Editing Associative array

{{Short description|Data structure that associates keys with values}}
{{Redirect-distinguish|Dictionary (data structure)|data dictionary}}
{{Redirect|Associative container|their C++ implementation|associative containers (C++)}}
{{Redirect|Map (computer science)|the higher-order function|Map (higher-order function)}}
{{Redirect|Associative table|the relation used in database systems to resolve many-to-many relationship|Associative entity}}
In [[computer science]], an '''associative array''', '''key-value store''', '''map''', '''symbol table''', or '''dictionary''' is an [[abstract data type]] that stores a [[collection (abstract data type)|collection]] of [[attribute–value pair|(key, value) pairs]], such that each possible key appears at most once in the collection. In mathematical terms, an associative array is a [[Function (mathematics)|function]] with ''finite'' [[Domain of a function|domain]].<ref>{{cite book |last1=Collins |first1=Graham |last2=Syme |first2=Donald |chapter=A theory of finite maps |title=Higher Order Logic Theorem Proving and Its Applications |series=Lecture Notes in Computer Science |date=1995 |volume=971 |pages=122–137 |doi=10.1007/3-540-60275-5_61|isbn=978-3-540-60275-0 }}</ref> It supports 'lookup', 'remove', and 'insert' operations. 

The '''dictionary problem''' is the classic problem of designing efficient [[data structure]]s that implement associative arrays.<ref>{{cite book|chapter=Optimal Bounds on the Dictionary Problem|last=Andersson|first=Arne|title= Proc. Symposium on Optimal Algorithms|series=Lecture Notes in Computer Science|date=1989|volume=401|pages=106–114|publisher=Springer Verlag|doi=10.1007/3-540-51859-2_10|isbn=978-3-540-51859-4}}</ref>
The two major solutions to the dictionary problem are [[hash table]]s and [[search tree]]s.<ref name="gt"/><ref name="ms"/><ref name="clrs">{{citation | last1 = Cormen | first1 = Thomas H. | author1-link=Thomas H. Cormen | author2-link = Charles E. Leiserson|last2=Leiserson|first2=Charles E.|author3-link=Ron Rivest|last3=Rivest|first3=Ronald L.|author4-link=Clifford Stein|last4=Stein|first4=Clifford | title = [[Introduction to Algorithms]] | edition = 2nd | year = 2001 | publisher = [[MIT Press]] and [[McGraw-Hill]] | isbn=0-262-03293-7 | chapter = 11 Hash Tables | pages = 221–252}}.</ref><ref name="dietzfelbinger">Dietzfelbinger, M., Karlin, A., Mehlhorn, K., Meyer auf der Heide, F., Rohnert, H., and Tarjan, R. E. 1994.
[http://www.arl.wustl.edu/~sailesh/download_files/Limited_Edition/hash/Dynamic%20Perfect%20Hashing-%20Upper%20and%20Lower%20Bounds.pdf "Dynamic Perfect Hashing: Upper and Lower Bounds"] {{Webarchive|url=https://web.archive.org/web/20160304094014/http://www.arl.wustl.edu/~sailesh/download_files/Limited_Edition/hash/Dynamic%20Perfect%20Hashing-%20Upper%20and%20Lower%20Bounds.pdf |date=2016-03-04 }}.
SIAM J. Comput. 23, 4 (Aug. 1994), 738-761.
http://portal.acm.org/citation.cfm?id=182370
{{doi|10.1137/S0097539791194094}}</ref>
It is sometimes also possible to solve the problem using directly addressed [[Array data structure|array]]s, [[binary search tree]]s, or other more specialized structures.

Many programming languages include associative arrays as [[primitive data type]]s, while many other languages provide [[software library|software libraries]] that support associative arrays. [[Content-addressable memory]] is a form of direct hardware-level support for associative arrays.

Associative arrays have many applications including such fundamental [[software design pattern|programming pattern]]s as [[memoization]]<!--NOT a misspelling--><ref name="Michie1968">{{cite journal |last=Michie |first=Donald |url=https://www.cs.utexas.edu/users/hunt/research/hash-cons/hash-cons-papers/michie-memo-nature-1968.pdf |title='Memo' Functions and Machine Learning |journal=[[Nature (journal)|Nature]] |volume=218 |issue= 5136|pages=19–22 |year=1968 |doi=10.1038/218019a0 |bibcode=1968Natur.218...19M |s2cid=4265138 }}</ref> and the [[decorator pattern]].<ref name="decorator">{{harvtxt|Goodrich|Tamassia|2006}}, pp. 597–599.</ref>

The name does not come from the [[associative property]] known in mathematics. Rather, it arises from the association of values with keys. It is not to be confused with [[Flynn's taxonomy#Associative processor|associative processors]].

==Operations==
In an associative array, the association between [[attribute–value pair|a key and a value]] is often known as a "mapping"; the same word may also be used to refer to the process of creating a new association.

The operations that are usually defined for an associative array are:<ref name="gt">{{citation|contribution=9.1 The Map Abstract Data Type|title=Data Structures & Algorithms in Java|last1=Goodrich|first1=Michael T.|author1-link=Michael T. Goodrich|last2=Tamassia|first2=Roberto|author2-link=Roberto Tamassia|publisher=Wiley|edition=4th|year=2006|pages=368–371}}</ref><ref name="ms">{{citation|contribution=4 Hash Tables and Associative Arrays|title=Algorithms and Data Structures: The Basic Toolbox|first1=Kurt|last1=Mehlhorn|author1-link=Kurt Mehlhorn|first2=Peter|last2=Sanders|author2-link=Peter Sanders (computer scientist)|publisher=Springer|year=2008|pages=81–98 |url=http://people.mpi-inf.mpg.de/~mehlhorn/ftp/Toolbox/HashTables.pdf |archive-url=https://web.archive.org/web/20140802025330/http://people.mpi-inf.mpg.de/~mehlhorn/ftp/Toolbox/HashTables.pdf |archive-date=2014-08-02 |url-status=live}}</ref><ref name="Black"/>

;Insert or put
: add a new <math>(key, value)</math> pair to the collection, mapping the key to its new value. Any existing mapping is overwritten. The arguments to this operation are the key and the value.
;Remove or delete
: remove a <math>(key, value)</math> pair from the collection, unmapping a given key from its value. The argument to this operation is the key.
;Lookup, find, or get
: find the value (if any) that is bound to a given key. The argument to this operation is the key, and the value is returned from the operation. If no value is found, some lookup functions raise an [[Exception handling|exception]], while others return a default value (such as zero, null, or a specific value passed to the constructor).

Associative arrays may also include other operations such as determining the number of mappings or constructing an [[iterator]] to loop over all the mappings. For such operations, the order in which the mappings are returned is usually implementation-defined.

A [[multimap]] generalizes an associative array by allowing multiple values to be associated with a single key.<ref>{{harvtxt|Goodrich|Tamassia|2006}}, pp. 389–397.</ref> A [[bidirectional map]] is a related abstract data type in which the mappings operate in both directions: each value must be associated with a unique key, and a second lookup operation takes a value as an argument and looks up the key associated with that value.

===Properties===

The operations of the associative array should satisfy various properties:<ref name="Black">{{cite web |last1=Black |first1=Paul E. |last2=Stewart |first2=Rob |title=dictionary |url=https://xlinux.nist.gov/dads/HTML/dictionary.html |website=Dictionary of Algorithms and Data Structures |access-date=26 January 2022 |date=2 November 2020}}</ref>
* <code>lookup(k, insert(j, v, D)) = if k == j then v else lookup(k, D)</code>
* <code>lookup(k, new()) = fail</code>, where <code>fail</code> is an exception or default value
* <code>remove(k, insert(j, v, D)) = if k == j then remove(k, D) else insert(j, v, remove(k, D))</code>
* <code>remove(k, new()) = new()</code>
where <code>k</code> and <code>j</code> are keys, <code>v</code> is a value, <code>D</code> is an associative array, and <code>new()</code> creates a new, empty associative array.

===Example===
Suppose that the set of loans made by a library is represented in a data structure. Each book in a library may be checked out by one patron at a time. However, a single patron may be able to check out multiple books. Therefore, the information about which books are checked out to which patrons may be represented by an associative array, in which the books are the keys and the patrons are the values. Using notation from [[Python (programming language)|Python]] or [[JSON]], the data structure would be:

<syntaxhighlight lang="javascript">{
    "Pride and Prejudice": "Alice",
    "Wuthering Heights": "Alice",
    "Great Expectations": "John"
}</syntaxhighlight>

A lookup operation on the key "Great Expectations" would return "John". If John returns his book, that would cause a deletion operation, and if Pat checks out a book, that would cause an insertion operation, leading to a different state:

<syntaxhighlight lang="javascript">{
    "Pride and Prejudice": "Alice",
    "The Brothers Karamazov": "Pat",
    "Wuthering Heights": "Alice"
}</syntaxhighlight>

==Implementation==

For dictionaries with very few mappings, it may make sense to implement the dictionary using an [[association list]], which is a [[linked list]] of mappings. With this implementation, the time to perform the basic dictionary operations is linear in the total number of mappings. However, it is easy to implement and the constant factors in its running time are small.<ref name="gt"/><ref>{{cite web |url=http://www.faqs.org/faqs/lisp-faq/part2/section-2.html
|title=When should I use a hash table instead of an association list?
|publisher=lisp-faq/part2
|date=1996-02-20}}</ref>

Another very simple implementation technique, usable when the keys are restricted to a narrow range, is direct addressing into an array: the value for a given key ''k'' is stored at the array cell ''A''[''k''], or if there is no mapping for ''k'' then the cell stores a special [[sentinel value]] that indicates the lack of a mapping. This technique is simple and fast, with each dictionary operation taking constant time. However, the space requirement for this structure is the size of the entire keyspace, making it impractical unless the keyspace is small.<ref name="clrs" />

The two major approaches for implementing dictionaries are a [[hash table]] or a [[search tree]].<ref name="gt"/><ref name="ms"/><ref name="clrs"/><ref name="dietzfelbinger"/>

=== Hash table implementations ===
{{main | Hash table}}
[[File:Hash table average insertion time.png|thumb|right|362px|This graph compares the average number of [[CPU cache]] misses required to look up elements in large hash tables (far exceeding size of the cache) with chaining and [[linear probing]]. Linear probing performs better due to better [[locality of reference]], though as the table gets full, its performance degrades drastically.]]
The most frequently used general-purpose implementation of an associative array is with a [[hash table]]: an [[Array data structure|array]] combined with a [[hash function]] that separates each key into a separate "bucket" of the array. The basic idea behind a hash table is that accessing an element of an array via its index is a simple, constant-time operation. Therefore, the average overhead of an operation for a hash table is only the computation of the key's hash, combined with accessing the corresponding bucket within the array. As such, hash tables usually perform in O(1) time, and usually outperform alternative implementations.

Hash tables must be able to handle [[hash collision|collisions]]: the mapping by the hash function of two different keys to the same bucket of the array. The two most widespread approaches to this problem are [[separate chaining]] and [[open addressing]].<ref name="gt"/><ref name="ms"/><ref name="clrs"/><ref name="fklm">{{citation|contribution=Pathfinders for associative maps|title=Ext. Abstracts GIS-l 2006|last1=Klammer|first1=F.|author1-link=F. Klammer|last2=Mazzolini|first2=L.|author2-link=L. Mazzolini|publisher=GIS-I|year=2006|pages=71–74}}.</ref> In separate chaining, the array does not store the value itself but stores a [[Pointer (computer programming)|pointer]] to another container, usually an [[association list]], that stores all the values matching the hash. By contrast, in open addressing, if a hash collision is found, the table seeks an empty spot in an array to store the value in a deterministic manner, usually by looking at the next immediate position in the array.

Open addressing has a lower [[cache miss]] ratio than separate chaining when the table is mostly empty. However, as the table becomes filled with more elements, open addressing's performance degrades exponentially. Additionally, separate chaining uses less memory in most cases, unless the entries are very small (less than four times the size of a pointer).

=== Tree implementations ===
{{main | Search tree }}

==== Self-balancing binary search trees ====
Another common approach is to implement an associative array with a [[self-balancing binary search tree]], such as an [[AVL tree]] or a [[red–black tree]].<ref>
Joel Adams and Larry Nyhoff.
[http://cs.calvin.edu/books/c++/intro/3e/WebItems/Ch15-Web/STL-Trees.pdf "Trees in STL"].
Quote:

"The Standard Template library ... some of its containers -- the set<T>, map<T1, T2>, multiset<T>, and multimap<T1, T2> templates -- are generally built using a special kind of ''self-balancing binary search tree'' called a ''red–black tree''."
</ref>

Compared to hash tables, these structures have both strengths and weaknesses. The worst-case performance of self-balancing binary search trees is significantly better than that of a hash table, with a time complexity in [[big O notation]] of O(log ''n''). This is in contrast to hash tables, whose worst-case performance involves all elements sharing a single bucket, resulting in O(''n'') time complexity. In addition, and like all binary search trees, self-balancing binary search trees keep their elements in order. Thus, traversing its elements follows a least-to-greatest pattern, whereas traversing a hash table can result in elements being in seemingly random order. Because they are in order, tree-based maps can also satisfy range queries (find all values between two bounds) whereas a hashmap can only find exact values. However, hash tables have a much better average-case time complexity than self-balancing binary search trees of O(1), and their worst-case performance is highly unlikely when a good [[hash function]] is used.

A self-balancing binary search tree can be used to implement the buckets for a hash table that uses separate chaining. This allows for average-case constant lookup, but assures a worst-case performance of O(log ''n''). However, this introduces extra complexity into the implementation and may cause even worse performance for smaller hash tables, where the time spent inserting into and balancing the tree is greater than the time needed to perform a [[linear search]] on all elements of a linked list or similar data structure.<ref name="knuth">{{cite book| first=Donald |last=Knuth |author1-link=Donald Knuth| title = The Art of Computer Programming| volume = 3: ''Sorting and Searching''| edition = 2nd| publisher = Addison-Wesley| year = 1998| isbn = 0-201-89685-0| pages = 513–558}}</ref><ref>{{cite web |url=https://schani.wordpress.com/2010/04/30/linear-vs-binary-search/ |title=Linear vs Binary Search |last=Probst |first=Mark |date=2010-04-30 |access-date=2016-11-20 }}</ref>

==== Other trees ====
Associative arrays may also be stored in unbalanced [[binary search tree]]s or in data structures specialized to a particular type of keys such as [[radix tree]]s, [[trie]]s, [[Judy array]]s, or [[van Emde Boas tree]]s, though the relative performance of these implementations varies. For instance, Judy trees have been found to perform less efficiently than hash tables, while carefully selected hash tables generally perform more efficiently than adaptive radix trees, with potentially greater restrictions on the data types they can handle.<ref>{{Cite book|last1=Alvarez|first1=Victor|last2=Richter|first2=Stefan|last3=Chen|first3=Xiao|last4=Dittrich|first4=Jens|title=2015 IEEE 31st International Conference on Data Engineering |chapter=A comparison of adaptive radix trees and hash tables |date=April 2015|location=Seoul, South Korea|publisher=IEEE|pages=1227–1238|doi=10.1109/ICDE.2015.7113370|isbn=978-1-4799-7964-6|s2cid=17170456}}</ref> The advantages of these alternative structures come from their ability to handle additional associative array operations, such as finding the mapping whose key is the closest to a queried key when the query is absent in the set of mappings.

=== Comparison ===
{| class="wikitable"
! rowspan="2" | Underlying data structure
! colspan="2" | Lookup or Removal
! colspan="2" | Insertion
! rowspan="2" | Ordered
|-
! average
! worst case
! average
! worst case
|-
! [[Hash table]]
|style="background:#ddffdd"|O(1)
|style="background:#ffdddd"|O(''n'')
|style="background:#ddffdd"|O(1)
|style="background:#ffdddd"|O(''n'')
|{{no}}
|-
! [[Self-balancing binary search tree]]
|style="background:#ffffdd"|O(log ''n'')
|style="background:#ffffdd"|O(log ''n'')
|style="background:#ffffdd"|O(log ''n'')
|style="background:#ffffdd"|O(log ''n'')
|{{yes}}
|-
! unbalanced [[binary search tree]]
|style="background:#ffffdd"|O(log ''n'')
|style="background:#ffdddd"|O(''n'')
|style="background:#ffffdd"|O(log ''n'')
|style="background:#ffdddd"|O(''n'')
|{{yes}}
|-
! Sequential container of [[attribute–value pair|key–value pair]]s<br />(e.g. [[association list]])
|style="background:#ffdddd"|O(''n'')
|style="background:#ffdddd"|O(''n'')
|style="background:#ddffdd"|O(1)
|style="background:#ddffdd"|O(1)
|{{no}}
|}

== Ordered dictionary ==
The basic definition of a dictionary does not mandate an order. To guarantee a fixed order of enumeration, ordered versions of the associative array are often used. There are two senses of an ordered dictionary:

* The order of enumeration is always deterministic for a given set of keys by sorting. This is the case for tree-based implementations, one representative being the {{code|<map>}} container of C++.<ref>{{cite web |title=std::map |url=https://en.cppreference.com/w/cpp/container/map |website=en.cppreference.com}}</ref>
* The order of enumeration is key-independent and is instead based on the order of insertion. This is the case for the "ordered dictionary" in [[.NET Framework]], the LinkedHashMap of [[Java (programming language)|Java]] and [[Python (programming language)|Python]].<ref>{{cite web |title=OrderedDictionary Class (System.Collections.Specialized) |url=https://docs.microsoft.com/en-us/dotnet/api/system.collections.specialized.ordereddictionary?view=netframework-4.8 |website=MS Docs |language=en-us}}</ref><ref>{{cite web |title=LinkedHashMap |url=https://docs.oracle.com/en/java/javase/19/docs/api/java.base/java/util/LinkedHashMap.html}}</ref><ref>{{cite web |title=collections — Container datatypes — Python 3.9.0a3 documentation |url=https://docs.python.org/3.9/library/collections.html#collections.OrderedDict |website=docs.python.org}}</ref>

The latter is more common. Such ordered dictionaries can be implemented using an [[association list]], by overlaying a [[doubly linked list]] on top of a normal dictionary, or by moving the actual data out of the sparse (unordered) array and into a dense insertion-ordered one.

==Language support==
{{Main|Comparison of programming languages (associative array)}}

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 using array-like subscripting.

Built-in syntactic support for associative arrays was introduced in 1969 by [[SNOBOL|SNOBOL4]], under the name "table". [[TMG (language)|TMG]] offered tables with string keys and integer values. [[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 [[Rexx]], [[Perl]], [[PHP]], [[Tcl]], [[JavaScript]], [[Maple (software)|Maple]], [[Python (programming language)|Python]], [[Ruby (programming language)|Ruby]], [[Wolfram Language]], [[Go (programming language)|Go]], and [[Lua (programming language)|Lua]], support associative arrays as a primary container type. In many more languages, they are available as library functions without special syntax.

In [[Smalltalk]], [[Objective-C]], [[.NET Framework|.NET]],<ref>{{cite web |url=http://msdn.microsoft.com/en-us/library/xfhwa508.aspx |title=Dictionary<TKey, TValue> Class |publisher=MSDN}}</ref> [[Python (programming language)|Python]], [[REALbasic]], [[Swift (programming language)|Swift]], [[Visual Basic for Applications|VBA]] and [[Delphi (programming language)|Delphi]]<ref>{{Cite web|url=http://docwiki.embarcadero.com/Libraries/Tokyo/en/System.Generics.Collections.TDictionary|title=System.Generics.Collections.TDictionary - RAD Studio API Documentation|website=docwiki.embarcadero.com|language=en|access-date=2017-04-18}}</ref> they are called ''dictionaries''; in [[Perl]], [[Ruby (programming language)|Ruby]] and [[Seed7]] they are called ''hashes''; in [[C++]], [[C_Sharp_(programming_language)|C#]], [[Java (programming language)|Java]], [[Go (programming language)|Go]], [[Clojure]], [[Scala (programming language)|Scala]], [[OCaml]], [[Haskell (programming language)|Haskell]] they are called ''maps'' (see [[map (C++)]], [[unordered_map (C++)]], and {{Javadoc:SE|java/util|Map}}); in [[Common Lisp]] and [[Windows PowerShell]], they are called ''hash tables'' (since both typically use this implementation); in [[Maple (software)|Maple]] and Lua, they are called ''tables''. In [[PHP]] and [[R (programming language)|R]], all arrays can be associative, except that the keys are limited to integers and strings. In JavaScript (see also [[JSON]]), all objects behave as associative arrays with string-valued keys, while the Map and WeakMap types take arbitrary objects as keys. In Lua, they are used as the primitive building block for all data structures. In [[Visual FoxPro]], they are called ''Collections''. The [[D (programming language)|D language]] also supports associative arrays.<ref>{{cite web |url=http://dlang.org/hash-map.html|title=Associative Arrays, the D programming language|publisher=Digital Mars}}</ref>

==Permanent storage==
{{Main|Key–value store}}

Many programs using associative arrays will need to store that data in a more permanent form, such as a [[computer file]]. A common solution to this problem is a generalized concept known as ''archiving'' or ''[[serialization]]'', which produces a text or binary representation of the original objects that can be written directly to a file. This is most commonly implemented in the underlying object model, like .Net or Cocoa, which includes standard functions that convert the internal data into text. The program can create a complete text representation of any group of objects by calling these methods, which are almost always already implemented in the base associative array class.<ref>[https://developer.apple.com/library/prerelease/ios/documentation/Cocoa/Conceptual/Archiving/Archiving.html#//apple_ref/doc/uid/10000047i "Archives and Serializations Programming Guide"], Apple Inc., 2012</ref>

For programs that use very large data sets, this sort of individual file storage is not appropriate, and a [[database management system]] (DB) is required. Some DB systems natively store associative arrays by serializing the data and then storing that serialized data and the key. Individual arrays can then be loaded or saved from the database using the key to refer to them. These [[key–value database|key–value store]]s have been used for many years and have a history as long as that of the more common [[relational database]] (RDBs), but a lack of standardization, among other reasons, limited their use to certain niche roles. RDBs were used for these roles in most cases, although saving objects to a RDB can be complicated, a problem known as [[object-relational impedance mismatch]].

After approximately 2010, the need for high-performance databases suitable for [[cloud computing]] and more closely matching the internal structure of the programs using them led to a renaissance in the key–value store market. These systems can store and retrieve associative arrays in a native fashion, which can greatly improve performance in common web-related workflows.

==See also==
{{Portal|Computer programming}}
* [[Tuple]]
* [[Function (mathematics)]]

== References ==
{{Reflist}}

==External links==
{{wiktionary|associative array}}
* [https://xlinux.nist.gov/dads/HTML/assocarray.html NIST's Dictionary of Algorithms and Data Structures: Associative Array]

{{Data structures}}
{{Data types}}

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