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Fine-grained Reduction
In computational complexity theory, a fine-grained reduction is a transformation from one computational problem to another, used to relate the difficulty of improving the time bounds for the two problems. Intuitively, it provides a method for solving one problem efficiently by using the solution to the other problem as a subroutine. If problem A can be solved in time a(n) and problem B can be solved in time b(n), then the existence of an (a,b)-reduction from problem A to problem B implies that any significant speedup for problem B would also lead to a speedup for problem A. Definition Let A and B be computational problems, specified as the desired output for each possible input. Let a and b both be time-constructible functions that take an integer argument n and produce an integer result. Usually, a and b are the time bounds for known or naive algorithms for the two problems, and often they are monomials such as n^2. Then A is said to be (a,b)-reducible to B if, for every real numb ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Computational Complexity Theory
In theoretical computer science and mathematics, computational complexity theory focuses on classifying computational problems according to their resource usage, and relating these classes to each other. A computational problem is a task solved by a computer. A computation problem is solvable by mechanical application of mathematical steps, such as an algorithm. A problem is regarded as inherently difficult if its solution requires significant resources, whatever the algorithm used. The theory formalizes this intuition, by introducing mathematical models of computation to study these problems and quantifying their computational complexity, i.e., the amount of resources needed to solve them, such as time and storage. Other measures of complexity are also used, such as the amount of communication (used in communication complexity), the number of gates in a circuit (used in circuit complexity) and the number of processors (used in parallel computing). One of the roles of computationa ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Subroutine
In computer programming, a function or subroutine is a sequence of program instructions that performs a specific task, packaged as a unit. This unit can then be used in programs wherever that particular task should be performed. Functions may be defined within programs, or separately in libraries that can be used by many programs. In different programming languages, a function may be called a routine, subprogram, subroutine, method, or procedure. Technically, these terms all have different definitions, and the nomenclature varies from language to language. The generic umbrella term ''callable unit'' is sometimes used. A function is often coded so that it can be started several times and from several places during one execution of the program, including from other functions, and then branch back (''return'') to the next instruction after the ''call'', once the function's task is done. The idea of a subroutine was initially conceived by John Mauchly during his work on ENIAC, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Constructible Function
In complexity theory, a time-constructible function is a function ''f'' from natural numbers to natural numbers with the property that ''f''(''n'') can be constructed from ''n'' by a Turing machine in the time of order ''f''(''n''). The purpose of such a definition is to exclude functions that do not provide an upper bound on the runtime of some Turing machine. Time-constructible definitions There are two different definitions of a time-constructible function. In the first definition, a function ''f'' is called time-constructible if there exists a positive integer ''n''0 and Turing machine ''M'' which, given a string 1''n'' consisting of ''n'' ones, stops after exactly ''f''(''n'') steps for all ''n'' ≥ ''n''0. In the second definition, a function ''f'' is called time-constructible if there exists a Turing machine ''M'' which, given a string 1''n'', outputs the binary representation of ''f''(''n'') in '' O''(''f''(''n'')) time (a unary representation may be used instead, since th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Monomial
In mathematics, a monomial is, roughly speaking, a polynomial which has only one term. Two definitions of a monomial may be encountered: # A monomial, also called power product, is a product of powers of variables with nonnegative integer exponents, or, in other words, a product of variables, possibly with repetitions. For example, x^2yz^3=xxyzzz is a monomial. The constant 1 is a monomial, being equal to the empty product and to x^0 for any variable x. If only a single variable x is considered, this means that a monomial is either 1 or a power x^n of x, with n a positive integer. If several variables are considered, say, x, y, z, then each can be given an exponent, so that any monomial is of the form x^a y^b z^c with a,b,c non-negative integers (taking note that any exponent 0 makes the corresponding factor equal to 1). # A monomial is a monomial in the first sense multiplied by a nonzero constant, called the coefficient of the monomial. A monomial in the first sense is a special c ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Virginia Vassilevska Williams
Virginia Vassilevska Williams (née Virginia Panayotova Vassilevska) is a theoretical computer scientist and mathematician known for her research in computational complexity theory and algorithms. She is currently the Steven and Renee Finn Career Development Associate Professor of Electrical Engineering and Computer Science at the Massachusetts Institute of Technology. She is notable for her breakthrough results in fast matrix multiplication, for her work on dynamic algorithms, and for helping to develop the field of fine-grained complexity. Education and career Williams is originally from Bulgaria, and attended a German-language high school in Sofia. She graduated from the California Institute of Technology in 2003, and completed her Ph.D. at Carnegie Mellon University in 2008. Her dissertation, ''Efficient Algorithms for Path Problems in Weighted Graphs'', was supervised by Guy Blelloch. After postdoctoral research at the Institute for Advanced Study and University of Calif ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ryan Williams (computer Scientist)
Richard Ryan Williams, known as Ryan Williams (born 1979), is an American theoretical computer scientist working in computational complexity theory and algorithms. Education Williams graduated from the Alabama School of Mathematics and Science before receiving his bachelor's degree in math and computer science from Cornell University in 2001 and his Ph.D in computer science in 2007 from Carnegie Mellon University under the supervision of Manuel Blum. From 2010 to 2012, he was a member of the Theory Group of IBM Almaden Research Center. From Fall 2011 to Fall 2016, he was a professor at Stanford University. In January 2017, he joined the faculty at MIT. Research Williams has been a member of the program committee for the Symposium on Theory of Computing in 2011 and various other conferences. He won the Ron V. Book best student paper award at the IEEE Conference on Computational Complexity in 2005 and 2007, and at the best student paper award at the International Colloquium on A ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Shortest Path Problem
In graph theory, the shortest path problem is the problem of finding a path between two vertices (or nodes) in a graph such that the sum of the weights of its constituent edges is minimized. The problem of finding the shortest path between two intersections on a road map may be modeled as a special case of the shortest path problem in graphs, where the vertices correspond to intersections and the edges correspond to road segments, each weighted by the length of the segment. Definition The shortest path problem can be defined for graphs whether undirected, directed, or mixed. It is defined here for undirected graphs; for directed graphs the definition of path requires that consecutive vertices be connected by an appropriate directed edge. Two vertices are adjacent when they are both incident to a common edge. A path in an undirected graph is a sequence of vertices P = ( v_1, v_2, \ldots, v_n ) \in V \times V \times \cdots \times V such that v_i is adjacent to v_ for 1 \leq i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
K Shortest Path Routing
The ''k'' shortest path routing problem is a generalization of the shortest path routing problem in a given network. It asks not only about a shortest path but also about next ''k−1'' shortest paths (which may be longer than the shortest path). A variation of the problem is the loopless ''k'' shortest paths. Finding ''k'' shortest paths is possible by extending Dijkstra algorithm or Bellman-Ford algorithm. History Since 1957 many papers were published on the ''k'' shortest path routing problem. Most of the fundamental works were done between 1960s and 2001. Since then, most of the research has been on the problem's applications and its variants. In 2010, Michael Günther et al. published a book on ''Symbolic calculation of ''k''-shortest paths and related measures with the stochastic process algebra tool CASPA''. Algorithm The Dijkstra algorithm can be generalized to find the ''k'' shortest paths. Variations There are two main variations of the ''k'' shortest pa ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Distance Matrix
In mathematics, computer science and especially graph theory, a distance matrix is a square matrix (two-dimensional array) containing the distances, taken pairwise, between the elements of a set. Depending upon the application involved, the ''distance'' being used to define this matrix may or may not be a metric. If there are elements, this matrix will have size . In graph-theoretic applications the elements are more often referred to as points, nodes or vertices. Non-metric distance matrix In general, a distance matrix is a weighted adjacency matrix of some graph. In a network, a directed graph with weights assigned to the arcs, the distance between two nodes of the network can be defined as the minimum of the sums of the weights on the shortest paths joining the two nodes. This distance function, while well defined, is not a metric. There need be no restrictions on the weights other than the need to be able to combine and compare them, so negative weights are used in some appli ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Metric Space
In mathematics, a metric space is a set together with a notion of ''distance'' between its elements, usually called points. The distance is measured by a function called a metric or distance function. Metric spaces are the most general setting for studying many of the concepts of mathematical analysis and geometry. The most familiar example of a metric space is 3-dimensional Euclidean space with its usual notion of distance. Other well-known examples are a sphere equipped with the angular distance and the hyperbolic plane. A metric may correspond to a metaphorical, rather than physical, notion of distance: for example, the set of 100-character Unicode strings can be equipped with the Hamming distance, which measures the number of characters that need to be changed to get from one string to another. Since they are very general, metric spaces are a tool used in many different branches of mathematics. Many types of mathematical objects have a natural notion of distance and t ... [...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] |
Nondeterministic Algorithm
In computer programming, a nondeterministic algorithm is an algorithm that, even for the same input, can exhibit different behaviors on different runs, as opposed to a deterministic algorithm. There are several ways an algorithm may behave differently from run to run. A concurrent algorithm can perform differently on different runs due to a race condition. A probabilistic algorithm's behaviors depends on a random number generator. An algorithm that solves a problem in nondeterministic polynomial time can run in polynomial time or exponential time depending on the choices it makes during execution. The nondeterministic algorithms are often used to find an approximation to a solution, when the exact solution would be too costly to obtain using a deterministic one. The notion was introduced by Robert W. Floyd in 1967. Use Often in computational theory, the term "algorithm" refers to a deterministic algorithm. A nondeterministic algorithm is different from its more familiar determi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
