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diff --git a/libstdc++-v3/doc/html/ext/pb_ds/hash_based_containers.html b/libstdc++-v3/doc/html/ext/pb_ds/hash_based_containers.html new file mode 100644 index 000000000..21d092a76 --- /dev/null +++ b/libstdc++-v3/doc/html/ext/pb_ds/hash_based_containers.html @@ -0,0 +1,835 @@ +<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN" + "http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd"> + +<html xmlns="http://www.w3.org/1999/xhtml" xml:lang="en" lang="en"> +<head> + <meta name="generator" content= + "HTML Tidy for Linux/x86 (vers 12 April 2005), see www.w3.org" /> + + <title>Hash-Based Containers</title> + <meta http-equiv="Content-Type" content= + "text/html; charset=us-ascii" /> + </head> + +<body> + <div id="page"> + <h1>Hash Table Design</h1> + + <h2><a name="overview" id="overview">Overview</a></h2> + + <p>The collision-chaining hash-based container has the + following declaration.</p> + <pre> +<b>template</b>< + <b>typename</b> Key, + <b>typename</b> Mapped, + <b>typename</b> Hash_Fn = std::hash<Key>, + <b>typename</b> Eq_Fn = std::equal_to<Key>, + <b>typename</b> Comb_Hash_Fn = <a href= +"direct_mask_range_hashing.html">direct_mask_range_hashing</a><> + <b>typename</b> Resize_Policy = <i>default explained below.</i> + <b>bool</b> Store_Hash = <b>false</b>, + <b>typename</b> Allocator = std::allocator<<b>char</b>> > +<b>class</b> <a href= +"cc_hash_table.html">cc_hash_table</a>; +</pre> + + <p>The parameters have the following meaning:</p> + + <ol> + <li><tt>Key</tt> is the key type.</li> + + <li><tt>Mapped</tt> is the mapped-policy, and is explained in + <a href="tutorial.html#assoc_ms">Tutorial::Associative + Containers::Associative Containers Others than Maps</a>.</li> + + <li><tt>Hash_Fn</tt> is a key hashing functor.</li> + + <li><tt>Eq_Fn</tt> is a key equivalence functor.</li> + + <li><tt>Comb_Hash_Fn</tt> is a <i>range-hashing_functor</i>; + it describes how to translate hash values into positions + within the table. This is described in <a href= + "#hash_policies">Hash Policies</a>.</li> + + <li><tt>Resize_Policy</tt> describes how a container object + should change its internal size. This is described in + <a href="#resize_policies">Resize Policies</a>.</li> + + <li><tt>Store_Hash</tt> indicates whether the hash value + should be stored with each entry. This is described in + <a href="#policy_interaction">Policy Interaction</a>.</li> + + <li><tt>Allocator</tt> is an allocator + type.</li> + </ol> + + <p>The probing hash-based container has the following + declaration.</p> + <pre> +<b>template</b>< + <b>typename</b> Key, + <b>typename</b> Mapped, + <b>typename</b> Hash_Fn = std::hash<Key>, + <b>typename</b> Eq_Fn = std::equal_to<Key>, + <b>typename</b> Comb_Probe_Fn = <a href= +"direct_mask_range_hashing.html">direct_mask_range_hashing</a><> + <b>typename</b> Probe_Fn = <i>default explained below.</i> + <b>typename</b> Resize_Policy = <i>default explained below.</i> + <b>bool</b> Store_Hash = <b>false</b>, + <b>typename</b> Allocator = std::allocator<<b>char</b>> > +<b>class</b> <a href= +"gp_hash_table.html">gp_hash_table</a>; +</pre> + + <p>The parameters are identical to those of the + collision-chaining container, except for the following.</p> + + <ol> + <li><tt>Comb_Probe_Fn</tt> describes how to transform a probe + sequence into a sequence of positions within the table.</li> + + <li><tt>Probe_Fn</tt> describes a probe sequence policy.</li> + </ol> + + <p>Some of the default template values depend on the values of + other parameters, and are explained in <a href= + "#policy_interaction">Policy Interaction</a>.</p> + + <h2><a name="hash_policies" id="hash_policies">Hash + Policies</a></h2> + + <h3><a name="general_terms" id="general_terms">General + Terms</a></h3> + + <p>Following is an explanation of some functions which hashing + involves. Figure <a href= + "#hash_ranged_hash_range_hashing_fns">Hash functions, + ranged-hash functions, and range-hashing functions</a>) + illustrates the discussion.</p> + + <h6 class="c1"><a name="hash_ranged_hash_range_hashing_fns" id= + "hash_ranged_hash_range_hashing_fns"><img src= + "hash_ranged_hash_range_hashing_fns.png" alt= + "no image" /></a></h6> + + <h6 class="c1">Hash functions, ranged-hash functions, and + range-hashing functions.</h6> + + <p>Let <i>U</i> be a domain (<i>e.g.</i>, the integers, or the + strings of 3 characters). A hash-table algorithm needs to map + elements of <i>U</i> "uniformly" into the range <i>[0,..., m - + 1]</i> (where <i>m</i> is a non-negative integral value, and + is, in general, time varying). <i>I.e.</i>, the algorithm needs + a <i>ranged-hash</i> function</p> + + <p><i>f : U × Z<sub>+</sub> → Z<sub>+</sub></i> + ,</p> + + <p>such that for any <i>u</i> in <i>U</i> ,</p> + + <p><i>0 ≤ f(u, m) ≤ m - 1</i> ,</p> + + <p>and which has "good uniformity" properties [<a href= + "references.html#knuth98sorting">knuth98sorting</a>]. One + common solution is to use the composition of the hash + function</p> + + <p><i>h : U → Z<sub>+</sub></i> ,</p> + + <p>which maps elements of <i>U</i> into the non-negative + integrals, and</p> + + <p class="c2">g : Z<sub>+</sub> × Z<sub>+</sub> → + Z<sub>+</sub>,</p> + + <p>which maps a non-negative hash value, and a non-negative + range upper-bound into a non-negative integral in the range + between 0 (inclusive) and the range upper bound (exclusive), + <i>i.e.</i>, for any <i>r</i> in <i>Z<sub>+</sub></i>,</p> + + <p><i>0 ≤ g(r, m) ≤ m - 1</i> .</p> + + <p>The resulting ranged-hash function, is</p> + + <p><i><a name="ranged_hash_composed_of_hash_and_range_hashing" + id="ranged_hash_composed_of_hash_and_range_hashing">f(u , m) = + g(h(u), m)</a></i> (1) .</p> + + <p>From the above, it is obvious that given <i>g</i> and + <i>h</i>, <i>f</i> can always be composed (however the converse + is not true). The STL's hash-based containers allow specifying + a hash function, and use a hard-wired range-hashing function; + the ranged-hash function is implicitly composed.</p> + + <p>The above describes the case where a key is to be mapped + into a <i>single position</i> within a hash table, <i>e.g.</i>, + in a collision-chaining table. In other cases, a key is to be + mapped into a <i>sequence of positions</i> within a table, + <i>e.g.</i>, in a probing table. Similar terms apply in this + case: the table requires a <i>ranged probe</i> function, + mapping a key into a sequence of positions withing the table. + This is typically achieved by composing a <i>hash function</i> + mapping the key into a non-negative integral type, a + <i>probe</i> function transforming the hash value into a + sequence of hash values, and a <i>range-hashing</i> function + transforming the sequence of hash values into a sequence of + positions.</p> + + <h3><a name="range_hashing_fns" id= + "range_hashing_fns">Range-Hashing Functions</a></h3> + + <p>Some common choices for range-hashing functions are the + division, multiplication, and middle-square methods [<a href= + "references.html#knuth98sorting">knuth98sorting</a>], defined + as</p> + + <p><i><a name="division_method" id="division_method">g(r, m) = + r mod m</a></i> (2) ,</p> + + <p><i>g(r, m) = ⌈ u/v ( a r mod v ) ⌉</i> ,</p> + + <p>and</p> + + <p><i>g(r, m) = ⌈ u/v ( r<sup>2</sup> mod v ) ⌉</i> + ,</p> + + <p>respectively, for some positive integrals <i>u</i> and + <i>v</i> (typically powers of 2), and some <i>a</i>. Each of + these range-hashing functions works best for some different + setting.</p> + + <p>The division method <a href="#division_method">(2)</a> is a + very common choice. However, even this single method can be + implemented in two very different ways. It is possible to + implement <a href="#division_method">(2)</a> using the low + level <i>%</i> (modulo) operation (for any <i>m</i>), or the + low level <i>&</i> (bit-mask) operation (for the case where + <i>m</i> is a power of 2), <i>i.e.</i>,</p> + + <p><i><a name="division_method_prime_mod" id= + "division_method_prime_mod">g(r, m) = r % m</a></i> (3) ,</p> + + <p>and</p> + + <p><i><a name="division_method_bit_mask" id= + "division_method_bit_mask">g(r, m) = r & m - 1, (m = + 2<sup>k</sup>)</a></i> for some <i>k)</i> (4),</p> + + <p>respectively.</p> + + <p>The <i>%</i> (modulo) implementation <a href= + "#division_method_prime_mod">(3)</a> has the advantage that for + <i>m</i> a prime far from a power of 2, <i>g(r, m)</i> is + affected by all the bits of <i>r</i> (minimizing the chance of + collision). It has the disadvantage of using the costly modulo + operation. This method is hard-wired into SGI's implementation + [<a href="references.html#sgi_stl">sgi_stl</a>].</p> + + <p>The <i>&</i> (bit-mask) implementation <a href= + "#division_method_bit_mask">(4)</a> has the advantage of + relying on the fast bit-wise and operation. It has the + disadvantage that for <i>g(r, m)</i> is affected only by the + low order bits of <i>r</i>. This method is hard-wired into + Dinkumware's implementation [<a href= + "references.html#dinkumware_stl">dinkumware_stl</a>].</p> + + <h3><a name="hash_policies_ranged_hash_policies" id= + "hash_policies_ranged_hash_policies">Ranged-Hash + Functions</a></h3> + + <p>In cases it is beneficial to allow the + client to directly specify a ranged-hash hash function. It is + true, that the writer of the ranged-hash function cannot rely + on the values of <i>m</i> having specific numerical properties + suitable for hashing (in the sense used in [<a href= + "references.html#knuth98sorting">knuth98sorting</a>]), since + the values of <i>m</i> are determined by a resize policy with + possibly orthogonal considerations.</p> + + <p>There are two cases where a ranged-hash function can be + superior. The firs is when using perfect hashing [<a href= + "references.html#knuth98sorting">knuth98sorting</a>]; the + second is when the values of <i>m</i> can be used to estimate + the "general" number of distinct values required. This is + described in the following.</p> + + <p>Let</p> + + <p class="c2">s = [ s<sub>0</sub>,..., s<sub>t - 1</sub>]</p> + + <p>be a string of <i>t</i> characters, each of which is from + domain <i>S</i>. Consider the following ranged-hash + function:</p> + + <p><a name="total_string_dna_hash" id= + "total_string_dna_hash"><i>f<sub>1</sub>(s, m) = ∑ <sub>i = + 0</sub><sup>t - 1</sup> s<sub>i</sub> a<sup>i</sup></i> mod + <i>m</i></a> (5) ,</p> + + <p>where <i>a</i> is some non-negative integral value. This is + the standard string-hashing function used in SGI's + implementation (with <i>a = 5</i>) [<a href= + "references.html#sgi_stl">sgi_stl</a>]. Its advantage is that + it takes into account all of the characters of the string.</p> + + <p>Now assume that <i>s</i> is the string representation of a + of a long DNA sequence (and so <i>S = {'A', 'C', 'G', + 'T'}</i>). In this case, scanning the entire string might be + prohibitively expensive. A possible alternative might be to use + only the first <i>k</i> characters of the string, where</p> + + <p>|S|<sup>k</sup> ≥ m ,</p> + + <p><i>i.e.</i>, using the hash function</p> + + <p><a name="only_k_string_dna_hash" id= + "only_k_string_dna_hash"><i>f<sub>2</sub>(s, m) = ∑ <sub>i + = 0</sub><sup>k - 1</sup> s<sub>i</sub> a<sup>i</sup></i> mod + <i>m</i></a> , (6)</p> + + <p>requiring scanning over only</p> + + <p><i>k =</i> log<i><sub>4</sub>( m )</i></p> + + <p>characters.</p> + + <p>Other more elaborate hash-functions might scan <i>k</i> + characters starting at a random position (determined at each + resize), or scanning <i>k</i> random positions (determined at + each resize), <i>i.e.</i>, using</p> + + <p><i>f<sub>3</sub>(s, m) = ∑ <sub>i = + r</sub>0</i><sup>r<sub>0</sub> + k - 1</sup> s<sub>i</sub> + a<sup>i</sup> mod <i>m</i> ,</p> + + <p>or</p> + + <p><i>f<sub>4</sub>(s, m) = ∑ <sub>i = 0</sub><sup>k - + 1</sup> s<sub>r</sub>i</i> a<sup>r<sub>i</sub></sup> mod + <i>m</i> ,</p> + + <p>respectively, for <i>r<sub>0</sub>,..., r<sub>k-1</sub></i> + each in the (inclusive) range <i>[0,...,t-1]</i>.</p> + + <p>It should be noted that the above functions cannot be + decomposed as <a href= + "#ranged_hash_composed_of_hash_and_range_hashing">(1)</a> .</p> + + <h3><a name="pb_ds_imp" id="pb_ds_imp">Implementation</a></h3> + + <p>This sub-subsection describes the implementation of the + above in <tt>pb_ds</tt>. It first explains range-hashing + functions in collision-chaining tables, then ranged-hash + functions in collision-chaining tables, then probing-based + tables, and, finally, lists the relevant classes in + <tt>pb_ds</tt>.</p> + + <h4>Range-Hashing and Ranged-Hashes in Collision-Chaining + Tables</h4> + + <p><a href= + "cc_hash_table.html"><tt>cc_hash_table</tt></a> is + parametrized by <tt>Hash_Fn</tt> and <tt>Comb_Hash_Fn</tt>, a + hash functor and a combining hash functor, respectively.</p> + + <p>In general, <tt>Comb_Hash_Fn</tt> is considered a + range-hashing functor. <a href= + "cc_hash_table.html"><tt>cc_hash_table</tt></a> + synthesizes a ranged-hash function from <tt>Hash_Fn</tt> and + <tt>Comb_Hash_Fn</tt> (see <a href= + "#ranged_hash_composed_of_hash_and_range_hashing">(1)</a> + above). Figure <a href="#hash_range_hashing_seq_diagram">Insert + hash sequence diagram</a> shows an <tt>insert</tt> sequence + diagram for this case. The user inserts an element (point A), + the container transforms the key into a non-negative integral + using the hash functor (points B and C), and transforms the + result into a position using the combining functor (points D + and E).</p> + + <h6 class="c1"><a name="hash_range_hashing_seq_diagram" id= + "hash_range_hashing_seq_diagram"><img src= + "hash_range_hashing_seq_diagram.png" alt="no image" /></a></h6> + + <h6 class="c1">Insert hash sequence diagram.</h6> + + <p>If <a href= + "cc_hash_table.html"><tt>cc_hash_table</tt></a>'s + hash-functor, <tt>Hash_Fn</tt> is instantiated by <a href= + "null_hash_fn.html"><tt>null_hash_fn</tt></a> (see <a href= + "concepts.html#concepts_null_policies">Interface::Concepts::Null + Policy Classes</a>), then <tt>Comb_Hash_Fn</tt> is taken to be + a ranged-hash function. Figure <a href= + "#hash_range_hashing_seq_diagram2">Insert hash sequence diagram + with a null hash policy</a> shows an <tt>insert</tt> sequence + diagram. The user inserts an element (point A), the container + transforms the key into a position using the combining functor + (points B and C).</p> + + <h6 class="c1"><a name="hash_range_hashing_seq_diagram2" id= + "hash_range_hashing_seq_diagram2"><img src= + "hash_range_hashing_seq_diagram2.png" alt= + "no image" /></a></h6> + + <h6 class="c1">Insert hash sequence diagram with a null hash + policy.</h6> + + <h4>Probing Tables</h4> + + <p><a href= + "gp_hash_table.html"></a><tt>gp_hash_table</tt> is + parametrized by <tt>Hash_Fn</tt>, <tt>Probe_Fn</tt>, and + <tt>Comb_Probe_Fn</tt>. As before, if <tt>Hash_Fn</tt> and + <tt>Probe_Fn</tt> are, respectively, <a href= + "null_hash_fn.html"><tt>null_hash_fn</tt></a> and <a href= + "null_probe_fn.html"><tt>null_probe_fn</tt></a>, then + <tt>Comb_Probe_Fn</tt> is a ranged-probe functor. Otherwise, + <tt>Hash_Fn</tt> is a hash functor, <tt>Probe_Fn</tt> is a + functor for offsets from a hash value, and + <tt>Comb_Probe_Fn</tt> transforms a probe sequence into a + sequence of positions within the table.</p> + + <h4>Pre-Defined Policies</h4> + + <p><tt>pb_ds</tt> contains some pre-defined classes + implementing range-hashing and probing functions:</p> + + <ol> + <li><a href= + "direct_mask_range_hashing.html"><tt>direct_mask_range_hashing</tt></a> + and <a href= + "direct_mod_range_hashing.html"><tt>direct_mod_range_hashing</tt></a> + are range-hashing functions based on a bit-mask and a modulo + operation, respectively.</li> + + <li><a href= + "linear_probe_fn.html"><tt>linear_probe_fn</tt></a>, and + <a href= + "quadratic_probe_fn.html"><tt>quadratic_probe_fn</tt></a> are + a linear probe and a quadratic probe function, + respectively.</li> + </ol>Figure <a href="#hash_policy_cd">Hash policy class + diagram</a> shows a class diagram. + + <h6 class="c1"><a name="hash_policy_cd" id= + "hash_policy_cd"><img src="hash_policy_cd.png" alt= + "no image" /></a></h6> + + <h6 class="c1">Hash policy class diagram.</h6> + + <h2><a name="resize_policies" id="resize_policies">Resize + Policies</a></h2> + + <h3><a name="general" id="general">General Terms</a></h3> + + <p>Hash-tables, as opposed to trees, do not naturally grow or + shrink. It is necessary to specify policies to determine how + and when a hash table should change its size. Usually, resize + policies can be decomposed into orthogonal policies:</p> + + <ol> + <li>A <i>size policy</i> indicating <i>how</i> a hash table + should grow (<i>e.g.,</i> it should multiply by powers of + 2).</li> + + <li>A <i>trigger policy</i> indicating <i>when</i> a hash + table should grow (<i>e.g.,</i> a load factor is + exceeded).</li> + </ol> + + <h3><a name="size_policies" id="size_policies">Size + Policies</a></h3> + + <p>Size policies determine how a hash table changes size. These + policies are simple, and there are relatively few sensible + options. An exponential-size policy (with the initial size and + growth factors both powers of 2) works well with a mask-based + range-hashing function (see <a href= + "#hash_policies">Range-Hashing Policies</a>), and is the + hard-wired policy used by Dinkumware [<a href= + "references.html#dinkumware_stl">dinkumware_stl</a>]. A + prime-list based policy works well with a modulo-prime range + hashing function (see <a href="#hash_policies">Range-Hashing + Policies</a>), and is the hard-wired policy used by SGI's + implementation [<a href= + "references.html#sgi_stl">sgi_stl</a>].</p> + + <h3><a name="trigger_policies" id="trigger_policies">Trigger + Policies</a></h3> + + <p>Trigger policies determine when a hash table changes size. + Following is a description of two policies: <i>load-check</i> + policies, and collision-check policies.</p> + + <p>Load-check policies are straightforward. The user specifies + two factors, <i>α<sub>min</sub></i> and + <i>α<sub>max</sub></i>, and the hash table maintains the + invariant that</p> + + <p><i><a name="load_factor_min_max" id= + "load_factor_min_max">α<sub>min</sub> ≤ (number of + stored elements) / (hash-table size) ≤ + α<sub>max</sub></a></i> (1) .</p> + + <p>Collision-check policies work in the opposite direction of + load-check policies. They focus on keeping the number of + collisions moderate and hoping that the size of the table will + not grow very large, instead of keeping a moderate load-factor + and hoping that the number of collisions will be small. A + maximal collision-check policy resizes when the longest + probe-sequence grows too large.</p> + + <p>Consider Figure <a href="#balls_and_bins">Balls and + bins</a>. Let the size of the hash table be denoted by + <i>m</i>, the length of a probe sequence be denoted by + <i>k</i>, and some load factor be denoted by α. We would + like to calculate the minimal length of <i>k</i>, such that if + there were <i>α m</i> elements in the hash table, a probe + sequence of length <i>k</i> would be found with probability at + most <i>1/m</i>.</p> + + <h6 class="c1"><a name="balls_and_bins" id= + "balls_and_bins"><img src="balls_and_bins.png" alt= + "no image" /></a></h6> + + <h6 class="c1">Balls and bins.</h6> + + <p>Denote the probability that a probe sequence of length + <i>k</i> appears in bin <i>i</i> by <i>p<sub>i</sub></i>, the + length of the probe sequence of bin <i>i</i> by + <i>l<sub>i</sub></i>, and assume uniform distribution. Then</p> + + <p><a name="prob_of_p1" id= + "prob_of_p1"><i>p<sub>1</sub></i></a> = (3)</p> + + <p class="c2"><b>P</b>(l<sub>1</sub> ≥ k) =</p> + + <p><i><b>P</b>(l<sub>1</sub> ≥ α ( 1 + k / α - 1 + ) ≤</i> (a)</p> + + <p><i>e ^ ( - ( α ( k / α - 1 )<sup>2</sup> ) /2 + )</i> ,</p> + + <p>where (a) follows from the Chernoff bound [<a href= + "references.html#motwani95random">motwani95random</a>]. To + calculate the probability that <i>some</i> bin contains a probe + sequence greater than <i>k</i>, we note that the + <i>l<sub>i</sub></i> are negatively-dependent [<a href= + "references.html#dubhashi98neg">dubhashi98neg</a>]. Let + <i><b>I</b>(.)</i> denote the indicator function. Then</p> + + <p><a name="at_least_k_i_n_some_bin" id= + "at_least_k_i_n_some_bin"><i><b>P</b>( exists<sub>i</sub> + l<sub>i</sub> ≥ k ) =</i> (3)</a></p> + + <p class="c2"><b>P</b> ( ∑ <sub>i = 1</sub><sup>m</sup> + <b>I</b>(l<sub>i</sub> ≥ k) ≥ 1 ) =</p> + + <p><i><b>P</b> ( ∑ <sub>i = 1</sub><sup>m</sup> <b>I</b> ( + l<sub>i</sub> ≥ k ) ≥ m p<sub>1</sub> ( 1 + 1 / (m + p<sub>1</sub>) - 1 ) ) ≤</i> (a)</p> + + <p class="c2">e ^ ( ( - m p<sub>1</sub> ( 1 / (m p<sub>1</sub>) + - 1 ) <sup>2</sup> ) / 2 ) ,</p> + + <p>where (a) follows from the fact that the Chernoff bound can + be applied to negatively-dependent variables [<a href= + "references.html#dubhashi98neg">dubhashi98neg</a>]. Inserting + <a href="#prob_of_p1">(2)</a> into <a href= + "#at_least_k_i_n_some_bin">(3)</a>, and equating with + <i>1/m</i>, we obtain</p> + + <p><i>k ~ √ ( 2 α</i> ln <i>2 m</i> ln<i>(m) ) + )</i> .</p> + + <h3><a name="imp_pb_ds" id="imp_pb_ds">Implementation</a></h3> + + <p>This sub-subsection describes the implementation of the + above in <tt>pb_ds</tt>. It first describes resize policies and + their decomposition into trigger and size policies, then + describes pre-defined classes, and finally discusses controlled + access the policies' internals.</p> + + <h4>Resize Policies and Their Decomposition</h4> + + <p>Each hash-based container is parametrized by a + <tt>Resize_Policy</tt> parameter; the container derives + <tt><b>public</b></tt>ly from <tt>Resize_Policy</tt>. For + example:</p> + <pre> +<a href="cc_hash_table.html">cc_hash_table</a>< + <b>typename</b> Key, + <b>typename</b> Mapped, + ... + <b>typename</b> Resize_Policy + ...> : + <b>public</b> Resize_Policy +</pre> + + <p>As a container object is modified, it continuously notifies + its <tt>Resize_Policy</tt> base of internal changes + (<i>e.g.</i>, collisions encountered and elements being + inserted). It queries its <tt>Resize_Policy</tt> base whether + it needs to be resized, and if so, to what size.</p> + + <p>Figure <a href="#insert_resize_sequence_diagram1">Insert + resize sequence diagram</a> shows a (possible) sequence diagram + of an insert operation. The user inserts an element; the hash + table notifies its resize policy that a search has started + (point A); in this case, a single collision is encountered - + the table notifies its resize policy of this (point B); the + container finally notifies its resize policy that the search + has ended (point C); it then queries its resize policy whether + a resize is needed, and if so, what is the new size (points D + to G); following the resize, it notifies the policy that a + resize has completed (point H); finally, the element is + inserted, and the policy notified (point I).</p> + + <h6 class="c1"><a name="insert_resize_sequence_diagram1" id= + "insert_resize_sequence_diagram1"><img src= + "insert_resize_sequence_diagram1.png" alt= + "no image" /></a></h6> + + <h6 class="c1">Insert resize sequence diagram.</h6> + + <p>In practice, a resize policy can be usually orthogonally + decomposed to a size policy and a trigger policy. Consequently, + the library contains a single class for instantiating a resize + policy: <a href= + "hash_standard_resize_policy.html"><tt>hash_standard_resize_policy</tt></a> + is parametrized by <tt>Size_Policy</tt> and + <tt>Trigger_Policy</tt>, derives <tt><b>public</b></tt>ly from + both, and acts as a standard delegate [<a href= + "references.html#gamma95designpatterns">gamma95designpatterns</a>] + to these policies.</p> + + <p>Figures <a href="#insert_resize_sequence_diagram2">Standard + resize policy trigger sequence diagram</a> and <a href= + "#insert_resize_sequence_diagram3">Standard resize policy size + sequence diagram</a> show sequence diagrams illustrating the + interaction between the standard resize policy and its trigger + and size policies, respectively.</p> + + <h6 class="c1"><a name="insert_resize_sequence_diagram2" id= + "insert_resize_sequence_diagram2"><img src= + "insert_resize_sequence_diagram2.png" alt= + "no image" /></a></h6> + + <h6 class="c1">Standard resize policy trigger sequence + diagram.</h6> + + <h6 class="c1"><a name="insert_resize_sequence_diagram3" id= + "insert_resize_sequence_diagram3"><img src= + "insert_resize_sequence_diagram3.png" alt= + "no image" /></a></h6> + + <h6 class="c1">Standard resize policy size sequence + diagram.</h6> + + <h4>Pre-Defined Policies</h4> + + <p>The library includes the following + instantiations of size and trigger policies:</p> + + <ol> + <li><a href= + "hash_load_check_resize_trigger.html"><tt>hash_load_check_resize_trigger</tt></a> + implements a load check trigger policy.</li> + + <li><a href= + "cc_hash_max_collision_check_resize_trigger.html"><tt>cc_hash_max_collision_check_resize_trigger</tt></a> + implements a collision check trigger policy.</li> + + <li><a href= + "hash_exponential_size_policy.html"><tt>hash_exponential_size_policy</tt></a> + implements an exponential-size policy (which should be used + with mask range hashing).</li> + + <li><a href= + "hash_prime_size_policy.html"><tt>hash_prime_size_policy</tt></a> + implementing a size policy based on a sequence of primes + [<a href="references.html#sgi_stl">sgi_stl</a>] (which should + be used with mod range hashing</li> + </ol> + + <p>Figure <a href="#resize_policy_cd">Resize policy class + diagram</a> gives an overall picture of the resize-related + classes. <a href= + "basic_hash_table.html"><tt>basic_hash_table</tt></a> + is parametrized by <tt>Resize_Policy</tt>, which it subclasses + publicly. This class is currently instantiated only by <a href= + "hash_standard_resize_policy.html"><tt>hash_standard_resize_policy</tt></a>. + <a href= + "hash_standard_resize_policy.html"><tt>hash_standard_resize_policy</tt></a> + itself is parametrized by <tt>Trigger_Policy</tt> and + <tt>Size_Policy</tt>. Currently, <tt>Trigger_Policy</tt> is + instantiated by <a href= + "hash_load_check_resize_trigger.html"><tt>hash_load_check_resize_trigger</tt></a>, + or <a href= + "cc_hash_max_collision_check_resize_trigger.html"><tt>cc_hash_max_collision_check_resize_trigger</tt></a>; + <tt>Size_Policy</tt> is instantiated by <a href= + "hash_exponential_size_policy.html"><tt>hash_exponential_size_policy</tt></a>, + or <a href= + "hash_prime_size_policy.html"><tt>hash_prime_size_policy</tt></a>.</p> + + <h6 class="c1"><a name="resize_policy_cd" id= + "resize_policy_cd"><img src="resize_policy_cd.png" alt= + "no image" /></a></h6> + + <h6 class="c1">Resize policy class diagram.</h6> + + <h4>Controlled Access to Policies' Internals</h4> + + <p>There are cases where (controlled) access to resize + policies' internals is beneficial. <i>E.g.</i>, it is sometimes + useful to query a hash-table for the table's actual size (as + opposed to its <tt>size()</tt> - the number of values it + currently holds); it is sometimes useful to set a table's + initial size, externally resize it, or change load factors.</p> + + <p>Clearly, supporting such methods both decreases the + encapsulation of hash-based containers, and increases the + diversity between different associative-containers' interfaces. + Conversely, omitting such methods can decrease containers' + flexibility.</p> + + <p>In order to avoid, to the extent possible, the above + conflict, the hash-based containers themselves do not address + any of these questions; this is deferred to the resize policies, + which are easier to change or replace. Thus, for example, + neither <a href= + "cc_hash_table.html"><tt>cc_hash_table</tt></a> nor + <a href= + "gp_hash_table.html"><tt>gp_hash_table</tt></a> + contain methods for querying the actual size of the table; this + is deferred to <a href= + "hash_standard_resize_policy.html"><tt>hash_standard_resize_policy</tt></a>.</p> + + <p>Furthermore, the policies themselves are parametrized by + template arguments that determine the methods they support + ([<a href= + "references.html#alexandrescu01modern">alexandrescu01modern</a>] + shows techniques for doing so). <a href= + "hash_standard_resize_policy.html"><tt>hash_standard_resize_policy</tt></a> + is parametrized by <tt>External_Size_Access</tt> that + determines whether it supports methods for querying the actual + size of the table or resizing it. <a href= + "hash_load_check_resize_trigger.html"><tt>hash_load_check_resize_trigger</tt></a> + is parametrized by <tt>External_Load_Access</tt> that + determines whether it supports methods for querying or + modifying the loads. <a href= + "cc_hash_max_collision_check_resize_trigger.html"><tt>cc_hash_max_collision_check_resize_trigger</tt></a> + is parametrized by <tt>External_Load_Access</tt> that + determines whether it supports methods for querying the + load.</p> + + <p>Some operations, for example, resizing a container at + run time, or changing the load factors of a load-check trigger + policy, require the container itself to resize. As mentioned + above, the hash-based containers themselves do not contain + these types of methods, only their resize policies. + Consequently, there must be some mechanism for a resize policy + to manipulate the hash-based container. As the hash-based + container is a subclass of the resize policy, this is done + through virtual methods. Each hash-based container has a + <tt><b>private</b></tt> <tt><b>virtual</b></tt> method:</p> + <pre> +<b>virtual void</b> + do_resize + (size_type new_size); +</pre> + + <p>which resizes the container. Implementations of + <tt>Resize_Policy</tt> can export public methods for resizing + the container externally; these methods internally call + <tt>do_resize</tt> to resize the table.</p> + + <h2><a name="policy_interaction" id="policy_interaction">Policy + Interaction</a></h2> + + <p>Hash-tables are unfortunately especially susceptible to + choice of policies. One of the more complicated aspects of this + is that poor combinations of good policies can form a poor + container. Following are some considerations.</p> + + <h3><a name="policy_interaction_probe_size_trigger" id= + "policy_interaction_probe_size_trigger">Probe Policies, Size + Policies, and Trigger Policies</a></h3> + + <p>Some combinations do not work well for probing containers. + For example, combining a quadratic probe policy with an + exponential size policy can yield a poor container: when an + element is inserted, a trigger policy might decide that there + is no need to resize, as the table still contains unused + entries; the probe sequence, however, might never reach any of + the unused entries.</p> + + <p>Unfortunately, <tt>pb_ds</tt> cannot detect such problems at + compilation (they are halting reducible). It therefore defines + an exception class <a href= + "insert_error.html"><tt>insert_error</tt></a> to throw an + exception in this case.</p> + + <h3><a name="policy_interaction_hash_trigger" id= + "policy_interaction_hash_trigger">Hash Policies and Trigger + Policies</a></h3> + + <p>Some trigger policies are especially susceptible to poor + hash functions. Suppose, as an extreme case, that the hash + function transforms each key to the same hash value. After some + inserts, a collision detecting policy will always indicate that + the container needs to grow.</p> + + <p>The library, therefore, by design, limits each operation to + one resize. For each <tt>insert</tt>, for example, it queries + only once whether a resize is needed.</p> + + <h3><a name="policy_interaction_eq_sth_hash" id= + "policy_interaction_eq_sth_hash">Equivalence Functors, Storing + Hash Values, and Hash Functions</a></h3> + + <p><a href= + "cc_hash_table.html"><tt>cc_hash_table</tt></a> and + <a href= + "gp_hash_table.html"><tt>gp_hash_table</tt></a> are + parametrized by an equivalence functor and by a + <tt>Store_Hash</tt> parameter. If the latter parameter is + <tt><b>true</b></tt>, then the container stores with each entry + a hash value, and uses this value in case of collisions to + determine whether to apply a hash value. This can lower the + cost of collision for some types, but increase the cost of + collisions for other types.</p> + + <p>If a ranged-hash function or ranged probe function is + directly supplied, however, then it makes no sense to store the + hash value with each entry. <tt>pb_ds</tt>'s container will + fail at compilation, by design, if this is attempted.</p> + + <h3><a name="policy_interaction_size_load_check" id= + "policy_interaction_size_load_check">Size Policies and + Load-Check Trigger Policies</a></h3> + + <p>Assume a size policy issues an increasing sequence of sizes + <i>a, a q, a q<sup>1</sup>, a q<sup>2</sup>, ...</i> For + example, an exponential size policy might issue the sequence of + sizes <i>8, 16, 32, 64, ...</i></p> + + <p>If a load-check trigger policy is used, with loads + <i>α<sub>min</sub></i> and <i>α<sub>max</sub></i>, + respectively, then it is a good idea to have:</p> + + <ol> + <li><i>α<sub>max</sub> ~ 1 / q</i></li> + + <li><i>α<sub>min</sub> < 1 / (2 q)</i></li> + </ol> + + <p>This will ensure that the amortized hash cost of each + modifying operation is at most approximately 3.</p> + + <p><i>α<sub>min</sub> ~ α<sub>max</sub></i> is, in + any case, a bad choice, and <i>α<sub>min</sub> > + α<sub>max</sub></i> is horrendous.</p> + </div> +</body> +</html> |