Pseudo-polynomial time
In computational complexity theory, a numeric algorithm runs in pseudo-polynomial time if its running time is a polynomial in the numeric value of the input (the largest integer present in the input)—
Proof (truth)
A proof is sufficient evidence or a sufficient argument for the truth of a proposition. The concept applies in a variety of disciplines,with both the nature of the evidence or justification and the cr
Log-space computable function
In computational complexity theory, a log-space computable function is a function that requires only memory to be computed (this restriction does not apply to the size of the output). The computation
Circuits over sets of natural numbers
Circuits over natural numbers are a mathematical model used in studying computational complexity theory. They are a special case of circuits. The object is a labeled directed acyclic graph the nodes o
Transdichotomous model
In computational complexity theory, and more specifically in the analysis of algorithms with integer data, the transdichotomous model is a variation of the random access machine in which the machine w
Complete (complexity)
In computational complexity theory, a computational problem is complete for a complexity class if it is, in a technical sense, among the "hardest" (or "most expressive") problems in the complexity cla
Circuit complexity
In theoretical computer science, circuit complexity is a branch of computational complexity theory in which Boolean functions are classified according to the size or depth of the Boolean circuits that
Hardness of approximation
In computer science, hardness of approximation is a field that studies the algorithmic complexity of finding near-optimal solutions to optimization problems.
Information-based complexity
Information-based complexity (IBC) studies optimal algorithms and computational complexity for the continuous problems that arise in physical science, economics, engineering, and mathematical finance.
Computational resource
In computational complexity theory, a computational resource is a resource used by some computational models in the solution of computational problems. The simplest computational resources are computa
Combinatorial search
In computer science and artificial intelligence, combinatorial search studies search algorithms for solving instances of problems that are believed to be hard in general, by efficiently exploring the
Integer circuit
In computational complexity theory, an integer circuit is a circuit model of computation in which inputs to the circuit are sets of integers and each gate of the circuit computes either a set operatio
Bernstein–Vazirani algorithm
The Bernstein–Vazirani algorithm, which solves the Bernstein–Vazirani problem, is a quantum algorithm invented by and Umesh Vazirani in 1992. It is a restricted version of the Deutsch–Jozsa algorithm
Averaging argument
In computational complexity theory and cryptography, averaging argument is a standard argument for proving theorems. It usually allows us to convert probabilistic polynomial-time algorithms into non-u
Smoothed analysis
In theoretical computer science, smoothed analysis is a way of measuring the complexity of an algorithm. Since its introduction in 2001, smoothed analysis has been used as a basis for considerable res
Randomness merger
In extractor theory, a randomness merger is a function which extracts randomness out of a set of random variables, provided that at least one of them is uniformly random. Its name stems from the fact
Mahaney's theorem
Mahaney's theorem is a theorem in computational complexity theory proven by Stephen Mahaney that states that if any sparse language is NP-complete, then P = NP. Also, if any sparse language is NP-comp
Computing the permanent
In linear algebra, the computation of the permanent of a matrix is a problem that is thought to be more difficult than the computation of the determinant of a matrix despite the apparent similarity of
Propositional proof system
In propositional calculus and proof complexity a propositional proof system (pps), also called a Cook–Reckhow propositional proof system, is a system for proving classical propositional tautologies.
Complexity and Real Computation
Complexity and Real Computation is a book on the computational complexity theory of real computation. It studies algorithms whose inputs and outputs are real numbers, using the Blum–Shub–Smale machine
Symmetric Turing machine
A symmetric Turing machine is a Turing machine which has a configuration graph that is undirected (that is, configuration i yields configuration j if and only if j yields i).
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 ord
Exact algorithm
In computer science and operations research, exact algorithms are algorithms that always solve an optimization problem to optimality. Unless P = NP, an exact algorithm for an NP-hard optimization prob
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 oth
Weak NP-completeness
In computational complexity, an NP-complete (or NP-hard) problem is weakly NP-complete (or weakly NP-hard) if there is an algorithm for the problem whose running time is polynomial in the dimension of
Decision tree model
In computational complexity the decision tree model is the model of computation in which an algorithm is considered to be basically a decision tree, i.e., a sequence of queries or tests that are done
Analysis of algorithms
In computer science, the analysis of algorithms is the process of finding the computational complexity of algorithms—the amount of time, storage, or other resources needed to execute them. Usually, th
Quantum computing
Quantum computing is a type of computation whose operations can harness the phenomena of quantum mechanics, such as superposition, interference, and entanglement. Devices that perform quantum computat
Quantum supremacy
In quantum computing, quantum supremacy or quantum advantage is the goal of demonstrating that a programmable quantum device can solve a problem that no classical computer can solve in any feasible am
Generic-case complexity
Generic-case complexity is a subfield of computational complexity theory that studies the complexity of computational problems on "most inputs". Generic-case complexity is a way of measuring the compl
Computational topology
Algorithmic topology, or computational topology, is a subfield of topology with an overlap with areas of computer science, in particular, computational geometry and computational complexity theory. A
Unary language
In computational complexity theory, a unary language or tally language is a formal language (a set of strings) where all strings have the form 1k, where "1" can be any fixed symbol. For example, the l
Complement (complexity)
In computational complexity theory, the complement of a decision problem is the decision problem resulting from reversing the yes and no answers. Equivalently, if we define decision problems as sets o
Geometric complexity theory
Geometric complexity theory (GCT), is a research program in computational complexity theory proposed by Ketan Mulmuley and Milind Sohoni. The goal of the program is to answer the most famous open prob
Log-rank conjecture
In theoretical computer science, the log-rank conjecture states that the deterministic communication complexity of a two-party Boolean function is polynomially related to the logarithm of the rank of
Low (complexity)
In computational complexity theory, a language B (or a complexity class B) is said to be low for a complexity class A (with some reasonable relativized version of A) if AB = A; that is, A with an orac
Model of computation
In computer science, and more specifically in computability theory and computational complexity theory, a model of computation is a model which describes how an output of a mathematical function is co
Proof of knowledge
In cryptography, a proof of knowledge is an interactive proof in which the prover succeeds in 'convincing' a verifier that the prover knows something. What it means for a machine to 'know something' i
Sparse language
In computational complexity theory, a sparse language is a formal language (a set of strings) such that the complexity function, counting the number of strings of length n in the language, is bounded
Communication complexity
In theoretical computer science, communication complexity studies the amount of communication required to solve a problem when the input to the problem is distributed among two or more parties. The st
Pebble game
In mathematics and computer science, a pebble game is a type of mathematical game played by placing "pebbles" or "markers" on a directed acyclic graph according to certain rules:
* A given step of th
Glossary of quantum computing
This glossary of quantum computing is a list of definitions of terms and concepts used in quantum computing, its sub-disciplines, and related fields. Bacon–Shor_codeis a Subsystem error correcting cod
Quantum complexity theory
Quantum complexity theory is the subfield of computational complexity theory that deals with complexity classes defined using quantum computers, a computational model based on quantum mechanics. It st
Leaf language
In computational complexity theory, a leaf language is a method of characterizing a complexity class by formalizing what it means for a machine to "accept" an input. Several complexity classes are typ
Kolmogorov complexity
In algorithmic information theory (a subfield of computer science and mathematics), the Kolmogorov complexity of an object, such as a piece of text, is the length of a shortest computer program (in a
Advice (complexity)
In computational complexity theory, an advice string is an extra input to a Turing machine that is allowed to depend on the length n of the input, but not on the input itself. A decision problem is in
Unique games conjecture
In computational complexity theory, the unique games conjecture (often referred to as UGC) is a conjecture made by Subhash Khot in 2002. The conjecture postulates that the problem of determining the a
Descriptive complexity theory
Descriptive complexity is a branch of computational complexity theory and of finite model theory that characterizes complexity classes by the type of logic needed to express the languages in them. For
Space complexity
The space complexity of an algorithm or a computer program is the amount of memory space required to solve an instance of the computational problem as a function of characteristics of the input. It is
Approximation algorithm
In computer science and operations research, approximation algorithms are efficient algorithms that find approximate solutions to optimization problems (in particular NP-hard problems) with provable g
Proper complexity function
A proper complexity function is a function f mapping a natural number to a natural number such that:
* f is nondecreasing;
* there exists a k-string Turing machine M such that on any input of length
Graph isomorphism problem
Unsolved problem in computer science: Can the graph isomorphism problem be solved in polynomial time? (more unsolved problems in computer science) The graph isomorphism problem is the computational pr
Strong NP-completeness
In computational complexity, strong NP-completeness is a property of computational problems that is a special case of NP-completeness. A general computational problem may have numerical parameters. Fo
Gap-Hamming problem
In communication complexity, the gap-Hamming problem asks, if Alice and Bob are each given a (potentially different) string, what is the minimal number of bits that they need to exchange in order for
Best, worst and average case
In computer science, best, worst, and average cases of a given algorithm express what the resource usage is at least, at most and on average, respectively. Usually the resource being considered is run
Log-space transducer
In computational complexity theory, a log space transducer (LST) is a type of Turing machine used for log-space reductions. A log space transducer, , has three tapes:
* A read-only input tape.
* A r
Computational complexity of matrix multiplication
In theoretical computer science, the computational complexity of matrix multiplication dictates how quickly the operation of matrix multiplication can be performed. Matrix multiplication algorithms ar
Dynamic problem (algorithms)
Dynamic problems in computational complexity theory are problems stated in terms of the changing input data. In the most general form a problem in this category is usually stated as follows:
* Given
Klee–Minty cube
The Klee–Minty cube or Klee–Minty polytope (named after Victor Klee and George J. Minty) is a unit hypercube of variable dimension whose corners have been perturbed. Klee and Minty demonstrated that G
Implicit computational complexity
Implicit computational complexity (ICC) is a subfield of computational complexity theory that characterizes algorithms by constraints on the way in which they are constructed, without reference to a s
Complexity class
In computational complexity theory, a complexity class is a set of computational problems of related resource-based complexity. The two most commonly analyzed resources are time and memory. In general
Randomness extractor
A randomness extractor, often simply called an "extractor", is a function, which being applied to output from a weakly random entropy source, together with a short, uniformly random seed, generates a
Effective complexity
Effective complexity is a measure of complexity defined in a 1996 paper by Murray Gell-Mann and Seth Lloyd that attempts to measure the amount of non-random information in a system. It has been critic
Generalized game
In computational complexity theory, a generalized game is a game or puzzle that has been generalized so that it can be played on a board or grid of any size. For example, generalized chess is the game
Compression theorem
In computational complexity theory, the compression theorem is an important theorem about the complexity of computable functions. The theorem states that there exists no largest complexity class, with
Noisy intermediate-scale quantum era
In the noisy intermediate-scale quantum (NISQ) era the leading quantum processors contain about 50 to a few hundred qubits, but are not advanced enough to reach fault-tolerance nor large enough to pro
Pseudo-polynomial transformation
In computational complexity theory, a pseudo-polynomial transformation is a function which maps instances of one strongly NP-complete problem into another and is computable in pseudo-polynomial time.
Existential theory of the reals
In mathematical logic, computational complexity theory, and computer science, the existential theory of the reals is the set of all true sentences of the form where the variables are interpreted as ha
Asymptotic computational complexity
In computational complexity theory, asymptotic computational complexity is the usage of asymptotic analysis for the estimation of computational complexity of algorithms and computational problems, com
Computational complexity
In computer science, the computational complexity or simply complexity of an algorithm is the amount of resources required to run it. Particular focus is given to computation time (generally measured
Universal hashing
In mathematics and computing, universal hashing (in a randomized algorithm or data structure) refers to selecting a hash function at random from a family of hash functions with a certain mathematical
List decoding
In coding theory, list decoding is an alternative to unique decoding of error-correcting codes for large error rates. The notion was proposed by Elias in the 1950s. The main idea behind list decoding
Claw finding problem
The claw finding problem is a classical problem in complexity theory, with several applications in cryptography. In short, given two functions f, g, viewed as oracles, the problem is to find x and y s
Natural proof
In computational complexity theory, a natural proof is a certain kind of proof establishing that one complexity class differs from another one. While these proofs are in some sense "natural", it can b
Proof complexity
In logic and theoretical computer science, and specifically proof theory and computational complexity theory, proof complexity is the field aiming to understand and analyse the computational resources
Cobham's thesis
Cobham's thesis, also known as Cobham–Edmonds thesis (named after Alan Cobham and Jack Edmonds), asserts that computational problems can be feasibly computed on some computational device only if they
Lempel–Ziv complexity
The Lempel–Ziv complexity is a measure that was first presented in the article On the Complexity of Finite Sequences (IEEE Trans. On IT-22,1 1976), by two Israeli computer scientists, Abraham Lempel a
The Complexity of Songs
"The Complexity of Songs" is a scholarly article by computer scientist Donald Knuth in 1977, as an in-joke about computational complexity theory. The article capitalizes on the tendency of popular son
Half-exponential function
In mathematics, a half-exponential function is a functional square root of an exponential function. That is, a function such that composed with itself results in an exponential function: for some cons
Folded Reed–Solomon code
In coding theory, folded Reed–Solomon codes are like Reed–Solomon codes, which are obtained by mapping Reed–Solomon codewords over a larger alphabet by careful bundling of codeword symbols. Folded Ree
Semi-membership
In mathematics and theoretical computer science, the semi-membership problem for a set is the problem of deciding which of two possible elements is logically more likely to belong to that set; alterna
Complexity index
In modern computer science and statistics, the complexity index of a function denotes the level of informational content, which in turn affects the difficulty of learning the function from examples. T
Low-complexity art
Low-complexity art, first described by Jürgen Schmidhuber in 1997 and now established as a seminal topic within the larger field of computer science, is art that can be described by a short computer p
Computation tree
A computation tree is a representation for the computation steps of a non-deterministic Turing machine on a specified input. A computation tree is a rooted tree of nodes and edges. Each node in the tr
Time complexity
In 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 counting the
Interactive proof system
In computational complexity theory, an interactive proof system is an abstract machine that models computation as the exchange of messages between two parties: a prover and a verifier. The parties int
Switching lemma
In computational complexity theory, Håstad's switching lemma is a key tool for proving lower bounds on the size of constant-depth Boolean circuits.Using the switching lemma, Johan Håstad showed that B
Computationally bounded adversary
In information theory, the computationally bounded adversary problem is a different way of looking at the problem of sending data over a noisy channel. In previous models the best that could be done w
Logical depth
Logical depth is a measure of complexity for individual strings devised by Charles H. Bennett based on the computational complexity of an algorithm that can recreate a given piece of information. It d
Combinatorial optimization
Combinatorial optimization is a subfield of mathematical optimization that consists of finding an optimal object from a finite set of objects, where the set of feasible solutions is discrete or can be
Boolean circuit
In computational complexity theory and circuit complexity, a Boolean circuit is a mathematical model for combinational digital logic circuits. A formal language can be decided by a family of Boolean c
Quantum capacity
In the theory of quantum communication, the quantum capacity is the highest rate at which quantum information can be communicated over many independent uses of a noisy quantum channel from a sender to
Configuration graph
Configuration graphs are a theoretical tool used in computational complexity theory to prove a relation between graph reachability and complexity classes.
Aanderaa–Karp–Rosenberg conjecture
In theoretical computer science, the Aanderaa–Karp–Rosenberg conjecture (also known as the Aanderaa–Rosenberg conjecture or the evasiveness conjecture) is a group of related conjectures about the numb
Computable topology
Computable topology is a discipline in mathematics that studies the topological and algebraic structure of computation. Computable topology is not to be confused with algorithmic or computational topo
Non-constructive algorithm existence proofs
The vast majority of positive results about computational problems are constructive proofs, i.e., a computational problem is proved to be solvable by showing an algorithm that solves it; a computation
Certificate (complexity)
In computational complexity theory, a certificate (also called a witness) is a string that certifies the answer to a computation, or certifies the membership of some string in a language. A certificat
Electronic Colloquium on Computational Complexity
The Electronic Colloquium on Computational Complexity (ECCC) is an electronic archive of research papers in computational complexity theory, a branch of computer science. The intention of the ECCC is
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 a
Transcomputational problem
In computational complexity theory, a transcomputational problem is a problem that requires processing of more than 1093 bits of information. Any number greater than 1093 is called a transcomputationa
Hidden linear function problem
The hidden linear function problem, is a search problem that generalizes the Bernstein–Vazirani problem. In the Bernstein–Vazirani problem, the hidden function is implicitly specified in an oracle; wh
Hamiltonian complexity
Hamiltonian complexity or quantum Hamiltonian complexity is a topic which deals with problems in quantum complexity theory and condensed matter physics. It mostly studies constraint satisfaction probl
Introduction to the Theory of Computation
Introduction to the Theory of Computation (ISBN 0-534-95097-3) is a textbook in theoretical computer science, written by Michael Sipser and first published by PWS Publishing in 1997.
Padding argument
In computational complexity theory, the padding argument is a tool to conditionally prove that if some complexity classes are equal, then some other bigger classes are also equal.
L-notation
L-notation is an asymptotic notation analogous to big-O notation, denoted as for a bound variable tending to infinity. Like big-O notation, it is usually used to roughly convey the rate of growth of a
Games, Puzzles, and Computation
Games, Puzzles, and Computation is a book on game complexity, written by Robert Hearn and Erik Demaine, and published in 2009 by A K Peters. It is revised from Hearn's doctoral dissertation, which was
Yao's principle
In computational complexity theory, Yao's principle (also called Yao's minimax principle or Yao's lemma) is a way to prove lower bounds on the worst-case performance of randomized algorithms, by compa
Parameterized complexity
In computer science, parameterized complexity is a branch of computational complexity theory that focuses on classifying computational problems according to their inherent difficulty with respect to m