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STCON
In computer science, st-connectivity or STCON is a decision problem asking, for vertices ''s'' and ''t'' in a directed graph, if ''t'' is reachable from ''s''. Formally, the decision problem is given by :. Complexity On a sequential computer, st-connectivity can easily be solved in linear time by either depth-first search or breadth-first search. The interest in this problem in computational complexity concerns its complexity with respect to more limited forms of computation. For instance, the complexity class of problems that can be solved by a non-deterministic Turing machine using only a logarithmic amount of memory is called NL. The st-connectivity problem can be shown to be in NL, as a non-deterministic Turing machine can guess the next node of the path, while the only information which has to be stored is the total length of the path and which node is currently under consideration. The algorithm terminates if either the target node ''t'' is reached, or the length of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
SL (complexity)
In computational complexity theory, SL (Symmetric Logspace or Sym-L) is the complexity class of problems log-space reducible to USTCON (''undirected s-t connectivity''), which is the problem of determining whether there exists a path between two vertices in an undirected graph, otherwise described as the problem of determining whether two vertices are in the same connected component. This problem is also called the undirected reachability problem. It does not matter whether many-one reducibility or Turing reducibility is used. Although originally described in terms of symmetric Turing machines, that equivalent formulation is very complex, and the reducibility definition is what is used in practice. USTCON is a special case of STCON (''directed reachability''), the problem of determining whether a directed path between two vertices in a directed graph exists, which is complete for NL. Because USTCON is SL-complete, most advances that impact USTCON have also impacted SL. Thus they ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Savitch's Theorem
In computational complexity theory, Savitch's theorem, proved by Walter Savitch in 1970, gives a relationship between deterministic and non-deterministic space complexity. It states that for any function f\in\Omega(\log(n)), :\mathsf\left(f\left(n\right)\right) \subseteq \mathsf\left(f\left(n\right)^2\right). In other words, if a nondeterministic Turing machine can solve a problem using f(n) space, a deterministic Turing machine can solve the same problem in the square of that space bound.Arora & Barak (2009) p.86 Although it seems that nondeterminism may produce exponential gains in time (as formalized in the unproven exponential time hypothesis), Savitch's theorem shows that it has a markedly more limited effect on space requirements.Arora & Barak (2009) p.92 Proof The proof relies on an algorithm for STCON, the problem of determining whether there is a path between two vertices in a directed graph, which runs in O\left((\log n)^2\right) space for n vertices. The basic id ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
L (complexity)
In computational complexity theory, L (also known as LSPACE or DLOGSPACE) is the complexity class containing decision problems that can be solved by a deterministic Turing machine using a logarithmic amount of writable memory space., Definition 8.12, p. 295., p. 177. Formally, the Turing machine has two tapes, one of which encodes the input and can only be read, whereas the other tape has logarithmic size but can be read as well as written. Logarithmic space is sufficient to hold a constant number of pointers into the input and a logarithmic number of boolean flags, and many basic logspace algorithms use the memory in this way. Complete problems and logical characterization Every non-trivial problem in L is complete under log-space reductions, so weaker reductions are required to identify meaningful notions of L-completeness, the most common being first-order reductions. A 2004 result by Omer Reingold shows that USTCON, the problem of whether there exists a pat ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Immerman–Szelepcsényi Theorem
In computational complexity theory, the Immerman–Szelepcsényi theorem states that nondeterministic space complexity classes are closed under complementation. It was proven independently by Neil Immerman and Róbert Szelepcsényi in 1987, for which they shared the 1995 Gödel Prize. In its general form the theorem states that NSPACE(''s''(''n'')) = co-NSPACE(''s''(''n'')) for any function ''s''(''n'') ≥ log ''n''. The result is equivalently stated as NL = co-NL; although this is the special case when ''s''(''n'') = log ''n'', it implies the general theorem by a standard padding argument. The result solved the second LBA problem. In other words, if a nondeterministic machine can solve a problem, another machine with the same resource bounds can solve its complement problem (with the ''yes'' and ''no'' answers reversed) in the same asymptotic amount of space. No similar result is known for the time complexity classes, and indeed it is conjectured that NP is not equal to ... [...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 practical disciplines (including the design and implementation of hardware and software). Computer science is generally considered an area of 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 problems that can be solved using them. The fields of cryptography and computer security involve studying the means for secure communication and for preventing security vulnerabilities. Computer graphics and computational geometry address the generation of images. Programming language theory considers different ways to describe computational processes, and database theory concerns the management of repositories ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
First-order Reduction
In computer science, a first-order reduction is a very strong type of reduction between two computational problems in computational complexity theory. A first-order reduction is a reduction where each component is restricted to be in the class FO of problems calculable in first-order logic. Since we have \mbox \subsetneq \mbox, the first-order reductions are stronger reductions than the logspace reductions. Many important complexity classes are closed under first-order reductions, and many of the traditional complete problems are first-order complete as well (Immerman 1999 p. 49-50). For example, ST-connectivity In computer science, st-connectivity or STCON is a decision problem asking, for vertices ''s'' and ''t'' in a directed graph, if ''t'' is reachable from ''s''. Formally, the decision problem is given by :. Complexity On a sequential computer ... is FO-complete for NL, and NL is closed under FO reductions (Immerman 1999, p. 51) (as are P, NP, and most other " ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Graph Connectivity
In mathematics and computer science, connectivity is one of the basic concepts of graph theory: it asks for the minimum number of elements (nodes or edges) that need to be removed to separate the remaining nodes into two or more isolated subgraphs. It is closely related to the theory of network flow problems. The connectivity of a graph is an important measure of its resilience as a network. Connected vertices and graphs In an undirected graph , two '' vertices'' and are called connected if contains a path from to . Otherwise, they are called disconnected. If the two vertices are additionally connected by a path of length , i.e. by a single edge, the vertices are called adjacent. A graph is said to be connected if every pair of vertices in the graph is connected. This means that there is a path between every pair of vertices. An undirected graph that is not connected is called disconnected. An undirected graph ''G'' is therefore disconnected if there exist two vertic ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
P (complexity)
In computational complexity theory, P, also known as PTIME or DTIME(''n''O(1)), is a fundamental complexity class. It contains all decision problems that can be solved by a deterministic Turing machine using a polynomial amount of computation time, or polynomial time. Cobham's thesis holds that P is the class of computational problems that are "efficiently solvable" or " tractable". This is inexact: in practice, some problems not known to be in P have practical solutions, and some that are in P do not, but this is a useful rule of thumb. Definition A language ''L'' is in P if and only if there exists a deterministic Turing machine ''M'', such that * ''M'' runs for polynomial time on all inputs * For all ''x'' in ''L'', ''M'' outputs 1 * For all ''x'' not in ''L'', ''M'' outputs 0 P can also be viewed as a uniform family of boolean circuits. A language ''L'' is in P if and only if there exists a polynomial-time uniform family of boolean circuits \, such that * For all n ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Grace Murray Hopper Award
The Grace Murray Hopper Award (named for computer pioneer RADM Grace Hopper) has been awarded by the Association for Computing Machinery (ACM) since 1971. The award goes to a computer professional who makes a single, significant technical or service contribution at or before age 35. __TOC__ Recipients * 1971 Donald Knuth * 1972 Paul H. Dirksen * 1972 Paul H. Cress * 1973 Lawrence M. Breed * 1973 Richard H. Lathwell * 1973 Roger Moore * 1974 George N. Baird * 1975 Allan L. Scherr * 1976 Edward H. Shortliffe * 1977 ''no award'' * 1978 Ray Kurzweil * 1979 Steve Wozniak * 1980 Robert M. Metcalfe * 1981 Daniel S. Bricklin * 1982 Brian K. Reid * 1983 ''no award'' * 1984 Daniel Henry Holmes Ingalls, Jr. * 1985 Cordell Green * 1986 William Nelson "Bill" Joy * 1987 John Ousterhout * 1988 Guy L. Steele Jr. * 1989 W. Daniel Hillis * 1990 Richard Stallman * 1991 Feng-hsiung Hsu * 1992 ''no award'' * 1993 Bjarne Stroustrup * 1994–1995 ''no award'' * 1996 Shafrira Goldwasser * 199 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Omer Reingold
Omer Reingold ( he, עומר ריינגולד) is an Israeli computer scientist. He is the Rajeev Motwani professor of Computer Science in the Computer Science Department at Stanford University and the director of thSimons Collaboration on the Theory of Algorithmic Fairness He received a PhD in computer science at Weizmann in 1998 under Moni Naor. He received the 2005 Grace Murray Hopper Award for his work in finding a deterministic logarithmic-space algorithm for st-connectivity in undirected graphs. He, along with Avi Wigderson and Salil Vadhan, won the Gödel Prize (2009) for their work on the zig-zag product. He became a Fellow of the Association for Computing Machinery in 2014 ''"For contributions to the study of pseudorandomness, derandomization, and cryptography Cryptography, or cryptology (from grc, , translit=kryptós "hidden, secret"; and ''graphein'', "to write", or ''-logia'', "study", respectively), is the practice and study of techniques for secure com ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Undirected Graph
In discrete mathematics, and more specifically in graph theory, a graph is a structure amounting to a set of objects in which some pairs of the objects are in some sense "related". The objects correspond to mathematical abstractions called '' vertices'' (also called ''nodes'' or ''points'') and each of the related pairs of vertices is called an ''edge'' (also called ''link'' or ''line''). Typically, a graph is depicted in diagrammatic form as a set of dots or circles for the vertices, joined by lines or curves for the edges. Graphs are one of the objects of study in discrete mathematics. The edges may be directed or undirected. For example, if the vertices represent people at a party, and there is an edge between two people if they shake hands, then this graph is undirected because any person ''A'' can shake hands with a person ''B'' only if ''B'' also shakes hands with ''A''. In contrast, if an edge from a person ''A'' to a person ''B'' means that ''A'' owes money to ''B'', th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
NL-complete
In computational complexity theory, NL-complete is a complexity class containing the languages that are complete for NL, the class of decision problems that can be solved by a nondeterministic Turing machine using a logarithmic amount of memory space. The NL-complete languages are the most "difficult" or "expressive" problems in NL. If a deterministic algorithm exists for solving any one of the NL-complete problems in logarithmic memory space, then NL = L. Definitions NL consists of the decision problems that can be solved by a nondeterministic Turing machine with a read-only input tape and a separate read-write tape whose size is limited to be proportional to the logarithm of the input length. Similarly, L consists of the languages that can be solved by a deterministic Turing machine with the same assumptions about tape length. Because there are only a polynomial number of distinct configurations of these machines, both L and NL are subsets of the class P of de ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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