Associative array

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An associative array, also sometimes called a map, mapping, hash, dictionary, finite map, and lookup table, is an abstract data type composed of a collection of keys and a collection of values, where each key is associated with one value. The operation of finding the value associated with a key is called a lookup or indexing. The relationship between a key and its value is sometimes called a mapping or binding.

For example, if the value associated with the key "a" is 5, we say that our array maps "a" to 5.

Associative arrays are very closely related to the mathematical concept of a function with a finite domain. As a consequence, a common and important use of associative arrays is in memoization.

From the perspective of a programmer using an associative array, it can be viewed as a generalization of an array: While a regular array maps integers to arbitrarily typed objects (integers, strings, pointers, and, in an OO sense, objects), an associative array maps arbitrarily typed objects to arbitrarily typed objects. (Implementations of the two data structures, though, may be considerably different.)

The operations that are usually defined for an associative array are:

• Add: Bind a new key to a new value
• Reassign: Bind an old key to a new value
• Remove: Unbind a key from a value and remove the key from the key set
• Lookup: Find the value (if any) that is bound to a key

[edit] Examples

One can think of a telephone book as an example of an associative array, where names are the keys and phone numbers are the values. Using the usual array-like notation, we might write

telephone['ada']='01-1234-56'
telephone['bert']='02-4321-56'

These entries can be thought of as two records in a database table:

Another example would be a dictionary where words are the keys and definitions are the values.

dictionary['toad']='four legged amphibian'
dictionary['cow']='female four-legged domesticated mammalian ruminant'

Since a database equivalent is that of a table containing precisely two fields, key and value, we can use an associative array to store any information which can be held in this form.

capital['england']='london'
capital['france']='paris'
 
bossof['subeditor']='editor'
bossof['reporter']='subeditor'
 
salary['editor']=50000
salary['reporter']=30000

[edit] Data Structures for Associative Arrays

Associative arrays are usually used when lookup is the most frequent operation. For this reason, implementations are usually designed to allow speedy lookup, at the expense of slower insertion and a larger storage footprint than other data structures (such as association lists).

[edit] Efficient Representations

There are two main efficient data structures used to represent associative arrays, the hash table and the self-balancing binary search tree. Skip lists are also an alternative, though relatively new and not as widely used. Relative advantages and disadvantages include:

• Hash tables have faster average lookup and insertion time (O(1)), while some kinds of binary search tree have faster worst-case lookup and insertion time (O(log n) instead of O(n)). Hash tables have seen extensive use in realtime systems, but trees can be useful in high-security realtime systems where untrusted users may deliberately supply information that triggers worst-case performance in a hash table, although careful design can remove that issue. Hash tables shine in very large arrays, where O(1) performance is important. Skip lists have worst-case operation time of O(n), but average-case of O(log n), with much less insertion and deletion overhead than balanced binary trees.
• Hash tables can have more compact storage for small value types, especially when the values are bits.
• There are simple persistent versions of balanced binary trees, which are especially prominent in functional languages.
• Building a hash table requires a reasonable hash function for the key type, which can be difficult to write well, while balanced binary trees and skip lists only require a total ordering on the keys. On the other hand, with hash tables the data may be cyclically or partially ordered without any problems.
• Balanced binary trees and skip lists preserve ordering — allowing one to efficiently iterate over the keys in order or to efficiently locate an association whose key is nearest to a given value. Hash tables do not preserve ordering and therefore cannot perform these operations as efficiently.
• Balanced binary trees can be easily adapted to efficiently assign a single value to a large ordered range of keys, or to count the number of keys in an ordered range.

[edit] Association Lists

A simple but generally inefficient type of associative array is an association list, often called an alist for short, which simply stores a linked list of key-value pairs. Each lookup does a linear search through the list looking for a key match.

Strong advantages of association lists include:

• No knowledge is needed about the keys, such as an order or a hash function.
• For small associative arrays, common in some applications, association lists can take less time and space than other data structures.
• Insertions are done in constant time by cons'ing the new association to the head of the list.

[edit] Specialized Representations

If the keys have a specific type, one can often use specialized data structures to gain performance. For example, integer-keyed maps can be implemented using Patricia trees or Judy arrays, and are useful space-saving replacements for sparse arrays. Because this type of data structure can perform longest-prefix matching, they're particularly useful in applications where a single value is assigned to most of a large range of keys with a common prefix except for a few exceptions, such as in routing tables.

String-keyed maps can avoid extra comparisons during lookups by using tries.

[edit] Multimap

A variation of the map (associative array) is the multimap, which is the same as map data structures, but allows a key to be mapped to more than one value. Formally, a multimap can be thought of as a regular associative array that maps unique keys to nonempty multisets of values, although actual implementation may vary. C++'s Standard Template Library provides the "multimap" container for the sorted multimap, SGI's STL provides the "hash_multimap" container, which implements a multimap using a hash table, and some varieties of LPC have built-in multimap support.

[edit] Language Support

Associative arrays can be implemented in any programming language as a package and many language systems provide them as part of their standard library. In some languages, they are not only built into the standard system, but have special syntax, often array-like subscripting.

Built-in syntactic support for associative arrays was introduced by Snobol4, under the name "table". MUMPS made multi-dimensional associative arrays, optionally persistent, its key data structure. SETL supported them as one possible implementation of sets and maps. Most modern scripting languages, starting with awk and including Perl, PHP, tcl, Javascript, Python, and Ruby, support associative arrays as a primary container type.

In many more languages, they are available as library functions without special syntax.

Associative arrays have a variety of names. In Smalltalk, Objective-C and Python they are called dictionaries; in Perl and Ruby they are called hashes; in C++ and Java they are called maps and in Common Lisp and Windows PowerShell they are called hashtables (since both typically use this implementation). In PHP all arrays can be associative, except that the keys are limited to integers and strings and can only be a single level of subscripts.

In the scripting language Lua, associative arrays, called tables, are used as the primitive building block for all data structures, even arrays. Likewise, in JavaScript, all objects are associative arrays. In MUMPS, the associative arrays are typically stored as B-trees.

[edit] External Links

• TreeDictionary<K,V> Implementation in C#
• NIST's Dictionary of Algorithms and Data Structures: Associative Array
• NIST's Dictionary of Algorithms and Data Structures: Association List
• SGI STL page on Associative Containers
• SGI STL page on std::map
• SGI STL page on std::hash_map