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4 Hashing techniques
4.1 The idea of hashing
4.2 Hash functions
4.2-1 ChooseHashFunction
4.2-2 ChooseHashFunction
4.2-3 ChooseHashFunction
4.2-4 ChooseHashFunction
4.2-5 ChooseHashFunction
4.2-6 ChooseHashFunction
4.2-7 ChooseHashFunction
4.2-8 ChooseHashFunction
4.2-9 ChooseHashFunction
4.2-10 ChooseHashFunction
4.2-11 ChooseHashFunction
4.3 Using hash tables
4.3-1 NewHT
4.3-2 AddHT
4.3-3 ValueHT
4.3-4 InitHT
4.3-5 GrowHT
4.4 The data structures for hash tables
4.4-1 Memory requirements
4.4-2 Handling of collisions
4.4-3 Efficiency

### 4 Hashing techniques

#### 4.1 The idea of hashing

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 so-called "collision", that is, if more than one object with the same hash value has to be stored.

The basic functions to work with hash tables are `NewHT` (4.3-1), `AddHT` (4.3-2), and `ValueHT` (4.3-3). They are described in Section 4.3. In the next section, we first describe the infrastructure for hash functions.

#### 4.2 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:

##### 4.2-1 ChooseHashFunction

`> 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`.

##### 4.2-2 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.

##### 4.2-3 ChooseHashFunction

`> 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.

##### 4.2-4 ChooseHashFunction

`> 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.

##### 4.2-5 ChooseHashFunction

`> 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.

##### 4.2-6 ChooseHashFunction

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

Returns: a record

This method is for integers.

##### 4.2-7 ChooseHashFunction

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

Returns: a record

This method is for permutations.

##### 4.2-8 ChooseHashFunction

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

Returns: a record

This method is for lists of integers.

##### 4.2-9 ChooseHashFunction

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

Returns: a record

This method is for kernel Pc words.

##### 4.2-10 ChooseHashFunction

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

Returns: a record

This method is for lists of integers.

##### 4.2-11 ChooseHashFunction

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

Returns: a record

This method is for lists of matrices.

#### 4.3 Using hash tables

The following functions are needed to use hash tables. For details about the data structures see Section 4.4.

##### 4.3-1 NewHT

`> 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` (4.3-2) and `ValueHT` (4.3-3). It will start with length len but will grow as necessary.

##### 4.3-2 AddHT

`> 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 4.4 for details on the data structures and especially about memory requirements.

##### 4.3-3 ValueHT

`> 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` (4.3-1) tries to be too intelligent.

##### 4.3-4 InitHT

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

Returns: a new hash table object

This is usually only an internal function. It is called from `NewHT` (4.3-1). 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.

##### 4.3-5 GrowHT

`> 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. 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.

#### 4.4 The data structures for hash tables

A 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.

##### 4.4-1 Memory requirements

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.

##### 4.4-2 Handling of collisions

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.

##### 4.4-3 Efficiency

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.

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