<!-- 

         hash.xml            orb package documentation             
                                                               Juergen Mueller
                                                               Max Neunhoeffer
                                                                  Felix Noeske

         Copyright (C) 2005-2008 by the authors

This chapter explains functionality for hash lookups.

-->

<Chapter Label="hash">
<Heading>Hashing techniques</Heading>

<Section Label="hashidea">
<Heading>The idea of hashing</Heading>

If one wants to store
a certain set of similar objects and wants to quickly access a given
one (or come back with the result that it is unknown), the first idea
would be to store them in a list, possibly sorted for faster access.
This however still would need <M>\log(n)</M> comparisons to find
a given element or to decide that it is not yet stored. 

<P/>
Therefore one uses a much bigger array and uses a function on the
space of possible objects with integer values to decide, where in the
array to store a certain object. If this so called hash function
distributes the actually stored objects well enough over the array,
the access time is constant in average. Of course, a hash function
will usually not be injective, so one needs a strategy what to do
in case of so-called <Q>collision</Q>, that is, if more than one
object with the same hash value has to be stored.

<P/>
The basic functions to work with hash tables are <Ref Func="NewHT"/>,
<Ref Func="AddHT"/>, and <Ref Func="ValueHT"/>. They are described
in Section <Ref Sect="hashtables"/>. In the next section, we first
describe the infrastructure for hash functions.
</Section>

<Section Label="hashfunc">
<Heading>Hash functions</Heading>

In the <Package>orb</Package> package hash functions are chosen
automatically by giving a sample object together with the length
of the hash table. This is done with the following operation:

<ManSection>
<Oper Name="ChooseHashFunction" Arg="ob, len"/>
<Returns> a record </Returns>
<Description>
The first argument <A>ob</A> must be a sample object, that is, an object
like those we want to store in the hash table later on. The argument
<A>len</A> is an integer that gives the length of the hash table. Note
that this might be called later on automatically, when a hash table
is increased in size. The operation returns a record with two components.
The component <C>func</C> is a &GAP; function taking two arguments, see
below. The component <C>data</C> is some &GAP; object. Later on, the
hash function will be called with two arguments, the first is the
object for which it should call the hash value and the second argument
must be the data stored in the <C>data</C> component. 
<P/>
The hash function has to return values between <M>1</M> and the 
hash length <A>len</A> inclusively.
<P/>
This setup is chosen such that the hash functions can be global
objects that are not created during the execution of <Ref
Oper="ChooseHashFunction"/> but still can change their behaviour
depending on the data.
</Description>
</ManSection>

In the following we just document, for which types of objects there
are hash functions that can be found using <Ref
Oper="ChooseHashFunction"/>.

<ManSection>
<Meth Name="ChooseHashFunction" Arg="ob, len" Label="gf2vec"/>
<Returns> a record </Returns>
<Description>
This method is for compressed vectors over the field <C>GF(2)</C> of
two elements. Note that there is no hash function for non-compressed
vectors over <C>GF(2)</C> because those objects cannot efficiently
be recognised from their type.
<P/>
Note that you can only use the resulting hash functions for vectors
of the same length.
</Description>
</ManSection>

<ManSection>
<Meth Name="ChooseHashFunction" Arg="ob, len" Label="8bitvec"/>
<Returns> a record </Returns>
<Description>
This method is for compressed vectors over a finite field with up
to <M>256</M> elements. Note that there is no hash function for
non-compressed such vectors because those objects cannot efficiently be
recognised from their type.
<P/>
Note that you can only use the resulting hash functions for vectors
of the same length.
</Description>
</ManSection>

<ManSection>
<Meth Name="ChooseHashFunction" Arg="ob, len" Label="gf2mat"/>
<Returns> a record </Returns>
<Description>
This method is for compressed matrices over the field <C>GF(2)</C> of
two elements. Note that there is no hash function for non-compressed
matrices over <C>GF(2)</C> because those objects cannot efficiently
be recognised from their type.
<P/>
Note that you can only use the resulting hash functions for matrices
of the same size.
</Description>
</ManSection>

<ManSection>
<Meth Name="ChooseHashFunction" Arg="ob, len" Label="8bitmat"/>
<Returns> a record </Returns>
<Description>
This method is for compressed matrices over a finite field with up
to <M>256</M> elements. Note that there is no hash function for
non-compressed such vectors because those objects cannot efficiently be
recognised from their type.
<P/>
Note that you can only use the resulting hash functions for matrices
of the same size.
</Description>
</ManSection>

<ManSection>
<Meth Name="ChooseHashFunction" Arg="ob, len" Label="int"/>
<Returns> a record </Returns>
<Description>
This method is for integers.
</Description>
</ManSection>

<ManSection>
<Meth Name="ChooseHashFunction" Arg="ob, len" Label="perm"/>
<Returns> a record </Returns>
<Description>
This method is for permutations.
</Description>
</ManSection>

<ManSection>
<Meth Name="ChooseHashFunction" Arg="ob, len" Label="intlist"/>
<Returns> a record </Returns>
<Description>
This method is for lists of integers.
</Description>
</ManSection>

<ManSection>
<Meth Name="ChooseHashFunction" Arg="ob, len" Label="NBitsPcWord"/>
<Returns> a record </Returns>
<Description>
This method is for kernel Pc words.
</Description>
</ManSection>

<ManSection>
<Meth Name="ChooseHashFunction" Arg="ob, len" Label="IntLists"/>
<Returns> a record </Returns>
<Description>
This method is for lists of integers.
</Description>
</ManSection>

<ManSection>
<Meth Name="ChooseHashFunction" Arg="ob, len" Label="MatLists"/>
<Returns> a record </Returns>
<Description>
This method is for lists of matrices.
</Description>
</ManSection>

</Section>

<Section Label="hashtables">
<Heading>Using hash tables</Heading>

The following functions are needed to use hash tables. For details
about the data structures see Section <Ref Sect="hashdata"/>.

<ManSection>
<Func Name="NewHT" Arg="sample, len"/>
<Returns> a new hash table object </Returns>
<Description>
A new hash table for objects like <A>sample</A> of length <A>len</A> 
is created. Note that it is a good idea to choose a prime number as
the hash length due to the algorithm for collision handling which 
works particularly well in that case. The hash function is chosen
automatically. The resulting object can be used with the functions
<Ref Func="AddHT"/> and <Ref Func="ValueHT"/>. It will start with length
<A>len</A> but will grow as necessary.
</Description>
</ManSection>

<ManSection>
<Func Name="AddHT" Arg="ht, ob, val"/>
<Returns> an integer or fail </Returns>
<Description>
Stores the object <A>ob</A> into the hash table <A>ht</A> and stores
the value <A>val</A> together with <A>ob</A>. The result is
<K>fail</K> if an error occurred, which can only be that the hash
table is already full. This can only happen, if the hash table cannot
grow automatically.
<P/>
If no error occurs, the result is an integer indicating the place in the
hash table where the object is stored. Note that once the hash table
grows automatically this number is no longer the same!
<P/>
If the value <A>val</A> is <K>true</K> for all objects in the hash,
no extra memory is used for the values. All other values are stored
in the hash. The value <K>fail</K> cannot be stored as it indicates
that the object is not found in the hash.
<P/>
See Section <Ref Sect="hashdata"/> for details on the data structures and
especially about memory requirements.
</Description>
</ManSection>

<ManSection>
<Func Name="ValueHT" Arg="ht, ob"/>
<Returns> the stored value, <K>true</K>, or <K>fail</K> </Returns>
<Description>
Looks up the object <A>ob</A> in the hash table <A>ht</A>. If the object
is not found, <K>fail</K> is returned. Otherwise, the value stored with
the object is returned. Note that if this value was <K>true</K> 
no extra memory is used for this.
</Description>
</ManSection>

The following function is only documented for the sake of completeness
and for emergency situations, where <Ref Func="NewHT"/> tries to be
too intelligent.

<ManSection>
<Func Name="InitHT" Arg="len, hfun, eqfun"/>
<Returns> a new hash table object </Returns>
<Description>
This is usually only an internal function. It is called from 
<Ref Func="NewHT"/>. The argument <A>len</A> is the length of the
hash table, <A>hfun</A> is the hash function record as returned
by <Ref Oper="ChooseHashFunction"/> and <A>eqfun</A> is a comparison
function taking two arguments and returning <K>true</K> or <K>false</K>.
<P/>
Note that automatic growing is switched on for the new hash table which
means that if the hash table grows, a new hash function is chosen using
<Ref Oper="ChooseHashFunction"/>. If you do not want this, change the
component <C>cangrow</C> to <K>false</K> after creating the hash table.
</Description>
</ManSection>

<ManSection>
<Func Name="GrowHT" Arg="ht, ob"/>
<Returns> nothing </Returns>
<Description>
This is a more or less internal function. It is called when the space
in a hash table becomes scarce. The first argument <A>ht</A> must be a
hash table object, the second a sample point. 
The function increases the hash size
by a factor of 2. This makes it necessary to choose a new hash
function. Usually this is done with the usual
<C>ChooseHashFunction</C> method. However, one can assign the two
components <C>hfbig</C> and <C>hfdbig</C> to a function and a record
respectively. In that case, upon growing the hash, a new hash function
is created by taking the function <C>hfbig</C> together with
<C>hfdbig</C> as second data argument and reducing the resulting
integer modulo the hash length. In this way one can specify a hash
function suitable for all hash sizes by simply producing big enough
hash values.
</Description>
</ManSection>

</Section>

<Section Label="hashdata">
<Heading> The data structures for hash tables </Heading>

A hash table object is just a record with the following components:
<List>
<Mark><C>els</C></Mark>
<Item> A &GAP; list storing the elements. Its length can be as long
as the component <C>len</C> indicates but will only grow as necessary
when elements are stored in the hash.
</Item>
<Mark><C>vals</C></Mark>
<Item> A &GAP; list storing the corresponding values. If a value is
<K>true</K> nothing is stored here to save memory.
</Item>
<Mark><C>len</C></Mark>
<Item> Length of the hash table.</Item>
<Mark><C>nr</C></Mark>
<Item> Number of elements stored in the hash table.</Item>
<Mark><C>hf</C></Mark>
<Item> The hash function (value of the <C>func</C> component in the
record returned by <Ref Oper="ChooseHashFunction"/>). </Item>
<Mark><C>hfd</C></Mark>
<Item> The data for the second argument of the hash function (value of
the <C>data</C> component in the record returned by
<Ref Oper="ChooseHashFunction"/>). </Item>
<Mark><C>eqf</C></Mark>
<Item> A comparison function taking two arguments and returning
<K>true</K> for equality or <K>false</K> otherwise. </Item>
<Mark><C>collisions</C></Mark>
<Item> Number of collisions (see below). </Item>
<Mark><C>accesses</C></Mark>
<Item> Number of lookup or store accesses to the hash. </Item>
<Mark><C>cangrow</C></Mark>
<Item> A boolean value indicating whether the hash can grow automatically
or not.</Item>
<Mark><C>ishash</C></Mark>
<Item> Is <K>true</K> to indicate that this is a hash table record.</Item>
</List>

<Subsection>
<Heading> Memory requirements </Heading>

Due to the data structure defined above the hash table
will need one machine word (<M>4</M> bytes on 32bit machines and
<M>8</M> bytes on 64bit machines) per possible entry in the hash if all values
corresponding to objects in the hash are <K>true</K> and two machine
words otherwise. This means that the memory requirement for the hash
itself is proportional to the hash table length and not to the number
of objects actually stored in the hash!
<P/>
In addition one of course needs the memory to store
the objects themselves.
</Subsection>

<Subsection>
<Heading> Handling of collisions </Heading>

If two or more objects have the same hash value, the following is done:
If the hash value is coprime to the hash length, the hash value is 
taken as <Q>the
increment</Q>, otherwise <M>1</M> is taken. The code to find the proper
place for an object just repeatedly adds the increment to the current
position modulo the hash length. Due to the choice of the increment this
will eventually try all
places in the hash table. Every such increment step is counted as
a collision in the <C>collisions</C> component in the hash table.
This algorithm explains why it is sensible to
choose a prime number as the length of a hash table.
</Subsection>

<Subsection>
<Heading> Efficiency </Heading>

Hashing is efficient as long as there are not too many collisions.
It is not a problem if the number of collisions (counted in the
<C>collisions</C> component) is smaller than the number of accesses
(counted in the <C>accesses</C> component). 

<P/>
A high number of collisions can be caused by a bad hash function, 
because the hash table is too small (do not fill a hash table to
more than about 80&percent;), or because the objects to store
are just not well enough distributed. Hash tables will grow automatically
if too many collisions are detected.
</Subsection>

</Section>

<!-- ############################################################ -->

</Chapter>