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NC (complexity)
In computational complexity theory, the class NC (for "Nick's Class") is the set of decision problems decidable in polylogarithmic time on a parallel computer with a polynomial number of processors. In other words, a problem with input size ''n'' is in NC if there exist constants ''c'' and ''k'' such that it can be solved in time using parallel processors. Stephen Cook coined the name "Nick's class" after Nick Pippenger, who had done extensive research on circuits with polylogarithmic depth and polynomial size.Arora & Barak (2009) p.120 As in the case of circuit complexity theory, usually the class has an extra constraint that the circuit family must be ''uniform'' ( see below). Just as the class P can be thought of as the tractable problems ( Cobham's thesis), so NC can be thought of as the problems that can be efficiently solved on a parallel computer.Arora & Barak (2009) p.118 NC is a subset of P because polylogarithmic parallel computations can be simulated by polynomi ... [...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 explores the relationships between these classifications. 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 logic gate, gates in a circuit (used in circuit complexity) and the number of processors (used in parallel computing). O ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Matrix Inverse
In linear algebra, an invertible matrix (''non-singular'', ''non-degenarate'' or ''regular'') is a square matrix that has an inverse. In other words, if some other matrix is multiplied by the invertible matrix, the result can be multiplied by an inverse to undo the operation. An invertible matrix multiplied by its inverse yields the identity matrix. Invertible matrices are the same size as their inverse. Definition An -by- square matrix is called invertible if there exists an -by- square matrix such that\mathbf = \mathbf = \mathbf_n ,where denotes the -by- identity matrix and the multiplication used is ordinary matrix multiplication. If this is the case, then the matrix is uniquely determined by , and is called the (multiplicative) ''inverse'' of , denoted by . Matrix inversion is the process of finding the matrix which when multiplied by the original matrix gives the identity matrix. Over a field, a square matrix that is ''not'' invertible is called singular or degener ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
LOGCFL
In computational complexity theory, LOGCFL is the complexity class that contains all decision problems that can be reduced in logarithmic space to a context-free language. This class is closed under complementation. It is situated between NL and AC1, in the sense that it contains the former and is contained in the latter. Problems that are complete for LOGCFL include many problems that can be characterized by acyclic hypergraphs: * evaluating acyclic Boolean conjunctive queries * checking the existence of a homomorphism between two acyclic relational structures * checking the existence of solutions of acyclic constraint satisfaction problems LOGCFL is the set of decision problems solvable by nondeterministic auxiliary pushdown automata in log space and polynomial time In theoretical computer science, the time complexity is the computational complexity that describes the amount of computer time it takes to run an algorithm. Time complexity is commonly estimated by count ... [...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] |
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 th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
L (complexity)
In computational complexity theory, L (also known as LSPACE, LOGSPACE 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. 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 written as well as read. 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 path between two vertices in a given u ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Logical Operator
In logic, a logical connective (also called a logical operator, sentential connective, or sentential operator) is a logical constant. Connectives can be used to connect logical formulas. For instance in the syntax of propositional logic, the binary connective \lor can be used to join the two atomic formulas P and Q, rendering the complex formula P \lor Q . Common connectives include negation, disjunction, conjunction, implication, and equivalence. In standard systems of classical logic, these connectives are interpreted as truth functions, though they receive a variety of alternative interpretations in nonclassical logics. Their classical interpretations are similar to the meanings of natural language expressions such as English "not", "or", "and", and "if", but not identical. Discrepancies between natural language connectives and those of classical logic have motivated nonclassical approaches to natural language meaning as well as approaches which pair a classica ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Binary Tree
In computer science, a binary tree is a tree data structure in which each node has at most two children, referred to as the ''left child'' and the ''right child''. That is, it is a ''k''-ary tree with . A recursive definition using set theory is that a binary tree is a triple , where ''L'' and ''R'' are binary trees or the empty set and ''S'' is a singleton (a single–element set) containing the root. From a graph theory perspective, binary trees as defined here are arborescences. A binary tree may thus be also called a bifurcating arborescence, a term which appears in some early programming books before the modern computer science terminology prevailed. It is also possible to interpret a binary tree as an undirected, rather than directed graph, in which case a binary tree is an ordered, rooted tree. Some authors use rooted binary tree instead of ''binary tree'' to emphasize the fact that the tree is rooted, but as defined above, a binary tree is always rooted. In ma ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Addition
Addition (usually signified by the Plus and minus signs#Plus sign, plus symbol, +) is one of the four basic Operation (mathematics), operations of arithmetic, the other three being subtraction, multiplication, and Division (mathematics), division. The addition of two Natural number, whole numbers results in the total or ''summation, sum'' of those values combined. For example, the adjacent image shows two columns of apples, one with three apples and the other with two apples, totaling to five apples. This observation is expressed as , which is read as "three plus two Equality (mathematics), equals five". Besides counting items, addition can also be defined and executed without referring to concrete objects, using abstractions called numbers instead, such as integers, real numbers, and complex numbers. Addition belongs to arithmetic, a branch of mathematics. In algebra, another area of mathematics, addition can also be performed on abstract objects such as Euclidean vector, vec ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Clarkson University
Clarkson University is a private research university with its main campus in Potsdam, New York. Clarkson has additional graduate programs and research facilities in the New York Capital District. It was established in 1896 and enrolled over 4,000 students in 2024. Students pursue bachelor's, master's, and doctoral degrees in each of its schools and institutes: the David D. Reh School of Business, Wallace H. Coulter School of Engineering and Applied Sciences, Earl R. and Barbara D. Lewis School of Health Sciences, Institute for a Sustainable Environment, and Institute for STEM Education. The university is classified among "R2: Doctoral Universities – High research activity". Clarkson's athletic teams, known as the Golden Knights, compete with 20 varsity teams. While predominantly an NCAA Division III school competing in the Liberty League, both men's and women's ice hockey teams compete in Division I ECAC Hockey. The women's hockey program has won three NCAA championship ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Carry-lookahead Adder
A carry-lookahead adder (CLA) or fast adder is a type of Adder (electronics), electronics adder used in digital logic. A carry-lookahead adder improves speed by reducing the amount of time required to determine carry bits. It can be contrasted with the simpler, but usually slower, ripple-carry adder (RCA), for which the carry bit is calculated alongside the sum bit, and each stage must wait until the previous carry bit has been calculated to begin calculating its own sum bit and carry bit. The carry-lookahead adder calculates one or more carry bits before the sum, which reduces the wait time to calculate the result of the larger-value bits of the adder. Already in the mid-1800s, Charles Babbage recognized the performance penalty imposed by the ripple-carry used in his Difference engine, Difference Engine, and subsequently designed mechanisms for ''anticipating carriage'' for his never-built Analytical Engine. Konrad Zuse is thought to have implemented the first carry-lookahead ad ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ripple Carry Adder
An adder, or summer, is a digital circuit that performs addition of numbers. In many computers and other kinds of processors, adders are used in the arithmetic logic units (ALUs). They are also used in other parts of the processor, where they are used to calculate addresses, table indices, increment and decrement operators and similar operations. Although adders can be constructed for many number representations, such as binary-coded decimal or excess-3, the most common adders operate on binary numbers. In cases where two's complement or ones' complement is being used to represent negative numbers, it is trivial to modify an adder into an adder–subtractor. Other signed number representations require more logic around the basic adder. History George Stibitz invented the 2-bit binary adder (the Model K) in 1937. Binary adders Half adder The half adder adds two single binary digits A and B. It has two outputs, sum (S) and carry (C). The carry signal represents an overflow i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
