  
  [1X4 Hashing techniques[0X
  
  
  [1X4.1 The idea of hashing[0X
  
  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 log(n) comparisons to find a given
  element or to decide that it is not yet stored.
  
  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 a so-called "collision", that is, if
  more than one object with the same hash value has to be stored. This package
  provides  two  ways  to  deal  with collisions, one is implemented in the so
  called  "HashTabs" and another in the "TreeHashTabs". The former simply uses
  other  parts  of  the array to store the data involved in the collisions and
  the  latter  uses an AVL tree (see Chapter [14X8[0m) to store all data objects with
  the  same  hash value. Both are used basically in the same way but sometimes
  behave a bit differently.
  
  The  basic  functions  to  work with hash tables are [2XHTCreate[0m ([14X4.3-1[0m), [2XHTAdd[0m
  ([14X4.3-2[0m),  [2XHTValue[0m  ([14X4.3-3[0m),  [2XHTDelete[0m ([14X4.3-5[0m) and [2XHTUpdate[0m ([14X4.3-4[0m). They are
  described in Section [14X4.3[0m.
  
  The  legacy  functions from older versions of this package to work with hash
  tables  are  [2XNewHT[0m  ([14X4.4-1[0m),  [2XAddHT[0m  ([14X4.4-2[0m),  and [2XValueHT[0m ([14X4.4-3[0m). They are
  described  in  Section  [14X4.4[0m.  In  the  next  section,  we first describe the
  infrastructure for hash functions.
  
  
  [1X4.2 Hash functions[0X
  
  In  the  [5Xorb[0m  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:
  
  [1X4.2-1 ChooseHashFunction[0m
  
  [2X> ChooseHashFunction( [0X[3Xob, len[0X[2X ) ___________________________________[0Xoperation
  [6XReturns:[0X  a record
  
  The first argument [3Xob[0m must be a sample object, that is, an object like those
  we  want to store in the hash table later on. The argument [3Xlen[0m 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 [10Xfunc[0m is a [5XGAP[0m
  function  taking  two  arguments,  see below. The component [10Xdata[0m is some [5XGAP[0m
  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 [10Xdata[0m component.
  
  The  hash  function  has  to return values between 1 and the hash length [3Xlen[0m
  inclusively.
  
  This setup is chosen such that the hash functions can be global objects that
  are  not  created  during  the execution of [2XChooseHashFunction[0m but still can
  change their behaviour depending on the data.
  
  In the following we just document, for which types of objects there are hash
  functions that can be found using [2XChooseHashFunction[0m.
  
  [1X4.2-2 ChooseHashFunction[0m
  
  [2X> ChooseHashFunction( [0X[3Xob, len[0X[2X ) ______________________________________[0Xmethod
  [6XReturns:[0X  a record
  
  This  method is for compressed vectors over the field [10XGF(2)[0m of two elements.
  Note  that  there  is no hash function for non-compressed vectors over [10XGF(2)[0m
  because those objects cannot efficiently be recognised from their type.
  
  Note  that  you can only use the resulting hash functions for vectors of the
  same length.
  
  [1X4.2-3 ChooseHashFunction[0m
  
  [2X> ChooseHashFunction( [0X[3Xob, len[0X[2X ) ______________________________________[0Xmethod
  [6XReturns:[0X  a record
  
  This  method  is  for  compressed vectors over a finite field with up to 256
  elements.  Note  that  there  is  no  hash  function for non-compressed such
  vectors  because  those  objects cannot efficiently be recognised from their
  type.
  
  Note  that  you can only use the resulting hash functions for vectors of the
  same length.
  
  [1X4.2-4 ChooseHashFunction[0m
  
  [2X> ChooseHashFunction( [0X[3Xob, len[0X[2X ) ______________________________________[0Xmethod
  [6XReturns:[0X  a record
  
  This method is for compressed matrices over the field [10XGF(2)[0m of two elements.
  Note  that  there is no hash function for non-compressed matrices over [10XGF(2)[0m
  because those objects cannot efficiently be recognised from their type.
  
  Note  that you can only use the resulting hash functions for matrices of the
  same size.
  
  [1X4.2-5 ChooseHashFunction[0m
  
  [2X> ChooseHashFunction( [0X[3Xob, len[0X[2X ) ______________________________________[0Xmethod
  [6XReturns:[0X  a record
  
  This  method  is  for compressed matrices over a finite field with up to 256
  elements.  Note  that  there  is  no  hash  function for non-compressed such
  vectors  because  those  objects cannot efficiently be recognised from their
  type.
  
  Note  that you can only use the resulting hash functions for matrices of the
  same size.
  
  [1X4.2-6 ChooseHashFunction[0m
  
  [2X> ChooseHashFunction( [0X[3Xob, len[0X[2X ) ______________________________________[0Xmethod
  [6XReturns:[0X  a record
  
  This method is for integers.
  
  [1X4.2-7 ChooseHashFunction[0m
  
  [2X> ChooseHashFunction( [0X[3Xob, len[0X[2X ) ______________________________________[0Xmethod
  [6XReturns:[0X  a record
  
  This method is for permutations.
  
  [1X4.2-8 ChooseHashFunction[0m
  
  [2X> ChooseHashFunction( [0X[3Xob, len[0X[2X ) ______________________________________[0Xmethod
  [6XReturns:[0X  a record
  
  This method is for lists of integers.
  
  [1X4.2-9 ChooseHashFunction[0m
  
  [2X> ChooseHashFunction( [0X[3Xob, len[0X[2X ) ______________________________________[0Xmethod
  [6XReturns:[0X  a record
  
  This method is for kernel Pc words.
  
  [1X4.2-10 ChooseHashFunction[0m
  
  [2X> ChooseHashFunction( [0X[3Xob, len[0X[2X ) ______________________________________[0Xmethod
  [6XReturns:[0X  a record
  
  This method is for lists of integers.
  
  [1X4.2-11 ChooseHashFunction[0m
  
  [2X> ChooseHashFunction( [0X[3Xob, len[0X[2X ) ______________________________________[0Xmethod
  [6XReturns:[0X  a record
  
  This method is for lists of matrices.
  
  
  [1X4.3 Using hash tables[0X
  
  [1X4.3-1 HTCreate[0m
  
  [2X> HTCreate( [0X[3Xsample[, opt][0X[2X ) _______________________________________[0Xoperation
  [6XReturns:[0X  a new hash table object
  
  A new hash table for objects like [3Xsample[0m is created. The second argument [3Xopt[0m
  is an optional options record, which will supplied in most cases, if only to
  specify  the  length  and  type  of the hash table to be used. The following
  components in this record can be bound:
  
  [8X[10Xtreehashsize[0m[8X[0m
        If  this  component  is  bound  the  type  of  the  hash  table  is  a
        TreeHashTab. The value must be a positive integer and will be the size
        of  the  hash table. Note that for this type of hash table the keys to
        be  stored  in  the  hash  must  be  comparable  using  <. A three-way
        comparison  function  can be supplied using the component [10Xcmpfunc[0m (see
        below).
  
  [8X[10Xtreehashtab[0m[8X[0m
        If  this  component  is  bound  the  type  of  the  hash  table  is  a
        TreeHashTab. This option is superfluous if [10Xtreehashsize[0m is used.
  
  [8X[10Xhf[0m[8X and [10Xhfd[0m[8X[0m
        If these components are bound, they are used as the hash function. The
        value  of  [10Xhf[0m must be a function taking two arguments, the first being
        the  object  for  which  the  hash  function shall be computed and the
        second  being  the value of [10Xhfd[0m. The returned value must be an integer
        in  the  range  from  1  to the length of the hash. If either of these
        components  is not bound, an automatic choice for the hash function is
        done  using  [2XChooseHashFunction[0m ([14X4.2-1[0m) and the supplied sample object
        [3Xsample[0m.
  
        Note  that if you specify these two components and are using a HashTab
        table  then this table cannot grow unless you also bind the components
        [10Xhfbig[0m, [10Xhfdbig[0m and [10Xcangrow[0m.
  
  [8X[10Xcmpfunc[0m[8X[0m
        This  component can be bound to a three-way comparison function taking
        two  arguments  [3Xa[0m  and  [3Xb[0m (which will be keys for the TreeHashTab) and
        returns -1 if [3Xa[0m<[3Xb[0m, 0 if [3Xa[0m = [3Xb[0m and 1 if [3Xa[0m > [3Xb[0m. If this component is not
        bound  the  function  [2XAVLCmp[0m  ([14X8.2-2[0m) is taken, which simply calls the
        generic operations [10X<[0m and [10X=[0m to do the job.
  
  [8X[10Xhashlen[0m[8X[0m
        If  this  component  is bound the type of the hash table is a standard
        HashTab   table.  That  is,  collisions  are  dealt  with  by  storing
        additional  entries  in  other  slots.  This is the traditional way to
        implement a hash table. Note that currently deleting entries in such a
        hash  table is not implemented, since it could only be done by leaving
        a "deleted" mark which could pollute that hash table. Use TreeHashTabs
        instead  if  you  need  deletion. The value bound to [10Xhashlen[0m must be a
        positive integer and will be the initial length of the hash table.
  
        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.
  
  [8X[10Xhashtab[0m[8X[0m
        If  this  component  is bound the type of the hash table is a standard
        HashTab table. This component is superfluous if [10Xhashlen[0m is bound.
  
  [8X[10Xeqf[0m[8X[0m
        For  HashTab  tables  the  function taking two arguments bound to this
        component is used to compare keys in the hash table. If this component
        is not bound the usual [10X=[0m operation is taken.
  
  [8X[10Xhfbig[0m[8X and [10Xhfdbig[0m[8X and [10Xcangrow[0m[8X[0m
        If you have used the components [10Xhf[0m and [10Xhfd[0m then your hash table cannot
        automatically grow when it fills up. This is because the length of the
        table  is  built  into  the hash function. If you still want your hash
        table  to  be  able  to  grow automatically, then bind a hash function
        returning  arbitrary integers to [10Xhfbig[0m, the corresponding data for the
        second  argument  to  [10Xhfdbig[0m  and  bind [10Xcangrow[0m to [9Xtrue[0m. Then the hash
        table  will  automatically grow and take this new hash function modulo
        the new length of the hash table as hash function.
  
  [1X4.3-2 HTAdd[0m
  
  [2X> HTAdd( [0X[3Xht, key, value[0X[2X ) _________________________________________[0Xoperation
  [6XReturns:[0X  a hash value
  
  Stores  the  object  [3Xkey[0m  into  the  hash  table [3Xht[0m and stores the value [3Xval[0m
  together with [3Xob[0m. The result is [9Xfail[0m if an error occurred, which can be that
  an  object equal to [3Xkey[0m is already stored in the hash table or that the hash
  table  is  already full. The latter can only happen, if the hash table is no
  TreeHashTab and cannot grow automatically.
  
  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!
  
  If  the  value  [3Xval[0m  is [9Xtrue[0m for all objects in the hash, no extra memory is
  used for the values. All other values are stored in the hash. The value [9Xfail[0m
  cannot be stored as it indicates that the object is not found in the hash.
  
  See  Section  [14X4.5[0m  for  details  on the data structures and especially about
  memory requirements.
  
  [1X4.3-3 HTValue[0m
  
  [2X> HTValue( [0X[3Xht, key[0X[2X ) ______________________________________________[0Xoperation
  [6XReturns:[0X  [9Xfail[0m or [9Xtrue[0m or a value
  
  Looks  up  the  object [3Xkey[0m in the hash table [3Xht[0m. If the object is not found,
  [9Xfail[0m  is  returned. Otherwise, the value stored with the object is returned.
  Note that if this value was [9Xtrue[0m no extra memory is used for this.
  
  [1X4.3-4 HTUpdate[0m
  
  [2X> HTUpdate( [0X[3Xht, key, value[0X[2X ) ______________________________________[0Xoperation
  [6XReturns:[0X  [9Xfail[0m or [9Xtrue[0m or a value
  
  The  object  [3Xkey[0m must already be stored in the hash table [3Xht[0m, otherwise this
  operation returns [9Xfail[0m. The value stored with [3Xkey[0m in the hash is replaced by
  [3Xvalue[0m and the previously stored value is returned.
  
  [1X4.3-5 HTDelete[0m
  
  [2X> HTDelete( [0X[3Xht, key[0X[2X ) _____________________________________________[0Xoperation
  [6XReturns:[0X  [9Xfail[0m or [9Xtrue[0m or a value
  
  The  object  [3Xkey[0m  along with its stored value is removed from the hash table
  [3Xht[0m. Note that this currently only works for TreeHashTabs and not for HashTab
  tables.  It  is  an  error if [3Xkey[0m is not found in the hash table and [9Xfail[0m is
  returned in this case.
  
  [1X4.3-6 HTGrow[0m
  
  [2X> HTGrow( [0X[3Xht, ob[0X[2X ) _________________________________________________[0Xfunction
  [6XReturns:[0X  nothing
  
  This  is a more or less internal operation. It is called when the space in a
  hash  table  becomes  scarce.  The  first  argument  [3Xht[0m 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 [10XChooseHashFunction[0m method. However, one can bind
  the  two  components  [10Xhfbig[0m  and  [10Xhfdbig[0m  in  the options record of [2XHTCreate[0m
  ([14X4.3-1[0m) to a function and a record respectively and bind [10Xcangrow[0m to [9Xtrue[0m. In
  that  case,  upon growing the hash, a new hash function is created by taking
  the function [10Xhfbig[0m together with [10Xhfdbig[0m 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.
  
  
  [1X4.4 Using hash tables (legacy code)[0X
  
  Note that the functions described in this section are obsolete since version
  3.0  of  [5Xorb[0m  and  are  only kept for backward compatibility. Please use the
  functions in Section [14X4.3[0m in new code.
  
  The following functions are needed to use hash tables. For details about the
  data structures see Section [14X4.5[0m.
  
  [1X4.4-1 NewHT[0m
  
  [2X> NewHT( [0X[3Xsample, len[0X[2X ) _____________________________________________[0Xfunction
  [6XReturns:[0X  a new hash table object
  
  A new hash table for objects like [3Xsample[0m of length [3Xlen[0m 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  [2XAddHT[0m  ([14X4.4-2[0m) and [2XValueHT[0m ([14X4.4-3[0m). It will start with
  length [3Xlen[0m but will grow as necessary.
  
  [1X4.4-2 AddHT[0m
  
  [2X> AddHT( [0X[3Xht, ob, val[0X[2X ) _____________________________________________[0Xfunction
  [6XReturns:[0X  an integer or fail
  
  Stores  the  object  [3Xob[0m  into  the  hash  table  [3Xht[0m and stores the value [3Xval[0m
  together with [3Xob[0m. The result is [9Xfail[0m 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.
  
  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!
  
  If  the  value  [3Xval[0m  is [9Xtrue[0m for all objects in the hash, no extra memory is
  used for the values. All other values are stored in the hash. The value [9Xfail[0m
  cannot be stored as it indicates that the object is not found in the hash.
  
  See  Section  [14X4.5[0m  for  details  on the data structures and especially about
  memory requirements.
  
  [1X4.4-3 ValueHT[0m
  
  [2X> ValueHT( [0X[3Xht, ob[0X[2X ) ________________________________________________[0Xfunction
  [6XReturns:[0X  the stored value, [9Xtrue[0m, or [9Xfail[0m
  
  Looks  up  the  object  [3Xob[0m in the hash table [3Xht[0m. If the object is not found,
  [9Xfail[0m  is  returned. Otherwise, the value stored with the object is returned.
  Note that if this value was [9Xtrue[0m no extra memory is used for this.
  
  The  following  function is only documented for the sake of completeness and
  for emergency situations, where [2XNewHT[0m ([14X4.4-1[0m) tries to be too intelligent.
  
  [1X4.4-4 InitHT[0m
  
  [2X> InitHT( [0X[3Xlen, hfun, eqfun[0X[2X ) _______________________________________[0Xfunction
  [6XReturns:[0X  a new hash table object
  
  This  is usually only an internal function. It is called from [2XNewHT[0m ([14X4.4-1[0m).
  The  argument [3Xlen[0m is the length of the hash table, [3Xhfun[0m is the hash function
  record  as  returned by [2XChooseHashFunction[0m ([14X4.2-1[0m) and [3Xeqfun[0m is a comparison
  function taking two arguments and returning [9Xtrue[0m or [9Xfalse[0m.
  
  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
  [2XChooseHashFunction[0m  ([14X4.2-1[0m).  If  you do not want this, change the component
  [10Xcangrow[0m to [9Xfalse[0m after creating the hash table.
  
  [1X4.4-5 GrowHT[0m
  
  [2X> GrowHT( [0X[3Xht, ob[0X[2X ) _________________________________________________[0Xfunction
  [6XReturns:[0X  nothing
  
  This  is  a more or less internal function. It is called when the space in a
  hash  table  becomes  scarce.  The  first  argument  [3Xht[0m 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 [10XChooseHashFunction[0m method. However, one can
  assign  the  two  components  [10Xhfbig[0m  and  [10Xhfdbig[0m  to a function and a record
  respectively.  In  that  case, upon growing the hash, a new hash function is
  created  by  taking  the  function [10Xhfbig[0m together with [10Xhfdbig[0m 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.
  
  
  [1X4.5 The data structures for hash tables[0X
  
  A legacy hash table object is just a record with the following components:
  
  [8X[10Xels[0m[8X[0m
        A  [5XGAP[0m  list  storing  the  elements. Its length can be as long as the
        component  [10Xlen[0m indicates but will only grow as necessary when elements
        are stored in the hash.
  
  [8X[10Xvals[0m[8X[0m
        A  [5XGAP[0m  list  storing  the  corresponding  values.  If a value is [9Xtrue[0m
        nothing is stored here to save memory.
  
  [8X[10Xlen[0m[8X[0m
        Length of the hash table.
  
  [8X[10Xnr[0m[8X[0m
        Number of elements stored in the hash table.
  
  [8X[10Xhf[0m[8X[0m
        The  hash function (value of the [10Xfunc[0m component in the record returned
        by [2XChooseHashFunction[0m ([14X4.2-1[0m)).
  
  [8X[10Xhfd[0m[8X[0m
        The  data  for  the second argument of the hash function (value of the
        [10Xdata[0m component in the record returned by [2XChooseHashFunction[0m ([14X4.2-1[0m)).
  
  [8X[10Xeqf[0m[8X[0m
        A  comparison  function  taking  two  arguments and returning [9Xtrue[0m for
        equality or [9Xfalse[0m otherwise.
  
  [8X[10Xcollisions[0m[8X[0m
        Number of collisions (see below).
  
  [8X[10Xaccesses[0m[8X[0m
        Number of lookup or store accesses to the hash.
  
  [8X[10Xcangrow[0m[8X[0m
        A  boolean value indicating whether the hash can grow automatically or
        not.
  
  [8X[10Xishash[0m[8X[0m
        Is [9Xtrue[0m to indicate that this is a hash table record.
  
  [8X[10Xhfbig[0m[8X and [10Xhfdbig[0m[8X[0m
        Used  for hash tables which need to be able to grow but where the user
        supplied  the  hash  function.  See  Section [2XHTCreate[0m ([14X4.3-1[0m) for more
        details.
  
  A  new  style HashTab objects are component objects with the same components
  except  that there is no component [10Xishash[0m since these objects are recognised
  by their type.
  
  A  TreeHashTab is very similar. It is a positional object with basically the
  same  components,  except  that  [10Xeqf[0m is replaced by the three-way comparison
  function  [10Xcmpfunc[0m.  Since  TreeHashTabs  do  not grow, the components [10Xhfbig[0m,
  [10Xhfdbig[0m and [10Xcangrow[0m are never bound. Each slot in the [10Xels[0m component is either
  unbound  (empty), or bound to the only key stored in the hash which has this
  hash  value  or, if there is more than one key for that hash value, the slot
  is bound to an AVL tree containing all such keys (and values).
  
  
  [1X4.5-1 Memory requirements[0X
  
  Due to the data structure defined above the hash table will need one machine
  word  (4 bytes on 32bit machines and 8 bytes on 64bit machines) per possible
  entry  in  the  hash  if all values corresponding to objects in the hash are
  [9Xtrue[0m 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!
  
  In addition one of course needs the memory to store the objects themselves.
  
  For  TreeHashTabs there are additional memory requirements. As soon as there
  are  more than one key hashing to the same value, the memory for an AVL tree
  object  is  needed  in  addition. An AVL tree objects needs about 10 machine
  words  for  the  tree object and then another 4 machine words for each entry
  stored  in the tree. Note that for many collisions this can be significantly
  more than for HashTab tables. However, the advantage of TreeHashTabs is that
  even  for a bad hash function the performance is never worse than log(n) for
  each  operation where n is the number of keys in the hash with the same hash
  value.
  
  
  [1X4.5-2 Handling of collisions[0X
  
  This section is only relevant for HashTab objects.
  
  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
  "the increment", otherwise 1 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  [10Xcollisions[0m  component  in the hash table. This algorithm
  explains why it is sensible to choose a prime number as the length of a hash
  table.
  
  
  [1X4.5-3 Efficiency[0X
  
  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 [10Xcollisions[0m component)
  is smaller than the number of accesses (counted in the [10Xaccesses[0m component).
  
  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%),  or because the objects to store are just not well enough distributed.
  Hash  tables  will grow automatically if too many collisions are detected or
  if they are filled to 80%.
  
  The  advantage  of  TreeHashTabs  is  that  even for a bad hash function the
  performance  is  never  worse  than log(n) for each operation where n is the
  number  of  keys  in the hash with the same hash value. However, they need a
  bit more memory.
  
