Sorting refers to ordering data in an increasing or decreasing manner according to some linear relationship among the data items.
72-405: Ordering items is the combination of categorizing them based on equivalent order, and ordering the categories themselves. In computer science , arranging in an ordered sequence is called "sorting". Sorting is a common operation in many applications, and efficient algorithms have been developed to perform it. The most common uses of sorted sequences are: The opposite of sorting, rearranging
144-413: A search problem , the answers can be arbitrary strings. For example, factoring is a search problem where the instances are (string representations of) positive integers and the solutions are (string representations of) collections of primes. A search problem is represented as a relation consisting of all the instance-solution pairs, called a search relation . For example, factoring can be represented as
216-405: A set of instances or cases together with a, possibly empty, set of solutions for every instance/case. The question then is, whether there exists an algorithm that maps instances to solutions. For example, in the factoring problem , the instances are the integers n , and solutions are prime numbers p that are the nontrivial prime factors of n . An example of a computational problem without
288-585: A discipline, computer science spans a range of topics from theoretical studies of algorithms and the limits of computation to the practical issues of implementing computing systems in hardware and software. CSAB , formerly called Computing Sciences Accreditation Board—which is made up of representatives of the Association for Computing Machinery (ACM), and the IEEE Computer Society (IEEE CS) —identifies four areas that it considers crucial to
360-753: 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 department 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 first proposed in 1956,
432-464: A mathematical discipline argue that computer programs are physical realizations of mathematical entities and programs that can be deductively reasoned through mathematical formal methods . Computer scientists Edsger W. Dijkstra and Tony Hoare regard instructions for computer programs as mathematical sentences and interpret formal semantics for programming languages as mathematical axiomatic systems . A number of computer scientists have argued for
504-443: A mathematics emphasis and with a numerical orientation consider alignment with computational science . Both types of departments tend to make efforts to bridge the field educationally if not across all research. Despite the word science in its name, there is debate over whether or not computer science is a discipline of science, mathematics, or engineering. Allen Newell and Herbert A. Simon argued in 1975, Computer science
576-463: A network while using concurrency, this is known as a distributed system. Computers within that distributed system have their own private memory, and information can be exchanged to achieve common goals. This branch of computer science aims to manage networks between computers worldwide. Computer security is a branch of computer technology with the objective of protecting information from unauthorized access, disruption, or modification while maintaining
648-550: A number of terms for the practitioners of the field of computing were suggested in the Communications of the ACM — turingineer , turologist , flow-charts-man , applied meta-mathematician , and applied epistemologist . Three months later in the same journal, comptologist was suggested, followed next year by hypologist . The term computics has also been suggested. In Europe, terms derived from contracted translations of
720-495: A particular kind of mathematically based technique for the specification , development and verification of software and hardware systems. The use of formal methods for software and hardware design is motivated by the expectation that, as in other engineering disciplines, performing appropriate mathematical analysis can contribute to the reliability and robustness of a design. They form an important theoretical underpinning for software engineering, especially where safety or security
792-416: A sequence of items in a random or meaningless order, is called shuffling . For sorting, either a weak order, "should not come after", can be specified, or a strict weak order , "should come before" (specifying one defines also the other, the two are the complement of the inverse of each other, see operations on binary relations ). For the sorting to be unique, these two are restricted to a total order and
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#1732801242194864-512: A significant amount of computer science does not involve the study of computers themselves. Because of this, several alternative names have been proposed. Certain departments of major universities prefer the term computing science , to emphasize precisely that difference. Danish scientist Peter Naur suggested the term datalogy , to reflect the fact that the scientific discipline revolves around data and data treatment, while not necessarily involving computers. The first scientific institution to use
936-626: A solution is the Halting problem . Computational problems are one of the main objects of study in theoretical computer science. One is often interested not only in mere existence of an algorithm, but also how efficient the algorithm can be. The field of computational complexity theory addresses such questions by determining the amount of resources ( computational complexity ) solving a given problem will require, and explain why some problems are intractable or undecidable . Solvable computational problems belong to complexity classes that define broadly
1008-410: A specific application. Codes are used for data compression , cryptography , error detection and correction , and more recently also for network coding . Codes are studied for the purpose of designing efficient and reliable data transmission methods. Data structures and algorithms are the studies of commonly used computational methods and their computational efficiency. Programming language theory
1080-502: A strict total order, respectively. Sorting n-tuples (depending on context also called e.g. records consisting of fields) can be done based on one or more of its components. More generally objects can be sorted based on a property. Such a component or property is called a sort key . For example, the items are books, the sort key is the title, subject or author, and the order is alphabetical. A new sort key can be created from two or more sort keys by lexicographical order . The first
1152-457: A unique order of the items. A standard order is often called ascending (corresponding to the fact that the standard order of numbers is ascending, i.e. A to Z, 0 to 9), the reverse order descending (Z to A, 9 to 0). For dates and times, ascending means that earlier values precede later ones e.g. 1/1/2000 will sort ahead of 1/1/2001. Various sorting tasks are essential in industrial processes, such as mineral processing . For example, during
1224-415: Is a branch of computer science that deals with the design, implementation, analysis, characterization, and classification of programming languages and their individual features . It falls within the discipline of computer science, both depending on and affecting mathematics, software engineering, and linguistics . It is an active research area, with numerous dedicated academic journals. Formal methods are
1296-422: Is an empirical discipline. We would have called it an experimental science, but like astronomy, economics, and geology, some of its unique forms of observation and experience do not fit a narrow stereotype of the experimental method. Nonetheless, they are experiments. Each new machine that is built is an experiment. Actually constructing the machine poses a question to nature; and we listen for the answer by observing
1368-484: Is an open problem in the theory of computation. Information theory, closely related to probability and statistics , is related to the quantification of information. This was developed by Claude Shannon to find fundamental limits on signal processing operations such as compressing data and on reliably storing and communicating data. Coding theory is the study of the properties of codes (systems for converting information from one form to another) and their fitness for
1440-479: Is associated in the popular mind with robotic development , but the main field of practical application has been as an embedded component in areas of software development , which require computational understanding. The starting point in the late 1940s was Alan Turing's question " Can computers think? ", and the question remains effectively unanswered, although the Turing test is still used to assess computer output on
1512-548: Is connected to many other fields in computer science, including computer vision , image processing , and computational geometry , and is heavily applied in the fields of special effects and video games . Information can take the form of images, sound, video or other multimedia. Bits of information can be streamed via signals . Its processing is the central notion of informatics, the European view on computing, which studies information processing algorithms independently of
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#17328012421941584-409: Is considered by some to have a much closer relationship with mathematics than many scientific disciplines, with some observers saying that computing is a mathematical science. Early computer science was strongly influenced by the work of mathematicians such as Kurt Gödel , Alan Turing , John von Neumann , Rózsa Péter and Alonzo Church and there continues to be a useful interchange of ideas between
1656-508: Is determining what can and cannot be automated. The Turing Award is generally recognized as the highest distinction in computer science. 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, even before
1728-415: Is expressed as a function of the length of the input representation. A decision problem is a computational problem where the answer for every instance is either yes or no. An example of a decision problem is primality testing : A decision problem is typically represented as the set of all instances for which the answer is yes . For example, primality testing can be represented as the infinite set In
1800-630: Is generally considered the province of disciplines other than computer science. For example, the study of computer hardware is usually considered part of computer engineering , while the study of commercial computer systems and their deployment is often called information technology or information systems . However, there has been exchange of ideas between the various computer-related disciplines. Computer science research also often intersects other disciplines, such as cognitive science , linguistics , mathematics , physics , biology , Earth science , statistics , philosophy , and logic . Computer science
1872-584: Is intended to organize, store, and retrieve large amounts of data easily. Digital databases are managed using database management systems to store, create, maintain, and search data, through database models and query languages . Data mining is a process of discovering patterns in large data sets. The philosopher of computing Bill Rapaport noted three Great Insights of Computer Science : Programming languages can be used to accomplish different tasks in different ways. Common programming paradigms include: Many languages offer support for multiple paradigms, making
1944-426: Is involved. Formal methods are a useful adjunct to software testing since they help avoid errors and can also give a framework for testing. For industrial use, tool support is required. However, the high cost of using formal methods means that they are usually only used in the development of high-integrity and life-critical systems , where safety or security is of utmost importance. Formal methods are best described as
2016-545: Is mathematical and abstract in spirit, but it derives its motivation from practical and everyday computation. It aims to understand the nature of computation and, as a consequence of this understanding, provide more efficient methodologies. According to Peter Denning, the fundamental question underlying computer science is, "What can be automated?" Theory of computation is focused on answering fundamental questions about what can be computed and what amount of resources are required to perform those computations. In an effort to answer
2088-419: Is more complex than that of a decision problem , that is, it isn't just "yes" or "no". One of the most famous examples is the traveling salesman problem: It is an NP-hard problem in combinatorial optimization , important in operations research and theoretical computer science . In computational complexity theory , it is usually implicitly assumed that any string in {0, 1} represents an instance of
2160-519: Is of high quality, affordable, maintainable, and fast to build. It is a systematic approach to software design, involving the application of engineering practices to software. Software engineering deals with the organizing and analyzing of software—it does not just deal with the creation or manufacture of new software, but its internal arrangement and maintenance. For example software testing , systems engineering , technical debt and software development processes . Artificial intelligence (AI) aims to or
2232-584: Is required to synthesize goal-orientated processes such as problem-solving, decision-making, environmental adaptation, learning, and communication found in humans and animals. From its origins in cybernetics and in the Dartmouth Conference (1956), artificial intelligence research has been necessarily cross-disciplinary, drawing on areas of expertise such as applied mathematics , symbolic logic, semiotics , electrical engineering , philosophy of mind , neurophysiology , and social intelligence . AI
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2304-432: Is the field of study and research concerned with the design and use of computer systems , mainly based on the analysis of the interaction between humans and computer interfaces . HCI has several subfields that focus on the relationship between emotions , social behavior and brain activity with computers . Software engineering is the study of designing, implementing, and modifying the software in order to ensure it
2376-783: Is the field of study concerned with constructing mathematical models and quantitative analysis techniques and using computers to analyze and solve scientific problems. A major usage of scientific computing is simulation of various processes, including computational fluid dynamics , physical, electrical, and electronic systems and circuits, as well as societies and social situations (notably war games) along with their habitats, among many others. Modern computers enable optimization of such designs as complete aircraft. Notable in electrical and electronic circuit design are SPICE, as well as software for physical realization of new (or modified) designs. The latter includes essential design software for integrated circuits . Human–computer interaction (HCI)
2448-403: Is the function An optimization problem asks for finding a "best possible" solution among the set of all possible solutions to a search problem. One example is the maximum independent set problem: Optimization problems are represented by their objective function and their constraints. In a function problem a single output (of a total function ) is expected for every input, but the output
2520-505: Is the quick development of this relatively new field requires rapid review and distribution of results, a task better handled by conferences than by journals. Computational problem In theoretical computer science , a computational problem is one that asks for a solution in terms of an algorithm . For example, the problem of factoring is a computational problem that has a solution, as there are many known integer factorization algorithms. A computational problem can be viewed as
2592-457: Is then called the primary sort key , the second the secondary sort key , etc. For example, addresses could be sorted using the city as primary sort key, and the street as secondary sort key. If the sort key values are totally ordered , the sort key defines a weak order of the items: items with the same sort key are equivalent with respect to sorting. See also stable sorting . If different items have different sort key values then this defines
2664-475: The "technocratic paradigm" (which might be found in engineering approaches, most prominently in software engineering), and the "scientific paradigm" (which approaches computer-related artifacts from the empirical perspective of natural sciences , identifiable in some branches of artificial intelligence ). Computer science focuses on methods involved in design, specification, programming, verification, implementation and testing of human-made computing systems. As
2736-570: The 100th anniversary of the invention of the arithmometer, Torres presented in Paris the Electromechanical Arithmometer , a prototype that demonstrated the feasibility of an electromechanical analytical engine, on which commands could be typed and the results printed automatically. In 1937, one hundred years after Babbage's impossible dream, Howard Aiken convinced IBM, which was making all kinds of punched card equipment and
2808-456: The 2nd of the only two designs for mechanical analytical engines in history. In 1914, the Spanish engineer Leonardo Torres Quevedo published his Essays on Automatics , and designed, inspired by Babbage, a theoretical electromechanical calculating machine which was to be controlled by a read-only program. The paper also introduced the idea of floating-point arithmetic . In 1920, to celebrate
2880-631: The Analytical Engine, Ada Lovelace wrote, in one of the many notes she included, an algorithm to compute the Bernoulli numbers , which is considered to be the first published algorithm ever specifically tailored for implementation on a computer. Around 1885, Herman Hollerith invented the tabulator , which used punched cards to process statistical information; eventually his company became part of IBM . Following Babbage, although unaware of his earlier work, Percy Ludgate in 1909 published
2952-559: The Machine Organization department in IBM's main research center in 1959. Concurrency is a property of systems in which several computations are executing simultaneously, and potentially interacting with each other. A number of mathematical models have been developed for general concurrent computation including Petri nets , process calculi and the parallel random access machine model. When multiple computers are connected in
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3024-553: The UK (as in the School of Informatics, University of Edinburgh ). "In the U.S., however, informatics is linked with applied computing, or computing in the context of another domain." A folkloric quotation, often attributed to—but almost certainly not first formulated by— Edsger Dijkstra , states that "computer science is no more about computers than astronomy is about telescopes." The design and deployment of computers and computer systems
3096-516: The accessibility and usability of the system for its intended users. Historical cryptography is the art of writing and deciphering secret messages. Modern cryptography is the scientific study of problems relating to distributed computations that can be attacked. Technologies studied in modern cryptography include symmetric and asymmetric encryption , digital signatures , cryptographic hash functions , key-agreement protocols , blockchain , zero-knowledge proofs , and garbled circuits . A database
3168-433: The application of a fairly broad variety of theoretical computer science fundamentals, in particular logic calculi, formal languages , automata theory , and program semantics , but also type systems and algebraic data types to problems in software and hardware specification and verification. Computer graphics is the study of digital visual contents and involves the synthesis and manipulation of image data. The study
3240-410: The binary number system. In 1820, Thomas de Colmar launched the mechanical calculator industry when he invented 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 eventually gave him the idea of
3312-642: The computational problem in question. However, sometimes not all strings {0, 1} represent valid instances, and one specifies a proper subset of {0, 1} as the set of "valid instances". Computational problems of this type are called promise problems . The following is an example of a (decision) promise problem: Here, the valid instances are those graphs whose maximum independent set size is either at most 5 or at least 10. Decision promise problems are usually represented as pairs of disjoint subsets ( L yes , L no ) of {0, 1} . The valid instances are those in L yes ∪ L no . L yes and L no represent
3384-471: The design and implementation of hardware and software ). Algorithms and data structures are central to computer science. The theory of computation concerns abstract models of computation and general classes of problems that can be solved using them. The fields of cryptography and computer security involve studying the means for secure communication and preventing security vulnerabilities . Computer graphics and computational geometry address
3456-475: 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 . Leibniz may be considered the first computer scientist and information theorist, because of various reasons, including the fact that he documented
3528-583: The discipline of computer science: theory of computation , algorithms and data structures , programming methodology and languages , and computer elements and architecture . In addition to these four areas, CSAB also identifies fields such as software engineering, artificial intelligence, computer networking and communication, database systems, parallel computation, distributed computation, human–computer interaction, computer graphics, operating systems, and numerical and symbolic computation as being important areas of computer science. Theoretical computer science
3600-424: The distinction more a matter of style than of technical capabilities. Conferences are important events for computer science research. During these conferences, researchers from the public and private sectors present their recent work and meet. Unlike in most other academic fields, in computer science, the prestige of conference papers is greater than that of journal publications. One proposed explanation for this
3672-459: The distinction of three separate paradigms in computer science. Peter Wegner argued that those paradigms are science, technology, and mathematics. Peter Denning 's working group argued that they are theory, abstraction (modeling), and design. Amnon H. Eden described them as the "rationalist paradigm" (which treats computer science as a branch of mathematics, which is prevalent in theoretical computer science, and mainly employs deductive reasoning),
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#17328012421943744-520: The expression "automatic information" (e.g. "informazione automatica" in Italian) or "information and mathematics" are often used, e.g. informatique (French), Informatik (German), informatica (Italian, Dutch), informática (Spanish, Portuguese), informatika ( Slavic languages and Hungarian ) or pliroforiki ( πληροφορική , which means informatics) in Greek . Similar words have also been adopted in
3816-510: The extraction of gold from ore, a device called a shaker table uses gravity , vibration , and flow to separate gold from lighter materials in the ore (sorting by size and weight). Sorting is also a naturally occurring process that results in the concentration of ore or sediment . Sorting results from the application of some criterion or differential stressors to a mass to separate it into its components based on some variable quality. Materials that are different, but only slightly so, such as
3888-462: The first programmable mechanical calculator , his Analytical Engine . He started developing this machine in 1834, and "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
3960-488: The first question, computability theory examines which computational problems are solvable on various theoretical models of computation . The second question is addressed by computational complexity theory , which studies the time and space costs associated with different approaches to solving a multitude of computational problems. The famous P = NP? problem, one of the Millennium Prize Problems ,
4032-461: The generation of images. Programming language theory considers different ways to describe computational processes, and database theory concerns the management of repositories of data. Human–computer interaction investigates the interfaces through which humans and computers interact, and software engineering focuses on the design and principles behind developing software. Areas such as operating systems , networks and embedded systems investigate
4104-530: The isotopes of uranium, are very difficult to separate. Optical sorting is an automated process of sorting solid products using cameras and/or lasers and has widespread use in the food industry. Sensor-based sorting is used in mineral processing. Computer science Computer science is the study of computation , information , and automation . Computer science spans theoretical disciplines (such as algorithms , theory of computation , and information theory ) to applied disciplines (including
4176-502: The machine in operation and analyzing it by all analytical and measurement means available. It has since been argued that computer science can be classified as an empirical science since it makes use of empirical testing to evaluate the correctness of programs , but a problem remains in defining the laws and theorems of computer science (if any exist) and defining the nature of experiments in computer science. Proponents of classifying computer science as an engineering discipline argue that
4248-478: The principal focus of computer science is studying the properties of computation in general, while the principal focus of software engineering is the design of specific computations to achieve practical goals, making the two separate but complementary disciplines. The academic, political, and funding aspects of computer science tend to depend on whether a department is formed with a mathematical emphasis or with an engineering emphasis. Computer science departments with
4320-615: The principles and design behind complex systems . Computer architecture describes the construction of computer components and computer-operated equipment. Artificial intelligence and machine learning aim to synthesize goal-orientated processes such as problem-solving, decision-making, environmental adaptation, planning and learning found in humans and animals. Within artificial intelligence, computer vision aims to understand and process image and video data, while natural language processing aims to understand and process textual and linguistic data. The fundamental concern of computer science
4392-409: The relation which consist of all pairs of numbers ( n , p ), where p is a prime factor of n . A counting problem asks for the number of solutions to a given search problem. For example, a counting problem associated with factoring is A counting problem can be represented by a function f from {0, 1} to the nonnegative integers. For a search relation R , the counting problem associated to R
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#17328012421944464-484: The reliability of computational systems is investigated in the same way as bridges in civil engineering and airplanes in aerospace engineering . They also argue that while empirical sciences observe what presently exists, computer science observes what is possible to exist and while scientists discover laws from observation, no proper laws have been found in computer science and it is instead concerned with creating phenomena. Proponents of classifying computer science as
4536-399: The resources (e.g. time, space/memory, energy, circuit depth) it takes to compute (solve) them with various abstract machines . For example, the complexity classes Both instances and solutions are represented by binary strings , namely elements of {0, 1} . For example, natural numbers are usually represented as binary strings using binary encoding . This is important since the complexity
4608-409: The scale of human intelligence. But the automation of evaluative and predictive tasks has been increasingly successful as a substitute for human monitoring and intervention in domains of computer application involving complex real-world data. Computer architecture, or digital computer organization, is the conceptual design and fundamental operational structure of a computer system. It focuses largely on
4680-581: 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
4752-758: The term "computer science" appears in a 1959 article in Communications of the ACM , in which Louis Fein argues for the creation of a Graduate School in Computer Sciences analogous to the creation of Harvard Business School in 1921. Louis justifies the name by arguing that, like management science , the subject is applied and interdisciplinary in nature, while having the characteristics typical of an academic discipline. His efforts, and those of others such as numerical analyst George Forsythe , were rewarded: universities went on to create such departments, starting with Purdue in 1962. Despite its name,
4824-579: The term was the Department of Datalogy at the University of Copenhagen, founded in 1969, with Peter Naur being the first professor in datalogy. The term is used mainly in the Scandinavian countries. An alternative term, also proposed by Naur, is data science ; this is now used for a multi-disciplinary field of data analysis, including statistics and databases. In the early days of computing,
4896-443: The two fields in areas such as mathematical logic , category theory , domain theory , and algebra . The relationship between computer science and software engineering is a contentious issue, which is further muddied by disputes over what the term "software engineering" means, and how computer science is defined. David Parnas , taking a cue from the relationship between other engineering and science disciplines, has claimed that
4968-481: The type of information carrier – whether it is electrical, mechanical or biological. This field plays important role in information theory , telecommunications , information engineering and has applications in medical image computing and speech synthesis , among others. What is the lower bound on the complexity of fast Fourier transform algorithms? is one of the unsolved problems in theoretical computer science . Scientific computing (or computational science)
5040-438: The way by which the central processing unit performs internally and accesses addresses in memory. Computer engineers study computational logic and design of computer hardware, from individual processor components, microcontrollers , personal computers to supercomputers and embedded systems . The term "architecture" in computer literature can be traced to the work of Lyle R. Johnson and Frederick P. Brooks Jr. , members of
5112-440: 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. Ultimately, the close relationship between IBM and Columbia 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
5184-544: Was also 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, with the development of new and more powerful computing machines such as the Atanasoff–Berry computer and ENIAC ,
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