(data structure)

Definition: An abstract data type storing items, or values. A value is accessed by an associated key. Basic operations are new, insert, find and delete.

Formal Definition: The operations new(), insert(k, v, D), and find(k, D) may be defined with axiomatic semantics as follows.

  1. new() returns a dictionary
  2. find(k, insert(k, v, D)) = v
  3. find(k, insert(j, v, D)) = find(k, D) if k ≠ j
where k and j are keys, v is a value, and D is a dictionary.

The modifier function delete(k, D) may be defined as follows.

  1. delete(k, new()) = new()
  2. delete(k, insert(k, v, D)) = delete(k, D)
  3. delete(k, insert(j, v, D)) = insert(j, v, delete(k, D)) if k ≠ j

If we want find to be a total function, we could define find(k, new()) using a special value: fail. This only changes the return type of find.

  1. find(k, new()) = fail

Also known as association list, map, property list.

Generalization (I am a kind of ...)
binary relation, abstract data type.

Specialization (... is a kind of me.)
associative array.

See also total order, set Some implementations: linked list, hash table, B-tree, jump list, directed acyclic word graph.

Note: The terms "association list" and "property list" are used with LISP-like languages and in the area of Artificial Intelligence. These suggest a relatively small number of items, whereas a dictionary may be quite large. Professionals in the Data Management area have specialized semantics for "dictionary" and related terms.

A dictionary defines a binary relation that maps keys to values. The keys of a dictionary are a set.

Contributions by Rob Stewart 16 March 2004.

Author: PEB


(C++, Pascal, and Fortran). Maksim Goleta's Collections (C#) implementing stacks, queues, linked lists, binary search trees, AVL trees, and dictionaries.
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If you have suggestions, corrections, or comments, please get in touch with Paul E. Black.

Entry modified 23 May 2011.
HTML page formatted Tue Dec 6 16:16:32 2011.

Cite this as:
Paul E. Black, "dictionary", in Dictionary of Algorithms and Data Structures [online], Paul E. Black, ed., U.S. National Institute of Standards and Technology. 23 May 2011. (accessed TODAY) Available from:

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