1065 to the first and last element of a key.</p><p>The graphic below shows a
1066 (PATRICIA) trie resulting from inserting the words: "I wish
1067 that I could ever see a poem lovely as a trie" (which,
1068 unfortunately, does not rhyme).</p><p>The leaf nodes contain values; each internal node contains
1069 two <code class="classname">typename E_Access_Traits::const_iterator</code>
1070 objects, indicating the maximal common prefix of all keys in
1071 the sub-tree. For example, the shaded internal node roots a
1072 sub-tree with leafs "a" and "as". The maximal common prefix is
1073 "a". The internal node contains, consequently, to const
1074 iterators, one pointing to <code class="varname">'a'</code>, and the other to
1075 <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
1076 tree-based containers. There are two minor
1077 differences, though, which, unfortunately, thwart sharing them
1078 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
1079 parameter is parametrized by <code class="classname">E_Access_Traits</code>, while
1080 a tree's <code class="classname">Node_Update</code> template-template parameter is
1081 parametrized by <code class="classname">Cmp_Fn</code>.</p></li><li class="listitem"><p>Tree-based containers store values in all nodes, while
1082 trie-based containers (at least in this implementation) store
1083 values in leafs.</p></li></ol></div><p>The graphic below shows the scheme, as well as some predefined
1084 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
1085 updating policies:</p><div class="orderedlist"><ol class="orderedlist"><li class="listitem"><p>
1086 <code class="classname">trie_order_statistics_node_update</code>
1087 supports order statistics.
1088 </p></li><li class="listitem"><p><code class="classname">trie_prefix_search_node_update</code>
1089 supports searching for ranges that match a given prefix.</p></li><li class="listitem"><p><code class="classname">null_node_update</code>
1090 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
1091 rationale is equal to that of tree-based containers supporting
1092 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">
1093 template<typename Key,
1095 typename Eq_Fn = std::equal_to<Key>,
1096 typename Update_Policy = move_to_front_lu_policy<>,
1097 typename Allocator = std::allocator<char> >
1099 </pre><p>The parameters have the following meaning:</p><div class="orderedlist"><ol class="orderedlist"><li class="listitem"><p>
1100 <code class="classname">Key</code> is the key type.
1101 </p></li><li class="listitem"><p>
1102 <code class="classname">Mapped</code> is the mapped-policy.
1103 </p></li><li class="listitem"><p>
1104 <code class="classname">Eq_Fn</code> is a key equivalence functor.
1105 </p></li><li class="listitem"><p>
1106 <code class="classname">Update_Policy</code> is a policy updating positions in
1107 the list based on access patterns. It is described in the
1108 following subsection.
1109 </p></li><li class="listitem"><p>
1110 <code class="classname">Allocator</code> is an allocator type.
1111 </p></li></ol></div><p>A list-based associative container is a container that
1112 stores elements in a linked-list. It does not order the elements
1113 by any particular order related to the keys. List-based
1114 containers are primarily useful for creating "multimaps". In fact,
1115 list-based containers are designed in this library expressly for
1116 this purpose.</p><p>List-based containers might also be useful for some rare
1117 cases, where a key is encapsulated to the extent that only
1118 key-equivalence can be tested. Hash-based containers need to know
1119 how to transform a key into a size type, and tree-based containers
1120 need to know if some key is larger than another. List-based
1121 associative containers, conversely, only need to know if two keys
1122 are equivalent.</p><p>Since a list-based associative container does not order
1123 elements by keys, is it possible to order the list in some
1124 useful manner? Remarkably, many on-line competitive
1125 algorithms exist for reordering lists to reflect access
1126 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>).
1127 </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>
1128 </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
1129 simple list of integer keys. If we search for the integer 6, we
1130 are paying an overhead: the link with key 6 is only the fifth
1131 link; if it were the first link, it could be accessed
1132 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
1133 accessed. They try to determine, by the access history, which
1134 keys to move to the front of the list. Some of these algorithms
1135 require adding some metadata alongside each entry.</p><p>For example, in the graphic below label A shows the counter
1136 algorithm. Each node contains both a key and a count metadata
1137 (shown in bold). When an element is accessed (e.g. 6) its count is
1138 incremented, as shown in label B. If the count reaches some
1139 predetermined value, say 10, as shown in label C, the count is set
1140 to 0 and the node is moved to the front of the list, as in label
1142 </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
1143 implementing any algorithm moving nodes to the front of the
1144 list (policies implementing algorithms interchanging nodes are
1145 unsupported).</p><p>Associative containers based on lists are parametrized by a
1146 <code class="classname">Update_Policy</code> parameter. This parameter defines the
1147 type of metadata each node contains, how to create the
1148 metadata, and how to decide, using this metadata, whether to
1149 move a node to the front of the list. A list-based associative
1150 container object derives (publicly) from its update policy.
1151 </p><p>An instantiation of <code class="classname">Update_Policy</code> must define
1152 internally <code class="classname">update_metadata</code> as the metadata it
1153 requires. Internally, each node of the list contains, besides
1154 the usual key and data, an instance of <code class="classname">typename
1155 Update_Policy::update_metadata</code>.</p><p>An instantiation of <code class="classname">Update_Policy</code> must define
1156 internally two operators:</p><pre class="programlisting">
1161 operator()(update_metadata &);
1162 </pre><p>The first is called by the container object, when creating a
1163 new node, to create the node's metadata. The second is called
1164 by the container object, when a node is accessed (
1165 when a find operation's key is equivalent to the key of the
1166 node), to determine whether to move the node to the front of
1168 </p><p>The library contains two predefined implementations of
1169 list-update policies. The first
1170 is <code class="classname">lu_counter_policy</code>, which implements the
1171 counter algorithm described above. The second is
1172 <code class="classname">lu_move_to_front_policy</code>,
1173 which unconditionally move an accessed element to the front of
1174 the list. The latter type is very useful in this library,
1175 since there is no need to associate metadata with each element.
1176 (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>
1177 </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
1178 multimaps and multisets; instead one uses an associative
1179 container mapping primary keys to secondary keys.</p><p>List-based containers are especially useful as associative
1180 containers for secondary keys. In fact, they are implemented
1181 here expressly for this purpose.</p><p>To begin with, these containers use very little per-entry
1182 structure memory overhead, since they can be implemented as
1183 singly-linked lists. (Arrays use even lower per-entry memory
1184 overhead, but they are less flexible in moving around entries,
1185 and have weaker invalidation guarantees).</p><p>More importantly, though, list-based containers use very
1186 little per-container memory overhead. The memory overhead of an
1187 empty list-based container is practically that of a pointer.
1188 This is important for when they are used as secondary
1189 associative-containers in situations where the average ratio of
1190 secondary keys to primary keys is low (or even 1).</p><p>In order to reduce the per-container memory overhead as much
1191 as possible, they are implemented as closely as possible to
1192 singly-linked lists.</p><div class="orderedlist"><ol class="orderedlist"><li class="listitem"><p>
1193 List-based containers do not store internally the number
1194 of values that they hold. This means that their <code class="function">size</code>
1195 method has linear complexity (just like <code class="classname">std::list</code>).
1196 Note that finding the number of equivalent-key values in a
1197 standard multimap also has linear complexity (because it must be
1198 done, via <code class="function">std::distance</code> of the
1199 multimap's <code class="function">equal_range</code> method), but usually with
1201 </p></li><li class="listitem"><p>
1202 Most associative-container objects each hold a policy
1203 object (a hash-based container object holds a
1204 hash functor). List-based containers, conversely, only have
1205 class-wide policy objects.
1206 </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
1208 </p><pre class="programlisting">
1209 template<typename Value_Type,
1210 typename Cmp_Fn = std::less<Value_Type>,
1211 typename Tag = pairing_heap_tag,
1212 typename Allocator = std::allocator<char > >
1213 class priority_queue;
1214 </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
1215 to use.</p></li><li class="listitem"><p><code class="classname">Allocator</code> is an allocator
1216 type.</p></li></ol></div><p>The <code class="classname">Tag</code> parameter specifies which underlying
1217 data structure to use. Instantiating it by<code class="classname">pairing_heap_tag</code>,<code class="classname">binary_heap_tag</code>,
1218 <code class="classname">binomial_heap_tag</code>,
1219 <code class="classname">rc_binomial_heap_tag</code>,
1220 or <code class="classname">thin_heap_tag</code>,
1221 specifies, respectively,
1222 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>),
1223 binary heap (<a class="xref" href="policy_data_structures.html#biblio.clrs2001" title="Introduction to Algorithms, 2nd edition">[biblio.clrs2001]</a>),
1224 binomial heap (<a class="xref" href="policy_data_structures.html#biblio.clrs2001" title="Introduction to Algorithms, 2nd edition">[biblio.clrs2001]</a>),
1225 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>),
1226 or a thin heap (<a class="xref" href="policy_data_structures.html#biblio.kt99fat_heaps" title="New Heap Data Structures">[biblio.kt99fat_heaps]</a>).
1228 As mentioned in the tutorial,
1229 <code class="classname">__gnu_pbds::priority_queue</code> shares most of the
1230 same interface with <code class="classname">std::priority_queue</code>.
1231 E.g. if <code class="varname">q</code> is a priority queue of type
1232 <code class="classname">Q</code>, then <code class="function">q.top()</code> will
1233 return the "largest" value in the container (according to
1234 <code class="classname">typename
1235 Q::cmp_fn</code>). <code class="classname">__gnu_pbds::priority_queue</code>
1236 has a larger (and very slightly different) interface than
1237 <code class="classname">std::priority_queue</code>, however, since typically
1238 <code class="classname">push</code> and <code class="classname">pop</code> are deemed
1239 insufficient for manipulating priority-queues. </p><p>Different settings require different priority-queue
1240 implementations which are described in later; see traits
1241 discusses ways to differentiate between the different traits of
1242 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
1243 implementing priority queues. Unfortunately, most such
1244 structures are oriented towards making <code class="function">push</code> and
1245 <code class="function">top</code> efficient, and consequently don't allow efficient
1246 access of other elements: for instance, they cannot support an efficient
1247 <code class="function">find</code> method. In the use case where it
1248 is important to both access and "do something with" an
1249 arbitrary value, one would be out of luck. For example, many graph algorithms require
1250 modifying a value (typically increasing it in the sense of the
1251 priority queue's comparison functor).</p><p>In order to access and manipulate an arbitrary value in a
1252 priority queue, one needs to reference the internals of the
1253 priority queue from some form of an associative container -
1254 this is unavoidable. Of course, in order to maintain the
1255 encapsulation of the priority queue, this needs to be done in a
1256 way that minimizes exposure to implementation internals.</p><p>In this library the priority queue's <code class="function">insert</code>
1257 method returns an iterator, which if valid can be used for subsequent <code class="function">modify</code> and
1258 <code class="function">erase</code> operations. This both preserves the priority
1259 queue's encapsulation, and allows accessing arbitrary values (since the
1260 returned iterators from the <code class="function">push</code> operation can be
1261 stored in some form of associative container).</p><p>Priority queues' iterators present a problem regarding their
1262 invalidation guarantees. One assumes that calling
1263 <code class="function">operator++</code> on an iterator will associate it
1264 with the "next" value. Priority-queues are
1265 self-organizing: each operation changes what the "next" value
1266 means. Consequently, it does not make sense that <code class="function">push</code>
1267 will return an iterator that can be incremented - this can have
1268 no possible use. Also, as in the case of hash-based containers,
1269 it is awkward to define if a subsequent <code class="function">push</code> operation
1270 invalidates a prior returned iterator: it invalidates it in the
1271 sense that its "next" value is not related to what it
1272 previously considered to be its "next" value. However, it might not
1273 invalidate it, in the sense that it can be
1274 de-referenced and used for <code class="function">modify</code> and <code class="function">erase</code>
1275 operations.</p><p>Similarly to the case of the other unordered associative
1276 containers, this library uses a distinction between
1277 point-type and range type iterators. A priority queue's <code class="classname">iterator</code> can always be
1278 converted to a <code class="classname">point_iterator</code>, and a
1279 <code class="classname">const_iterator</code> can always be converted to a
1280 <code class="classname">point_const_iterator</code>.</p><p>The following snippet demonstrates manipulating an arbitrary
1281 value:</p><pre class="programlisting">
1282 // A priority queue of integers.
1283 priority_queue<int > p;
1285 // Insert some values into the priority queue.
1286 priority_queue<int >::point_iterator it = p.push(0);
1291 // Now modify a value.
1294 assert(p.top() == 3);
1295 </pre><p>It should be noted that an alternative design could embed an
1296 associative container in a priority queue. Could, but most
1297 probably should not. To begin with, it should be noted that one
1298 could always encapsulate a priority queue and an associative
1299 container mapping values to priority queue iterators with no
1300 performance loss. One cannot, however, "un-encapsulate" a priority
1301 queue embedding an associative container, which might lead to
1302 performance loss. Assume, that one needs to associate each value
1303 with some data unrelated to priority queues. Then using
1304 this library's design, one could use an
1305 associative container mapping each value to a pair consisting of
1306 this data and a priority queue's iterator. Using the embedded
1307 method would need to use two associative containers. Similar
1308 problems might arise in cases where a value can reside
1309 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
1310 first employs a binary heap, typically one which uses a
1311 sequence; the second uses a tree (or forest of trees), which is
1312 typically less structured than an associative container's tree;
1313 the third simply uses an associative container. These are
1314 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
1315 from a priority queue must incur a logarithmic expense (in the
1316 amortized sense). Any priority queue implementation that would
1317 avoid this, would violate known bounds on comparison-based
1318 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>).
1319 </p><p>Most implementations do
1320 not differ in the asymptotic amortized complexity of
1321 <code class="function">push</code> and <code class="function">pop</code> operations, but they differ in
1322 the constants involved, in the complexity of other operations
1323 (e.g., <code class="function">modify</code>), and in the worst-case
1324 complexity of single operations. In general, the more
1325 "structured" an implementation (i.e., the more internal
1326 invariants it possesses) - the higher its amortized complexity
1327 of <code class="function">push</code> and <code class="function">pop</code> operations.</p><p>This library implements different algorithms using a
1328 single class: <code class="classname">priority_queue</code>.
1329 Instantiating the <code class="classname">Tag</code> template parameter, "selects"
1330 the implementation:</p><div class="orderedlist"><ol class="orderedlist"><li class="listitem"><p>
1331 Instantiating <code class="classname">Tag = binary_heap_tag</code> creates
1332 a binary heap of the form in represented in the graphic with labels A1 or A2. The former is internally
1333 selected by priority_queue
1334 if <code class="classname">Value_Type</code> is instantiated by a primitive type
1335 (e.g., an <span class="type">int</span>); the latter is
1336 internally selected for all other types (e.g.,
1337 <code class="classname">std::string</code>). This implementations is relatively
1338 unstructured, and so has good <code class="classname">push</code> and <code class="classname">pop</code>
1339 performance; it is the "best-in-kind" for primitive
1340 types, e.g., <span class="type">int</span>s. Conversely, it has
1341 high worst-case performance, and can support only linear-time
1342 <code class="function">modify</code> and <code class="function">erase</code> operations.</p></li><li class="listitem"><p>Instantiating <code class="classname">Tag =
1343 pairing_heap_tag</code> creates a pairing heap of the form
1344 in represented by label B in the graphic above. This
1345 implementations too is relatively unstructured, and so has good
1346 <code class="function">push</code> and <code class="function">pop</code>
1347 performance; it is the "best-in-kind" for non-primitive types,
1348 e.g., <code class="classname">std:string</code>s. It also has very good
1349 worst-case <code class="function">push</code> and
1350 <code class="function">join</code> performance (O(1)), but has high
1351 worst-case <code class="function">pop</code>
1352 complexity.</p></li><li class="listitem"><p>Instantiating <code class="classname">Tag =
1353 binomial_heap_tag</code> creates a binomial heap of the
1354 form repsented by label B in the graphic above. This
1355 implementations is more structured than a pairing heap, and so
1356 has worse <code class="function">push</code> and <code class="function">pop</code>
1357 performance. Conversely, it has sub-linear worst-case bounds for
1358 <code class="function">pop</code>, e.g., and so it might be preferred in
1359 cases where responsiveness is important.</p></li><li class="listitem"><p>Instantiating <code class="classname">Tag =
1360 rc_binomial_heap_tag</code> creates a binomial heap of the
1361 form represented in label B above, accompanied by a redundant
1362 counter which governs the trees. This implementations is
1363 therefore more structured than a binomial heap, and so has worse
1364 <code class="function">push</code> and <code class="function">pop</code>
1365 performance. Conversely, it guarantees O(1)
1366 <code class="function">push</code> complexity, and so it might be
1367 preferred in cases where the responsiveness of a binomial heap
1368 is insufficient.</p></li><li class="listitem"><p>Instantiating <code class="classname">Tag =
1369 thin_heap_tag</code> creates a thin heap of the form
1370 represented by the label B in the graphic above. This
1371 implementations too is more structured than a pairing heap, and
1372 so has worse <code class="function">push</code> and
1373 <code class="function">pop</code> performance. Conversely, it has better
1374 worst-case and identical amortized complexities than a Fibonacci
1375 heap, and so might be more appropriate for some graph
1376 algorithms.</p></li></ol></div><p>Of course, one can use any order-preserving associative
1377 container as a priority queue, as in the graphic above label C, possibly by creating an adapter class
1378 over the associative container (much as
1379 <code class="classname">std::priority_queue</code> can adapt <code class="classname">std::vector</code>).
1380 This has the advantage that no cross-referencing is necessary
1381 at all; the priority queue itself is an associative container.
1382 Most associative containers are too structured to compete with
1383 priority queues in terms of <code class="function">push</code> and <code class="function">pop</code>
1384 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
1385 share exactly the same behavior regardless of implementation. Sadly, this is not possible. Just one for instance is in join operations: joining
1386 two binary heaps might throw an exception (not corrupt
1387 any of the heaps on which it operates), but joining two pairing
1388 heaps is exception free.</p><p>Tags and traits are very useful for manipulating generic
1389 types. <code class="classname">__gnu_pbds::priority_queue</code>
1390 publicly defines <code class="classname">container_category</code> as one of the tags. Given any
1391 container <code class="classname">Cntnr</code>, the tag of the underlying
1392 data structure can be found via <code class="classname">typename
1393 Cntnr::container_category</code>; this is one of the possible tags shown in the graphic below.
1394 </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
1395 container type for its attributes. Given any container
1396 <code class="classname">Cntnr</code>, then </p><pre class="programlisting">__gnu_pbds::container_traits<Cntnr></pre><p>
1397 is a traits class identifying the properties of the
1398 container.</p><p>To find if a container might throw if two of its objects are
1400 </p><pre class="programlisting">
1401 container_traits<Cntnr>::split_join_can_throw
1404 Different priority-queue implementations have different invalidation guarantees. This is
1405 especially important, since there is no way to access an arbitrary
1406 value of priority queues except for iterators. Similarly to
1407 associative containers, one can use
1408 </p><pre class="programlisting">
1409 container_traits<Cntnr>::invalidation_guarantee
1411 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>
1412 will be for different implementations. All implementations of
1413 type represented by label B have <code class="classname">point_invalidation_guarantee</code>:
1414 the container can freely internally reorganize the nodes -
1415 range-type iterators are invalidated, but point-type iterators
1416 are always valid. Implementations of type represented by labels A1 and A2 have <code class="classname">basic_invalidation_guarantee</code>:
1417 the container can freely internally reallocate the array - both
1418 point-type and range-type iterators might be invalidated.</p><p>
1419 This has major implications, and constitutes a good reason to avoid
1420 using binary heaps. A binary heap can perform <code class="function">modify</code>
1421 or <code class="function">erase</code> efficiently given a valid point-type
1422 iterator. However, in order to supply it with a valid point-type
1423 iterator, one needs to iterate (linearly) over all
1424 values, then supply the relevant iterator (recall that a
1425 range-type iterator can always be converted to a point-type
1426 iterator). This means that if the number of <code class="function">modify</code> or
1427 <code class="function">erase</code> operations is non-negligible (say
1428 super-logarithmic in the total sequence of operations) - binary
1429 heaps will perform badly.
1430 </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>
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