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 8) 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 `HTCreate`

(4.3-1), `HTAdd`

(4.3-2), `HTValue`

(4.3-3), `HTDelete`

(4.3-5) and `HTUpdate`

(4.3-4). They are described in Section 4.3.

The legacy functions from older versions of this package to work with hash tables are `NewHT`

(4.4-1), `AddHT`

(4.4-2), and `ValueHT`

(4.4-3). They are described in Section 4.4. In the next section, we first describe the infrastructure for hash functions.

In the **orb** 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:

`‣ ChooseHashFunction` ( ob, len ) | ( operation ) |

Returns: a record

The first argument `ob` must be a sample object, that is, an object like those we want to store in the hash table later on. The argument `len` 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 `func`

is a **GAP** function taking two arguments, see below. The component `data`

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 `data`

component.

The hash function has to return values between \(1\) and the hash length `len` inclusively.

This setup is chosen such that the hash functions can be global objects that are not created during the execution of `ChooseHashFunction`

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 `ChooseHashFunction`

.

`‣ ChooseHashFunction` ( ob, len ) | ( method ) |

Returns: a record

This method is for compressed vectors over the field `GF(2)`

of two elements. Note that there is no hash function for non-compressed vectors over `GF(2)`

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.

`‣ ChooseHashFunction` ( ob, len ) | ( method ) |

Returns: 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.

`‣ ChooseHashFunction` ( ob, len ) | ( method ) |

Returns: a record

This method is for compressed matrices over the field `GF(2)`

of two elements. Note that there is no hash function for non-compressed matrices over `GF(2)`

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.

`‣ ChooseHashFunction` ( ob, len ) | ( method ) |

Returns: 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.

`‣ ChooseHashFunction` ( ob, len ) | ( method ) |

Returns: a record

This method is for integers.

`‣ ChooseHashFunction` ( ob, len ) | ( method ) |

Returns: a record

This method is for permutations.

`‣ ChooseHashFunction` ( ob, len ) | ( method ) |

Returns: a record

This method is for lists of integers.

`‣ ChooseHashFunction` ( ob, len ) | ( method ) |

Returns: a record

This method is for kernel Pc words.

`‣ ChooseHashFunction` ( ob, len ) | ( method ) |

Returns: a record

This method is for lists of integers.

`‣ ChooseHashFunction` ( ob, len ) | ( method ) |

Returns: a record

This method is for lists of matrices.

`‣ HTCreate` ( sample[, opt] ) | ( operation ) |

Returns: a new hash table object

A new hash table for objects like `sample` is created. The second argument `opt` 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:

`treehashsize`

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

`cmpfunc`

(see below).`treehashtab`

If this component is bound the type of the hash table is a TreeHashTab. This option is superfluous if

`treehashsize`

is used.`forflatplainlists`

If this component is set to

`true`

then the user guarantees that all the elements in the hash will be flat plain lists, that is, plain lists with no subobjects. For example lists of immediate integers will fulfill this requirement, but ranges don't. In this case, a particularly good and efficient hash function will automatically be taken and the components`hashfunc`

,`hfbig`

and`hfdbig`

are ignored. Note that this cannot be automatically detected because it depends not only on the sample point but also potentially on all the other points to be stored in the hash table.`hf`

and`hfd`

If these components are bound, they are used as the hash function. The value of

`hf`

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`hfd`

. 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`ChooseHashFunction`

(4.2-1) and the supplied sample object`sample`.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

`hfbig`

,`hfdbig`

and`cangrow`

.`cmpfunc`

This component can be bound to a three-way comparison function taking two arguments

`a`and`b`(which will be keys for the TreeHashTab) and returns \(-1\) if \(\textit{a}<\textit{b}\), \(0\) if \(\textit{a} = \textit{b}\) and \(1\) if \(\textit{a} > \textit{b}\). If this component is not bound the function`AVLCmp`

(8.2-2) is taken, which simply calls the generic operations`<`

and`=`

to do the job.`hashlen`

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

`hashlen`

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.

`hashtab`

If this component is bound the type of the hash table is a standard HashTab table. This component is superfluous if

`hashlen`

is bound.`eqf`

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

`=`

operation is taken.`hfbig`

and`hfdbig`

and`cangrow`

If you have used the components

`hf`

and`hfd`

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`hfbig`

, the corresponding data for the second argument to`hfdbig`

and bind`cangrow`

to`true`

. Then the hash table will automatically grow and take this new hash function modulo the new length of the hash table as hash function.

`‣ HTAdd` ( ht, key, value ) | ( operation ) |

Returns: a hash value

Stores the object `key` into the hash table `ht` and stores the value `val` together with `ob`. The result is `fail`

if an error occurred, which can be that an object equal to `key` 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 `val` is `true`

for all objects in the hash, no extra memory is used for the values. All other values are stored in the hash. The value `fail`

cannot be stored as it indicates that the object is not found in the hash.

See Section 4.5 for details on the data structures and especially about memory requirements.

`‣ HTValue` ( ht, key ) | ( operation ) |

Returns: `fail`

or `true`

or a value

Looks up the object `key` in the hash table `ht`. If the object is not found, `fail`

is returned. Otherwise, the value stored with the object is returned. Note that if this value was `true`

no extra memory is used for this.

`‣ HTUpdate` ( ht, key, value ) | ( operation ) |

Returns: `fail`

or `true`

or a value

The object `key` must already be stored in the hash table `ht`, otherwise this operation returns `fail`

. The value stored with `key` in the hash is replaced by `value` and the previously stored value is returned.

`‣ HTDelete` ( ht, key ) | ( operation ) |

Returns: `fail`

or `true`

or a value

The object `key` along with its stored value is removed from the hash table `ht`. Note that this currently only works for TreeHashTabs and not for HashTab tables. It is an error if `key` is not found in the hash table and `fail`

is returned in this case.

`‣ HTGrow` ( ht, ob ) | ( function ) |

Returns: nothing

This is a more or less internal operation. It is called when the space in a hash table becomes scarce. The first argument `ht` 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 `ChooseHashFunction`

method. However, one can bind the two components `hfbig`

and `hfdbig`

in the options record of `HTCreate`

(4.3-1) to a function and a record respectively and bind `cangrow`

to `true`

. In that case, upon growing the hash, a new hash function is created by taking the function `hfbig`

together with `hfdbig`

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.

Note that the functions described in this section are obsolete since version 3.0 of **orb** and are only kept for backward compatibility. Please use the functions in Section 4.3 in new code.

The following functions are needed to use hash tables. For details about the data structures see Section 4.5.

`‣ NewHT` ( sample, len ) | ( function ) |

Returns: a new hash table object

A new hash table for objects like `sample` of length `len` 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 `AddHT`

(4.4-2) and `ValueHT`

(4.4-3). It will start with length `len` but will grow as necessary.

`‣ AddHT` ( ht, ob, val ) | ( function ) |

Returns: an integer or fail

Stores the object `ob` into the hash table `ht` and stores the value `val` together with `ob`. The result is `fail`

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 `val` is `true`

for all objects in the hash, no extra memory is used for the values. All other values are stored in the hash. The value `fail`

cannot be stored as it indicates that the object is not found in the hash.

See Section 4.5 for details on the data structures and especially about memory requirements.

`‣ ValueHT` ( ht, ob ) | ( function ) |

Returns: the stored value, `true`

, or `fail`

Looks up the object `ob` in the hash table `ht`. If the object is not found, `fail`

is returned. Otherwise, the value stored with the object is returned. Note that if this value was `true`

no extra memory is used for this.

The following function is only documented for the sake of completeness and for emergency situations, where `NewHT`

(4.4-1) tries to be too intelligent.

`‣ InitHT` ( len, hfun, eqfun ) | ( function ) |

Returns: a new hash table object

This is usually only an internal function. It is called from `NewHT`

(4.4-1). The argument `len` is the length of the hash table, `hfun` is the hash function record as returned by `ChooseHashFunction`

(4.2-1) and `eqfun` is a comparison function taking two arguments and returning `true`

or `false`

.

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 `ChooseHashFunction`

(4.2-1). If you do not want this, change the component `cangrow`

to `false`

after creating the hash table.

`‣ GrowHT` ( ht, ob ) | ( function ) |

Returns: nothing

This is a more or less internal function. It is called when the space in a hash table becomes scarce. The first argument `ht` must be a hash table object, the second a sample point. The function increases the hash size by a factor of 2 for hash tables and 20 for tree hash tables. This makes it necessary to choose a new hash function. Usually this is done with the usual `ChooseHashFunction`

method. However, one can assign the two components `hfbig`

and `hfdbig`

to a function and a record respectively. In that case, upon growing the hash, a new hash function is created by taking the function `hfbig`

together with `hfdbig`

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.

A legacy hash table object is just a record with the following components:

`els`

A

**GAP**list storing the elements. Its length can be as long as the component`len`

indicates but will only grow as necessary when elements are stored in the hash.`vals`

A

**GAP**list storing the corresponding values. If a value is`true`

nothing is stored here to save memory.`len`

Length of the hash table.

`nr`

Number of elements stored in the hash table.

`hf`

The hash function (value of the

`func`

component in the record returned by`ChooseHashFunction`

(4.2-1)).`hfd`

The data for the second argument of the hash function (value of the

`data`

component in the record returned by`ChooseHashFunction`

(4.2-1)).`eqf`

A comparison function taking two arguments and returning

`true`

for equality or`false`

otherwise.`collisions`

Number of collisions (see below).

`accesses`

Number of lookup or store accesses to the hash.

`cangrow`

A boolean value indicating whether the hash can grow automatically or not.

`ishash`

Is

`true`

to indicate that this is a hash table record.`hfbig`

and`hfdbig`

Used for hash tables which need to be able to grow but where the user supplied the hash function. See Section

`HTCreate`

(4.3-1) for more details.

A new style HashTab objects are component objects with the same components except that there is no component `ishash`

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 `eqf`

is replaced by the three-way comparison function `cmpfunc`

. Since TreeHashTabs do not grow, the components `hfbig`

, `hfdbig`

and `cangrow`

are never bound. Each slot in the `els`

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).

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 `true`

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.

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 `collisions`

component in the hash table. This algorithm explains why it is sensible to choose a prime number as the length of a hash table.

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 `collisions`

component) is smaller than the number of accesses (counted in the `accesses`

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.

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