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Isolation Lemma
In theoretical computer science, the term isolation lemma (or isolating lemma) refers to randomized algorithms that reduce the number of solutions to a problem to one, should a solution exist. This is achieved by constructing random constraints such that, with non-negligible probability, exactly one solution satisfies these additional constraints if the solution space is not empty. Isolation lemmas have important applications in computer science, such as the Valiant–Vazirani theorem and Toda's theorem in computational complexity theory. The first isolation lemma was introduced by , albeit not under that name. Their isolation lemma chooses a random number of random hyperplanes, and has the property that, with non-negligible probability, the intersection of any fixed non-empty solution space with the chosen hyperplanes contains exactly one element. This suffices to show the Valiant–Vazirani theorem: there exists a randomized polynomial-time reduction from the Boolean satisfiabilit ... [...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] |
Tutte Matrix
In graph theory, the Tutte matrix ''A'' of a graph ''G'' = (''V'', ''E'') is a matrix used to determine the existence of a perfect matching: that is, a set of edges which is incident with each vertex exactly once. If the set of vertices is V = \ then the Tutte matrix is an ''n'' × ''n'' matrix A with entries : A_ = \begin x_\;\;\mbox\;(i,j) \in E \mbox ij\\ 0\;\;\;\;\mbox \end where the ''x''''ij'' are indeterminates. The determinant of this skew-symmetric matrix is then a polynomial (in the variables ''xij'', ''i < j'' ): this coincides with the square of the pfaffian of the matrix ''A'' and is non-zero (as a polynomial) if and only if a perfect matching exists. (This polynomial is not the Tutte polynomial of ''G''.) The Tutte matrix is named after |
Probability Theorems
Probability is the branch of mathematics concerning numerical descriptions of how likely an event is to occur, or how likely it is that a proposition is true. The probability of an event is a number between 0 and 1, where, roughly speaking, 0 indicates impossibility of the event and 1 indicates certainty."Kendall's Advanced Theory of Statistics, Volume 1: Distribution Theory", Alan Stuart and Keith Ord, 6th Ed, (2009), .William Feller, ''An Introduction to Probability Theory and Its Applications'', (Vol 1), 3rd Ed, (1968), Wiley, . The higher the probability of an event, the more likely it is that the event will occur. A simple example is the tossing of a fair (unbiased) coin. Since the coin is fair, the two outcomes ("heads" and "tails") are both equally probable; the probability of "heads" equals the probability of "tails"; and since no other outcomes are possible, the probability of either "heads" or "tails" is 1/2 (which could also be written as 0.5 or 50%). These conce ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Richard J
Richard is a male given name. It originates, via Old French, from Old Frankish and is a compound of the words descending from Proto-Germanic ''*rīk-'' 'ruler, leader, king' and ''*hardu-'' 'strong, brave, hardy', and it therefore means 'strong in rule'. Nicknames include "Richie", "Dick", "Dickon", " Dickie", "Rich", "Rick", "Rico", "Ricky", and more. Richard is a common English, German and French male name. It's also used in many more languages, particularly Germanic, such as Norwegian, Danish, Swedish, Icelandic, and Dutch, as well as other languages including Irish, Scottish, Welsh and Finnish. Richard is cognate with variants of the name in other European languages, such as the Swedish "Rickard", the Catalan "Ricard" and the Italian "Riccardo", among others (see comprehensive variant list below). People named Richard Multiple people with the same name * Richard Andersen (other) * Richard Anderson (other) * Richard Cartwright (other) * Ri ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lance Fortnow
Lance Jeremy Fortnow (born August 15, 1963) is a computer scientist known for major results in computational complexity and interactive proof systems. He is currently Dean of the College of Computing at the Illinois Institute of Technology. Biography Lance Fortnow received a doctorate in applied mathematics from MIT in 1989, supervised by Michael Sipser. Since graduation, he has been on the faculty of the University of Chicago (1989–1999, 2003–2007), Northwestern University (2008–2012) and the Georgia Institute of Technology (2012–2019) as chair of the School of Computer Science. Fortnow was the founding editor-in-chief of the journal ''ACM Transactions on Computation Theory'' in 2009. He was the chair of ACM SIGACT and succeeded by Paul Beame. He was the chair of the IEEE Conference on Computational Complexity from 2000 to 2006. In 2002, he began one of the first blogs devoted to theoretical computer science and has written for it since then. Since 2007, he has had a co-b ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Digital Watermarking
A digital watermark is a kind of marker covertly embedded in a noise-tolerant signal such as audio, video or image data. It is typically used to identify ownership of the copyright of such signal. "Watermarking" is the process of hiding digital information in a carrier signal; the hidden information should,Ingemar J. Cox: ''Digital watermarking and steganography''. Morgan Kaufmann, Burlington, MA, USA, 2008 but does not need to, contain a relation to the carrier signal. Digital watermarks may be used to verify the authenticity or integrity of the carrier signal or to show the identity of its owners. It is prominently used for tracing copyright infringements and for banknote authentication. Like traditional physical watermarks, digital watermarks are often only perceptible under certain conditions, e.g. after using some algorithm.Frank Y. Shih: ''Digital watermarking and steganography: fundamentals and techniques''. Taylor & Francis, Boca Raton, FL, USA, 2008 If a digital watermark ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
NL (complexity)
In computational complexity theory, NL (Nondeterministic Logarithmic-space) is the complexity class containing decision problems that can be solved by a nondeterministic Turing machine using a logarithmic amount of memory space. NL is a generalization of L, the class for logspace problems on a deterministic Turing machine. Since any deterministic Turing machine is also a nondeterministic Turing machine, we have that L is contained in NL. NL can be formally defined in terms of the computational resource nondeterministic space (or NSPACE) as NL = NSPACE(log ''n''). Important results in complexity theory allow us to relate this complexity class with other classes, telling us about the relative power of the resources involved. Results in the field of algorithms, on the other hand, tell us which problems can be solved with this resource. Like much of complexity theory, many important questions about NL are still open (see Unsolved problems in computer science). Occasionally NL ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Avi Wigderson
Avi Wigderson ( he, אבי ויגדרזון; born 9 September 1956) is an Israeli mathematician and computer scientist. He is the Herbert H. Maass Professor in the school of mathematics at the Institute for Advanced Study in Princeton, New Jersey, United States of America. His research interests include complexity theory, parallel algorithms, graph theory, cryptography, distributed computing, and neural networks. Wigderson received the Abel Prize in 2021 for his work in theoretical computer science. Biography Avi Wigderson was born in Haifa, Israel, to Holocaust survivors. Wigderson is a graduate of the Hebrew Reali School in Haifa, and did his undergraduate studies at the Technion in Haifa, Israel, graduating in 1980, and went on to graduate study at Princeton University. He received his PhD in computer science in 1983 after completing a doctoral dissertation, titled "Studies in computational complexity", under the supervision of Richard Lipton. After short-term positions at t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Average-case Complexity
In computational complexity theory, the average-case complexity of an algorithm is the amount of some computational resource (typically time) used by the algorithm, averaged over all possible inputs. It is frequently contrasted with worst-case complexity which considers the maximal complexity of the algorithm over all possible inputs. There are three primary motivations for studying average-case complexity.O. Goldreich and S. Vadhan, Special issue on worst-case versus average-case complexity, Comput. Complex. 16, 325–330, 2007. First, although some problems may be intractable in the worst-case, the inputs which elicit this behavior may rarely occur in practice, so the average-case complexity may be a more accurate measure of an algorithm's performance. Second, average-case complexity analysis provides tools and techniques to generate hard instances of problems which can be utilized in areas such as cryptography and derandomization. Third, average-case complexity allows discrimin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Clique Problem
In computer science, the clique problem is the computational problem of finding cliques (subsets of vertices, all adjacent to each other, also called complete subgraphs) in a graph. It has several different formulations depending on which cliques, and what information about the cliques, should be found. Common formulations of the clique problem include finding a maximum clique (a clique with the largest possible number of vertices), finding a maximum weight clique in a weighted graph, listing all maximal cliques (cliques that cannot be enlarged), and solving the decision problem of testing whether a graph contains a clique larger than a given size. The clique problem arises in the following real-world setting. Consider a social network, where the graph's vertices represent people, and the graph's edges represent mutual acquaintance. Then a clique represents a subset of people who all know each other, and algorithms for finding cliques can be used to discover these groups of m ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Joel Spencer
Joel Spencer (born April 20, 1946) is an American mathematician. He is a combinatorialist who has worked on probabilistic methods in combinatorics and on Ramsey theory. He received his doctorate from Harvard University in 1970, under the supervision of Andrew Gleason. He is currently () a professor at the Courant Institute of Mathematical Sciences of New York University. Spencer's work was heavily influenced by Paul Erdős, with whom he coauthored many papers (giving him an Erdős number of 1). In 1963, while studying at the Massachusetts Institute of Technology, Spencer became a Putnam Fellow. In 1984 Spencer received a Lester R. Ford Award. He was an Erdős Lecturer at Hebrew University of Jerusalem in 2001. In 2012 he became a fellow of the American Mathematical Society. He was elected as a fellow of the Society for Industrial and Applied Mathematics in 2017, "for contributions to discrete mathematics and theory of computing, particularly random graphs and networks, Ramsey ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Randomized Algorithm
A randomized algorithm is an algorithm that employs a degree of randomness as part of its logic or procedure. The algorithm typically uses uniformly random bits as an auxiliary input to guide its behavior, in the hope of achieving good performance in the "average case" over all possible choices of random determined by the random bits; thus either the running time, or the output (or both) are random variables. One has to distinguish between algorithms that use the random input so that they always terminate with the correct answer, but where the expected running time is finite (Las Vegas algorithms, for example Quicksort), and algorithms which have a chance of producing an incorrect result (Monte Carlo algorithms, for example the Monte Carlo algorithm for the MFAS problem) or fail to produce a result either by signaling a failure or failing to terminate. In some cases, probabilistic algorithms are the only practical means of solving a problem. In common practice, randomized algor ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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