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For test cases, one source [65] uses and Knuth suggested , Another interesting case is the two relatively prime numbers and But "exceptional cases" [66] must be identified and tested. Yes to all. What happens when one number is zero, both numbers are zero? What happens if negative numbers are entered? Fractional numbers? If the input numbers, i. A notable failure due to exceptions is the Ariane 5 Flight rocket failure June 4, Proof of program correctness by use of mathematical induction : Knuth demonstrates the application of mathematical induction to an "extended" version of Euclid's algorithm, and he proposes "a general method applicable to proving the validity of any algorithm".

Elegance compactness versus goodness speed : With only six core instructions, "Elegant" is the clear winner, compared to "Inelegant" at thirteen instructions. Algorithm analysis [69] indicates why this is the case: "Elegant" does two conditional tests in every subtraction loop, whereas "Inelegant" only does one. Can the algorithms be improved? The compactness of "Inelegant" can be improved by the elimination of five steps.

But Chaitin proved that compacting an algorithm cannot be automated by a generalized algorithm; [70] rather, it can only be done heuristically ; i. Observe that steps 4, 5 and 6 are repeated in steps 11, 12 and Comparison with "Elegant" provides a hint that these steps, together with steps 2 and 3, can be eliminated. This reduces the number of core instructions from thirteen to eight, which makes it "more elegant" than "Elegant", at nine steps.

Now "Elegant" computes the example-numbers faster; whether this is always the case for any given A, B, and R, S would require a detailed analysis.

It is frequently important to know how much of a particular resource such as time or storage is theoretically required for a given algorithm. Methods have been developed for the analysis of algorithms to obtain such quantitative answers estimates ; for example, the sorting algorithm above has a time requirement of O n , using the big O notation with n as the length of the list. At all times the algorithm only needs to remember two values: the largest number found so far, and its current position in the input list.

Therefore, it is said to have a space requirement of O 1 , if the space required to store the input numbers is not counted, or O n if it is counted.

Different algorithms may complete the same task with a different set of instructions in less or more time, space, or ' effort ' than others. For example, a binary search algorithm with cost O log n outperforms a sequential search cost O n when used for table lookups on sorted lists or arrays. The analysis, and study of algorithms is a discipline of computer science , and is often practiced abstractly without the use of a specific programming language or implementation.

In this sense, algorithm analysis resembles other mathematical disciplines in that it focuses on the underlying properties of the algorithm and not on the specifics of any particular implementation.

Usually pseudocode is used for analysis as it is the simplest and most general representation. For the solution of a "one off" problem, the efficiency of a particular algorithm may not have significant consequences unless n is extremely large but for algorithms designed for fast interactive, commercial or long life scientific usage it may be critical.

Scaling from small n to Fully Automated Wood Designing Machine Number large n frequently exposes inefficient algorithms that are otherwise benign. Empirical testing is useful because it may uncover unexpected interactions that affect performance. Empirical tests cannot replace formal analysis, though, and are not trivial to perform in a fair manner. To illustrate the potential improvements possible even in well-established algorithms, a recent significant innovation, relating to FFT algorithms used heavily in the field of image processing , can decrease processing time up to 1, times for applications like medical imaging.

Another way of classifying algorithms is by their design methodology or paradigm. There is a certain number of paradigms, each different from the other. Furthermore, each of these categories includes many different types of algorithms. Some common paradigms are:. For optimization problems there is a more specific classification of algorithms; an algorithm for such problems may fall into one or more of the general categories described above as well as into one of the following:.

Every field of science has its own problems and needs efficient algorithms. Related problems in one field are often studied together. Some example classes are search algorithms , sorting algorithms , merge algorithms , numerical algorithms , graph algorithms , string algorithms , computational geometric algorithms , combinatorial algorithms , medical algorithms , machine learning , cryptography , data compression algorithms and parsing techniques.

Fields tend to overlap with each other, and algorithm advances in one field may improve those of other, sometimes completely unrelated, fields. For example, dynamic programming was invented for optimization of resource consumption in industry but is now used in solving a broad range of problems in many fields.

Algorithms can be classified by the amount of time they need to complete compared to their input size:. Some problems may have multiple algorithms of differing complexity, while other problems might have no algorithms or no known efficient algorithms. There are also mappings from some problems to other problems. Owing to this, it was found to be more suitable to classify the problems themselves instead of the algorithms into equivalence classes based on the complexity of the best possible algorithms for them.

Algorithms, by themselves, are not usually patentable. In the United States, a claim consisting solely of simple manipulations of abstract concepts, numbers, or signals does not constitute "processes" USPTO , and hence algorithms are not patentable as in Gottschalk v. However practical applications of algorithms are sometimes patentable. For example, in Diamond v. Diehr , the application of a simple feedback algorithm to aid in the curing of synthetic rubber was deemed patentable.

The patenting of software is highly controversial, and there are highly criticized patents involving algorithms, especially data compression algorithms, such as Unisys ' LZW patent.

Additionally, some cryptographic algorithms have export restrictions see export of cryptography. The earliest evidence of algorithms is found in the Babylonian mathematics of ancient Mesopotamia modern Iraq. A Sumerian clay tablet found in Shuruppak near Baghdad and dated to circa BC described the earliest division algorithm.

Babylonian clay tablets describe and employ algorithmic procedures to compute the time and place of significant astronomical events. Algorithms for arithmetic are also found in ancient Egyptian mathematics , dating back to the Rhind Mathematical Papyrus circa BC.

Two examples are the Sieve of Eratosthenes , which was described in the Introduction to Arithmetic by Nicomachus , [82] [12] : Ch 9. Tally-marks: To keep track of their flocks, their sacks of grain and their money the ancients used tallying: accumulating stones or marks scratched on sticks or making discrete symbols in clay.

Through the Babylonian and Egyptian use of marks and symbols, eventually Roman numerals and the abacus evolved Dilson, p. Tally marks appear prominently in unary numeral system arithmetic used in Turing machine and Post—Turing machine computations.

In Europe, the word "algorithm" was originally used to refer to the sets of rules and techniques used by Al-Khwarizmi to solve algebraic equations, before later being generalized to refer to any set of rules or techniques.

A good century and a half ahead of his time, Leibniz proposed an algebra of logic, an algebra that would specify the rules for manipulating logical concepts in the manner that ordinary algebra specifies the rules for manipulating numbers.

The first cryptographic algorithm for deciphering encrypted code was developed by Al-Kindi , a 9th-century Arab mathematician , in A Manuscript On Deciphering Cryptographic Messages. He gave the first description of cryptanalysis by frequency analysis , the earliest codebreaking algorithm. The clock : Bolter credits the invention of the weight-driven clock as "The key invention [of Europe in the Middle Ages]", in particular, the verge escapement [85] that provides us with the tick and tock of a mechanical clock.

Logical machines — Stanley Jevons ' "logical abacus" and "logical machine" : The technical problem was to reduce Boolean equations when presented in a form similar to what is now known as Karnaugh maps. Jevons describes first a simple "abacus" of "slips of wood furnished with pins, contrived so that any part or class of the [logical] combinations can be picked out mechanically More recently, however, I have reduced the system to a completely mechanical form, and have thus embodied the whole of the indirect process of inference in what may be called a Logical Machine " His machine came equipped with "certain moveable wooden rods" and "at the foot are 21 keys like those of a piano [etc] With this machine he could analyze a " syllogism or any other simple logical argument".

This machine he displayed in before the Fellows of the Royal Society. But not to be outdone he too presented "a plan somewhat analogous, I apprehend, to Prof. Jevon's abacus Jevons's logical machine, the following contrivance may be described.

I prefer to call it merely a logical-diagram machine Jacquard loom, Hollerith punch cards, telegraphy and telephony — the electromechanical relay : Bell and Newell indicate that the Jacquard loom , precursor to Hollerith cards punch cards, , and "telephone switching technologies" were the roots of a tree leading to the development of the first computers. By the late 19th century the ticker tape ca s was in use, as was the use of Hollerith cards in the U.

Then came the teleprinter ca. Telephone-switching networks of electromechanical relays invented was behind the work of George Stibitz , the inventor of the digital adding device.

As he worked in Bell Laboratories, he observed the "burdensome' use of mechanical calculators with gears. When the tinkering was over, Stibitz had constructed a binary adding device".

Davis observes the particular importance of the electromechanical relay with its two "binary states" open and closed :.

Symbols and rules : In rapid succession, the mathematics of George Boole , , Gottlob Frege , and Giuseppe Peano — reduced arithmetic to a sequence of symbols manipulated by rules.

Peano's The principles of arithmetic, presented by a new method was "the first attempt at an axiomatization of mathematics in a symbolic language ". But Heijenoort gives Frege this kudos: Frege's is "perhaps the most important single work ever written in logic.

The paradoxes : At the same time a number of disturbing paradoxes appeared in the literature, in particular, the Burali-Forti paradox , the Russell paradox —03 , and the Richard Paradox. Effective calculability : In an effort to solve the Entscheidungsproblem defined precisely by Hilbert in , mathematicians first set about to define what was meant by an "effective method" or "effective calculation" or "effective calculability" i.

Barkley Rosser 's definition of "effective method" in terms of "a machine". Kleene 's proposal of a precursor to " Church thesis " that he called "Thesis I", [] and a few years later Kleene's renaming his Thesis "Church's Thesis" [] and proposing "Turing's Thesis".

Emil Post described the actions of a "computer" human being as follows:. Alan Turing 's work [] preceded that of Stibitz ; it is unknown whether Stibitz knew of the work of Turing. Turing's biographer believed that Turing's use of a typewriter-like model derived from a youthful interest: "Alan had dreamt of inventing typewriters as a boy; Mrs. Turing had a typewriter, and he could well have begun by asking himself what was meant by calling a typewriter 'mechanical'".

Turing—his model of computation is now called a Turing machine —begins, as did Post, with an analysis of a human computer that he whittles down to a simple set of basic motions and "states of mind".

But he continues a step further and creates a machine as a model of computation of numbers. The most general single operation must, therefore, be taken to be one of the following:.

A few years later, Turing expanded his analysis thesis, definition with this forceful expression of it:. Barkley Rosser defined an 'effective [mathematical] method' in the following manner italicization added :. Rosser's footnote No. Stephen C. Kleene defined as his now-famous "Thesis I" known as the Church—Turing thesis. But he did this in the following context boldface in original :. A number of efforts have been directed toward further refinement of the definition of "algorithm", and activity is on-going because of issues surrounding, in particular, foundations of mathematics especially the Church—Turing thesis and philosophy of mind especially arguments about artificial intelligence.

For more, see Algorithm characterizations. From Wikipedia, the free encyclopedia. For other uses, see Algorithm disambiguation. Unambiguous specification of how to solve a class of problems.

For a detailed presentation of the various points of view on the definition of "algorithm", see Algorithm characterizations.

It has been suggested that this article be split into a new article titled Algorithm design. Discuss March Further information: List of algorithms. Output: The largest number in the list L. Further information: Euclid's algorithm.

Main article: Analysis of algorithms. Main articles: Empirical algorithmics , Profiling computer programming , and Program optimization. Main article: Algorithmic efficiency. See also: List of algorithms. See also: Complexity class and Parameterized complexity. See also: Software patent. Abstract machine Algorithm engineering Algorithm characterizations Algorithmic composition Algorithmic entities Algorithmic synthesis Algorithmic technique Algorithmic topology Garbage in, garbage out Introduction to Algorithms textbook List of algorithms List of algorithm general topics List of important publications in theoretical computer science — Algorithms Regulation of algorithms Theory of computation Computability theory Computational complexity theory.

Math Vault. August 1, Archived from the original on February 28, Retrieved November 14, Merriam-Webster Online Dictionary. Archived from the original on February 14, Rogers opines that: "a computation is carried out in a discrete stepwise fashion, without the use of continuous methods or analogue devices ISBN The Story of Mathematics.

Archived from the original on September 11, Archived from the original on July 25, Archived from the original on August 2, Retrieved May 3, Chambers Dictionary. Archived from the original on March 31, Retrieved December 13, Archived from the original on April 12, University of Indianapolis.

Archived from the original on July 18, Retrieved May 30, The Rosen Publishing Group. Olympiads in Informatics. Archived from the original on August 21, These programs prepare students for jobs in production, supervision, engineering, and management, and are becoming increasingly important as woodworking technology advances.

Education is helpful, but woodworkers are trained primarily on the job, where they learn skills from experienced workers. Beginning workers are given basic tasks, such as placing a piece of wood through a machine and stacking the finished product at the end of the process.

As they gain experience, new woodworkers perform more complex tasks with less supervision. In about 1 month, they learn basic machine operations and job tasks. Becoming a skilled woodworker often takes several months or even years. Skilled workers can read blueprints, set up machines, and plan work sequences. Although not required, becoming certified can demonstrate competence and professionalism. It also may help a candidate advance in the profession.

The Woodwork Career Alliance of North America offers a national certificate program, with five progressive credentials, which adds a level of credibility to the work of woodworkers.

Detail oriented. Woodworkers must pay attention to details in order to meet specifications and to keep themselves safe. Woodworkers must make precise cuts with a variety of hand tools and power tools, so they need a steady hand and good hand-eye coordination.

Math skills. Knowledge of basic math and computer skills are important, particularly for those who work in manufacturing, in which technology continues to advance. Woodworkers need to understand basic geometry in order to visualize how a three-dimensional wooden object, such as a cabinet or piece of furniture, will fit together.

Mechanical skills. The use of hand tools, such as screwdrivers and wrenches, is required to set up, adjust, and calibrate machines. Modern technology systems require woodworkers to be able to use computers and other programmable devices.

Physical stamina. The ability to endure long periods of standing and repetitive movements is crucial for woodworkers, who often stand all day performing many of the same functions. Physical strength. Woodworkers must be strong enough to lift bulky and heavy pieces of wood. Technical skills. Woodworkers must understand and interpret design drawings and technical manuals for a range of products and machines.

The median wage is the wage at which half the workers in an occupation earned more than that amount and half earned less. Some demand for woodworkers is expected in residential and commercial property repairs and renovations.

However, automation, especially the use of computerized numerical controlled machines in wood product manufacturing, should reduce the overall need for these workers over the decade. Woodworkers who know how to create and carry out custom designs on a computer should have the best job opportunities in manufacturing industries. Those who can demonstrate leadership, problem-solving, and advanced math skills should also have the best job prospects.

Some job openings will result from the need to replace those who retire or leave the occupation for another job. Carpenters construct, repair, and install building frameworks and structures made from wood and other materials. Computer programmers write and test code that allows computer applications and software programs to function properly.

They turn the program designs created by software developers and engineers into instructions that a computer can follow. Craft and fine artists use a variety of materials and techniques to create art for sale and exhibition.

Craft artists create handmade objects, such as pottery, glassware, textiles, and other objects that are designed to be functional. Fine artists, including painters, sculptors, and illustrators, create original works of art for their aesthetic value, rather than for a functional one. Ironworkers install structural and reinforcing iron and steel to form and support buildings, bridges, and roads. Jewelers and precious stone and metal workers design, construct, adjust, repair, appraise and sell jewelry.

Machinists and tool and die makers set up and operate a variety of computer-controlled and mechanically controlled machine tools to produce precision metal parts, instruments, and tools. Sheet metal workers fabricate or install products that are made from thin metal sheets, such as ducts used in heating and air conditioning systems. We provide users with convincing, interactive, often immersive experiences in a computer generated synthetic environment.

Some of the recent research topics include image-generation algorithms, geometric and physics-based modeling, computer animation, multi-modal interaction techniques including haptics, audio, and project-based rendering , model and motion acquisition, large-scale data management, analysis, and visualization, graphics hardware, display devices, and their applications.

Traditional computer science has assumed that a single user interacts with a computer program at any one time. A whole range of issues emerge when you decide to violate this fundamental assumption by allowing multiple, distributed users to simultaneously communicate with a program to collaborate with each other. Solutions to these issues come from a variety of fields, making this area — called computer-supported collaborative work — an interdisciplinary field.

At UNC, we are looking at graphics techniques to support telepresence; architectures and abstractions to support scalable, efficient, multi-device collaboration; data mining techniques to make collaboration-related inferences and recommendations; and environments to support collaborative software engineering and distance education.

The goal of computer vision is to extract information from visual data and help computers understand the visual world. Vision algorithms increasingly impact our everyday lives. They help to keep cars safely on the road, enable remote robotics operations in hazardous environments, reconstruct 3D models of cities, and organize photo collections, both personal and across the web. The research in computer vision at UNC spans the breadth of the field, and is a fertile ground for collaborations bringing together geometry and recognition, computer vision and natural language, sensing and display devices, theory, and system architectures for vision processing.

We also investigate frameworks to ensure that algorithms are robust to numerical error and to test geometric consistency for applications across the length scales: from molecular modeling to geographic information systems. By studying geometric problems in an abstract setting, we learn techniques that can be applied in many application domains. Parallel programming models and their embodiment in programming languages and runtime systems.

Programming models for nested parallelism, multi-core and many-core processors, and the parallel memory hierarchy.

Design and analysis of parallel algorithms. This includes head-mounted displays that lets our collaborators walk around inside their molecular data, ship designs, and architecture. It also includes force-feedback systems that let our collaborators touch molecules, feel the brush interacting with the canvas for virtual painting, and feel the 3D models they are sculpting.

It also includes the real-time generation of physically-based surround sound to enable designers to hear how a space will sound and to improve the realism and engagement of game play. Our focus on making complete tools for our collaborators continues to push us to develop both software, hardware graphics engines, trackers, hand-held interaction devices , and new interaction techniques to meet their needs.

Data Mining: Our group has a long history of developing data mining methods and has successfully applied them to solve problems in many other disciplines. Our interests include clustering and subspace clustering in high dimensional data, matrix factorization, graph mining and classification, efficient methods for large scale statistical tests.

Medical Image Analysis research in the Department of Computer Science focuses on many problems of extracting and displaying information from CT, MR, ultrasound, X-ray, nuclear medicine images, and microscopy images to help physicians plan and deliver therapy and diagnose disease. Most of the recent work has been done on 3D medical images, and much of it involves describing the shape and longitudinal changes of anatomic objects of individuals and populations of individuals.

We focus especially on applications in neuroscience, radiation oncology, and musculoskeletal diseases. Natural language processing NLP or computational linguistics is an area in machine learning and artificial intelligence that deals with understanding the meaning of human-style language and its interactions with machines. It also has strong connections with computer vision and machine learning research Prof.

Networking research at UNC is centered on how to provide better or more efficient communication services by making sure that the components of distributed computers and networks all work together seamlessly. Specifically, the research focus of Drs. Jeffay, Smith, and Aikat is on network management software, software for routers and experimental methods for networking research.

Kaur concentrates on the communications software found in PCs as well as large systems and Dr. Reiter and Monrose work on ways to make networks more secure. Nirjon works on low-power networks and radio frequency RF sensing techniques. Modern operating systems struggle to ensure security, efficiency, reliability, and usability in the face of increasingly complex hardware and software.

At a high-level, the OS group explores different system design points through building realistic prototypes and empirical measurement of the impact of these design choices.

The Real-Time Systems Group studies application systems in which timing constraints exist. Usually, such systems are structured as a set of tasks i. To ensure that timing constraints are met, offline validation algorithms are required that check whether deadlines will be met at runtime.

The design of such validation algorithms hinges on how tasks are scheduled and how various complexities such as synchronization are addressed. Accordingly, research involving the design of algorithms for validating, scheduling, and synchronizing real-time task systems has been a continual focus within the group.

Additionally, the development and evaluation of operating system infrastructure, in the form of scheduling and synchronization methods, has been an active topic of investigation. Robotics is increasingly affecting our daily lives and impacting healthcare, transportation, defense, manufacturing, and entertainment.

New robots and algorithms are enabling physicians to perform more precise surgical procedures, assisting individuals with tasks of daily living, enabling autonomous agents to maneuver through cluttered environments, and manipulating materials at large and small scales.

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