|
Divide And Conquer Algorithms
In computer science, divide and conquer is an algorithm design paradigm. A divide-and-conquer algorithm recursively breaks down a problem into two or more sub-problems of the same or related type, until these become simple enough to be solved directly. The solutions to the sub-problems are then combined to give a solution to the original problem. The divide-and-conquer technique is the basis of efficient algorithms for many problems, such as sorting (e.g., quicksort, merge sort), multiplying large numbers (e.g., the Karatsuba algorithm), finding the closest pair of points, syntactic analysis (e.g., top-down parsers), and computing the discrete Fourier transform (FFT). Designing efficient divide-and-conquer algorithms can be difficult. As in mathematical induction, it is often necessary to generalize the problem to make it amenable to a recursive solution. The correctness of a divide-and-conquer algorithm is usually proved by mathematical induction, and its computational cost is o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Computer Science
Computer science is the study of computation, automation, and information. Computer science spans theoretical disciplines (such as algorithms, theory of computation, information theory, and automation) to Applied science, practical disciplines (including the design and implementation of Computer architecture, hardware and Computer programming, software). Computer science is generally considered an area of research, academic research and distinct from computer programming. Algorithms and data structures are central to computer science. The theory of computation concerns abstract models of computation and general classes of computational problem, problems that can be solved using them. The fields of cryptography and computer security involve studying the means for secure communication and for preventing Vulnerability (computing), security vulnerabilities. Computer graphics (computer science), Computer graphics and computational geometry address the generation of images. Progr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Binary Search
In computer science, binary search, also known as half-interval search, logarithmic search, or binary chop, is a search algorithm that finds the position of a target value within a sorted array. Binary search compares the target value to the middle element of the array. If they are not equal, the half in which the target cannot lie is eliminated and the search continues on the remaining half, again taking the middle element to compare to the target value, and repeating this until the target value is found. If the search ends with the remaining half being empty, the target is not in the array. Binary search runs in logarithmic time in the worst case, making O(\log n) comparisons, where n is the number of elements in the array. Binary search is faster than linear search except for small arrays. However, the array must be sorted first to be able to apply binary search. There are specialized data structures designed for fast searching, such as hash tables, that can be searched mor ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cooley–Tukey FFT Algorithm
The Cooley–Tukey algorithm, named after J. W. Cooley and John Tukey, is the most common fast Fourier transform (FFT) algorithm. It re-expresses the discrete Fourier transform (DFT) of an arbitrary composite size N = N_1N_2 in terms of ''N''1 smaller DFTs of sizes ''N''2, recursively, to reduce the computation time to O(''N'' log ''N'') for highly composite ''N'' (smooth numbers). Because of the algorithm's importance, specific variants and implementation styles have become known by their own names, as described below. Because the Cooley–Tukey algorithm breaks the DFT into smaller DFTs, it can be combined arbitrarily with any other algorithm for the DFT. For example, Rader's or Bluestein's algorithm can be used to handle large prime factors that cannot be decomposed by Cooley–Tukey, or the prime-factor algorithm can be exploited for greater efficiency in separating out relatively prime factors. The algorithm, along with its recursive application, was invented by Carl ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Carl Friedrich Gauss
Johann Carl Friedrich Gauss (; german: Gauß ; la, Carolus Fridericus Gauss; 30 April 177723 February 1855) was a German mathematician and physicist who made significant contributions to many fields in mathematics and science. Sometimes referred to as the ''Princeps mathematicorum'' () and "the greatest mathematician since antiquity", Gauss had an exceptional influence in many fields of mathematics and science, and he is ranked among history's most influential mathematicians. Also available at Retrieved 23 February 2014. Comprehensive biographical article. Biography Early years Johann Carl Friedrich Gauss was born on 30 April 1777 in Brunswick (Braunschweig), in the Duchy of Brunswick-Wolfenbüttel (now part of Lower Saxony, Germany), to poor, working-class parents. His mother was illiterate and never recorded the date of his birth, remembering only that he had been born on a Wednesday, eight days before the Feast of the Ascension (which occurs 39 days after Easter). Ga ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Greatest Common Divisor
In mathematics, the greatest common divisor (GCD) of two or more integers, which are not all zero, is the largest positive integer that divides each of the integers. For two integers ''x'', ''y'', the greatest common divisor of ''x'' and ''y'' is denoted \gcd (x,y). For example, the GCD of 8 and 12 is 4, that is, \gcd (8, 12) = 4. In the name "greatest common divisor", the adjective "greatest" may be replaced by "highest", and the word "divisor" may be replaced by "factor", so that other names include highest common factor (hcf), etc. Historically, other names for the same concept have included greatest common measure. This notion can be extended to polynomials (see Polynomial greatest common divisor) and other commutative rings (see below). Overview Definition The ''greatest common divisor'' (GCD) of two nonzero integers and is the greatest positive integer such that is a divisor of both and ; that is, there are integers and such that and , and is the largest s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Euclidean Algorithm
In mathematics, the Euclidean algorithm,Some widely used textbooks, such as I. N. Herstein's ''Topics in Algebra'' and Serge Lang's ''Algebra'', use the term "Euclidean algorithm" to refer to Euclidean division or Euclid's algorithm, is an efficient method for computing the greatest common divisor (GCD) of two integers (numbers), the largest number that divides them both without a remainder. It is named after the ancient Greek mathematician Euclid, who first described it in Euclid's Elements, his ''Elements'' (c. 300 BC). It is an example of an ''algorithm'', a step-by-step procedure for performing a calculation according to well-defined rules, and is one of the oldest algorithms in common use. It can be used to reduce Fraction (mathematics), fractions to their Irreducible fraction, simplest form, and is a part of many other number-theoretic and cryptographic calculations. The Euclidean algorithm is based on the principle that the greatest common divisor of two numbers does not ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Babylonia
Babylonia (; Akkadian: , ''māt Akkadī'') was an ancient Akkadian-speaking state and cultural area based in the city of Babylon in central-southern Mesopotamia (present-day Iraq and parts of Syria). It emerged as an Amorite-ruled state c. 1894 BCE. During the reign of Hammurabi and afterwards, Babylonia was called "the country of Akkad" (''Māt Akkadī'' in Akkadian), a deliberate archaism in reference to the previous glory of the Akkadian Empire. It was often involved in rivalry with the older state of Assyria to the north and Elam to the east in Ancient Iran. Babylonia briefly became the major power in the region after Hammurabi ( fl. c. 1792–1752 BCE middle chronology, or c. 1696–1654 BCE, short chronology) created a short-lived empire, succeeding the earlier Akkadian Empire, Third Dynasty of Ur, and Old Assyrian Empire. The Babylonian Empire rapidly fell apart after the death of Hammurabi and reverted to a small kingdom. Like Assyria, the Babylonian state retained ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
John Mauchly
John William Mauchly (August 30, 1907 – January 8, 1980) was an American physicist who, along with J. Presper Eckert, designed ENIAC, the first general-purpose electronic digital computer, as well as EDVAC, BINAC and UNIVAC I, the first commercial computer made in the United States. Together they started the first computer company, the Eckert–Mauchly Computer Corporation (EMCC), and pioneered fundamental computer concepts, including the stored program, subroutines, and programming languages. Their work, as exposed in the widely read ''First Draft of a Report on the EDVAC'' (1945) and as taught in the Moore School Lectures (1946), influenced an explosion of computer development in the late 1940s all over the world. Biography John W. Mauchly was born on August 30, 1907, to Sebastian and Rachel (Scheidemantel) Mauchly in Cincinnati, Ohio. He moved with his parents and sister, Helen Elizabeth (Betty), at an early age to Chevy Chase, Maryland, when Sebastian Mauchly obtained ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Prune And Search
Prune and search is a method of solving optimization problems suggested by Nimrod Megiddo in 1983.Nimrod Megiddo (1983) Linear-time algorithms for linear programming in R3 and related problems. SIAM J. Comput., 12:759–776 The basic idea of the method is a recursive procedure in which at each step the input size is reduced ("pruned") by a constant factor . As such, it is a form of decrease and conquer algorithm, where at each step the decrease is by a constant factor. Let be the input size, be the time complexity of the whole prune-and-search algorithm, and be the time complexity of the pruning step. Then obeys the following recurrence relation: : T(n) = S(n) + T(n(1-p)). This resembles the recurrence for binary search but has a larger term than the constant term of binary search. In prune and search algorithms S(n) is typically at least linear (since the whole input must be processed). With this assumption, the recurrence has the solution . This can be seen either by ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Geometric Series
In mathematics, a geometric series is the sum of an infinite number of terms that have a constant ratio between successive terms. For example, the series :\frac \,+\, \frac \,+\, \frac \,+\, \frac \,+\, \cdots is geometric, because each successive term can be obtained by multiplying the previous term by 1/2. In general, a geometric series is written as a + ar + ar^2 + ar^3 + ..., where a is the coefficient of each term and r is the common ratio between adjacent terms. The geometric series had an important role in the early development of calculus, is used throughout mathematics, and can serve as an introduction to frequently used mathematical tools such as the Taylor series, the complex Fourier series, and the matrix exponential. The name geometric series indicates each term is the geometric mean of its two neighboring terms, similar to how the name arithmetic series indicates each term is the arithmetic mean of its two neighboring terms. The sequence of geometric series term ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Loop (computing)
In computer science, control flow (or flow of control) is the order in which individual statements, instructions or function calls of an imperative program are executed or evaluated. The emphasis on explicit control flow distinguishes an ''imperative programming'' language from a ''declarative programming'' language. Within an imperative programming language, a ''control flow statement'' is a statement that results in a choice being made as to which of two or more paths to follow. For non-strict functional languages, functions and language constructs exist to achieve the same result, but they are usually not termed control flow statements. A set of statements is in turn generally structured as a block, which in addition to grouping, also defines a lexical scope. Interrupts and signals are low-level mechanisms that can alter the flow of control in a way similar to a subroutine, but usually occur as a response to some external stimulus or event (that can occur asynchronously), ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tail Recursion
In computer science, a tail call is a subroutine call performed as the final action of a procedure. If the target of a tail is the same subroutine, the subroutine is said to be tail recursive, which is a special case of direct recursion. Tail recursion (or tail-end recursion) is particularly useful, and is often easy to optimize in implementations. Tail calls can be implemented without adding a new stack frame to the call stack. Most of the frame of the current procedure is no longer needed, and can be replaced by the frame of the tail call, modified as appropriate (similar to overlay for processes, but for function calls). The program can then jump to the called subroutine. Producing such code instead of a standard call sequence is called tail-call elimination or tail-call optimization. Tail-call elimination allows procedure calls in tail position to be implemented as efficiently as goto statements, thus allowing efficient structured programming. In the words of Guy L. Steele, "i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
