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<!DOCTYPE HTML PUBLIC "-//W3C//DTD HTML 4.0 Transitional//EN">
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<html>
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<head>
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<title>Hash Policies</title>
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<meta name="GENERATOR" content="Microsoft Visual Studio .NET 7.1">
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<meta name="vs_targetSchema" content="http://schemas.microsoft.com/intellisense/ie5">
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</head>
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<body bgcolor="white">
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<h1>Hash Policies</h1>
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<p>
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This subsection describes hash policies. It is organized as follows:
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</p>
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<ol>
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<li> The <a href = "#general_terms">General Terms</a> Section describes
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some general terms.
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</li>
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<li> The <a href = "#range_hashing_fns">Range-Hashing Functions</a> Section
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describes range-hasing functions.</li>
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<li> The <a href = "#hash_policies_ranged_hash_policies">Ranged-Hash Functions</a> Section
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describes ranged-hash functions. </li>
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<li> The <a href = "#pb_assoc_imp">Implementation in <tt>pb_assoc</tt></a> Section
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describes the implementation of these concepts in <tt>pb_assoc</tt>.
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</li>
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</ol>
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<h2><a name="general_terms">General Terms</a></h2>
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<p>
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There
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are actually three functions involved in transforming a key into a hash-table's
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position (see Figure
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<a href = "#hash_ranged_hash_range_hashing_fns">
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Hash runctions, ranged-hash functions, and range-hashing functions
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</a>):
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</p>
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<ol>
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<li>
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A <i>ranged-hash</i> function, which maps keys into an interval of the
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non-negative integrals. This is the function actually required by the
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hash-table algorithm.
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</li>
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<li>
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A hash function, which maps keys into non-negative integral types. This is
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typically specified by the writer of the key class.
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</li>
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<li>
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A <i>range-hashing</i> function, which maps non-negative integral types into an
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interval of non-negative integral types.
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</li>
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</ol>
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<h6 align = "center">
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<a name = "hash_ranged_hash_range_hashing_fns">
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<img src = "hash_ranged_hash_range_hashing_fns.jpg" width = "40%" alt = "no image">
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</a>
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Hash runctions, ranged-hash functions, and range-hashing functions.
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</h6>
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<p>
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Let <i>U</i> be a domain (<i>e.g.</i>, the integers, or the strings of 3
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characters). A hash-table algorithm needs to map elements of <i>U</i> "uniformly"
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into the range <i>[0,..., m - 1]</i> (where <i>m</i> is a non-negative integral
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value, and is, in general, time varying). <i>I.e.</i>, the algorithm needs a <i>ranged-hash</i>
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function
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</p>
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<p>
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<i>f : U × Z<sub>+</sub> → Z<sub>+</sub> </i>,
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</p>
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<p>
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such that for any <i>u</i> in <i>U</i>
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,
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</p>
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<p>
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<i>0 ≤ f(u, m) ≤ m - 1 </i>,
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</p>
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<p>
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and which has "good uniformity" properties [<a href="references.html#knuth98sorting">knuth98sorting</a>].
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One common solution is to use the composition of the hash function
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</p>
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<p>
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<i>h : U → Z<sub>+</sub> </i>,
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</p>
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<p>
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which maps elements of <i>U</i> into the non-negative integrals, and
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</p>
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<p>
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<i>g : Z<sub>+</sub> × Z<sub>+</sub> → Z<sub>+</sub>, </i>
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</p>
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<p>
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which maps a non-negative hash value, and a non-negative range upper-bound into
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a non-negative integral in the range between 0 (inclusive) and the range upper
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bound (exclusive), <i>i.e.</i>, for any <i>r</i> in <i>Z<sub>+</sub></i>,
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</p>
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<p>
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<i>0 ≤ g(r, m) ≤ m - 1 </i>.
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</p>
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<p>
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The resulting ranged-hash function, is
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</p>
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<p>
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<i><a name="eqn:ranged_hash_composed_of_hash_and_range_hashing">f(u , m) = g(h(u), m) </a>
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</i>(1) .
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</p>
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<p>
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From the above, it is obvious that given <i>g</i> and <i>h</i>, <i>f</i> can
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always be composed (however the converse is not true).
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</p>
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<p>
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The above describes the case where a key is to be mapped into a <i>single
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position</i> within a hash table, <i>e.g.</i>, in a collision-chaining table.
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In other cases, a key is to be mapped into a <i>sequence of poisitions</i>
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within a table, <i>e.g.</i>, in a probing table.
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</p>
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<p>
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Similar terms apply in this case: the table requires a <i>ranged probe</i>
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function, mapping a key into a sequence of positions withing the table. This is
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typically acheived by composing a <i>hash function</i> mapping the key
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into a non-negative integral type, a <i>probe</i> function transforming the
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hash value into a sequence of hash values, and a <i>range-hashing</i> function
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transforming the sequence of hash values into a sequence of positions.
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</p>
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<h2><a name="range_hashing_fns">Range-Hashing Functions</a></h2>
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<p>
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Some common choices for range-hashing functions are the division,
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multiplication, and middle-square methods [<a href="references.html#knuth98sorting">knuth98sorting</a>],
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defined as
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</p>
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<p>
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<i><a name="eqn:division_method">g(r, m) = r mod m </a></i>(2) ,
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</p>
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<p>
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<i>g(r, m) = ⌈ u/v ( a r mod v ) ⌉ </i>,
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</p>
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<p>
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and
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</p>
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<p>
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<i>g(r, m) = ⌈ u/v ( r<sup>2</sup> mod v ) ⌉ </i>,
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</p>
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<p>
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respectively, for some positive integrals <i>u</i> and <i>v</i> (typically
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powers of 2), and some <i>a</i>. Each of these range-hashing functions works
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best for some different setting.
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</p>
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<p>
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The division method <a href="#division_method">(2)</a> is a very common
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choice. However, even this single method can be implemented in two very
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different ways. It is possible to implement <a href="#division_method">(2)</a>
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using the low level <i>%</i> (modulo) operation (for any <i>m</i>), or the low
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level <i>&</i> (bit-mask) operation (for the case where <i>m</i> is a power of
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2), <i>i.e.</i>,
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</p>
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<p>
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<i><a name="eqn:division_method_prime_mod">g(r, m) = r % m </a></i>(3) ,
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</p>
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<p>
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and
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</p>
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<p>
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<a name="eqn:division_method_bit_mask">
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<i>g(r, m) = r & m - 1, ( m = 2<sup>k</sup>
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</i>
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for some<i> k) </i></a>(4) ,
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</p>
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<p>
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respectively.
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</p>
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<p>
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The <i>%</i> (modulo) implementation <a href="#division_method_prime_mod">(3)</a>
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has the advantage that for <i>m</i> a prime far from a power of 2, <i>g(r, m)</i>
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is affected by all the bits of <i>r</i> (minimizing the chance of collision).
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It has the disadvantage of using the costly modulo operation. This method is
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hard-wired into SGI's implementation [<a href="references.html#sgi_stl">sgi_stl</a>].
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</p>
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<p>
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The <i>&</i> (bit-mask) implementation <a href="#division_method_bit_mask">(4)</a>
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has the advantage of relying on the fast bitwise and operation. It has the
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disadvantage that for <i>g(r, m)</i> is affected only by the low order bits of <i>r</i>.
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This method is hard-wired into Dinkumware's implementation [<a href="references.html#dinkumware_stl">dinkumware_stl</a>].
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</p>
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<h2><a name="hash_policies_ranged_hash_policies">Ranged-Hash Functions</a></h2>
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<p>
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Although rarer, there are cases where it is beneficial to allow the client to
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directly specify a ranged-hash hash function. It is true, that the writer of
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the ranged-hash function cannot rely on the values of <i>m</i> having specific
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numerical properties suitable for hashing (in the sense used in [<a href="references.html#knuth98sorting">knuth98sorting</a>]),
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since the values of <i>m</i> are determined by a resize policy with possibly
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orthogonal considerations [<a href="references.html#austern98segmented">austern98segmented</a>].
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The values of <i>m</i> can be used in some cases, though, to estimate the
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"general" number of distinct values required.
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</p>
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<p>
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Let
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</p>
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<p>
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<i>s = [ s<sub>0</sub>,..., s<sub>t - 1</sub>] </i>
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</p>
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<p>
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be a string of <i>t</i> characters, each of which is from domain <i>S</i>.
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Consider the following ranged-hash function:
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</p>
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<p>
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<a name="eqn:total_string_dna_hash">
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<i>
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f<sub>1</sub>(s, m) =
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∑ <sub>i =
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0</sub><sup>t - 1</sup> s<sub>i</sub> a<sup>i</sup> </i>mod<i> m </i>
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</a> (5) ,
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</p>
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<p>
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where <i>a</i> is some non-negative integral value. This is the standard
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string-hashing function used in SGI's implementation (with <i>a = 5</i>) [<a href="references.html#sgi_stl">sgi_stl</a>].
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Its advantage is that it takes into account all of the characters of the
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string.
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</p>
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<p>
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Now assume that <i>s</i> is the string representation of a of a long DNA
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sequence (and so <i>S = {'A', 'C', 'G', 'T'}</i>). In this case, scanning the
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entire string might be prohibitively expensive. A possible alternative might be
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to use only the first <i>k</i> characters of the string, where
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</p>
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<p>
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k <sup>|S|</sup> ≥ m ,
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</p>
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<p>
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<i>i.e.</i>, using the hash function
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</p>
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<p>
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<a name="eqn:only_k_string_dna_hash"><i>f<sub>2</sub>(s, m) = ∑ <sub>i = 0</sub><sup>k
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- 1</sup> s<sub>i</sub> a<sup>i</sup> </i>mod <i>m </i></a>, (6)
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</p>
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<p>
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requiring scanning over only
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</p>
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<p>
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<i>k = </i>log<i><sub>4</sub>( m ) </i>
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</p>
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<p>
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characters.
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</p>
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<p>
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Other more elaborate hash-functions might scan <i>k</i> characters starting at
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a random position (determined at each resize), or scanning <i>k</i> random
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positions (determined at each resize), <i>i.e.</i>, using
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</p>
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<p>
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<i>f<sub>3</sub>(s, m) = ∑ <sub>i = r<sub>0</sub></sub><sup>r<sub>0</sub> + k - 1</sup>
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s<sub>i</sub> a<sup>i</sup> </i>mod <i>m </i>,
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</p>
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<p>
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or
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</p>
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<p>
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<i>f<sub>4</sub>(s, m) = ∑ <sub>i = 0</sub><sup>k - 1</sup> s<sub>r<sub>i</sub></sub>
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a<sup>r<sub>i</sub></sup> </i>mod <i>m </i>,
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</p>
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<p>
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<p>
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respectively, for <i>r<sub>0</sub>,..., r<sub>k-1</sub></i> each in the
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(inclusive) range <i>[0,...,t-1]</i>.
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</p>
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<h2><a name="pb_assoc_imp">Implementation in <tt>pb_assoc</tt></a></h2>
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<p>
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Containers based on collision-chaining hash tables in <tt>pb_assoc</tt>
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are parameterized by the functors <tt>Hash_Fn</tt>, and <tt>Comb_Hash_Fn</tt>.
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</p>
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<p>
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If such a container is instantiated with any hash functor and
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range-hashing functor, the container will synthesize a ranged-hash functor
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automatically. For example, Figure
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<a href = "#hash_range_hashing_seq_diagram">
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Insert hash sequence diagram
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</a>
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shows an <tt>insert</tt> sequence diagram. The user inserts an element (point A),
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the container transforms the key into a non-negative integral using the hash
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functor (points B and C), and transforms the result into a position
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using the combining functor (points D and E).
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</p>
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<h6 align = "center">
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<a name = "hash_range_hashing_seq_diagram">
|
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|
<img src = "hash_range_hashing_seq_diagram.jpg" width = "50%" alt = "no image">
|
| 308 |
|
|
</a>
|
| 309 |
|
|
</h6>
|
| 310 |
|
|
<h6 align = "center">
|
| 311 |
|
|
Insert hash sequence diagram.
|
| 312 |
|
|
</h6>
|
| 313 |
|
|
|
| 314 |
|
|
|
| 315 |
|
|
<p>
|
| 316 |
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|
If such a container is instantiated with the
|
| 317 |
|
|
<a href = "concepts.html#concepts_null_policies">null policy</a>
|
| 318 |
|
|
hash functor,
|
| 319 |
|
|
<a href = "null_hash_fn.html"><tt>null_hash_fn</tt></a>,
|
| 320 |
|
|
and a combining-hash functor, the container treats
|
| 321 |
|
|
the combining hash functor as a ranged-hash function. For example, Figure
|
| 322 |
|
|
<a href = "#hash_range_hashing_seq_diagram2">
|
| 323 |
|
|
Insert hash sequence diagram with a null combination policy
|
| 324 |
|
|
</a>
|
| 325 |
|
|
shows an <tt>insert</tt> sequence diagram. The user inserts an element (point A),
|
| 326 |
|
|
the container transforms the key into a position
|
| 327 |
|
|
using the combining functor (points B and C).
|
| 328 |
|
|
</p>
|
| 329 |
|
|
|
| 330 |
|
|
|
| 331 |
|
|
<h6 align = "center">
|
| 332 |
|
|
<a name = "hash_range_hashing_seq_diagram2">
|
| 333 |
|
|
<img src = "hash_range_hashing_seq_diagram2.jpg" width = "50%" alt = "no image">
|
| 334 |
|
|
</a>
|
| 335 |
|
|
</h6>
|
| 336 |
|
|
<h6 align = "center">
|
| 337 |
|
|
Insert hash sequence diagram with a null combination policy.
|
| 338 |
|
|
</h6>
|
| 339 |
|
|
|
| 340 |
|
|
<p>
|
| 341 |
|
|
<tt>pb_assoc</tt> contains the following hash-related policies:
|
| 342 |
|
|
</p>
|
| 343 |
|
|
|
| 344 |
|
|
<ol>
|
| 345 |
|
|
<li>
|
| 346 |
|
|
<a href = "direct_mask_range_hashing.html"><tt>direct_mask_range_hashing</tt></a>
|
| 347 |
|
|
and
|
| 348 |
|
|
<a href = "direct_mod_range_hashing.html"><tt>direct_mod_range_hashing</tt></a>
|
| 349 |
|
|
are range-hashing functions based on a bit-mask and a modulo operation, respectively.
|
| 350 |
|
|
</li>
|
| 351 |
|
|
<li>
|
| 352 |
|
|
<a href = "linear_probe_fn.html"><tt>linear_probe_fn</tt></a> and
|
| 353 |
|
|
<a href = "quadratic_probe_fn.html"><tt>quadratic_probe_fn</tt></a> are probe
|
| 354 |
|
|
classes based on linear and quadratic increment, respectively.
|
| 355 |
|
|
</li>
|
| 356 |
|
|
<li>
|
| 357 |
|
|
<a href = "null_hash_fn.html"><tt>null_hash_fn</tt></a>
|
| 358 |
|
|
and
|
| 359 |
|
|
<a href = "null_probe_fn.html"><tt>null_probe_fn</tt></a>
|
| 360 |
|
|
are
|
| 361 |
|
|
<a href = "concepts.html#concepts_null_policies">null policy classes</a> for creating
|
| 362 |
|
|
ranged-hash and ranged-probe functions directly (<i>i.e.</i>, not through
|
| 363 |
|
|
composition).
|
| 364 |
|
|
</li>
|
| 365 |
|
|
</ol>
|
| 366 |
|
|
|
| 367 |
|
|
<p>
|
| 368 |
|
|
<tt>pb_assoc</tt> does not provide any hash functions (it relies on those
|
| 369 |
|
|
of the STL).
|
| 370 |
|
|
</p>
|
| 371 |
|
|
|
| 372 |
|
|
|
| 373 |
|
|
</body>
|
| 374 |
|
|
|
| 375 |
|
|
</html>
|
