1<?xml version="1.0" encoding="UTF-8" standalone="no"?> 2<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Transitional//EN" "http://www.w3.org/TR/xhtml1/DTD/xhtml1-transitional.dtd"><html xmlns="http://www.w3.org/1999/xhtml"><head><meta http-equiv="Content-Type" content="text/html; charset=UTF-8" /><title>Design</title><meta name="generator" content="DocBook XSL Stylesheets Vsnapshot" /><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 21. 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 width="20%" align="left"><a accesskey="p" href="policy_data_structures_using.html">Prev</a> </td><th width="60%" align="center">Chapter 21. Policy-Based Data Structures</th><td width="20%" align="right"> <a accesskey="n" href="policy_based_data_structures_test.html">Next</a></td></tr></table><hr /></div><div class="section"><div class="titlepage"><div><div><h2 class="title" style="clear: both"><a id="containers.pbds.design"></a>Design</h2></div></div></div><p></p><div class="section"><div class="titlepage"><div><div><h3 class="title"><a id="pbds.design.concepts"></a>Concepts</h3></div></div></div><div class="section"><div class="titlepage"><div><div><h4 class="title"><a id="pbds.design.concepts.null_type"></a>Null Policy Classes</h4></div></div></div><p> 3 Associative containers are typically parametrized by various 4 policies. For example, a hash-based associative container is 5 parametrized by a hash-functor, transforming each key into an 6 non-negative numerical type. Each such value is then further mapped 7 into a position within the table. The mapping of a key into a 8 position within the table is therefore a two-step process. 9 </p><p> 10 In some cases, instantiations are redundant. For example, when the 11 keys are integers, it is possible to use a redundant hash policy, 12 which transforms each key into its value. 13 </p><p> 14 In some other cases, these policies are irrelevant. For example, a 15 hash-based associative container might transform keys into positions 16 within a table by a different method than the two-step method 17 described above. In such a case, the hash functor is simply 18 irrelevant. 19 </p><p> 20 When a policy is either redundant or irrelevant, it can be replaced 21 by <code class="classname">null_type</code>. 22 </p><p> 23 For example, a <span class="emphasis"><em>set</em></span> is an associative 24 container with one of its template parameters (the one for the 25 mapped type) replaced with <code class="classname">null_type</code>. Other 26 places simplifications are made possible with this technique 27 include node updates in tree and trie data structures, and hash 28 and probe functions for hash data structures. 29 </p></div><div class="section"><div class="titlepage"><div><div><h4 class="title"><a id="pbds.design.concepts.associative_semantics"></a>Map and Set Semantics</h4></div></div></div><div class="section"><div class="titlepage"><div><div><h5 class="title"><a id="concepts.associative_semantics.set_vs_map"></a> 30 Distinguishing Between Maps and Sets 31 </h5></div></div></div><p> 32 Anyone familiar with the standard knows that there are four kinds 33 of associative containers: maps, sets, multimaps, and 34 multisets. The map datatype associates each key to 35 some data. 36 </p><p> 37 Sets are associative containers that simply store keys - 38 they do not map them to anything. In the standard, each map class 39 has a corresponding set class. E.g., 40 <code class="classname">std::map<int, char></code> maps each 41 <code class="classname">int</code> to a <code class="classname">char</code>, but 42 <code class="classname">std::set<int, char></code> simply stores 43 <code class="classname">int</code>s. In this library, however, there are no 44 distinct classes for maps and sets. Instead, an associative 45 container's <code class="classname">Mapped</code> template parameter is a policy: if 46 it is instantiated by <code class="classname">null_type</code>, then it 47 is a "set"; otherwise, it is a "map". E.g., 48 </p><pre class="programlisting"> 49 cc_hash_table<int, char> 50 </pre><p> 51 is a "map" mapping each <span class="type">int</span> value to a <span class="type"> 52 char</span>, but 53 </p><pre class="programlisting"> 54 cc_hash_table<int, null_type> 55 </pre><p> 56 is a type that uniquely stores <span class="type">int</span> values. 57 </p><p>Once the <code class="classname">Mapped</code> template parameter is instantiated 58 by <code class="classname">null_type</code>, then 59 the "set" acts very similarly to the standard's sets - it does not 60 map each key to a distinct <code class="classname">null_type</code> object. Also, 61 , the container's <span class="type">value_type</span> is essentially 62 its <span class="type">key_type</span> - just as with the standard's sets 63 .</p><p> 64 The standard's multimaps and multisets allow, respectively, 65 non-uniquely mapping keys and non-uniquely storing keys. As 66 discussed, the 67 reasons why this might be necessary are 1) that a key might be 68 decomposed into a primary key and a secondary key, 2) that a 69 key might appear more than once, or 3) any arbitrary 70 combination of 1)s and 2)s. Correspondingly, 71 one should use 1) "maps" mapping primary keys to secondary 72 keys, 2) "maps" mapping keys to size types, or 3) any arbitrary 73 combination of 1)s and 2)s. Thus, for example, an 74 <code class="classname">std::multiset<int></code> might be used to store 75 multiple instances of integers, but using this library's 76 containers, one might use 77 </p><pre class="programlisting"> 78 tree<int, size_t> 79 </pre><p> 80 i.e., a <code class="classname">map</code> of <span class="type">int</span>s to 81 <span class="type">size_t</span>s. 82 </p><p> 83 These "multimaps" and "multisets" might be confusing to 84 anyone familiar with the standard's <code class="classname">std::multimap</code> and 85 <code class="classname">std::multiset</code>, because there is no clear 86 correspondence between the two. For example, in some cases 87 where one uses <code class="classname">std::multiset</code> in the standard, one might use 88 in this library a "multimap" of "multisets" - i.e., a 89 container that maps primary keys each to an associative 90 container that maps each secondary key to the number of times 91 it occurs. 92 </p><p> 93 When one uses a "multimap," one should choose with care the 94 type of container used for secondary keys. 95 </p></div><div class="section"><div class="titlepage"><div><div><h5 class="title"><a id="concepts.associative_semantics.multi"></a>Alternatives to <code class="classname">std::multiset</code> and <code class="classname">std::multimap</code></h5></div></div></div><p> 96 Brace onself: this library does not contain containers like 97 <code class="classname">std::multimap</code> or 98 <code class="classname">std::multiset</code>. Instead, these data 99 structures can be synthesized via manipulation of the 100 <code class="classname">Mapped</code> template parameter. 101 </p><p> 102 One maps the unique part of a key - the primary key, into an 103 associative-container of the (originally) non-unique parts of 104 the key - the secondary key. A primary associative-container 105 is an associative container of primary keys; a secondary 106 associative-container is an associative container of 107 secondary keys. 108 </p><p> 109 Stepping back a bit, and starting in from the beginning. 110 </p><p> 111 Maps (or sets) allow mapping (or storing) unique-key values. 112 The standard library also supplies associative containers which 113 map (or store) multiple values with equivalent keys: 114 <code class="classname">std::multimap</code>, <code class="classname">std::multiset</code>, 115 <code class="classname">std::tr1::unordered_multimap</code>, and 116 <code class="classname">unordered_multiset</code>. We first discuss how these might 117 be used, then why we think it is best to avoid them. 118 </p><p> 119 Suppose one builds a simple bank-account application that 120 records for each client (identified by an <code class="classname">std::string</code>) 121 and account-id (marked by an <span class="type">unsigned long</span>) - 122 the balance in the account (described by a 123 <span class="type">float</span>). Suppose further that ordering this 124 information is not useful, so a hash-based container is 125 preferable to a tree based container. Then one can use 126 </p><pre class="programlisting"> 127 std::tr1::unordered_map<std::pair<std::string, unsigned long>, float, ...> 128 </pre><p> 129 which hashes every combination of client and account-id. This 130 might work well, except for the fact that it is now impossible 131 to efficiently list all of the accounts of a specific client 132 (this would practically require iterating over all 133 entries). Instead, one can use 134 </p><pre class="programlisting"> 135 std::tr1::unordered_multimap<std::pair<std::string, unsigned long>, float, ...> 136 </pre><p> 137 which hashes every client, and decides equivalence based on 138 client only. This will ensure that all accounts belonging to a 139 specific user are stored consecutively. 140 </p><p> 141 Also, suppose one wants an integers' priority queue 142 (a container that supports <code class="function">push</code>, 143 <code class="function">pop</code>, and <code class="function">top</code> operations, the last of which 144 returns the largest <span class="type">int</span>) that also supports 145 operations such as <code class="function">find</code> and <code class="function">lower_bound</code>. A 146 reasonable solution is to build an adapter over 147 <code class="classname">std::set<int></code>. In this adapter, 148 <code class="function">push</code> will just call the tree-based 149 associative container's <code class="function">insert</code> method; <code class="function">pop</code> 150 will call its <code class="function">end</code> method, and use it to return the 151 preceding element (which must be the largest). Then this might 152 work well, except that the container object cannot hold 153 multiple instances of the same integer (<code class="function">push(4)</code>, 154 will be a no-op if <code class="constant">4</code> is already in the 155 container object). If multiple keys are necessary, then one 156 might build the adapter over an 157 <code class="classname">std::multiset<int></code>. 158 </p><p> 159 The standard library's non-unique-mapping containers are useful 160 when (1) a key can be decomposed in to a primary key and a 161 secondary key, (2) a key is needed multiple times, or (3) any 162 combination of (1) and (2). 163 </p><p> 164 The graphic below shows how the standard library's container 165 design works internally; in this figure nodes shaded equally 166 represent equivalent-key values. Equivalent keys are stored 167 consecutively using the properties of the underlying data 168 structure: binary search trees (label A) store equivalent-key 169 values consecutively (in the sense of an in-order walk) 170 naturally; collision-chaining hash tables (label B) store 171 equivalent-key values in the same bucket, the bucket can be 172 arranged so that equivalent-key values are consecutive. 173 </p><div class="figure"><a id="id-1.3.5.8.4.3.3.3.14"></a><p class="title"><strong>Figure 21.8. Non-unique Mapping Standard Containers</strong></p><div class="figure-contents"><div class="mediaobject" align="center"><img src="../images/pbds_embedded_lists_1.png" align="middle" alt="Non-unique Mapping Standard Containers" /></div></div></div><br class="figure-break" /><p> 174 Put differently, the standards' non-unique mapping 175 associative-containers are associative containers that map 176 primary keys to linked lists that are embedded into the 177 container. The graphic below shows again the two 178 containers from the first graphic above, this time with 179 the embedded linked lists of the grayed nodes marked 180 explicitly. 181 </p><div class="figure"><a id="fig.pbds_embedded_lists_2"></a><p class="title"><strong>Figure 21.9. 182 Effect of embedded lists in 183 <code class="classname">std::multimap</code> 184 </strong></p><div class="figure-contents"><div class="mediaobject" align="center"><img src="../images/pbds_embedded_lists_2.png" align="middle" alt="Effect of embedded lists in std::multimap" /></div></div></div><br class="figure-break" /><p> 185 These embedded linked lists have several disadvantages. 186 </p><div class="orderedlist"><ol class="orderedlist" type="1"><li class="listitem"><p> 187 The underlying data structure embeds the linked lists 188 according to its own consideration, which means that the 189 search path for a value might include several different 190 equivalent-key values. For example, the search path for the 191 the black node in either of the first graphic, labels A or B, 192 includes more than a single gray node. 193 </p></li><li class="listitem"><p> 194 The links of the linked lists are the underlying data 195 structures' nodes, which typically are quite structured. In 196 the case of tree-based containers (the grapic above, label 197 B), each "link" is actually a node with three pointers (one 198 to a parent and two to children), and a 199 relatively-complicated iteration algorithm. The linked 200 lists, therefore, can take up quite a lot of memory, and 201 iterating over all values equal to a given key (through the 202 return value of the standard 203 library's <code class="function">equal_range</code>) can be 204 expensive. 205 </p></li><li class="listitem"><p> 206 The primary key is stored multiply; this uses more memory. 207 </p></li><li class="listitem"><p> 208 Finally, the interface of this design excludes several 209 useful underlying data structures. Of all the unordered 210 self-organizing data structures, practically only 211 collision-chaining hash tables can (efficiently) guarantee 212 that equivalent-key values are stored consecutively. 213 </p></li></ol></div><p> 214 The above reasons hold even when the ratio of secondary keys to 215 primary keys (or average number of identical keys) is small, but 216 when it is large, there are more severe problems: 217 </p><div class="orderedlist"><ol class="orderedlist" type="1"><li class="listitem"><p> 218 The underlying data structures order the links inside each 219 embedded linked-lists according to their internal 220 considerations, which effectively means that each of the 221 links is unordered. Irrespective of the underlying data 222 structure, searching for a specific value can degrade to 223 linear complexity. 224 </p></li><li class="listitem"><p> 225 Similarly to the above point, it is impossible to apply 226 to the secondary keys considerations that apply to primary 227 keys. For example, it is not possible to maintain secondary 228 keys by sorted order. 229 </p></li><li class="listitem"><p> 230 While the interface "understands" that all equivalent-key 231 values constitute a distinct list (through 232 <code class="function">equal_range</code>), the underlying data 233 structure typically does not. This means that operations such 234 as erasing from a tree-based container all values whose keys 235 are equivalent to a a given key can be super-linear in the 236 size of the tree; this is also true also for several other 237 operations that target a specific list. 238 </p></li></ol></div><p> 239 In this library, all associative containers map 240 (or store) unique-key values. One can (1) map primary keys to 241 secondary associative-containers (containers of 242 secondary keys) or non-associative containers (2) map identical 243 keys to a size-type representing the number of times they 244 occur, or (3) any combination of (1) and (2). Instead of 245 allowing multiple equivalent-key values, this library 246 supplies associative containers based on underlying 247 data structures that are suitable as secondary 248 associative-containers. 249 </p><p> 250 In the figure below, labels A and B show the equivalent 251 underlying data structures in this library, as mapped to the 252 first graphic above. Labels A and B, respectively. Each shaded 253 box represents some size-type or secondary 254 associative-container. 255 </p><div class="figure"><a id="id-1.3.5.8.4.3.3.3.23"></a><p class="title"><strong>Figure 21.10. Non-unique Mapping Containers</strong></p><div class="figure-contents"><div class="mediaobject" align="center"><img src="../images/pbds_embedded_lists_3.png" align="middle" alt="Non-unique Mapping Containers" /></div></div></div><br class="figure-break" /><p> 256 In the first example above, then, one would use an associative 257 container mapping each user to an associative container which 258 maps each application id to a start time (see 259 <code class="filename">example/basic_multimap.cc</code>); in the second 260 example, one would use an associative container mapping 261 each <code class="classname">int</code> to some size-type indicating the 262 number of times it logically occurs 263 (see <code class="filename">example/basic_multiset.cc</code>. 264 </p><p> 265 See the discussion in list-based container types for containers 266 especially suited as secondary associative-containers. 267 </p></div></div><div class="section"><div class="titlepage"><div><div><h4 class="title"><a id="pbds.design.concepts.iterator_semantics"></a>Iterator Semantics</h4></div></div></div><div class="section"><div class="titlepage"><div><div><h5 class="title"><a id="concepts.iterator_semantics.point_and_range"></a>Point and Range Iterators</h5></div></div></div><p> 268 Iterator concepts are bifurcated in this design, and are 269 comprised of point-type and range-type iteration. 270 </p><p> 271 A point-type iterator is an iterator that refers to a specific 272 element as returned through an 273 associative-container's <code class="function">find</code> method. 274 </p><p> 275 A range-type iterator is an iterator that is used to go over a 276 sequence of elements, as returned by a container's 277 <code class="function">find</code> method. 278 </p><p> 279 A point-type method is a method that 280 returns a point-type iterator; a range-type method is a method 281 that returns a range-type iterator. 282 </p><p>For most containers, these types are synonymous; for 283 self-organizing containers, such as hash-based containers or 284 priority queues, these are inherently different (in any 285 implementation, including that of C++ standard library 286 components), but in this design, it is made explicit. They are 287 distinct types. 288 </p></div><div class="section"><div class="titlepage"><div><div><h5 class="title"><a id="concepts.iterator_semantics.both"></a>Distinguishing Point and Range Iterators</h5></div></div></div><p>When using this library, is necessary to differentiate 289 between two types of methods and iterators: point-type methods and 290 iterators, and range-type methods and iterators. Each associative 291 container's interface includes the methods:</p><pre class="programlisting"> 292 point_const_iterator 293 find(const_key_reference r_key) const; 294 295 point_iterator 296 find(const_key_reference r_key); 297 298 std::pair<point_iterator,bool> 299 insert(const_reference r_val); 300 </pre><p>The relationship between these iterator types varies between 301 container types. The figure below 302 shows the most general invariant between point-type and 303 range-type iterators: In <span class="emphasis"><em>A</em></span> <code class="literal">iterator</code>, can 304 always be converted to <code class="literal">point_iterator</code>. In <span class="emphasis"><em>B</em></span> 305 shows invariants for order-preserving containers: point-type 306 iterators are synonymous with range-type iterators. 307 Orthogonally, <span class="emphasis"><em>C</em></span>shows invariants for "set" 308 containers: iterators are synonymous with const iterators.</p><div class="figure"><a id="id-1.3.5.8.4.3.4.3.5"></a><p class="title"><strong>Figure 21.11. Point Iterator Hierarchy</strong></p><div class="figure-contents"><div class="mediaobject" align="center"><img src="../images/pbds_point_iterator_hierarchy.png" align="middle" alt="Point Iterator Hierarchy" /></div></div></div><br class="figure-break" /><p>Note that point-type iterators in self-organizing containers 309 (hash-based associative containers) lack movement 310 operators, such as <code class="literal">operator++</code> - in fact, this 311 is the reason why this library differentiates from the standard C++ librarys 312 design on this point.</p><p>Typically, one can determine an iterator's movement 313 capabilities using 314 <code class="literal">std::iterator_traits<It>iterator_category</code>, 315 which is a <code class="literal">struct</code> indicating the iterator's 316 movement capabilities. Unfortunately, none of the standard predefined 317 categories reflect a pointer's <span class="emphasis"><em>not</em></span> having any 318 movement capabilities whatsoever. Consequently, 319 <code class="literal">pb_ds</code> adds a type 320 <code class="literal">trivial_iterator_tag</code> (whose name is taken from 321 a concept in C++ standardese, which is the category of iterators 322 with no movement capabilities.) All other standard C++ library 323 tags, such as <code class="literal">forward_iterator_tag</code> retain their 324 common use.</p></div><div class="section"><div class="titlepage"><div><div><h5 class="title"><a id="pbds.design.concepts.invalidation"></a>Invalidation Guarantees</h5></div></div></div><p> 325 If one manipulates a container object, then iterators previously 326 obtained from it can be invalidated. In some cases a 327 previously-obtained iterator cannot be de-referenced; in other cases, 328 the iterator's next or previous element might have changed 329 unpredictably. This corresponds exactly to the question whether a 330 point-type or range-type iterator (see previous concept) is valid or 331 not. In this design, one can query a container (in compile time) about 332 its invalidation guarantees. 333 </p><p> 334 Given three different types of associative containers, a modifying 335 operation (in that example, <code class="function">erase</code>) invalidated 336 iterators in three different ways: the iterator of one container 337 remained completely valid - it could be de-referenced and 338 incremented; the iterator of a different container could not even be 339 de-referenced; the iterator of the third container could be 340 de-referenced, but its "next" iterator changed unpredictably. 341 </p><p> 342 Distinguishing between find and range types allows fine-grained 343 invalidation guarantees, because these questions correspond exactly 344 to the question of whether point-type iterators and range-type 345 iterators are valid. The graphic below shows tags corresponding to 346 different types of invalidation guarantees. 347 </p><div class="figure"><a id="id-1.3.5.8.4.3.4.4.5"></a><p class="title"><strong>Figure 21.12. Invalidation Guarantee Tags Hierarchy</strong></p><div class="figure-contents"><div class="mediaobject" align="center"><img src="../images/pbds_invalidation_tag_hierarchy.png" align="middle" alt="Invalidation Guarantee Tags Hierarchy" /></div></div></div><br class="figure-break" /><div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; "><li class="listitem"><p> 348 <code class="classname">basic_invalidation_guarantee</code> 349 corresponds to a basic guarantee that a point-type iterator, 350 a found pointer, or a found reference, remains valid as long 351 as the container object is not modified. 352 </p></li><li class="listitem"><p> 353 <code class="classname">point_invalidation_guarantee</code> 354 corresponds to a guarantee that a point-type iterator, a 355 found pointer, or a found reference, remains valid even if 356 the container object is modified. 357 </p></li><li class="listitem"><p> 358 <code class="classname">range_invalidation_guarantee</code> 359 corresponds to a guarantee that a range-type iterator remains 360 valid even if the container object is modified. 361 </p></li></ul></div><p>To find the invalidation guarantee of a 362 container, one can use</p><pre class="programlisting"> 363 typename container_traits<Cntnr>::invalidation_guarantee 364 </pre><p>Note that this hierarchy corresponds to the logic it 365 represents: if a container has range-invalidation guarantees, 366 then it must also have find invalidation guarantees; 367 correspondingly, its invalidation guarantee (in this case 368 <code class="classname">range_invalidation_guarantee</code>) 369 can be cast to its base class (in this case <code class="classname">point_invalidation_guarantee</code>). 370 This means that this this hierarchy can be used easily using 371 standard metaprogramming techniques, by specializing on the 372 type of <code class="literal">invalidation_guarantee</code>.</p><p> 373 These types of problems were addressed, in a more general 374 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 375 our opinion, an invalidation-guarantee hierarchy would solve 376 these problems in all container types - not just associative 377 containers. 378 </p></div></div><div class="section"><div class="titlepage"><div><div><h4 class="title"><a id="pbds.design.concepts.genericity"></a>Genericity</h4></div></div></div><p> 379 The design attempts to address the following problem of 380 data-structure genericity. When writing a function manipulating 381 a generic container object, what is the behavior of the object? 382 Suppose one writes 383 </p><pre class="programlisting"> 384 template<typename Cntnr> 385 void 386 some_op_sequence(Cntnr &r_container) 387 { 388 ... 389 } 390 </pre><p> 391 then one needs to address the following questions in the body 392 of <code class="function">some_op_sequence</code>: 393 </p><div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; "><li class="listitem"><p> 394 Which types and methods does <code class="literal">Cntnr</code> support? 395 Containers based on hash tables can be queries for the 396 hash-functor type and object; this is meaningless for tree-based 397 containers. Containers based on trees can be split, joined, or 398 can erase iterators and return the following iterator; this 399 cannot be done by hash-based containers. 400 </p></li><li class="listitem"><p> 401 What are the exception and invalidation guarantees 402 of <code class="literal">Cntnr</code>? A container based on a probing 403 hash-table invalidates all iterators when it is modified; this 404 is not the case for containers based on node-based 405 trees. Containers based on a node-based tree can be split or 406 joined without exceptions; this is not the case for containers 407 based on vector-based trees. 408 </p></li><li class="listitem"><p> 409 How does the container maintain its elements? Tree-based and 410 Trie-based containers store elements by key order; others, 411 typically, do not. A container based on a splay trees or lists 412 with update policies "cache" "frequently accessed" elements; 413 containers based on most other underlying data structures do 414 not. 415 </p></li><li class="listitem"><p> 416 How does one query a container about characteristics and 417 capabilities? What is the relationship between two different 418 data structures, if anything? 419 </p></li></ul></div><p>The remainder of this section explains these issues in 420 detail.</p><div class="section"><div class="titlepage"><div><div><h5 class="title"><a id="concepts.genericity.tag"></a>Tag</h5></div></div></div><p> 421 Tags are very useful for manipulating generic types. For example, if 422 <code class="literal">It</code> is an iterator class, then <code class="literal">typename 423 It::iterator_category</code> or <code class="literal">typename 424 std::iterator_traits<It>::iterator_category</code> will 425 yield its category, and <code class="literal">typename 426 std::iterator_traits<It>::value_type</code> will yield its 427 value type. 428 </p><p> 429 This library contains a container tag hierarchy corresponding to the 430 diagram below. 431 </p><div class="figure"><a id="id-1.3.5.8.4.3.5.7.4"></a><p class="title"><strong>Figure 21.13. Container Tag Hierarchy</strong></p><div class="figure-contents"><div class="mediaobject" align="center"><img src="../images/pbds_container_tag_hierarchy.png" align="middle" alt="Container Tag Hierarchy" /></div></div></div><br class="figure-break" /><p> 432 Given any container <span class="type">Cntnr</span>, the tag of 433 the underlying data structure can be found via <code class="literal">typename 434 Cntnr::container_category</code>. 435 </p></div><div class="section"><div class="titlepage"><div><div><h5 class="title"><a id="concepts.genericity.traits"></a>Traits</h5></div></div></div><p></p><p>Additionally, a traits mechanism can be used to query a 436 container type for its attributes. Given any container 437 <code class="literal">Cntnr</code>, then <code class="literal"><Cntnr></code> 438 is a traits class identifying the properties of the 439 container.</p><p>To find if a container can throw when a key is erased (which 440 is true for vector-based trees, for example), one can 441 use 442 </p><pre class="programlisting">container_traits<Cntnr>::erase_can_throw</pre><p> 443 Some of the definitions in <code class="classname">container_traits</code> 444 are dependent on other 445 definitions. If <code class="classname">container_traits<Cntnr>::order_preserving</code> 446 is <code class="constant">true</code> (which is the case for containers 447 based on trees and tries), then the container can be split or 448 joined; in this 449 case, <code class="classname">container_traits<Cntnr>::split_join_can_throw</code> 450 indicates whether splits or joins can throw exceptions (which is 451 true for vector-based trees); 452 otherwise <code class="classname">container_traits<Cntnr>::split_join_can_throw</code> 453 will yield a compilation error. (This is somewhat similar to a 454 compile-time version of the COM model). 455 </p></div></div></div><div class="section"><div class="titlepage"><div><div><h3 class="title"><a id="pbds.design.container"></a>By Container</h3></div></div></div><div class="section"><div class="titlepage"><div><div><h4 class="title"><a id="pbds.design.container.hash"></a>hash</h4></div></div></div><div class="section"><div class="titlepage"><div><div><h5 class="title"><a id="container.hash.interface"></a>Interface</h5></div></div></div><p> 456 The collision-chaining hash-based container has the 457 following declaration.</p><pre class="programlisting"> 458 template< 459 typename Key, 460 typename Mapped, 461 typename Hash_Fn = std::hash<Key>, 462 typename Eq_Fn = std::equal_to<Key>, 463 typename Comb_Hash_Fn = direct_mask_range_hashing<> 464 typename Resize_Policy = default explained below. 465 bool Store_Hash = false, 466 typename Allocator = std::allocator<char> > 467 class cc_hash_table; 468 </pre><p>The parameters have the following meaning:</p><div class="orderedlist"><ol class="orderedlist" type="1"><li class="listitem"><p><code class="classname">Key</code> is the key type.</p></li><li class="listitem"><p><code class="classname">Mapped</code> is the mapped-policy.</p></li><li class="listitem"><p><code class="classname">Hash_Fn</code> is a key hashing functor.</p></li><li class="listitem"><p><code class="classname">Eq_Fn</code> is a key equivalence functor.</p></li><li class="listitem"><p><code class="classname">Comb_Hash_Fn</code> is a range-hashing_functor; 469 it describes how to translate hash values into positions 470 within the table. </p></li><li class="listitem"><p><code class="classname">Resize_Policy</code> describes how a container object 471 should change its internal size. </p></li><li class="listitem"><p><code class="classname">Store_Hash</code> indicates whether the hash value 472 should be stored with each entry. </p></li><li class="listitem"><p><code class="classname">Allocator</code> is an allocator 473 type.</p></li></ol></div><p>The probing hash-based container has the following 474 declaration.</p><pre class="programlisting"> 475 template< 476 typename Key, 477 typename Mapped, 478 typename Hash_Fn = std::hash<Key>, 479 typename Eq_Fn = std::equal_to<Key>, 480 typename Comb_Probe_Fn = direct_mask_range_hashing<> 481 typename Probe_Fn = default explained below. 482 typename Resize_Policy = default explained below. 483 bool Store_Hash = false, 484 typename Allocator = std::allocator<char> > 485 class gp_hash_table; 486 </pre><p>The parameters are identical to those of the 487 collision-chaining container, except for the following.</p><div class="orderedlist"><ol class="orderedlist" type="1"><li class="listitem"><p><code class="classname">Comb_Probe_Fn</code> describes how to transform a probe 488 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 489 other parameters, and are explained below.</p></div><div class="section"><div class="titlepage"><div><div><h5 class="title"><a id="container.hash.details"></a>Details</h5></div></div></div><div class="section"><div class="titlepage"><div><div><h6 class="title"><a id="container.hash.details.hash_policies"></a>Hash Policies</h6></div></div></div><div class="section"><div class="titlepage"><div><div><h6 class="title"><a id="details.hash_policies.general"></a>General</h6></div></div></div><p>Following is an explanation of some functions which hashing 490 involves. The graphic below illustrates the discussion.</p><div class="figure"><a id="id-1.3.5.8.4.4.2.3.2.2.3"></a><p class="title"><strong>Figure 21.14. Hash functions, ranged-hash functions, and 491 range-hashing functions</strong></p><div class="figure-contents"><div class="mediaobject" align="center"><img src="../images/pbds_hash_ranged_hash_range_hashing_fns.png" align="middle" alt="Hash functions, ranged-hash functions, and range-hashing functions" /></div></div></div><br class="figure-break" /><p>Let U be a domain (e.g., the integers, or the 492 strings of 3 characters). A hash-table algorithm needs to map 493 elements of U "uniformly" into the range [0,..., m - 494 1] (where m is a non-negative integral value, and 495 is, in general, time varying). I.e., the algorithm needs 496 a ranged-hash function</p><p> 497 f : U × Z<sub>+</sub> → Z<sub>+</sub> 498 </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 499 <a class="xref" href="policy_data_structures.html#biblio.knuth98sorting" title="The Art of Computer Programming - Sorting and Searching">[biblio.knuth98sorting]</a>.) 500 One 501 common solution is to use the composition of the hash 502 function</p><p>h : U → Z<sub>+</sub> ,</p><p>which maps elements of U into the non-negative 503 integrals, and</p><p>g : Z<sub>+</sub> × Z<sub>+</sub> → 504 Z<sub>+</sub>,</p><p>which maps a non-negative hash value, and a non-negative 505 range upper-bound into a non-negative integral in the range 506 between 0 (inclusive) and the range upper bound (exclusive), 507 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="id-1.3.5.8.4.4.2.3.2.2.15"></a><p class="title"><strong>Equation 21.1. Ranged Hash Function</strong></p><div class="equation-contents"><span class="mathphrase"> 508 f(u , m) = g(h(u), m) 509 </span></div></div><br class="equation-break" /><p>From the above, it is obvious that given g and 510 h, f can always be composed (however the converse 511 is not true). The standard's hash-based containers allow specifying 512 a hash function, and use a hard-wired range-hashing function; 513 the ranged-hash function is implicitly composed.</p><p>The above describes the case where a key is to be mapped 514 into a single position within a hash table, e.g., 515 in a collision-chaining table. In other cases, a key is to be 516 mapped into a sequence of positions within a table, 517 e.g., in a probing table. Similar terms apply in this 518 case: the table requires a ranged probe function, 519 mapping a key into a sequence of positions withing the table. 520 This is typically achieved by composing a hash function 521 mapping the key into a non-negative integral type, a 522 probe function transforming the hash value into a 523 sequence of hash values, and a range-hashing function 524 transforming the sequence of hash values into a sequence of 525 positions.</p></div><div class="section"><div class="titlepage"><div><div><h6 class="title"><a id="details.hash_policies.range"></a>Range Hashing</h6></div></div></div><p>Some common choices for range-hashing functions are the 526 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 527 as</p><div class="equation"><a id="id-1.3.5.8.4.4.2.3.2.3.3"></a><p class="title"><strong>Equation 21.2. Range-Hashing, Division Method</strong></p><div class="equation-contents"><span class="mathphrase"> 528 g(r, m) = r mod m 529 </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 530 v (typically powers of 2), and some a. Each of 531 these range-hashing functions works best for some different 532 setting.</p><p>The division method (see above) is a 533 very common choice. However, even this single method can be 534 implemented in two very different ways. It is possible to 535 implement using the low 536 level % (modulo) operation (for any m), or the 537 low level & (bit-mask) operation (for the case where 538 m is a power of 2), i.e.,</p><div class="equation"><a id="id-1.3.5.8.4.4.2.3.2.3.9"></a><p class="title"><strong>Equation 21.3. Division via Prime Modulo</strong></p><div class="equation-contents"><span class="mathphrase"> 539 g(r, m) = r % m 540 </span></div></div><br class="equation-break" /><p>and</p><div class="equation"><a id="id-1.3.5.8.4.4.2.3.2.3.11"></a><p class="title"><strong>Equation 21.4. Division via Bit Mask</strong></p><div class="equation-contents"><span class="mathphrase"> 541 g(r, m) = r & m - 1, (with m = 542 2<sup>k</sup> for some k) 543 </span></div></div><br class="equation-break" /><p>respectively.</p><p>The % (modulo) implementation has the advantage that for 544 m a prime far from a power of 2, g(r, m) is 545 affected by all the bits of r (minimizing the chance of 546 collision). It has the disadvantage of using the costly modulo 547 operation. This method is hard-wired into SGI's implementation 548 .</p><p>The & (bit-mask) implementation has the advantage of 549 relying on the fast bit-wise and operation. It has the 550 disadvantage that for g(r, m) is affected only by the 551 low order bits of r. This method is hard-wired into 552 Dinkumware's implementation.</p></div><div class="section"><div class="titlepage"><div><div><h6 class="title"><a id="details.hash_policies.ranged"></a>Ranged Hash</h6></div></div></div><p>In cases it is beneficial to allow the 553 client to directly specify a ranged-hash hash function. It is 554 true, that the writer of the ranged-hash function cannot rely 555 on the values of m having specific numerical properties 556 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 557 the values of m are determined by a resize policy with 558 possibly orthogonal considerations.</p><p>There are two cases where a ranged-hash function can be 559 superior. The firs is when using perfect hashing: the 560 second is when the values of m can be used to estimate 561 the "general" number of distinct values required. This is 562 described in the following.</p><p>Let</p><p> 563 s = [ s<sub>0</sub>,..., s<sub>t - 1</sub>] 564 </p><p>be a string of t characters, each of which is from 565 domain S. Consider the following ranged-hash 566 function:</p><div class="equation"><a id="id-1.3.5.8.4.4.2.3.2.4.7"></a><p class="title"><strong>Equation 21.5. 567 A Standard String Hash Function 568 </strong></p><div class="equation-contents"><span class="mathphrase"> 569 f<sub>1</sub>(s, m) = ∑ <sub>i = 570 0</sub><sup>t - 1</sup> s<sub>i</sub> a<sup>i</sup> mod m 571 </span></div></div><br class="equation-break" /><p>where a is some non-negative integral value. This is 572 the standard string-hashing function used in SGI's 573 implementation (with a = 5). Its advantage is that 574 it takes into account all of the characters of the string.</p><p>Now assume that s is the string representation of a 575 of a long DNA sequence (and so S = {'A', 'C', 'G', 576 'T'}). In this case, scanning the entire string might be 577 prohibitively expensive. A possible alternative might be to use 578 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="id-1.3.5.8.4.4.2.3.2.4.12"></a><p class="title"><strong>Equation 21.6. 579 Only k String DNA Hash 580 </strong></p><div class="equation-contents"><span class="mathphrase"> 581 f<sub>2</sub>(s, m) = ∑ <sub>i 582 = 0</sub><sup>k - 1</sup> s<sub>i</sub> a<sup>i</sup> mod m 583 </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 584 characters starting at a random position (determined at each 585 resize), or scanning k random positions (determined at 586 each resize), i.e., using</p><p>f<sub>3</sub>(s, m) = ∑ <sub>i = 587 r</sub>0<sup>r<sub>0</sub> + k - 1</sup> s<sub>i</sub> 588 a<sup>i</sup> mod m ,</p><p>or</p><p>f<sub>4</sub>(s, m) = ∑ <sub>i = 0</sub><sup>k - 589 1</sup> s<sub>r</sub>i a<sup>r<sub>i</sub></sup> mod 590 m ,</p><p>respectively, for r<sub>0</sub>,..., r<sub>k-1</sub> 591 each in the (inclusive) range [0,...,t-1].</p><p>It should be noted that the above functions cannot be 592 decomposed as per a ranged hash composed of hash and range hashing.</p></div><div class="section"><div class="titlepage"><div><div><h6 class="title"><a id="details.hash_policies.implementation"></a>Implementation</h6></div></div></div><p>This sub-subsection describes the implementation of 593 the above in this library. It first explains range-hashing 594 functions in collision-chaining tables, then ranged-hash 595 functions in collision-chaining tables, then probing-based 596 tables, and finally lists the relevant classes in this 597 library.</p><div class="section"><div class="titlepage"><div><div><h6 class="title"><a id="hash_policies.implementation.collision-chaining"></a> 598 Range-Hashing and Ranged-Hashes in Collision-Chaining Tables 599 </h6></div></div></div><p><code class="classname">cc_hash_table</code> is 600 parametrized by <code class="classname">Hash_Fn</code> and <code class="classname">Comb_Hash_Fn</code>, a 601 hash functor and a combining hash functor, respectively.</p><p>In general, <code class="classname">Comb_Hash_Fn</code> is considered a 602 range-hashing functor. <code class="classname">cc_hash_table</code> 603 synthesizes a ranged-hash function from <code class="classname">Hash_Fn</code> and 604 <code class="classname">Comb_Hash_Fn</code>. The figure below shows an <code class="classname">insert</code> sequence 605 diagram for this case. The user inserts an element (point A), 606 the container transforms the key into a non-negative integral 607 using the hash functor (points B and C), and transforms the 608 result into a position using the combining functor (points D 609 and E).</p><div class="figure"><a id="id-1.3.5.8.4.4.2.3.2.5.3.4"></a><p class="title"><strong>Figure 21.15. Insert hash sequence diagram</strong></p><div class="figure-contents"><div class="mediaobject" align="center"><img src="../images/pbds_hash_range_hashing_seq_diagram.png" align="middle" alt="Insert hash sequence diagram" /></div></div></div><br class="figure-break" /><p>If <code class="classname">cc_hash_table</code>'s 610 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 611 a ranged-hash function. The graphic below shows an <code class="function">insert</code> sequence 612 diagram. The user inserts an element (point A), the container 613 transforms the key into a position using the combining functor 614 (points B and C).</p><div class="figure"><a id="id-1.3.5.8.4.4.2.3.2.5.3.6"></a><p class="title"><strong>Figure 21.16. Insert hash sequence diagram with a null policy</strong></p><div class="figure-contents"><div class="mediaobject" align="center"><img src="../images/pbds_hash_range_hashing_seq_diagram2.png" align="middle" alt="Insert hash sequence diagram with a null policy" /></div></div></div><br class="figure-break" /></div><div class="section"><div class="titlepage"><div><div><h6 class="title"><a id="hash_policies.implementation.probe"></a> 615 Probing tables 616 </h6></div></div></div><p><code class="classname">gp_hash_table</code> is parametrized by 617 <code class="classname">Hash_Fn</code>, <code class="classname">Probe_Fn</code>, 618 and <code class="classname">Comb_Probe_Fn</code>. As before, if 619 <code class="classname">Hash_Fn</code> and <code class="classname">Probe_Fn</code> 620 are both <code class="classname">null_type</code>, then 621 <code class="classname">Comb_Probe_Fn</code> is a ranged-probe 622 functor. Otherwise, <code class="classname">Hash_Fn</code> is a hash 623 functor, <code class="classname">Probe_Fn</code> is a functor for offsets 624 from a hash value, and <code class="classname">Comb_Probe_Fn</code> 625 transforms a probe sequence into a sequence of positions within 626 the table.</p></div><div class="section"><div class="titlepage"><div><div><h6 class="title"><a id="hash_policies.implementation.predefined"></a> 627 Pre-Defined Policies 628 </h6></div></div></div><p>This library contains some pre-defined classes 629 implementing range-hashing and probing functions:</p><div class="orderedlist"><ol class="orderedlist" type="1"><li class="listitem"><p><code class="classname">direct_mask_range_hashing</code> 630 and <code class="classname">direct_mod_range_hashing</code> 631 are range-hashing functions based on a bit-mask and a modulo 632 operation, respectively.</p></li><li class="listitem"><p><code class="classname">linear_probe_fn</code>, and 633 <code class="classname">quadratic_probe_fn</code> are 634 a linear probe and a quadratic probe function, 635 respectively.</p></li></ol></div><p> 636 The graphic below shows the relationships. 637 </p><div class="figure"><a id="id-1.3.5.8.4.4.2.3.2.5.5.5"></a><p class="title"><strong>Figure 21.17. Hash policy class diagram</strong></p><div class="figure-contents"><div class="mediaobject" align="center"><img src="../images/pbds_hash_policy_cd.png" align="middle" alt="Hash policy class diagram" /></div></div></div><br class="figure-break" /></div></div></div><div class="section"><div class="titlepage"><div><div><h6 class="title"><a id="container.hash.details.resize_policies"></a>Resize Policies</h6></div></div></div><div class="section"><div class="titlepage"><div><div><h6 class="title"><a id="resize_policies.general"></a>General</h6></div></div></div><p>Hash-tables, as opposed to trees, do not naturally grow or 638 shrink. It is necessary to specify policies to determine how 639 and when a hash table should change its size. Usually, resize 640 policies can be decomposed into orthogonal policies:</p><div class="orderedlist"><ol class="orderedlist" type="1"><li class="listitem"><p>A size policy indicating how a hash table 641 should grow (e.g., it should multiply by powers of 642 2).</p></li><li class="listitem"><p>A trigger policy indicating when a hash 643 table should grow (e.g., a load factor is 644 exceeded).</p></li></ol></div></div><div class="section"><div class="titlepage"><div><div><h6 class="title"><a id="resize_policies.size"></a>Size Policies</h6></div></div></div><p>Size policies determine how a hash table changes size. These 645 policies are simple, and there are relatively few sensible 646 options. An exponential-size policy (with the initial size and 647 growth factors both powers of 2) works well with a mask-based 648 range-hashing function, and is the 649 hard-wired policy used by Dinkumware. A 650 prime-list based policy works well with a modulo-prime range 651 hashing function and is the hard-wired policy used by SGI's 652 implementation.</p></div><div class="section"><div class="titlepage"><div><div><h6 class="title"><a id="resize_policies.trigger"></a>Trigger Policies</h6></div></div></div><p>Trigger policies determine when a hash table changes size. 653 Following is a description of two policies: load-check 654 policies, and collision-check policies.</p><p>Load-check policies are straightforward. The user specifies 655 two factors, Α<sub>min</sub> and 656 Α<sub>max</sub>, and the hash table maintains the 657 invariant that</p><p>Α<sub>min</sub> ≤ (number of 658 stored elements) / (hash-table size) ≤ 659 Α<sub>max</sub> 660 661 </p><p>Collision-check policies work in the opposite direction of 662 load-check policies. They focus on keeping the number of 663 collisions moderate and hoping that the size of the table will 664 not grow very large, instead of keeping a moderate load-factor 665 and hoping that the number of collisions will be small. A 666 maximal collision-check policy resizes when the longest 667 probe-sequence grows too large.</p><p>Consider the graphic below. Let the size of the hash table 668 be denoted by m, the length of a probe sequence be denoted by k, 669 and some load factor be denoted by Α. We would like to 670 calculate the minimal length of k, such that if there were Α 671 m elements in the hash table, a probe sequence of length k would 672 be found with probability at most 1/m.</p><div class="figure"><a id="id-1.3.5.8.4.4.2.3.3.4.7"></a><p class="title"><strong>Figure 21.18. Balls and bins</strong></p><div class="figure-contents"><div class="mediaobject" align="center"><img src="../images/pbds_balls_and_bins.png" align="middle" alt="Balls and bins" /></div></div></div><br class="figure-break" /><p>Denote the probability that a probe sequence of length 673 k appears in bin i by p<sub>i</sub>, the 674 length of the probe sequence of bin i by 675 l<sub>i</sub>, and assume uniform distribution. Then</p><div class="equation"><a id="id-1.3.5.8.4.4.2.3.3.4.9"></a><p class="title"><strong>Equation 21.7. 676 Probability of Probe Sequence of Length k 677 </strong></p><div class="equation-contents"><span class="mathphrase"> 678 p<sub>1</sub> = 679 </span></div></div><br class="equation-break" /><p>P(l<sub>1</sub> ≥ k) =</p><p> 680 P(l<sub>1</sub> ≥ α ( 1 + k / α - 1) ≤ (a) 681 </p><p> 682 e ^ ( - ( α ( k / α - 1 )<sup>2</sup> ) /2) 683 </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 684 calculate the probability that some bin contains a probe 685 sequence greater than k, we note that the 686 l<sub>i</sub> are negatively-dependent 687 (<a class="xref" href="policy_data_structures.html#biblio.dubhashi98neg" title="Balls and bins: A study in negative dependence">[biblio.dubhashi98neg]</a>) 688 . Let 689 I(.) denote the indicator function. Then</p><div class="equation"><a id="id-1.3.5.8.4.4.2.3.3.4.14"></a><p class="title"><strong>Equation 21.8. 690 Probability Probe Sequence in Some Bin 691 </strong></p><div class="equation-contents"><span class="mathphrase"> 692 P( exists<sub>i</sub> l<sub>i</sub> ≥ k ) = 693 </span></div></div><br class="equation-break" /><p>P ( ∑ <sub>i = 1</sub><sup>m</sup> 694 I(l<sub>i</sub> ≥ k) ≥ 1 ) =</p><p>P ( ∑ <sub>i = 1</sub><sup>m</sup> I ( 695 l<sub>i</sub> ≥ k ) ≥ m p<sub>1</sub> ( 1 + 1 / (m 696 p<sub>1</sub>) - 1 ) ) ≤ (a)</p><p>e ^ ( ( - m p<sub>1</sub> ( 1 / (m p<sub>1</sub>) 697 - 1 ) <sup>2</sup> ) / 2 ) ,</p><p>where (a) follows from the fact that the Chernoff bound can 698 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 699 equation into the second one, and equating with 1/m, we 700 obtain</p><p>k ~ √ ( 2 α ln 2 m ln(m) ) 701 ) .</p></div><div class="section"><div class="titlepage"><div><div><h6 class="title"><a id="resize_policies.impl"></a>Implementation</h6></div></div></div><p>This sub-subsection describes the implementation of the 702 above in this library. It first describes resize policies and 703 their decomposition into trigger and size policies, then 704 describes pre-defined classes, and finally discusses controlled 705 access the policies' internals.</p><div class="section"><div class="titlepage"><div><div><h6 class="title"><a id="resize_policies.impl.decomposition"></a>Decomposition</h6></div></div></div><p>Each hash-based container is parametrized by a 706 <code class="classname">Resize_Policy</code> parameter; the container derives 707 <code class="classname">public</code>ly from <code class="classname">Resize_Policy</code>. For 708 example:</p><pre class="programlisting"> 709 cc_hash_table<typename Key, 710 typename Mapped, 711 ... 712 typename Resize_Policy 713 ...> : public Resize_Policy 714 </pre><p>As a container object is modified, it continuously notifies 715 its <code class="classname">Resize_Policy</code> base of internal changes 716 (e.g., collisions encountered and elements being 717 inserted). It queries its <code class="classname">Resize_Policy</code> base whether 718 it needs to be resized, and if so, to what size.</p><p>The graphic below shows a (possible) sequence diagram 719 of an insert operation. The user inserts an element; the hash 720 table notifies its resize policy that a search has started 721 (point A); in this case, a single collision is encountered - 722 the table notifies its resize policy of this (point B); the 723 container finally notifies its resize policy that the search 724 has ended (point C); it then queries its resize policy whether 725 a resize is needed, and if so, what is the new size (points D 726 to G); following the resize, it notifies the policy that a 727 resize has completed (point H); finally, the element is 728 inserted, and the policy notified (point I).</p><div class="figure"><a id="id-1.3.5.8.4.4.2.3.3.5.3.6"></a><p class="title"><strong>Figure 21.19. Insert resize sequence diagram</strong></p><div class="figure-contents"><div class="mediaobject" align="center"><img src="../images/pbds_insert_resize_sequence_diagram1.png" align="middle" alt="Insert resize sequence diagram" /></div></div></div><br class="figure-break" /><p>In practice, a resize policy can be usually orthogonally 729 decomposed to a size policy and a trigger policy. Consequently, 730 the library contains a single class for instantiating a resize 731 policy: <code class="classname">hash_standard_resize_policy</code> 732 is parametrized by <code class="classname">Size_Policy</code> and 733 <code class="classname">Trigger_Policy</code>, derives <code class="classname">public</code>ly from 734 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>) 735 to these policies.</p><p>The two graphics immediately below show sequence diagrams 736 illustrating the interaction between the standard resize policy 737 and its trigger and size policies, respectively.</p><div class="figure"><a id="id-1.3.5.8.4.4.2.3.3.5.3.9"></a><p class="title"><strong>Figure 21.20. Standard resize policy trigger sequence 738 diagram</strong></p><div class="figure-contents"><div class="mediaobject" align="center"><img src="../images/pbds_insert_resize_sequence_diagram2.png" align="middle" alt="Standard resize policy trigger sequence diagram" /></div></div></div><br class="figure-break" /><div class="figure"><a id="id-1.3.5.8.4.4.2.3.3.5.3.10"></a><p class="title"><strong>Figure 21.21. Standard resize policy size sequence 739 diagram</strong></p><div class="figure-contents"><div class="mediaobject" align="center"><img src="../images/pbds_insert_resize_sequence_diagram3.png" align="middle" alt="Standard resize policy size sequence diagram" /></div></div></div><br class="figure-break" /></div><div class="section"><div class="titlepage"><div><div><h6 class="title"><a id="resize_policies.impl.predefined"></a>Predefined Policies</h6></div></div></div><p>The library includes the following 740 instantiations of size and trigger policies:</p><div class="orderedlist"><ol class="orderedlist" type="1"><li class="listitem"><p><code class="classname">hash_load_check_resize_trigger</code> 741 implements a load check trigger policy.</p></li><li class="listitem"><p><code class="classname">cc_hash_max_collision_check_resize_trigger</code> 742 implements a collision check trigger policy.</p></li><li class="listitem"><p><code class="classname">hash_exponential_size_policy</code> 743 implements an exponential-size policy (which should be used 744 with mask range hashing).</p></li><li class="listitem"><p><code class="classname">hash_prime_size_policy</code> 745 implementing a size policy based on a sequence of primes 746 (which should 747 be used with mod range hashing</p></li></ol></div><p>The graphic below gives an overall picture of the resize-related 748 classes. <code class="classname">basic_hash_table</code> 749 is parametrized by <code class="classname">Resize_Policy</code>, which it subclasses 750 publicly. This class is currently instantiated only by <code class="classname">hash_standard_resize_policy</code>. 751 <code class="classname">hash_standard_resize_policy</code> 752 itself is parametrized by <code class="classname">Trigger_Policy</code> and 753 <code class="classname">Size_Policy</code>. Currently, <code class="classname">Trigger_Policy</code> is 754 instantiated by <code class="classname">hash_load_check_resize_trigger</code>, 755 or <code class="classname">cc_hash_max_collision_check_resize_trigger</code>; 756 <code class="classname">Size_Policy</code> is instantiated by <code class="classname">hash_exponential_size_policy</code>, 757 or <code class="classname">hash_prime_size_policy</code>.</p></div><div class="section"><div class="titlepage"><div><div><h6 class="title"><a id="resize_policies.impl.internals"></a>Controling Access to Internals</h6></div></div></div><p>There are cases where (controlled) access to resize 758 policies' internals is beneficial. E.g., it is sometimes 759 useful to query a hash-table for the table's actual size (as 760 opposed to its <code class="function">size()</code> - the number of values it 761 currently holds); it is sometimes useful to set a table's 762 initial size, externally resize it, or change load factors.</p><p>Clearly, supporting such methods both decreases the 763 encapsulation of hash-based containers, and increases the 764 diversity between different associative-containers' interfaces. 765 Conversely, omitting such methods can decrease containers' 766 flexibility.</p><p>In order to avoid, to the extent possible, the above 767 conflict, the hash-based containers themselves do not address 768 any of these questions; this is deferred to the resize policies, 769 which are easier to change or replace. Thus, for example, 770 neither <code class="classname">cc_hash_table</code> nor 771 <code class="classname">gp_hash_table</code> 772 contain methods for querying the actual size of the table; this 773 is deferred to <code class="classname">hash_standard_resize_policy</code>.</p><p>Furthermore, the policies themselves are parametrized by 774 template arguments that determine the methods they support 775 ( 776 <a class="xref" href="policy_data_structures.html#biblio.alexandrescu01modern" title="Modern C++ Design: Generic Programming and Design Patterns Applied">[biblio.alexandrescu01modern]</a> 777 shows techniques for doing so). <code class="classname">hash_standard_resize_policy</code> 778 is parametrized by <code class="classname">External_Size_Access</code> that 779 determines whether it supports methods for querying the actual 780 size of the table or resizing it. <code class="classname">hash_load_check_resize_trigger</code> 781 is parametrized by <code class="classname">External_Load_Access</code> that 782 determines whether it supports methods for querying or 783 modifying the loads. <code class="classname">cc_hash_max_collision_check_resize_trigger</code> 784 is parametrized by <code class="classname">External_Load_Access</code> that 785 determines whether it supports methods for querying the 786 load.</p><p>Some operations, for example, resizing a container at 787 run time, or changing the load factors of a load-check trigger 788 policy, require the container itself to resize. As mentioned 789 above, the hash-based containers themselves do not contain 790 these types of methods, only their resize policies. 791 Consequently, there must be some mechanism for a resize policy 792 to manipulate the hash-based container. As the hash-based 793 container is a subclass of the resize policy, this is done 794 through virtual methods. Each hash-based container has a 795 <code class="classname">private</code> <code class="classname">virtual</code> method:</p><pre class="programlisting"> 796 virtual void 797 do_resize 798 (size_type new_size); 799 </pre><p>which resizes the container. Implementations of 800 <code class="classname">Resize_Policy</code> can export public methods for resizing 801 the container externally; these methods internally call 802 <code class="classname">do_resize</code> to resize the table.</p></div></div></div><div class="section"><div class="titlepage"><div><div><h6 class="title"><a id="container.hash.details.policy_interaction"></a>Policy Interactions</h6></div></div></div><p> 803 </p><p>Hash-tables are unfortunately especially susceptible to 804 choice of policies. One of the more complicated aspects of this 805 is that poor combinations of good policies can form a poor 806 container. Following are some considerations.</p><div class="section"><div class="titlepage"><div><div><h6 class="title"><a id="policy_interaction.probesizetrigger"></a>probe/size/trigger</h6></div></div></div><p>Some combinations do not work well for probing containers. 807 For example, combining a quadratic probe policy with an 808 exponential size policy can yield a poor container: when an 809 element is inserted, a trigger policy might decide that there 810 is no need to resize, as the table still contains unused 811 entries; the probe sequence, however, might never reach any of 812 the unused entries.</p><p>Unfortunately, this library cannot detect such problems at 813 compilation (they are halting reducible). It therefore defines 814 an exception class <code class="classname">insert_error</code> to throw an 815 exception in this case.</p></div><div class="section"><div class="titlepage"><div><div><h6 class="title"><a id="policy_interaction.hashtrigger"></a>hash/trigger</h6></div></div></div><p>Some trigger policies are especially susceptible to poor 816 hash functions. Suppose, as an extreme case, that the hash 817 function transforms each key to the same hash value. After some 818 inserts, a collision detecting policy will always indicate that 819 the container needs to grow.</p><p>The library, therefore, by design, limits each operation to 820 one resize. For each <code class="classname">insert</code>, for example, it queries 821 only once whether a resize is needed.</p></div><div class="section"><div class="titlepage"><div><div><h6 class="title"><a id="policy_interaction.eqstorehash"></a>equivalence functors/storing hash values/hash</h6></div></div></div><p><code class="classname">cc_hash_table</code> and 822 <code class="classname">gp_hash_table</code> are 823 parametrized by an equivalence functor and by a 824 <code class="classname">Store_Hash</code> parameter. If the latter parameter is 825 <code class="classname">true</code>, then the container stores with each entry 826 a hash value, and uses this value in case of collisions to 827 determine whether to apply a hash value. This can lower the 828 cost of collision for some types, but increase the cost of 829 collisions for other types.</p><p>If a ranged-hash function or ranged probe function is 830 directly supplied, however, then it makes no sense to store the 831 hash value with each entry. This library's container will 832 fail at compilation, by design, if this is attempted.</p></div><div class="section"><div class="titlepage"><div><div><h6 class="title"><a id="policy_interaction.sizeloadtrigger"></a>size/load-check trigger</h6></div></div></div><p>Assume a size policy issues an increasing sequence of sizes 833 a, a q, a q<sup>1</sup>, a q<sup>2</sup>, ... For 834 example, an exponential size policy might issue the sequence of 835 sizes 8, 16, 32, 64, ...</p><p>If a load-check trigger policy is used, with loads 836 α<sub>min</sub> and α<sub>max</sub>, 837 respectively, then it is a good idea to have:</p><div class="orderedlist"><ol class="orderedlist" type="1"><li class="listitem"><p>α<sub>max</sub> ~ 1 / q</p></li><li class="listitem"><p>α<sub>min</sub> < 1 / (2 q)</p></li></ol></div><p>This will ensure that the amortized hash cost of each 838 modifying operation is at most approximately 3.</p><p>α<sub>min</sub> ~ α<sub>max</sub> is, in 839 any case, a bad choice, and α<sub>min</sub> > 840 α <sub>max</sub> is horrendous.</p></div></div></div></div><div class="section"><div class="titlepage"><div><div><h4 class="title"><a id="pbds.design.container.tree"></a>tree</h4></div></div></div><div class="section"><div class="titlepage"><div><div><h5 class="title"><a id="container.tree.interface"></a>Interface</h5></div></div></div><p>The tree-based container has the following declaration:</p><pre class="programlisting"> 841 template< 842 typename Key, 843 typename Mapped, 844 typename Cmp_Fn = std::less<Key>, 845 typename Tag = rb_tree_tag, 846 template< 847 typename Const_Node_Iterator, 848 typename Node_Iterator, 849 typename Cmp_Fn_, 850 typename Allocator_> 851 class Node_Update = null_node_update, 852 typename Allocator = std::allocator<char> > 853 class tree; 854 </pre><p>The parameters have the following meaning:</p><div class="orderedlist"><ol class="orderedlist" type="1"><li class="listitem"><p><code class="classname">Key</code> is the key type.</p></li><li class="listitem"><p><code class="classname">Mapped</code> is the mapped-policy.</p></li><li class="listitem"><p><code class="classname">Cmp_Fn</code> is a key comparison functor</p></li><li class="listitem"><p><code class="classname">Tag</code> specifies which underlying data structure 855 to use.</p></li><li class="listitem"><p><code class="classname">Node_Update</code> is a policy for updating node 856 invariants.</p></li><li class="listitem"><p><code class="classname">Allocator</code> is an allocator 857 type.</p></li></ol></div><p>The <code class="classname">Tag</code> parameter specifies which underlying 858 data structure to use. Instantiating it by <code class="classname">rb_tree_tag</code>, <code class="classname">splay_tree_tag</code>, or 859 <code class="classname">ov_tree_tag</code>, 860 specifies an underlying red-black tree, splay tree, or 861 ordered-vector tree, respectively; any other tag is illegal. 862 Note that containers based on the former two contain more types 863 and methods than the latter (e.g., 864 <code class="classname">reverse_iterator</code> and <code class="classname">rbegin</code>), and different 865 exception and invalidation guarantees.</p></div><div class="section"><div class="titlepage"><div><div><h5 class="title"><a id="container.tree.details"></a>Details</h5></div></div></div><div class="section"><div class="titlepage"><div><div><h6 class="title"><a id="container.tree.node"></a>Node Invariants</h6></div></div></div><p>Consider the two trees in the graphic below, labels A and B. The first 866 is a tree of floats; the second is a tree of pairs, each 867 signifying a geometric line interval. Each element in a tree is referred to as a node of the tree. Of course, each of 868 these trees can support the usual queries: the first can easily 869 search for <code class="classname">0.4</code>; the second can easily search for 870 <code class="classname">std::make_pair(10, 41)</code>.</p><p>Each of these trees can efficiently support other queries. 871 The first can efficiently determine that the 2rd key in the 872 tree is <code class="constant">0.3</code>; the second can efficiently determine 873 whether any of its intervals overlaps 874 </p><pre class="programlisting">std::make_pair(29,42)</pre><p> (useful in geometric 875 applications or distributed file systems with leases, for 876 example). It should be noted that an <code class="classname">std::set</code> can 877 only solve these types of problems with linear complexity.</p><p>In order to do so, each tree stores some metadata in 878 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 879 each node the size of the sub-tree rooted at the node; the 880 second stores at each node the maximal endpoint of the 881 intervals at the sub-tree rooted at the node.</p><div class="figure"><a id="id-1.3.5.8.4.4.3.3.2.5"></a><p class="title"><strong>Figure 21.22. Tree node invariants</strong></p><div class="figure-contents"><div class="mediaobject" align="center"><img src="../images/pbds_tree_node_invariants.png" align="middle" alt="Tree node invariants" /></div></div></div><br class="figure-break" /><p>Supporting such trees is difficult for a number of 882 reasons:</p><div class="orderedlist"><ol class="orderedlist" type="1"><li class="listitem"><p>There must be a way to specify what a node's metadata 883 should be (if any).</p></li><li class="listitem"><p>Various operations can invalidate node 884 invariants. The graphic below shows how a right rotation, 885 performed on A, results in B, with nodes x and y having 886 corrupted invariants (the grayed nodes in C). The graphic shows 887 how an insert, performed on D, results in E, with nodes x and y 888 having corrupted invariants (the grayed nodes in F). It is not 889 feasible to know outside the tree the effect of an operation on 890 the nodes of the tree.</p></li><li class="listitem"><p>The search paths of standard associative containers are 891 defined by comparisons between keys, and not through 892 metadata.</p></li><li class="listitem"><p>It is not feasible to know in advance which methods trees 893 can support. Besides the usual <code class="classname">find</code> method, the 894 first tree can support a <code class="classname">find_by_order</code> method, while 895 the second can support an <code class="classname">overlaps</code> method.</p></li></ol></div><div class="figure"><a id="id-1.3.5.8.4.4.3.3.2.8"></a><p class="title"><strong>Figure 21.23. Tree node invalidation</strong></p><div class="figure-contents"><div class="mediaobject" align="center"><img src="../images/pbds_tree_node_invalidations.png" align="middle" alt="Tree node invalidation" /></div></div></div><br class="figure-break" /><p>These problems are solved by a combination of two means: 896 node iterators, and template-template node updater 897 parameters.</p><div class="section"><div class="titlepage"><div><div><h6 class="title"><a id="container.tree.node.iterators"></a>Node Iterators</h6></div></div></div><p>Each tree-based container defines two additional iterator 898 types, <code class="classname">const_node_iterator</code> 899 and <code class="classname">node_iterator</code>. 900 These iterators allow descending from a node to one of its 901 children. Node iterator allow search paths different than those 902 determined by the comparison functor. The <code class="classname">tree</code> 903 supports the methods:</p><pre class="programlisting"> 904 const_node_iterator 905 node_begin() const; 906 907 node_iterator 908 node_begin(); 909 910 const_node_iterator 911 node_end() const; 912 913 node_iterator 914 node_end(); 915 </pre><p>The first pairs return node iterators corresponding to the 916 root node of the tree; the latter pair returns node iterators 917 corresponding to a just-after-leaf node.</p></div><div class="section"><div class="titlepage"><div><div><h6 class="title"><a id="container.tree.node.updator"></a>Node Updator</h6></div></div></div><p>The tree-based containers are parametrized by a 918 <code class="classname">Node_Update</code> template-template parameter. A 919 tree-based container instantiates 920 <code class="classname">Node_Update</code> to some 921 <code class="classname">node_update</code> class, and publicly subclasses 922 <code class="classname">node_update</code>. The graphic below shows this 923 scheme, as well as some predefined policies (which are explained 924 below).</p><div class="figure"><a id="id-1.3.5.8.4.4.3.3.2.11.3"></a><p class="title"><strong>Figure 21.24. A tree and its update policy</strong></p><div class="figure-contents"><div class="mediaobject" align="center"><img src="../images/pbds_tree_node_updator_policy_cd.png" align="middle" alt="A tree and its update policy" /></div></div></div><br class="figure-break" /><p><code class="classname">node_update</code> (an instantiation of 925 <code class="classname">Node_Update</code>) must define <code class="classname">metadata_type</code> as 926 the type of metadata it requires. For order statistics, 927 e.g., <code class="classname">metadata_type</code> might be <code class="classname">size_t</code>. 928 The tree defines within each node a <code class="classname">metadata_type</code> 929 object.</p><p><code class="classname">node_update</code> must also define the following method 930 for restoring node invariants:</p><pre class="programlisting"> 931 void 932 operator()(node_iterator nd_it, const_node_iterator end_nd_it) 933 </pre><p>In this method, <code class="varname">nd_it</code> is a 934 <code class="classname">node_iterator</code> corresponding to a node whose 935 A) all descendants have valid invariants, and B) its own 936 invariants might be violated; <code class="classname">end_nd_it</code> is 937 a <code class="classname">const_node_iterator</code> corresponding to a 938 just-after-leaf node. This method should correct the node 939 invariants of the node pointed to by 940 <code class="classname">nd_it</code>. For example, say node x in the 941 graphic below label A has an invalid invariant, but its' children, 942 y and z have valid invariants. After the invocation, all three 943 nodes should have valid invariants, as in label B.</p><div class="figure"><a id="id-1.3.5.8.4.4.3.3.2.11.8"></a><p class="title"><strong>Figure 21.25. Restoring node invariants</strong></p><div class="figure-contents"><div class="mediaobject" align="center"><img src="../images/pbds_restoring_node_invariants.png" align="middle" alt="Restoring node invariants" /></div></div></div><br class="figure-break" /><p>When a tree operation might invalidate some node invariant, 944 it invokes this method in its <code class="classname">node_update</code> base to 945 restore the invariant. For example, the graphic below shows 946 an <code class="function">insert</code> operation (point A); the tree performs some 947 operations, and calls the update functor three times (points B, 948 C, and D). (It is well known that any <code class="function">insert</code>, 949 <code class="function">erase</code>, <code class="function">split</code> or <code class="function">join</code>, can restore 950 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>) 951 .</p><div class="figure"><a id="id-1.3.5.8.4.4.3.3.2.11.10"></a><p class="title"><strong>Figure 21.26. Insert update sequence</strong></p><div class="figure-contents"><div class="mediaobject" align="center"><img src="../images/pbds_update_seq_diagram.png" align="middle" alt="Insert update sequence" /></div></div></div><br class="figure-break" /><p>To complete the description of the scheme, three questions 952 need to be answered:</p><div class="orderedlist"><ol class="orderedlist" type="1"><li class="listitem"><p>How can a tree which supports order statistics define a 953 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 954 tree?</p></li><li class="listitem"><p>How can the following cyclic dependency be resolved? 955 <code class="classname">node_update</code> is a base class of the tree, yet it 956 uses node iterators defined in the tree (its child).</p></li></ol></div><p>The first two questions are answered by the fact that 957 <code class="classname">node_update</code> (an instantiation of 958 <code class="classname">Node_Update</code>) is a <span class="emphasis"><em>public</em></span> base class 959 of the tree. Consequently:</p><div class="orderedlist"><ol class="orderedlist" type="1"><li class="listitem"><p>Any public methods of 960 <code class="classname">node_update</code> are automatically methods of 961 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>). 962 Thus an order-statistics node updater, 963 <code class="classname">tree_order_statistics_node_update</code> defines 964 the <code class="function">find_by_order</code> method; any tree 965 instantiated by this policy consequently supports this method as 966 well.</p></li><li class="listitem"><p>In C++, if a base class declares a method as 967 <code class="literal">virtual</code>, it is 968 <code class="literal">virtual</code> in its subclasses. If 969 <code class="classname">node_update</code> needs to access one of the 970 tree's methods, say the member function 971 <code class="function">end</code>, it simply declares that method as 972 <code class="literal">virtual</code> abstract.</p></li></ol></div><p>The cyclic dependency is solved through template-template 973 parameters. <code class="classname">Node_Update</code> is parametrized by 974 the tree's node iterators, its comparison functor, and its 975 allocator type. Thus, instantiations of 976 <code class="classname">Node_Update</code> have all information 977 required.</p><p>This library assumes that constructing a metadata object and 978 modifying it are exception free. Suppose that during some method, 979 say <code class="classname">insert</code>, a metadata-related operation 980 (e.g., changing the value of a metadata) throws an exception. Ack! 981 Rolling back the method is unusually complex.</p><p>Previously, a distinction was made between redundant 982 policies and null policies. Node invariants show a 983 case where null policies are required.</p><p>Assume a regular tree is required, one which need not 984 support order statistics or interval overlap queries. 985 Seemingly, in this case a redundant policy - a policy which 986 doesn't affect nodes' contents would suffice. This, would lead 987 to the following drawbacks:</p><div class="orderedlist"><ol class="orderedlist" type="1"><li class="listitem"><p>Each node would carry a useless metadata object, wasting 988 space.</p></li><li class="listitem"><p>The tree cannot know if its 989 <code class="classname">Node_Update</code> policy actually modifies a 990 node's metadata (this is halting reducible). In the graphic 991 below, assume the shaded node is inserted. The tree would have 992 to traverse the useless path shown to the root, applying 993 redundant updates all the way.</p></li></ol></div><div class="figure"><a id="id-1.3.5.8.4.4.3.3.2.11.20"></a><p class="title"><strong>Figure 21.27. Useless update path</strong></p><div class="figure-contents"><div class="mediaobject" align="center"><img src="../images/pbds_rationale_null_node_updator.png" align="middle" alt="Useless update path" /></div></div></div><br class="figure-break" /><p>A null policy class, <code class="classname">null_node_update</code> 994 solves both these problems. The tree detects that node 995 invariants are irrelevant, and defines all accordingly.</p></div></div><div class="section"><div class="titlepage"><div><div><h6 class="title"><a id="container.tree.details.split"></a>Split and Join</h6></div></div></div><p>Tree-based containers support split and join methods. 996 It is possible to split a tree so that it passes 997 all nodes with keys larger than a given key to a different 998 tree. These methods have the following advantages over the 999 alternative of externally inserting to the destination 1000 tree and erasing from the source tree:</p><div class="orderedlist"><ol class="orderedlist" type="1"><li class="listitem"><p>These methods are efficient - red-black trees are split 1001 and joined in poly-logarithmic complexity; ordered-vector 1002 trees are split and joined at linear complexity. The 1003 alternatives have super-linear complexity.</p></li><li class="listitem"><p>Aside from orders of growth, these operations perform 1004 few allocations and de-allocations. For red-black trees, allocations are not performed, 1005 and the methods are exception-free. </p></li></ol></div></div></div></div><div class="section"><div class="titlepage"><div><div><h4 class="title"><a id="pbds.design.container.trie"></a>Trie</h4></div></div></div><div class="section"><div class="titlepage"><div><div><h5 class="title"><a id="container.trie.interface"></a>Interface</h5></div></div></div><p>The trie-based container has the following declaration:</p><pre class="programlisting"> 1006 template<typename Key, 1007 typename Mapped, 1008 typename Cmp_Fn = std::less<Key>, 1009 typename Tag = pat_trie_tag, 1010 template<typename Const_Node_Iterator, 1011 typename Node_Iterator, 1012 typename E_Access_Traits_, 1013 typename Allocator_> 1014 class Node_Update = null_node_update, 1015 typename Allocator = std::allocator<char> > 1016 class trie; 1017 </pre><p>The parameters have the following meaning:</p><div class="orderedlist"><ol class="orderedlist" type="1"><li class="listitem"><p><code class="classname">Key</code> is the key type.</p></li><li class="listitem"><p><code class="classname">Mapped</code> is the mapped-policy.</p></li><li class="listitem"><p><code class="classname">E_Access_Traits</code> is described in below.</p></li><li class="listitem"><p><code class="classname">Tag</code> specifies which underlying data structure 1018 to use, and is described shortly.</p></li><li class="listitem"><p><code class="classname">Node_Update</code> is a policy for updating node 1019 invariants. This is described below.</p></li><li class="listitem"><p><code class="classname">Allocator</code> is an allocator 1020 type.</p></li></ol></div><p>The <code class="classname">Tag</code> parameter specifies which underlying 1021 data structure to use. Instantiating it by <code class="classname">pat_trie_tag</code>, specifies an 1022 underlying PATRICIA trie (explained shortly); any other tag is 1023 currently illegal.</p><p>Following is a description of a (PATRICIA) trie 1024 (this implementation follows <a class="xref" href="policy_data_structures.html#biblio.okasaki98mereable" title="Fast mergeable integer maps">[biblio.okasaki98mereable]</a> and 1025 <a class="xref" href="policy_data_structures.html#biblio.filliatre2000ptset" title="Ptset: Sets of integers implemented as Patricia trees">[biblio.filliatre2000ptset]</a>). 1026 </p><p>A (PATRICIA) trie is similar to a tree, but with the 1027 following differences:</p><div class="orderedlist"><ol class="orderedlist" type="1"><li class="listitem"><p>It explicitly views keys as a sequence of elements. 1028 E.g., a trie can view a string as a sequence of 1029 characters; a trie can view a number as a sequence of 1030 bits.</p></li><li class="listitem"><p>It is not (necessarily) binary. Each node has fan-out n 1031 + 1, where n is the number of distinct 1032 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 1033 least two children, and B) each shares the same prefix with 1034 any of its descendant.</p></li></ol></div><p>A (PATRICIA) trie has some useful properties:</p><div class="orderedlist"><ol class="orderedlist" type="1"><li class="listitem"><p>It can be configured to use large node fan-out, giving it 1035 very efficient find performance (albeit at insertion 1036 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 1037 keys match a certain prefix. This is sometimes useful in file 1038 systems and routers, and for "type-ahead" aka predictive text matching 1039 on mobile devices.</p></li></ol></div></div><div class="section"><div class="titlepage"><div><div><h5 class="title"><a id="container.trie.details"></a>Details</h5></div></div></div><div class="section"><div class="titlepage"><div><div><h6 class="title"><a id="container.trie.details.etraits"></a>Element Access Traits</h6></div></div></div><p>A trie inherently views its keys as sequences of elements. 1040 For example, a trie can view a string as a sequence of 1041 characters. A trie needs to map each of n elements to a 1042 number in {0, n - 1}. For example, a trie can map a 1043 character <code class="varname">c</code> to 1044 </p><pre class="programlisting">static_cast<size_t>(c)</pre><p>.</p><p>Seemingly, then, a trie can assume that its keys support 1045 (const) iterators, and that the <code class="classname">value_type</code> of this 1046 iterator can be cast to a <code class="classname">size_t</code>. There are several 1047 reasons, though, to decouple the mechanism by which the trie 1048 accesses its keys' elements from the trie:</p><div class="orderedlist"><ol class="orderedlist" type="1"><li class="listitem"><p>In some cases, the numerical value of an element is 1049 inappropriate. Consider a trie storing DNA strings. It is 1050 logical to use a trie with a fan-out of 5 = 1 + |{'A', 'C', 1051 '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 1052 is needed. For example, a trie can be used to search for 1053 common suffixes, by using strings' 1054 <code class="classname">reverse_iterator</code>. As another example, a trie mapping 1055 UNICODE strings would have a huge fan-out if each node would 1056 branch on a UNICODE character; instead, one can define an 1057 iterator iterating over 8-bit (or less) groups.</p></li></ol></div><p>trie is, 1058 consequently, parametrized by <code class="classname">E_Access_Traits</code> - 1059 traits which instruct how to access sequences' elements. 1060 <code class="classname">string_trie_e_access_traits</code> 1061 is a traits class for strings. Each such traits define some 1062 types, like:</p><pre class="programlisting"> 1063 typename E_Access_Traits::const_iterator 1064 </pre><p>is a const iterator iterating over a key's elements. The 1065 traits class must also define methods for obtaining an iterator 1066 to the first and last element of a key.</p><p>The graphic below shows a 1067 (PATRICIA) trie resulting from inserting the words: "I wish 1068 that I could ever see a poem lovely as a trie" (which, 1069 unfortunately, does not rhyme).</p><p>The leaf nodes contain values; each internal node contains 1070 two <code class="classname">typename E_Access_Traits::const_iterator</code> 1071 objects, indicating the maximal common prefix of all keys in 1072 the sub-tree. For example, the shaded internal node roots a 1073 sub-tree with leafs "a" and "as". The maximal common prefix is 1074 "a". The internal node contains, consequently, to const 1075 iterators, one pointing to <code class="varname">'a'</code>, and the other to 1076 <code class="varname">'s'</code>.</p><div class="figure"><a id="id-1.3.5.8.4.4.4.3.2.10"></a><p class="title"><strong>Figure 21.28. A PATRICIA trie</strong></p><div class="figure-contents"><div class="mediaobject" align="center"><img src="../images/pbds_pat_trie.png" align="middle" alt="A PATRICIA trie" /></div></div></div><br class="figure-break" /></div><div class="section"><div class="titlepage"><div><div><h6 class="title"><a id="container.trie.details.node"></a>Node Invariants</h6></div></div></div><p>Trie-based containers support node invariants, as do 1077 tree-based containers. There are two minor 1078 differences, though, which, unfortunately, thwart sharing them 1079 sharing the same node-updating policies:</p><div class="orderedlist"><ol class="orderedlist" type="1"><li class="listitem"><p>A trie's <code class="classname">Node_Update</code> template-template 1080 parameter is parametrized by <code class="classname">E_Access_Traits</code>, while 1081 a tree's <code class="classname">Node_Update</code> template-template parameter is 1082 parametrized by <code class="classname">Cmp_Fn</code>.</p></li><li class="listitem"><p>Tree-based containers store values in all nodes, while 1083 trie-based containers (at least in this implementation) store 1084 values in leafs.</p></li></ol></div><p>The graphic below shows the scheme, as well as some predefined 1085 policies (which are explained below).</p><div class="figure"><a id="id-1.3.5.8.4.4.4.3.3.5"></a><p class="title"><strong>Figure 21.29. A trie and its update policy</strong></p><div class="figure-contents"><div class="mediaobject" align="center"><img src="../images/pbds_trie_node_updator_policy_cd.png" align="middle" alt="A trie and its update policy" /></div></div></div><br class="figure-break" /><p>This library offers the following pre-defined trie node 1086 updating policies:</p><div class="orderedlist"><ol class="orderedlist" type="1"><li class="listitem"><p> 1087 <code class="classname">trie_order_statistics_node_update</code> 1088 supports order statistics. 1089 </p></li><li class="listitem"><p><code class="classname">trie_prefix_search_node_update</code> 1090 supports searching for ranges that match a given prefix.</p></li><li class="listitem"><p><code class="classname">null_node_update</code> 1091 is the null node updater.</p></li></ol></div></div><div class="section"><div class="titlepage"><div><div><h6 class="title"><a id="container.trie.details.split"></a>Split and Join</h6></div></div></div><p>Trie-based containers support split and join methods; the 1092 rationale is equal to that of tree-based containers supporting 1093 these methods.</p></div></div></div><div class="section"><div class="titlepage"><div><div><h4 class="title"><a id="pbds.design.container.list"></a>List</h4></div></div></div><div class="section"><div class="titlepage"><div><div><h5 class="title"><a id="container.list.interface"></a>Interface</h5></div></div></div><p>The list-based container has the following declaration:</p><pre class="programlisting"> 1094 template<typename Key, 1095 typename Mapped, 1096 typename Eq_Fn = std::equal_to<Key>, 1097 typename Update_Policy = move_to_front_lu_policy<>, 1098 typename Allocator = std::allocator<char> > 1099 class list_update; 1100 </pre><p>The parameters have the following meaning:</p><div class="orderedlist"><ol class="orderedlist" type="1"><li class="listitem"><p> 1101 <code class="classname">Key</code> is the key type. 1102 </p></li><li class="listitem"><p> 1103 <code class="classname">Mapped</code> is the mapped-policy. 1104 </p></li><li class="listitem"><p> 1105 <code class="classname">Eq_Fn</code> is a key equivalence functor. 1106 </p></li><li class="listitem"><p> 1107 <code class="classname">Update_Policy</code> is a policy updating positions in 1108 the list based on access patterns. It is described in the 1109 following subsection. 1110 </p></li><li class="listitem"><p> 1111 <code class="classname">Allocator</code> is an allocator type. 1112 </p></li></ol></div><p>A list-based associative container is a container that 1113 stores elements in a linked-list. It does not order the elements 1114 by any particular order related to the keys. List-based 1115 containers are primarily useful for creating "multimaps". In fact, 1116 list-based containers are designed in this library expressly for 1117 this purpose.</p><p>List-based containers might also be useful for some rare 1118 cases, where a key is encapsulated to the extent that only 1119 key-equivalence can be tested. Hash-based containers need to know 1120 how to transform a key into a size type, and tree-based containers 1121 need to know if some key is larger than another. List-based 1122 associative containers, conversely, only need to know if two keys 1123 are equivalent.</p><p>Since a list-based associative container does not order 1124 elements by keys, is it possible to order the list in some 1125 useful manner? Remarkably, many on-line competitive 1126 algorithms exist for reordering lists to reflect access 1127 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>). 1128 </p></div><div class="section"><div class="titlepage"><div><div><h5 class="title"><a id="container.list.details"></a>Details</h5></div></div></div><p> 1129 </p><div class="section"><div class="titlepage"><div><div><h6 class="title"><a id="container.list.details.ds"></a>Underlying Data Structure</h6></div></div></div><p>The graphic below shows a 1130 simple list of integer keys. If we search for the integer 6, we 1131 are paying an overhead: the link with key 6 is only the fifth 1132 link; if it were the first link, it could be accessed 1133 faster.</p><div class="figure"><a id="id-1.3.5.8.4.4.5.3.3.3"></a><p class="title"><strong>Figure 21.30. A simple list</strong></p><div class="figure-contents"><div class="mediaobject" align="center"><img src="../images/pbds_simple_list.png" align="middle" alt="A simple list" /></div></div></div><br class="figure-break" /><p>List-update algorithms reorder lists as elements are 1134 accessed. They try to determine, by the access history, which 1135 keys to move to the front of the list. Some of these algorithms 1136 require adding some metadata alongside each entry.</p><p>For example, in the graphic below label A shows the counter 1137 algorithm. Each node contains both a key and a count metadata 1138 (shown in bold). When an element is accessed (e.g. 6) its count is 1139 incremented, as shown in label B. If the count reaches some 1140 predetermined value, say 10, as shown in label C, the count is set 1141 to 0 and the node is moved to the front of the list, as in label 1142 D. 1143 </p><div class="figure"><a id="id-1.3.5.8.4.4.5.3.3.6"></a><p class="title"><strong>Figure 21.31. The counter algorithm</strong></p><div class="figure-contents"><div class="mediaobject" align="center"><img src="../images/pbds_list_update.png" align="middle" alt="The counter algorithm" /></div></div></div><br class="figure-break" /></div><div class="section"><div class="titlepage"><div><div><h6 class="title"><a id="container.list.details.policies"></a>Policies</h6></div></div></div><p>this library allows instantiating lists with policies 1144 implementing any algorithm moving nodes to the front of the 1145 list (policies implementing algorithms interchanging nodes are 1146 unsupported).</p><p>Associative containers based on lists are parametrized by a 1147 <code class="classname">Update_Policy</code> parameter. This parameter defines the 1148 type of metadata each node contains, how to create the 1149 metadata, and how to decide, using this metadata, whether to 1150 move a node to the front of the list. A list-based associative 1151 container object derives (publicly) from its update policy. 1152 </p><p>An instantiation of <code class="classname">Update_Policy</code> must define 1153 internally <code class="classname">update_metadata</code> as the metadata it 1154 requires. Internally, each node of the list contains, besides 1155 the usual key and data, an instance of <code class="classname">typename 1156 Update_Policy::update_metadata</code>.</p><p>An instantiation of <code class="classname">Update_Policy</code> must define 1157 internally two operators:</p><pre class="programlisting"> 1158 update_metadata 1159 operator()(); 1160 1161 bool 1162 operator()(update_metadata &); 1163 </pre><p>The first is called by the container object, when creating a 1164 new node, to create the node's metadata. The second is called 1165 by the container object, when a node is accessed ( 1166 when a find operation's key is equivalent to the key of the 1167 node), to determine whether to move the node to the front of 1168 the list. 1169 </p><p>The library contains two predefined implementations of 1170 list-update policies. The first 1171 is <code class="classname">lu_counter_policy</code>, which implements the 1172 counter algorithm described above. The second is 1173 <code class="classname">lu_move_to_front_policy</code>, 1174 which unconditionally move an accessed element to the front of 1175 the list. The latter type is very useful in this library, 1176 since there is no need to associate metadata with each element. 1177 (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> 1178 </p></div><div class="section"><div class="titlepage"><div><div><h6 class="title"><a id="container.list.details.mapped"></a>Use in Multimaps</h6></div></div></div><p>In this library, there are no equivalents for the standard's 1179 multimaps and multisets; instead one uses an associative 1180 container mapping primary keys to secondary keys.</p><p>List-based containers are especially useful as associative 1181 containers for secondary keys. In fact, they are implemented 1182 here expressly for this purpose.</p><p>To begin with, these containers use very little per-entry 1183 structure memory overhead, since they can be implemented as 1184 singly-linked lists. (Arrays use even lower per-entry memory 1185 overhead, but they are less flexible in moving around entries, 1186 and have weaker invalidation guarantees).</p><p>More importantly, though, list-based containers use very 1187 little per-container memory overhead. The memory overhead of an 1188 empty list-based container is practically that of a pointer. 1189 This is important for when they are used as secondary 1190 associative-containers in situations where the average ratio of 1191 secondary keys to primary keys is low (or even 1).</p><p>In order to reduce the per-container memory overhead as much 1192 as possible, they are implemented as closely as possible to 1193 singly-linked lists.</p><div class="orderedlist"><ol class="orderedlist" type="1"><li class="listitem"><p> 1194 List-based containers do not store internally the number 1195 of values that they hold. This means that their <code class="function">size</code> 1196 method has linear complexity (just like <code class="classname">std::list</code>). 1197 Note that finding the number of equivalent-key values in a 1198 standard multimap also has linear complexity (because it must be 1199 done, via <code class="function">std::distance</code> of the 1200 multimap's <code class="function">equal_range</code> method), but usually with 1201 higher constants. 1202 </p></li><li class="listitem"><p> 1203 Most associative-container objects each hold a policy 1204 object (a hash-based container object holds a 1205 hash functor). List-based containers, conversely, only have 1206 class-wide policy objects. 1207 </p></li></ol></div></div></div></div><div class="section"><div class="titlepage"><div><div><h4 class="title"><a id="pbds.design.container.priority_queue"></a>Priority Queue</h4></div></div></div><div class="section"><div class="titlepage"><div><div><h5 class="title"><a id="container.priority_queue.interface"></a>Interface</h5></div></div></div><p>The priority queue container has the following 1208 declaration: 1209 </p><pre class="programlisting"> 1210 template<typename Value_Type, 1211 typename Cmp_Fn = std::less<Value_Type>, 1212 typename Tag = pairing_heap_tag, 1213 typename Allocator = std::allocator<char > > 1214 class priority_queue; 1215 </pre><p>The parameters have the following meaning:</p><div class="orderedlist"><ol class="orderedlist" type="1"><li class="listitem"><p><code class="classname">Value_Type</code> is the value type.</p></li><li class="listitem"><p><code class="classname">Cmp_Fn</code> is a value comparison functor</p></li><li class="listitem"><p><code class="classname">Tag</code> specifies which underlying data structure 1216 to use.</p></li><li class="listitem"><p><code class="classname">Allocator</code> is an allocator 1217 type.</p></li></ol></div><p>The <code class="classname">Tag</code> parameter specifies which underlying 1218 data structure to use. Instantiating it by<code class="classname">pairing_heap_tag</code>,<code class="classname">binary_heap_tag</code>, 1219 <code class="classname">binomial_heap_tag</code>, 1220 <code class="classname">rc_binomial_heap_tag</code>, 1221 or <code class="classname">thin_heap_tag</code>, 1222 specifies, respectively, 1223 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>), 1224 binary heap (<a class="xref" href="policy_data_structures.html#biblio.clrs2001" title="Introduction to Algorithms, 2nd edition">[biblio.clrs2001]</a>), 1225 binomial heap (<a class="xref" href="policy_data_structures.html#biblio.clrs2001" title="Introduction to Algorithms, 2nd edition">[biblio.clrs2001]</a>), 1226 a binomial heap with a redundant binary counter (<a class="xref" href="policy_data_structures.html#biblio.maverick_lowerbounds" title="Deamortization - Part 2: Binomial Heaps">[biblio.maverick_lowerbounds]</a>), 1227 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 </p><p> 1229 As mentioned in the tutorial, 1230 <code class="classname">__gnu_pbds::priority_queue</code> shares most of the 1231 same interface with <code class="classname">std::priority_queue</code>. 1232 E.g. if <code class="varname">q</code> is a priority queue of type 1233 <code class="classname">Q</code>, then <code class="function">q.top()</code> will 1234 return the "largest" value in the container (according to 1235 <code class="classname">typename 1236 Q::cmp_fn</code>). <code class="classname">__gnu_pbds::priority_queue</code> 1237 has a larger (and very slightly different) interface than 1238 <code class="classname">std::priority_queue</code>, however, since typically 1239 <code class="classname">push</code> and <code class="classname">pop</code> are deemed 1240 insufficient for manipulating priority-queues. </p><p>Different settings require different priority-queue 1241 implementations which are described in later; see traits 1242 discusses ways to differentiate between the different traits of 1243 different implementations.</p></div><div class="section"><div class="titlepage"><div><div><h5 class="title"><a id="container.priority_queue.details"></a>Details</h5></div></div></div><div class="section"><div class="titlepage"><div><div><h6 class="title"><a id="container.priority_queue.details.iterators"></a>Iterators</h6></div></div></div><p>There are many different underlying-data structures for 1244 implementing priority queues. Unfortunately, most such 1245 structures are oriented towards making <code class="function">push</code> and 1246 <code class="function">top</code> efficient, and consequently don't allow efficient 1247 access of other elements: for instance, they cannot support an efficient 1248 <code class="function">find</code> method. In the use case where it 1249 is important to both access and "do something with" an 1250 arbitrary value, one would be out of luck. For example, many graph algorithms require 1251 modifying a value (typically increasing it in the sense of the 1252 priority queue's comparison functor).</p><p>In order to access and manipulate an arbitrary value in a 1253 priority queue, one needs to reference the internals of the 1254 priority queue from some form of an associative container - 1255 this is unavoidable. Of course, in order to maintain the 1256 encapsulation of the priority queue, this needs to be done in a 1257 way that minimizes exposure to implementation internals.</p><p>In this library the priority queue's <code class="function">insert</code> 1258 method returns an iterator, which if valid can be used for subsequent <code class="function">modify</code> and 1259 <code class="function">erase</code> operations. This both preserves the priority 1260 queue's encapsulation, and allows accessing arbitrary values (since the 1261 returned iterators from the <code class="function">push</code> operation can be 1262 stored in some form of associative container).</p><p>Priority queues' iterators present a problem regarding their 1263 invalidation guarantees. One assumes that calling 1264 <code class="function">operator++</code> on an iterator will associate it 1265 with the "next" value. Priority-queues are 1266 self-organizing: each operation changes what the "next" value 1267 means. Consequently, it does not make sense that <code class="function">push</code> 1268 will return an iterator that can be incremented - this can have 1269 no possible use. Also, as in the case of hash-based containers, 1270 it is awkward to define if a subsequent <code class="function">push</code> operation 1271 invalidates a prior returned iterator: it invalidates it in the 1272 sense that its "next" value is not related to what it 1273 previously considered to be its "next" value. However, it might not 1274 invalidate it, in the sense that it can be 1275 de-referenced and used for <code class="function">modify</code> and <code class="function">erase</code> 1276 operations.</p><p>Similarly to the case of the other unordered associative 1277 containers, this library uses a distinction between 1278 point-type and range type iterators. A priority queue's <code class="classname">iterator</code> can always be 1279 converted to a <code class="classname">point_iterator</code>, and a 1280 <code class="classname">const_iterator</code> can always be converted to a 1281 <code class="classname">point_const_iterator</code>.</p><p>The following snippet demonstrates manipulating an arbitrary 1282 value:</p><pre class="programlisting"> 1283 // A priority queue of integers. 1284 priority_queue<int > p; 1285 1286 // Insert some values into the priority queue. 1287 priority_queue<int >::point_iterator it = p.push(0); 1288 1289 p.push(1); 1290 p.push(2); 1291 1292 // Now modify a value. 1293 p.modify(it, 3); 1294 1295 assert(p.top() == 3); 1296 </pre><p>It should be noted that an alternative design could embed an 1297 associative container in a priority queue. Could, but most 1298 probably should not. To begin with, it should be noted that one 1299 could always encapsulate a priority queue and an associative 1300 container mapping values to priority queue iterators with no 1301 performance loss. One cannot, however, "un-encapsulate" a priority 1302 queue embedding an associative container, which might lead to 1303 performance loss. Assume, that one needs to associate each value 1304 with some data unrelated to priority queues. Then using 1305 this library's design, one could use an 1306 associative container mapping each value to a pair consisting of 1307 this data and a priority queue's iterator. Using the embedded 1308 method would need to use two associative containers. Similar 1309 problems might arise in cases where a value can reside 1310 simultaneously in many priority queues.</p></div><div class="section"><div class="titlepage"><div><div><h6 class="title"><a id="container.priority_queue.details.d"></a>Underlying Data Structure</h6></div></div></div><p>There are three main implementations of priority queues: the 1311 first employs a binary heap, typically one which uses a 1312 sequence; the second uses a tree (or forest of trees), which is 1313 typically less structured than an associative container's tree; 1314 the third simply uses an associative container. These are 1315 shown in the graphic below, in labels A1 and A2, label B, and label C.</p><div class="figure"><a id="id-1.3.5.8.4.4.6.3.3.3"></a><p class="title"><strong>Figure 21.32. Underlying Priority-Queue Data-Structures.</strong></p><div class="figure-contents"><div class="mediaobject" align="center"><img src="../images/pbds_priority_queue_different_underlying_dss.png" align="middle" alt="Underlying Priority-Queue Data-Structures." /></div></div></div><br class="figure-break" /><p>Roughly speaking, any value that is both pushed and popped 1316 from a priority queue must incur a logarithmic expense (in the 1317 amortized sense). Any priority queue implementation that would 1318 avoid this, would violate known bounds on comparison-based 1319 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>). 1320 </p><p>Most implementations do 1321 not differ in the asymptotic amortized complexity of 1322 <code class="function">push</code> and <code class="function">pop</code> operations, but they differ in 1323 the constants involved, in the complexity of other operations 1324 (e.g., <code class="function">modify</code>), and in the worst-case 1325 complexity of single operations. In general, the more 1326 "structured" an implementation (i.e., the more internal 1327 invariants it possesses) - the higher its amortized complexity 1328 of <code class="function">push</code> and <code class="function">pop</code> operations.</p><p>This library implements different algorithms using a 1329 single class: <code class="classname">priority_queue</code>. 1330 Instantiating the <code class="classname">Tag</code> template parameter, "selects" 1331 the implementation:</p><div class="orderedlist"><ol class="orderedlist" type="1"><li class="listitem"><p> 1332 Instantiating <code class="classname">Tag = binary_heap_tag</code> creates 1333 a binary heap of the form in represented in the graphic with labels A1 or A2. The former is internally 1334 selected by priority_queue 1335 if <code class="classname">Value_Type</code> is instantiated by a primitive type 1336 (e.g., an <span class="type">int</span>); the latter is 1337 internally selected for all other types (e.g., 1338 <code class="classname">std::string</code>). This implementations is relatively 1339 unstructured, and so has good <code class="classname">push</code> and <code class="classname">pop</code> 1340 performance; it is the "best-in-kind" for primitive 1341 types, e.g., <span class="type">int</span>s. Conversely, it has 1342 high worst-case performance, and can support only linear-time 1343 <code class="function">modify</code> and <code class="function">erase</code> operations.</p></li><li class="listitem"><p>Instantiating <code class="classname">Tag = 1344 pairing_heap_tag</code> creates a pairing heap of the form 1345 in represented by label B in the graphic above. This 1346 implementations too is relatively unstructured, and so has good 1347 <code class="function">push</code> and <code class="function">pop</code> 1348 performance; it is the "best-in-kind" for non-primitive types, 1349 e.g., <code class="classname">std:string</code>s. It also has very good 1350 worst-case <code class="function">push</code> and 1351 <code class="function">join</code> performance (O(1)), but has high 1352 worst-case <code class="function">pop</code> 1353 complexity.</p></li><li class="listitem"><p>Instantiating <code class="classname">Tag = 1354 binomial_heap_tag</code> creates a binomial heap of the 1355 form repsented by label B in the graphic above. This 1356 implementations is more structured than a pairing heap, and so 1357 has worse <code class="function">push</code> and <code class="function">pop</code> 1358 performance. Conversely, it has sub-linear worst-case bounds for 1359 <code class="function">pop</code>, e.g., and so it might be preferred in 1360 cases where responsiveness is important.</p></li><li class="listitem"><p>Instantiating <code class="classname">Tag = 1361 rc_binomial_heap_tag</code> creates a binomial heap of the 1362 form represented in label B above, accompanied by a redundant 1363 counter which governs the trees. This implementations is 1364 therefore more structured than a binomial heap, and so has worse 1365 <code class="function">push</code> and <code class="function">pop</code> 1366 performance. Conversely, it guarantees O(1) 1367 <code class="function">push</code> complexity, and so it might be 1368 preferred in cases where the responsiveness of a binomial heap 1369 is insufficient.</p></li><li class="listitem"><p>Instantiating <code class="classname">Tag = 1370 thin_heap_tag</code> creates a thin heap of the form 1371 represented by the label B in the graphic above. This 1372 implementations too is more structured than a pairing heap, and 1373 so has worse <code class="function">push</code> and 1374 <code class="function">pop</code> performance. Conversely, it has better 1375 worst-case and identical amortized complexities than a Fibonacci 1376 heap, and so might be more appropriate for some graph 1377 algorithms.</p></li></ol></div><p>Of course, one can use any order-preserving associative 1378 container as a priority queue, as in the graphic above label C, possibly by creating an adapter class 1379 over the associative container (much as 1380 <code class="classname">std::priority_queue</code> can adapt <code class="classname">std::vector</code>). 1381 This has the advantage that no cross-referencing is necessary 1382 at all; the priority queue itself is an associative container. 1383 Most associative containers are too structured to compete with 1384 priority queues in terms of <code class="function">push</code> and <code class="function">pop</code> 1385 performance.</p></div><div class="section"><div class="titlepage"><div><div><h6 class="title"><a id="container.priority_queue.details.traits"></a>Traits</h6></div></div></div><p>It would be nice if all priority queues could 1386 share exactly the same behavior regardless of implementation. Sadly, this is not possible. Just one for instance is in join operations: joining 1387 two binary heaps might throw an exception (not corrupt 1388 any of the heaps on which it operates), but joining two pairing 1389 heaps is exception free.</p><p>Tags and traits are very useful for manipulating generic 1390 types. <code class="classname">__gnu_pbds::priority_queue</code> 1391 publicly defines <code class="classname">container_category</code> as one of the tags. Given any 1392 container <code class="classname">Cntnr</code>, the tag of the underlying 1393 data structure can be found via <code class="classname">typename 1394 Cntnr::container_category</code>; this is one of the possible tags shown in the graphic below. 1395 </p><div class="figure"><a id="id-1.3.5.8.4.4.6.3.4.4"></a><p class="title"><strong>Figure 21.33. Priority-Queue Data-Structure Tags.</strong></p><div class="figure-contents"><div class="mediaobject" align="center"><img src="../images/pbds_priority_queue_tag_hierarchy.png" align="middle" alt="Priority-Queue Data-Structure Tags." /></div></div></div><br class="figure-break" /><p>Additionally, a traits mechanism can be used to query a 1396 container type for its attributes. Given any container 1397 <code class="classname">Cntnr</code>, then </p><pre class="programlisting">__gnu_pbds::container_traits<Cntnr></pre><p> 1398 is a traits class identifying the properties of the 1399 container.</p><p>To find if a container might throw if two of its objects are 1400 joined, one can use 1401 </p><pre class="programlisting"> 1402 container_traits<Cntnr>::split_join_can_throw 1403 </pre><p> 1404 </p><p> 1405 Different priority-queue implementations have different invalidation guarantees. This is 1406 especially important, since there is no way to access an arbitrary 1407 value of priority queues except for iterators. Similarly to 1408 associative containers, one can use 1409 </p><pre class="programlisting"> 1410 container_traits<Cntnr>::invalidation_guarantee 1411 </pre><p> 1412 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> 1413 will be for different implementations. All implementations of 1414 type represented by label B have <code class="classname">point_invalidation_guarantee</code>: 1415 the container can freely internally reorganize the nodes - 1416 range-type iterators are invalidated, but point-type iterators 1417 are always valid. Implementations of type represented by labels A1 and A2 have <code class="classname">basic_invalidation_guarantee</code>: 1418 the container can freely internally reallocate the array - both 1419 point-type and range-type iterators might be invalidated.</p><p> 1420 This has major implications, and constitutes a good reason to avoid 1421 using binary heaps. A binary heap can perform <code class="function">modify</code> 1422 or <code class="function">erase</code> efficiently given a valid point-type 1423 iterator. However, in order to supply it with a valid point-type 1424 iterator, one needs to iterate (linearly) over all 1425 values, then supply the relevant iterator (recall that a 1426 range-type iterator can always be converted to a point-type 1427 iterator). This means that if the number of <code class="function">modify</code> or 1428 <code class="function">erase</code> operations is non-negligible (say 1429 super-logarithmic in the total sequence of operations) - binary 1430 heaps will perform badly. 1431 </p></div></div></div></div></div><div class="navfooter"><hr /><table width="100%" summary="Navigation footer"><tr><td width="40%" align="left"><a accesskey="p" href="policy_data_structures_using.html">Prev</a> </td><td width="20%" align="center"><a accesskey="u" href="policy_data_structures.html">Up</a></td><td width="40%" align="right"> <a accesskey="n" href="policy_based_data_structures_test.html">Next</a></td></tr><tr><td width="40%" align="left" valign="top">Using </td><td width="20%" align="center"><a accesskey="h" href="../index.html">Home</a></td><td width="40%" align="right" valign="top"> Testing</td></tr></table></div></body></html>
