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Computational Complexity Of Matrix Multiplication
In theoretical computer science, the computational complexity of matrix multiplication dictates how quickly the operation of matrix multiplication can be performed. Matrix multiplication algorithms are a central subroutine in theoretical and numerical algorithms for numerical linear algebra and optimization, so finding the right amount of time it should take is of major practical relevance. Directly applying the mathematical definition of matrix multiplication gives an algorithm that requires field operations to multiply two matrices over that field ( in big O notation). Surprisingly, algorithms exist that provide better running times than this straightforward "schoolbook algorithm". The first to be discovered was Strassen's algorithm, devised by Volker Strassen in 1969 and often referred to as "fast matrix multiplication". The optimal number of field operations needed to multiply two square matrices up to constant factors is still unknown. This is a major open question in t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Theoretical Computer Science
Theoretical computer science (TCS) is a subset of general computer science and mathematics that focuses on mathematical aspects of computer science such as the theory of computation, lambda calculus, and type theory. It is difficult to circumscribe the theoretical areas precisely. The Association for Computing Machinery, ACM's ACM SIGACT, Special Interest Group on Algorithms and Computation Theory (SIGACT) provides the following description: History While logical inference and mathematical proof had existed previously, in 1931 Kurt Gödel proved with his incompleteness theorem that there are fundamental limitations on what statements could be proved or disproved. Information theory was added to the field with a 1948 mathematical theory of communication by Claude Shannon. In the same decade, Donald Hebb introduced a mathematical model of Hebbian learning, learning in the brain. With mounting biological data supporting this hypothesis with some modification, the fields of n ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Computational Complexity
In computer science, the computational complexity or simply complexity of an algorithm is the amount of resources required to run it. Particular focus is given to computation time (generally measured by the number of needed elementary operations) and memory storage requirements. The complexity of a problem is the complexity of the best algorithms that allow solving the problem. The study of the complexity of explicitly given algorithms is called analysis of algorithms, while the study of the complexity of problems is called computational complexity theory. Both areas are highly related, as the complexity of an algorithm is always an upper bound on the complexity of the problem solved by this algorithm. Moreover, for designing efficient algorithms, it is often fundamental to compare the complexity of a specific algorithm to the complexity of the problem to be solved. Also, in most cases, the only thing that is known about the complexity of a problem is that it is lower than the c ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Robert Kleinberg
Robert David Kleinberg (also referred to as Bobby Kleinberg) is an American theoretical computer scientist and professor of Computer Science at Cornell University. Early life Robert Kleinberg was one of the finalists at the 1989 Mathcounts. He was a member of the 1991 and 1992 USA teams in the International Mathematical Olympiad, winning a silver medal and a gold medal, respectively. He was also a Putnam Fellow in 1996. He graduated from Iroquois Central High School in Elma, NY, where he was valedictorian. He is the younger brother of fellow Cornell computer scientist Jon Kleinberg. Research Robert Kleinberg is known for his research work on group theoretic algorithms for matrix multiplication, online learning, network coding and greedy embedding, social networks and algorithmic game theory. Career Robert Kleinberg received a B.A. in mathematics from Cornell University in 1997 and a Ph.D. in mathematics under Tom Leighton from MIT The Massachusetts Institute of Technolo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Henry Cohn
Henry Cohn is an American mathematician. He is a principal researcher at Microsoft Research and an adjunct professor at MIT. In collaboration with Abhinav Kumar, Stephen D. Miller, Danylo Radchenko, and Maryna Viazovska, he solved the sphere packing problem in 24 dimensions. Cohn graduated from Harvard University in 2000 with a doctorate in mathematics. Cohn was an Erdős Lecturer at Hebrew University of Jerusalem in 2008. In 2016, he became a Fellow of the American Mathematical Society "for contributions to discrete mathematics, including applications to computer science and physics." In 2018, he was awarded the Levi L. Conant Prize for his article “A Conceptual Breakthrough in Sphere Packing,” published in 2017 in the ''Notices of the AMS ''Notices of the American Mathematical Society'' is the membership journal of the American Mathematical Society (AMS), published monthly except for the combined June/July issue. The first volume appeared in 1953. Each issue of the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Symposium On Theory Of Computing
The Annual ACM Symposium on Theory of Computing (STOC) is an academic conference in the field of theoretical computer science. STOC has been organized annually since 1969, typically in May or June; the conference is sponsored by the Association for Computing Machinery special interest group SIGACT. Acceptance rate of STOC, averaged from 1970 to 2012, is 31%, with the rate of 29% in 2012. As writes, STOC and its annual IEEE counterpart FOCS (the Symposium on Foundations of Computer Science) are considered the two top conferences in theoretical computer science, considered broadly: they “are forums for some of the best work throughout theory of computing that promote breadth among theory of computing researchers and help to keep the community together.” includes regular attendance at STOC and FOCS as one of several defining characteristics of theoretical computer scientists. Awards The Gödel Prize for outstanding papers in theoretical computer science is presented alternately a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Shmuel Winograd
__NOTOC__ Shmuel Winograd ( he, שמואל וינוגרד; January 4, 1936 – March 25, 2019) was an Israeli-American computer scientist, noted for his contributions to computational complexity. He has proved several major results regarding the computational aspects of arithmetic; his contributions include the Coppersmith–Winograd algorithm and an algorithm for the fast Fourier transform. Winograd studied Electrical Engineering at the Massachusetts Institute of Technology, receiving his B.S. and M.S. degrees in 1959. He received his Ph.D. from the Courant Institute of Mathematical Sciences at New York University in 1968. He joined the research staff at IBM in 1961, eventually becoming director of the Mathematical Sciences Department there from 1970 to 1974 and 1980 to 1994. Honors *IBM Fellow (1972) *Fellow of the Institute of Electrical and Electronics Engineers (1974) * W. Wallace McDowell Award (1974) *Member, National Academy of Sciences (1978) *Member, American Academy ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Don Coppersmith
Don Coppersmith (born 1950) is a cryptographer and mathematician. He was involved in the design of the Data Encryption Standard block cipher at IBM, particularly the design of the S-boxes, strengthening them against differential cryptanalysis. He also improved the quantum Fourier transform discovered by Peter Shor in the same year (1994). He has also worked on algorithms for computing discrete logarithms, the cryptanalysis of RSA, methods for rapid matrix multiplication (see Coppersmith–Winograd algorithm) and IBM's MARS cipher. Don is also a co-designer of the SEAL and Scream ciphers. In 1972, Coppersmith obtained a bachelor's degree in mathematics at the Massachusetts Institute of Technology, and a Masters and Ph.D. in mathematics from Harvard University in 1975 and 1977 respectively. He was a Putnam Fellow each year from 1968–1971, becoming the first four-time Putnam Fellow in history. In 1998, he started ''Ponder This'', an online monthly column on mathematical puz ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Finite Field
In mathematics, a finite field or Galois field (so-named in honor of Évariste Galois) is a field that contains a finite number of elements. As with any field, a finite field is a set on which the operations of multiplication, addition, subtraction and division are defined and satisfy certain basic rules. The most common examples of finite fields are given by the integers mod when is a prime number. The ''order'' of a finite field is its number of elements, which is either a prime number or a prime power. For every prime number and every positive integer there are fields of order p^k, all of which are isomorphic. Finite fields are fundamental in a number of areas of mathematics and computer science, including number theory, algebraic geometry, Galois theory, finite geometry, cryptography and coding theory. Properties A finite field is a finite set which is a field; this means that multiplication, addition, subtraction and division (excluding division by zero) are ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cambridge University Press
Cambridge University Press is the university press of the University of Cambridge. Granted letters patent by Henry VIII of England, King Henry VIII in 1534, it is the oldest university press A university press is an academic publishing house specializing in monographs and scholarly journals. Most are nonprofit organizations and an integral component of a large research university. They publish work that has been reviewed by schola ... in the world. It is also the King's Printer. Cambridge University Press is a department of the University of Cambridge and is both an academic and educational publisher. It became part of Cambridge University Press & Assessment, following a merger with Cambridge Assessment in 2021. With a global sales presence, publishing hubs, and offices in more than 40 Country, countries, it publishes over 50,000 titles by authors from over 100 countries. Its publishing includes more than 380 academic journals, monographs, reference works, school and uni ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Basic Linear Algebra Subprograms
Basic Linear Algebra Subprograms (BLAS) is a specification that prescribes a set of low-level routines for performing common linear algebra operations such as vector addition, scalar multiplication, dot products, linear combinations, and matrix multiplication. They are the ''de facto'' standard low-level routines for linear algebra libraries; the routines have bindings for both C ("CBLAS interface") and Fortran ("BLAS interface"). Although the BLAS specification is general, BLAS implementations are often optimized for speed on a particular machine, so using them can bring substantial performance benefits. BLAS implementations will take advantage of special floating point hardware such as vector registers or SIMD instructions. It originated as a Fortran library in 1979* and its interface was standardized by the BLAS Technical (BLAST) Forum, whose latest BLAS report can be found on the netlib website. This Fortran library is known as the ''reference implementation'' (sometimes co ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Numerical Stability
In the mathematical subfield of numerical analysis, numerical stability is a generally desirable property of numerical algorithms. The precise definition of stability depends on the context. One is numerical linear algebra and the other is algorithms for solving ordinary and partial differential equations by discrete approximation. In numerical linear algebra, the principal concern is instabilities caused by proximity to singularities of various kinds, such as very small or nearly colliding eigenvalues. On the other hand, in numerical algorithms for differential equations the concern is the growth of round-off errors and/or small fluctuations in initial data which might cause a large deviation of final answer from the exact solution. Some numerical algorithms may damp out the small fluctuations (errors) in the input data; others might magnify such errors. Calculations that can be proven not to magnify approximation errors are called ''numerically stable''. One of the common task ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
