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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 satisfiability prob ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Theoretical Computer Science
Theoretical computer science is a subfield of computer science and mathematics that focuses on the Abstraction, abstract and mathematical foundations of computation. It is difficult to circumscribe the theoretical areas precisely. The Association for Computing Machinery, ACM's 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 Mathematical Theory of Communication, 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 neural networks and para ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Perfect Matching
In graph theory, a perfect matching in a graph is a matching that covers every vertex of the graph. More formally, given a graph with edges and vertices , a perfect matching in is a subset of , such that every vertex in is adjacent to exactly one edge in . The adjacency matrix of a perfect matching is a symmetric permutation matrix. A perfect matching is also called a 1-factor; see Graph factorization for an explanation of this term. In some literature, the term complete matching is used. Every perfect matching is a maximum-cardinality matching, but the opposite is not true. For example, consider the following graphs:Alan Gibbons, Algorithmic Graph Theory, Cambridge University Press, 1985, Chapter 5. : In graph (b) there is a perfect matching (of size 3) since all 6 vertices are matched; in graphs (a) and (c) there is a maximum-cardinality matching (of size 2) which is not perfect, since some vertices are unmatched. A perfect matching is also a minimum-size edge cov ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Theorems In Probability Theory
In mathematics and formal logic, a theorem is a statement that has been proven, or can be proven. The ''proof'' of a theorem is a logical argument that uses the inference rules of a deductive system to establish that the theorem is a logical consequence of the axioms and previously proved theorems. In mainstream mathematics, the axioms and the inference rules are commonly left implicit, and, in this case, they are almost always those of Zermelo–Fraenkel set theory with the axiom of choice (ZFC), or of a less powerful theory, such as Peano arithmetic. Generally, an assertion that is explicitly called a theorem is a proved result that is not an immediate consequence of other known theorems. Moreover, many authors qualify as ''theorems'' only the most important results, and use the terms ''lemma'', ''proposition'' and ''corollary'' for less important theorems. In mathematical logic, the concepts of theorems and proofs have been formalized in order to allow mathematical reason ... [...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 language">Proto-Germanic ''*rīk-'' 'ruler, leader, king' and ''*hardu-'' 'strong, brave, hardy', and it therefore means 'strong in rule'. Nicknames include "Richie", "Dick (nickname), Dick", "Dickon", "Dickie (name), Dickie", "Rich (given name), Rich", "Rick (given name), Rick", "Rico (name), Rico", "Ricky (given name), Ricky", and more. Richard is a common English (the name was introduced into England by the Normans), 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 Portuguese and Spanish "Ricardo" and the Italian "Riccardo" (see comprehensive variant list belo ... [...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 theory, computational complexity and interactive proof systems. Since 2019, he has been at the Illinois Institute of Technology, where he is currently the Dean of the College of Computing. 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 Georgia Institute of Technology School of Computer Science, School of Computer Science. From 1999-2003 he was a Senior Research Scientist at the NEC Research Institute. 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 ... [...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.H.T. Sencar, M. Ramkumar and A.N. Akansu: ''Data Hiding Fundamentals and Applications: Content Security in Digital Multimedia''. Academic Press, San Diego, CA, USA, 2004. It is typically used to identify ownership of the copyright of such a signal. Digital 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. af ... [...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 N ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Avi Wigderson
Avi Wigderson (; born 9 September 1956) is an Israeli computer scientist and mathematician. 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, and distributed computing. Wigderson received the Abel Prize in 2021 for his work in theoretical computer science. He also received the 2023 Turing Award for his contributions to the understanding of randomness in the theory of computation. Early life and studies Avi Wigderson was born in Haifa, Israel, to Holocaust survivors. Wigderson is a graduate of the Hebrew Reali School in Haifa. He began his undergraduate studies at the Technion in 1977 in Haifa, graduating in 1980. Heidelberg Laureate Foundation Portraits, interview with Avi Wigderson, 2017. In the Technion he met his wife Edna. He went on to graduate study at Princeton Univ ... [...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. 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 discriminating the most efficient algorithm in practice among algorithms of equivalent best case complexity (for instance Quicksort#For ... [...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 ... [...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''-by-''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) a perfect matching exists. (This polynomial is not the [...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] |
