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Hopcroft's Problem
In computational geometry, Hopcroft's problem is the problem of testing, for a given system of points and lines in the Euclidean plane, whether at least one of the points lies on at least one of the lines. More generally, one may ask for the number of point–line incidences. Both versions of the problem can be solved in time O(n^), where n is the total number of points and lines. This time bound matches the bound of O(n^) on the total number of point-line incidences given by the Szemerédi–Trotter theorem. Hopcroft's problem is named after John Hopcroft, who posed it in the early 1980s. Its computational complexity is closely connected to the complexity of several other problems in computational geometry, including that of three-dimensional Euclidean minimum spanning trees. Algorithms One way of solving the problem involves a geometric divide-and-conquer algorithm. For a given system of points and lines, it is possible to use the theory of epsilon-nets to subdivide the plan ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Discrete & Computational Geometry
'' Discrete & Computational Geometry'' is a peer-reviewed mathematics journal published quarterly by Springer. Founded in 1986 by Jacob E. Goodman and Richard M. Pollack, the journal publishes articles on discrete geometry and computational geometry. Abstracting and indexing The journal is indexed in: * ''Mathematical Reviews'' * '' Zentralblatt MATH'' * ''Science Citation Index'' * ''Current Contents ''Current Contents'' is a rapid alerting service database from Clarivate, formerly the Institute for Scientific Information and Thomson Reuters. It is published online and in several different printed subject sections. History ''Current Contents ...'' Notable articles Two articles published in ''Discrete & Computational Geometry'', one by Gil Kalai in 1992 with a proof of a subexponential upper bound on the diameter of a polytope and another by Samuel Ferguson in 2006 on the Kepler conjecture on optimal three-dimensional sphere packing, earned their authors the Fulk ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
ACM Transactions On Algorithms
''ACM Transactions on Algorithms'' (''TALG'') is a quarterly peer-reviewed scientific journal covering the field of algorithms. It was established in 2005 and is published by the Association for Computing Machinery. The editor-in-chief is Edith Cohen. The journal was created when the editorial board of the ''Journal of Algorithms'' resigned out of protest to the pricing policies of the publisher, Elsevier. Apart from regular submissions, the journal also invites selected papers from the ''ACM-SIAM Symposium on Discrete Algorithms (SODA)''. Abstracting and indexing The journal is abstracted and indexed in the Science Citation Index Expanded, Current Contents/Engineering, Computing & Technology, and Scopus. According to the ''Journal Citation Reports'', the journal has a 2023 impact factor The impact factor (IF) or journal impact factor (JIF) of an academic journal is a type of journal ranking. Journals with higher impact factor values are considered more prestigious or import ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lower Bound
In mathematics, particularly in order theory, an upper bound or majorant of a subset of some preordered set is an element of that is every element of . Dually, a lower bound or minorant of is defined to be an element of that is less than or equal to every element of . A set with an upper (respectively, lower) bound is said to be bounded from above or majorized (respectively bounded from below or minorized) by that bound. The terms bounded above (bounded below) are also used in the mathematical literature for sets that have upper (respectively lower) bounds. Examples For example, is a lower bound for the set (as a subset of the integers or of the real numbers, etc.), and so is . On the other hand, is not a lower bound for since it is not smaller than every element in . and other numbers ''x'' such that would be an upper bound for ''S''. The set has as both an upper bound and a lower bound; all other numbers are either an upper bound or a lower bound for tha ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quantum Algorithm
In quantum computing, a quantum algorithm is an algorithm that runs on a realistic model of quantum computation, the most commonly used model being the quantum circuit model of computation. A classical (or non-quantum) algorithm is a finite sequence of instructions, or a step-by-step procedure for solving a problem, where each step or instruction can be performed on a classical computer. Similarly, a quantum algorithm is a step-by-step procedure, where each of the steps can be performed on a quantum computer. Although all classical algorithms can also be performed on a quantum computer, the term quantum algorithm is generally reserved for algorithms that seem inherently quantum, or use some essential feature of quantum computation such as quantum superposition or quantum entanglement. Problems that are undecidable using classical computers remain undecidable using quantum computers. What makes quantum algorithms interesting is that they might be able to solve some problems fa ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Brute-force Search
In computer science, brute-force search or exhaustive search, also known as generate and test, is a very general problem-solving technique and algorithmic paradigm that consists of Iteration#Computing, systematically checking all possible candidates for whether or not each candidate satisfies the problem's statement. A brute-force algorithm that finds the divisors of a natural number ''n'' would enumerate all integers from 1 to n, and check whether each of them divides ''n'' without remainder. A brute-force approach for the eight queens puzzle would examine all possible arrangements of 8 pieces on the 64-square chessboard and for each arrangement, check whether each (queen) piece can attack any other. While a brute-force search is simple to implement and will always find a solution if it exists, implementation costs are proportional to the number of candidate solutionswhich in many practical problems tends to grow very quickly as the size of the problem increases (#Combinatorial ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Jiří Matoušek (mathematician)
Jiří (Jirka) Matoušek (10 March 1963 – 9 March 2015) was a Czech mathematician working in computational geometry and algebraic topology. He was a professor at Charles University in Prague and the author of several textbooks and research monographs. Biography Matoušek was born in Prague. In 1986, he received his Master's degree at Charles University under Miroslav Katětov. From 1986 until his death he was employed at the Department of Applied Mathematics of Charles University, holding a professor position since 2000. He was also a visiting and later full professor at ETH Zurich. In 1996, he won the European Mathematical Society prize and in 2000 he won the Scientist award of the Learned Society of the Czech Republic. In 1998 he was an Invited Speaker of the International Congress of Mathematicians in Berlin. He became a fellow of the Learned Society of the Czech Republic in 2005. Matoušek's paper on computational aspects of algebraic topology won the Best Paper award ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
David Eppstein
David Arthur Eppstein (born 1963) is an American computer scientist and mathematician. He is a distinguished professor of computer science at the University of California, Irvine. He is known for his work in computational geometry, graph algorithms, and recreational mathematics. In 2011, he was named an ACM Fellow. Biography Born in Windsor, England, in 1963, Eppstein received a B.S. in mathematics from Stanford University in 1984, and later an M.S. (1985) and Ph.D. (1989) in computer science from Columbia University, after which he took a postdoctoral position at Xerox's Palo Alto Research Center. He joined the UC Irvine faculty in 1990, and was co-chair of the Computer Science Department there from 2002 to 2005. In 2014, he was named a Chancellor's Professor. In October 2017, Eppstein was one of 396 members elected as fellows of the American Association for the Advancement of Science. Eppstein is an amateur digital photographer. He is also a Wikipedia editor and admi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Algebraic Decision Tree
In computational complexity theory, the decision tree model is the model of computation in which an algorithm can be considered to be a decision tree, i.e. a sequence of ''queries'' or ''tests'' that are done adaptively, so the outcome of previous tests can influence the tests performed next. Typically, these tests have a small number of outcomes (such as a yes–no question) and can be performed quickly (say, with unit computational cost), so the worst-case time complexity of an algorithm in the decision tree model corresponds to the depth of the corresponding tree. This notion of computational complexity of a problem or an algorithm in the decision tree model is called its decision tree complexity or query complexity. Decision tree models are instrumental in establishing lower bounds for the complexity of certain classes of computational problems and algorithms. Several variants of decision tree models have been introduced, depending on the computational model and type of quer ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Recursion (computer Science)
In computer science, recursion is a method of solving a computational problem where the solution depends on solutions to smaller instances of the same problem. Recursion solves such recursion, recursive problems by using function (computer science), functions that call themselves from within their own code. The approach can be applied to many types of problems, and recursion is one of the central ideas of computer science. Most computer programming languages support recursion by allowing a function to call itself from within its own code. Some functional programming languages (for instance, Clojure) do not define any looping constructs but rely solely on recursion to repeatedly call code. It is proved in computability theory that these recursive-only languages are Turing complete; this means that they are as powerful (they can be used to solve the same problems) as imperative languages based on control structures such as and . Repeatedly calling a function from within itse ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Euclidean Plane
In mathematics, a Euclidean plane is a Euclidean space of Two-dimensional space, dimension two, denoted \textbf^2 or \mathbb^2. It is a geometric space in which two real numbers are required to determine the position (geometry), position of each point (mathematics), point. It is an affine space, which includes in particular the concept of parallel lines. It has also measurement, metrical properties induced by a Euclidean distance, distance, which allows to define circles, and angle, angle measurement. A Euclidean plane with a chosen Cartesian coordinate system is called a ''Cartesian plane''. The set \mathbb^2 of the ordered pairs of real numbers (the real coordinate plane), equipped with the dot product, is often called ''the'' Euclidean plane or ''standard Euclidean plane'', since every Euclidean plane is isomorphic to it. History Books I through IV and VI of Euclid's Elements dealt with two-dimensional geometry, developing such notions as similarity of shapes, the Pythagor ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
