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+ <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>
projects / msp430-gcc.git / blobdiff