In computer science, a binary tree is a tree data structure in which each node has at most two children, which are referred to as the left child and the right child. A recursive definition using just set theory notions is that a binary tree is a tuple, where L and R are binary trees or the empty set and S is a singleton set; some authors allow the binary tree to be the empty set as well. From a graph theory perspective, binary trees as defined here are arborescences. A binary tree may thus be called a bifurcating arborescence—a term which appears in some old programming books, before the modern computer science terminology prevailed, it is possible to interpret a binary tree as an undirected, rather than a directed graph, in which case a binary tree is an ordered, rooted tree. Some authors use rooted binary tree instead of binary tree to emphasize the fact that the tree is rooted, but as defined above, a binary tree is always rooted. A binary tree is a special case of an ordered K-ary tree, where k is 2.
In mathematics, what is termed binary tree can vary from author to author. Some use the definition used in computer science, but others define it as every non-leaf having two children and don't order the children either. In computing, binary trees are used in two different ways: First, as a means of accessing nodes based on some value or label associated with each node. Binary trees labelled this way are used to implement binary search trees and binary heaps, are used for efficient searching and sorting; the designation of non-root nodes as left or right child when there is only one child present matters in some of these applications, in particular it is significant in binary search trees. However, the arrangement of particular nodes into the tree is not part of the conceptual information. For example, in a normal binary search tree the placement of nodes depends entirely on the order in which they were added, can be re-arranged without changing the meaning. Second, as a representation of data with a relevant bifurcating structure.
In such cases the particular arrangement of nodes under and/or to the left or right of other nodes is part of the information. Common examples occur with Huffman coding and cladograms; the everyday division of documents into chapters, paragraphs, so on is an analogous example with n-ary rather than binary trees. To define a binary tree in general, we must allow for the possibility that only one of the children may be empty. An artifact, which in some textbooks is called an extended binary tree is needed for that purpose. An extended binary tree is thus recursively defined as: the empty set is an extended binary tree if T1 and T2 are extended binary trees denote by T1 • T2 the extended binary tree obtained by adding a root r connected to the left to T1 and to the right to T2 by adding edges when these sub-trees are non-empty. Another way of imagining this construction is to consider instead of the empty set a different type of node—for instance square nodes if the regular ones are circles. A binary tree is a rooted tree, an ordered tree in which every node has at most two children.
A rooted tree imparts a notion of levels, thus for every node a notion of children may be defined as the nodes connected to it a level below. Ordering of these children makes possible to distinguish left child from right child, but this still doesn't distinguish between a node with left but not a right child from a one with right but no left child. The necessary distinction can be made by first partitioning the edges, i.e. defining the binary tree as triplet, where is a rooted tree and E1 ∩ E2 is empty, requiring that for all j ∈ every node has at most one Ej child. A more informal way of making the distinction is to say, quoting the Encyclopedia of Mathematics, that "every node has a left child, a right child, neither, or both" and to specify that these "are all different" binary trees. Tree terminology so varies in the literature. A rooted binary tree has a root node and every node has at most two children. A full binary tree is a tree in which every node has 2 children. Another way of defining a full binary tree is a recursive definition.
A full binary tree is either:A single vertex. A tree whose root note has two subtrees. In a complete binary tree every level, except the last, is filled, all nodes in the last level are as far left as possible, it can have between 2h nodes at the last level h. An alternative definition is a perfect tree; some authors use the term complete to refer instead to a perfect binary tree as defined above, in which case they call this type of tree an complete binary tree or nearly complete binary tree. A complete binary tree can be efficiently represented using an array. A perfect binary tree is a binary tree in which all interior nodes have two children and all leaves have the same depth or same level. An example of a perfect binary tree is the ancestry chart of a person to a given depth, as each person has two biological parents. In the infinite complete binary tree, every node has two children; the set of all nodes is countably infinite, but the set of all infinite paths from the root
A dictionary, sometimes known as a wordbook, is a collection of words in one or more specific languages arranged alphabetically, which may include information on definitions, etymologies, translation, etc. or a book of words in one language with their equivalents in another, sometimes known as a lexicon. It is a lexicographical reference. A broad distinction is made between specialized dictionaries. Specialized dictionaries include words in specialist fields, rather than a complete range of words in the language. Lexical items that describe concepts in specific fields are called terms instead of words, although there is no consensus whether lexicology and terminology are two different fields of study. In theory, general dictionaries are supposed to be semasiological, mapping word to definition, while specialized dictionaries are supposed to be onomasiological, first identifying concepts and establishing the terms used to designate them. In practice, the two approaches are used for both types.
There are other types of dictionaries that do not fit neatly into the above distinction, for instance bilingual dictionaries, dictionaries of synonyms, rhyming dictionaries. The word dictionary is understood to refer to a general purpose monolingual dictionary. There is a contrast between prescriptive or descriptive dictionaries. Stylistic indications in many modern dictionaries are considered by some to be less than objectively descriptive. Although the first recorded dictionaries date back to Sumerian times, the systematic study of dictionaries as objects of scientific interest themselves is a 20th-century enterprise, called lexicography, initiated by Ladislav Zgusta; the birth of the new discipline was not without controversy, the practical dictionary-makers being sometimes accused by others of "astonishing" lack of method and critical-self reflection. The oldest known dictionaries were Akkadian Empire cuneiform tablets with bilingual Sumerian–Akkadian wordlists, discovered in Ebla and dated 2300 BCE.
The early 2nd millennium BCE Urra=hubullu glossary is the canonical Babylonian version of such bilingual Sumerian wordlists. A Chinese dictionary, the c. 3rd century BCE Erya, was the earliest surviving monolingual dictionary. Philitas of Cos wrote a pioneering vocabulary Disorderly Words which explained the meanings of rare Homeric and other literary words, words from local dialects, technical terms. Apollonius the Sophist wrote the oldest surviving Homeric lexicon; the first Sanskrit dictionary, the Amarakośa, was written by Amara Sinha c. 4th century CE. Written in verse, it listed around 10,000 words. According to the Nihon Shoki, the first Japanese dictionary was the long-lost 682 CE Niina glossary of Chinese characters; the oldest existing Japanese dictionary, the c. 835 CE Tenrei Banshō Meigi, was a glossary of written Chinese. In Frahang-i Pahlavig, Aramaic heterograms are listed together with their translation in Middle Persian language and phonetic transcription in Pazand alphabet. A 9th-century CE Irish dictionary, Sanas Cormaic, contained etymologies and explanations of over 1,400 Irish words.
In India around 1320, Amir Khusro compiled the Khaliq-e-bari which dealt with Hindustani and Persian words. Arabic dictionaries were compiled between the 8th and 14th centuries CE, organizing words in rhyme order, by alphabetical order of the radicals, or according to the alphabetical order of the first letter; the modern system was used in specialist dictionaries, such as those of terms from the Qur'an and hadith, while most general use dictionaries, such as the Lisan al-`Arab and al-Qamus al-Muhit listed words in the alphabetical order of the radicals. The Qamus al-Muhit is the first handy dictionary in Arabic, which includes only words and their definitions, eliminating the supporting examples used in such dictionaries as the Lisan and the Oxford English Dictionary. In medieval Europe, glossaries with equivalents for Latin words in vernacular or simpler Latin were in use; the Catholicon by Johannes Balbus, a large grammatical work with an alphabetical lexicon, was adopted. It served as the basis for several bilingual dictionaries and was one of the earliest books to be printed.
In 1502 Ambrogio Calepino's Dictionarium was published a monolingual Latin dictionary, which over the course of the 16th century was enlarged to become a multilingual glossary. In 1532 Robert Estienne published the Thesaurus linguae latinae and in 1572 his son Henri Estienne published the Thesaurus linguae graecae, which served up to the 19th century as the basis of Greek lexicography; the first monolingual dictionary written in Europe was the Spanish, written by Sebastián Covarrubias' Tesoro de la lengua castellana o española, published in 1611 in Madrid, Spain. In 1612 the first edition of the Vocabolario degli Accademici della Crusca, for Italian, was published, it served as the model for similar works in English. In 1690 in Rotterdam was published, the Dictionnaire Universel by
In software, a spell checker is a software feature that checks for misspellings in a text. Features are in software, such as a word processor, email client, electronic dictionary, or search engine. A basic spell checker carries out the following processes: It scans the text and extracts the words contained in it, it compares each word with a known list of spelled words. This might contain just a list of words, or it might contain additional information, such as hyphenation points or lexical and grammatical attributes. An additional step is a language-dependent algorithm for handling morphology. For a inflected language like English, the spell-checker will need to consider different forms of the same word, such as plurals, verbal forms and possessives. For many other languages, such as those featuring agglutination and more complex declension and conjugation, this part of the process is more complicated, it is unclear whether morphological analysis—allowing for many different forms of a word depending on its grammatical role—provides a significant benefit for English, though its benefits for synthetic languages such as German, Hungarian or Turkish are clear.
As an adjunct to these components, the program's user interface will allow users to approve or reject replacements and modify the program's operation. An alternative type of spell checker uses statistical information, such as n-grams, to recognize errors instead of correctly-spelled words; this approach requires a lot of effort to obtain sufficient statistical information. Key advantages include needing less runtime storage and the ability to correct errors in words that are not included in a dictionary. In some cases spell checkers use a fixed list of misspellings and suggestions for those misspellings. Clustering algorithms have been used for spell checking combined with phonetic information. In 1961, Les Earnest, who headed the research on this budding technology, saw it necessary to include the first spell checker that accessed a list of 10,000 acceptable words. Ralph Gorin, a graduate student under Earnest at the time, created the first true spelling checker program written as an applications program for general English text: SPELL for the DEC PDP-10 at Stanford University's Artificial Intelligence Laboratory, in February 1971.
Gorin wrote SPELL in assembly language, for faster action. Gorin made SPELL publicly accessible, as was done with most SAIL programs, it soon spread around the world via the new ARPAnet, about ten years before personal computers came into general use. SPELL, its algorithms and data structures inspired the Unix ispell program; the first spell checkers were available on mainframe computers in the late 1970s. A group of six linguists from Georgetown University developed the first spell-check system for the IBM corporation. Henry Kučera invented one for the VAX machines of Digital Equipment Corp in 1981; the first spell checkers for personal computers appeared in 1980, such as "WordCheck" for Commodore systems, released in late 1980 in time for advertisements to go to print in January 1981. Developers such as Maria Mariani and Random House rushed OEM packages or end-user products into the expanding software market for the PC but for Apple Macintosh, VAX, Unix. On the PCs, these spell checkers were standalone programs, many of which could be run in TSR mode from within word-processing packages on PCs with sufficient memory.
However, the market for standalone packages was short-lived, as by the mid-1980s developers of popular word-processing packages like WordStar and WordPerfect had incorporated spell checkers in their packages licensed from the above companies, who expanded support from just English to European and even Asian languages. However, this required increasing sophistication in the morphology routines of the software with regard to heavily-agglutinative languages like Hungarian and Finnish. Although the size of the word-processing market in a country like Iceland might not have justified the investment of implementing a spell checker, companies like WordPerfect nonetheless strove to localize their software for as many national markets as possible as part of their global marketing strategy. Firefox 2.0, a web browser, has spell check support for user-written content, such as when editing Wikitext, writing on many webmail sites and social networking websites. The web browsers Google Chrome and Opera, the email client Kmail and the instant messaging client Pidgin offer spell checking support, transparently using GNU Aspell and Hunspell as their engine.
Mac OS X now has spell check system-wide, extending the service to all bundled and third party applications. Some spell checkers have separate support for medical dictionaries to help prevent medical errors; the first spell checkers were "verifiers" instead of "correctors." They offered no suggestions for incorrectly spelled words. This was helpful for typos but it was not so helpful for logical or phonetic errors; the challenge the developers faced was the difficulty in offering useful suggestions for misspelled words. This requires applying pattern-matching algorithms, it might seem logical that where spell-checking dictionaries are concerned, "the bigger, the better," so that
A database is an organized collection of data stored and accessed electronically from a computer system. Where databases are more complex they are developed using formal design and modeling techniques; the database management system is the software that interacts with end users and the database itself to capture and analyze the data. The DBMS software additionally encompasses; the sum total of the database, the DBMS and the associated applications can be referred to as a "database system". The term "database" is used to loosely refer to any of the DBMS, the database system or an application associated with the database. Computer scientists may classify database-management systems according to the database models that they support. Relational databases became dominant in the 1980s; these model data as rows and columns in a series of tables, the vast majority use SQL for writing and querying data. In the 2000s, non-relational databases became popular, referred to as NoSQL because they use different query languages.
Formally, a "database" refers to the way it is organized. Access to this data is provided by a "database management system" consisting of an integrated set of computer software that allows users to interact with one or more databases and provides access to all of the data contained in the database; the DBMS provides various functions that allow entry and retrieval of large quantities of information and provides ways to manage how that information is organized. Because of the close relationship between them, the term "database" is used casually to refer to both a database and the DBMS used to manipulate it. Outside the world of professional information technology, the term database is used to refer to any collection of related data as size and usage requirements necessitate use of a database management system. Existing DBMSs provide various functions that allow management of a database and its data which can be classified into four main functional groups: Data definition – Creation and removal of definitions that define the organization of the data.
Update – Insertion and deletion of the actual data. Retrieval – Providing information in a form directly usable or for further processing by other applications; the retrieved data may be made available in a form the same as it is stored in the database or in a new form obtained by altering or combining existing data from the database. Administration – Registering and monitoring users, enforcing data security, monitoring performance, maintaining data integrity, dealing with concurrency control, recovering information, corrupted by some event such as an unexpected system failure. Both a database and its DBMS conform to the principles of a particular database model. "Database system" refers collectively to the database model, database management system, database. Physically, database servers are dedicated computers that hold the actual databases and run only the DBMS and related software. Database servers are multiprocessor computers, with generous memory and RAID disk arrays used for stable storage.
RAID is used for recovery of data. Hardware database accelerators, connected to one or more servers via a high-speed channel, are used in large volume transaction processing environments. DBMSs are found at the heart of most database applications. DBMSs may be built around a custom multitasking kernel with built-in networking support, but modern DBMSs rely on a standard operating system to provide these functions. Since DBMSs comprise a significant market and storage vendors take into account DBMS requirements in their own development plans. Databases and DBMSs can be categorized according to the database model that they support, the type of computer they run on, the query language used to access the database, their internal engineering, which affects performance, scalability and security; the sizes and performance of databases and their respective DBMSs have grown in orders of magnitude. These performance increases were enabled by the technology progress in the areas of processors, computer memory, computer storage, computer networks.
The development of database technology can be divided into three eras based on data model or structure: navigational, SQL/relational, post-relational. The two main early navigational data models were the hierarchical model and the CODASYL model The relational model, first proposed in 1970 by Edgar F. Codd, departed from this tradition by insisting that applications should search for data by content, rather than by following links; the relational model employs sets of ledger-style tables, each used for a different type of entity. Only in the mid-1980s did computing hardware become powerful enough to allow the wide deployment of relational systems. By the early 1990s, relational systems dominated in all large-scale data processing applications, as of 2018 they remain dominant: IBM DB2, Oracle, MySQL, Microsoft SQL Server are the most searched DBMS; the dominant database language, standardised SQL for the relational model, has influenced database languages for other data models. Object databases were developed in the 1980s to overcome the inconvenience of object-relational impedance mismatch, which led to the coining of the term "post-relational" and the development of hybrid object-relational databas
Computer science is the study of processes that interact with data and that can be represented as data in the form of programs. It enables the use of algorithms to manipulate and communicate digital information. A computer scientist studies the theory of computation and the practice of designing software systems, its fields can be divided into practical disciplines. Computational complexity theory is abstract, while computer graphics emphasizes real-world applications. Programming language theory considers approaches to the description of computational processes, while computer programming itself involves the use of programming languages and complex systems. Human–computer interaction considers the challenges in making computers useful and accessible; the earliest foundations of what would become computer science predate the invention of the modern digital computer. Machines for calculating fixed numerical tasks such as the abacus have existed since antiquity, aiding in computations such as multiplication and division.
Algorithms for performing computations have existed since antiquity before the development of sophisticated computing equipment. Wilhelm Schickard designed and constructed the first working mechanical calculator in 1623. In 1673, Gottfried Leibniz demonstrated a digital mechanical calculator, called the Stepped Reckoner, he may be considered the first computer scientist and information theorist, among other reasons, documenting the binary number system. In 1820, Thomas de Colmar launched the mechanical calculator industry when he released his simplified arithmometer, the first calculating machine strong enough and reliable enough to be used daily in an office environment. Charles Babbage started the design of the first automatic mechanical calculator, his Difference Engine, in 1822, which gave him the idea of the first programmable mechanical calculator, his Analytical Engine, he started developing this machine in 1834, "in less than two years, he had sketched out many of the salient features of the modern computer".
"A crucial step was the adoption of a punched card system derived from the Jacquard loom" making it infinitely programmable. In 1843, during the translation of a French article on the Analytical Engine, Ada Lovelace wrote, in one of the many notes she included, an algorithm to compute the Bernoulli numbers, considered to be the first computer program. Around 1885, Herman Hollerith invented the tabulator, which used punched cards to process statistical information. In 1937, one hundred years after Babbage's impossible dream, Howard Aiken convinced IBM, making all kinds of punched card equipment and was in the calculator business to develop his giant programmable calculator, the ASCC/Harvard Mark I, based on Babbage's Analytical Engine, which itself used cards and a central computing unit; when the machine was finished, some hailed it as "Babbage's dream come true". During the 1940s, as new and more powerful computing machines were developed, the term computer came to refer to the machines rather than their human predecessors.
As it became clear that computers could be used for more than just mathematical calculations, the field of computer science broadened to study computation in general. In 1945, IBM founded the Watson Scientific Computing Laboratory at Columbia University in New York City; the renovated fraternity house on Manhattan's West Side was IBM's first laboratory devoted to pure science. The lab is the forerunner of IBM's Research Division, which today operates research facilities around the world; the close relationship between IBM and the university was instrumental in the emergence of a new scientific discipline, with Columbia offering one of the first academic-credit courses in computer science in 1946. Computer science began to be established as a distinct academic discipline in the 1950s and early 1960s; the world's first computer science degree program, the Cambridge Diploma in Computer Science, began at the University of Cambridge Computer Laboratory in 1953. The first computer science degree program in the United States was formed at Purdue University in 1962.
Since practical computers became available, many applications of computing have become distinct areas of study in their own rights. Although many believed it was impossible that computers themselves could be a scientific field of study, in the late fifties it became accepted among the greater academic population, it is the now well-known IBM brand that formed part of the computer science revolution during this time. IBM released the IBM 704 and the IBM 709 computers, which were used during the exploration period of such devices. "Still, working with the IBM was frustrating if you had misplaced as much as one letter in one instruction, the program would crash, you would have to start the whole process over again". During the late 1950s, the computer science discipline was much in its developmental stages, such issues were commonplace. Time has seen significant improvements in the effectiveness of computing technology. Modern society has seen a significant shift in the users of computer technology, from usage only by experts and professionals, to a near-ubiquitous user base.
Computers were quite costly, some degree of humanitarian aid was needed for efficient use—in part from professional computer operators. As computer adoption became more widespread and affordable, less human assistance was needed for common usage. Despite its short history as a formal academic discipline, computer science has made a number of fundamental contributions to science and society—in fact, along with electronics, it is
In computer science, the Bx tree is a query, used to update efficient B+ tree-based index structures for moving objects. The base structure of the Bx-tree is a B+ tree in which the internal nodes serve as a directory, each containing a pointer to its right sibling. In the earlier version of the Bx-tree, the leaf nodes contained the moving-object locations being indexed and corresponding index time. In the optimized version, each leaf node entry contains the id, single-dimensional mapping value and the latest update time of the object; the fanout is increased by not storing the locations of moving objects, as these can be derived from the mapping values. As for many other moving objects indexes, a two-dimensional moving object is modeled as a linear function as O =, where and are location and velocity of the object at a given time instance t, i.e. the time of last update. The B+ tree is a structure for indexing single-dimensional data. In order to adopt the B+ tree as a moving object index, the Bx-tree uses a linearization technique which helps to integrate objects' location at time t into single dimensional value.
Objects are first partitioned according to their update time. For objects within the same partition, the Bx-tree stores their locations at a given time which are estimated by linear interpolation. By doing so, the Bx-tree keeps a consistent view of all objects within the same partition without storing the update time of an objects. Secondly, the space is partitioned by a grid and the location of an object is linearized within the partitions according to a space-filling curve, e.g. the Peano or Hilbert curves. With the combination of the partition number and the linear order, an object is indexed in Bx-tree with a one-dimensional index key Bxvalue: B x v a l u e = 2 + 2 Here index-partition is an index partition determined by the update time and xrep is the space-filling curve value of the object position at the indexed time, 2 denotes the binary value of x, “+” means concatenation. Given an object O, tmu = 120, the Bxvalue for O can be computed. O is indexed in partition 0 as mentioned. Therefore, indexpartition = 2.
O’s position at the label timestamp of partition 0 is. Using Z-curve with order = 3, the Z-value of O, i.e. xrep is 2. Concatenating indexpartition and xrep, Bxvalue 2=19. Example O and tmu=120 O's position at the label timestamp of partition:??? Given a new object, its index key is computed and the object is inserted into the Bx-tree as in the B+ tree. An update consists of a deletion followed by an insertion. An auxiliary structure is employed to keep the latest key of each index so that an object can be deleted by searching for the key; the indexing key is computed before affecting the tree. In this way, the Bx-tree directly inherits the good properties of the B+ tree, achieves efficient update performance. A range query retrieves all objects whose location falls within the rectangular range q = at time t q not prior to the current time; the Bx-tree uses query-window enlargement technique to answer queries. Since the Bx-tree stores an object's location as of sometime after its update time, the enlargement involves two cases: a location must either be brought back to an earlier time or forward to a time.
The main idea is to enlarge the query window so that it encloses all objects whose positions are not within query window at its label timestamp but will enter the query window at the query timestamp. After the enlargement, the partitions of the Bx-tree need to be traversed to find objects falling in the enlarged query window. In each partition, the use of a space-filling curve means that a range query in the native, two-dimensional space becomes a set of range queries in the transformed, one-dimensional space. To avoid excessively large query region after expansion in skewed datasets, an optimization of the query algorithm exists, which improves the query efficiency by avoiding unnecessary query enlargement. K nearest neighbor query is computed by iteratively performing range queries with an incrementally enlarged search region until k answers are obtained. Another possibility is to employ similar querying ideas in The iDistance Technique; the range query and K Nearest Neighbor query algorithms can be extended to support interval queries, continuous queries, etc.
Since the Bx-tree is an index built on top of a B+ tree index, all operations in the Bx-tree, including the insertion and search, are the same as those in the B
In computer science, an associative array, symbol table, or dictionary is an abstract data type composed of a collection of pairs, such that each possible key appears at most once in the collection. Operations associated with this data type allow: the addition of a pair to the collection the removal of a pair from the collection the modification of an existing pair the lookup of a value associated with a particular keyThe dictionary problem is a classic computer science problem: the task of designing a data structure that maintains a set of data during'search','delete', and'insert' operations; the two major solutions to the dictionary problem are a search tree. In some cases it is possible to solve the problem using directly addressed arrays, binary search trees, or other more specialized structures. Many programming languages include associative arrays as primitive data types, they are available in software libraries for many others. Content-addressable memory is a form of direct hardware-level support for associative arrays.
Associative arrays have many applications including such fundamental programming patterns as memoization and the decorator pattern. The name does not come from the associative property known in mathematics. Rather, it arises from the fact. In an associative array, the association between a key and a value is known as a "binding", the same word "binding" may be used to refer to the process of creating a new association; the operations that are defined for an associative array are: Add or insert: add a new pair to the collection, binding the new key to its new value. The arguments to this operation are the value. Reassign: replace the value in one of the pairs that are in the collection, binding an old key to a new value; as with an insertion, the arguments to this operation are the value. Remove or delete: remove a pair from the collection, unbinding a given key from its value; the argument to this operation is the key. Lookup: find the value, bound to a given key; the argument to this operation is the key, the value is returned from the operation.
If no value is found, some associative array implementations raise an exception, while others create a pair with the given key and the default value of the value type. Instead of add or reassign there is a single set operation that adds a new pair if one does not exist, otherwise reassigns it. In addition, associative arrays may include other operations such as determining the number of bindings or constructing an iterator to loop over all the bindings. For such an operation, the order in which the bindings are returned may be arbitrary. A multimap generalizes an associative array by allowing multiple values to be associated with a single key. A bidirectional map is a related abstract data type in which the bindings operate in both directions: each value must be associated with a unique key, a second lookup operation takes a value as argument and looks up the key associated with that value. 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 only by a single library 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 or JSON, the data structure would be: A lookup operation on the key "Great Expectations" would return "John". If John returns his book, that would cause a deletion operation, if Pat checks out a book, that would cause an insertion operation, leading to a different state: For dictionaries with small numbers of bindings, it may make sense to implement the dictionary using an association list, a linked list of bindings. With this implementation, the time to perform the basic dictionary operations is linear in the total number of bindings. Another simple implementation technique, usable when the keys are restricted to a narrow range of integers, is direct addressing into an array: the value for a given key k is stored at the array cell A, or if there is no binding for k the cell stores a special sentinel value that indicates the absence of a binding.
As well as being simple, this technique is fast: each dictionary operation takes constant time. However, the space requirement for this structure is the size of the entire keyspace, making it impractical unless the keyspace is small; the two major approaches to implementing dictionaries are a search tree. The most used general purpose implementation of an associative array is with a hash table: an 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