<!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>