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Length of expression as combination of 1s
In number theory, the complexity of an integer is the smallest number of ones that can be used to represent it using ones and any number of additions,
Integer_complexity
Decomposition of a number into a product
decomposition of a positive integer into a product of integers. Every positive integer greater than 1 is either the product of two or more integer factors greater
Integer_factorization
Mathematical optimization problem restricted to integers
An integer programming, also known as integer optimization, problem is a mathematical optimization or feasibility program in which some or all of the variables
Integer_programming
Inherent difficulty of computational problems
In theoretical computer science and mathematics, computational complexity theory focuses on classifying computational problems according to their resource
Computational complexity theory
Computational_complexity_theory
Number in {..., –2, –1, 0, 1, 2, ...}
factorization of a positive integer Complex integer Hyperinteger Integer complexity Integer lattice Integer part Integer sequence Integer-valued function Mathematical
Integer
Estimate of time taken for running an algorithm
the time complexity is the computational complexity that describes the amount of computer time it takes to run an algorithm. Time complexity is commonly
Time_complexity
Complexity class used to classify decision problems
problems in computer science In computational complexity theory, NP (nondeterministic polynomial time) is a complexity class used to classify decision problems
NP_(complexity)
Algorithmic runtime requirements for common math procedures
stands in for the complexity of the chosen multiplication algorithm. This table lists the complexity of mathematical operations on integers. On stronger computational
Computational complexity of mathematical operations
Computational_complexity_of_mathematical_operations
Algorithm to multiply two numbers
two n-bit integers. This is known as the computational complexity of multiplication. Usual algorithms done by hand have asymptotic complexity of O(n2)
Multiplication_algorithm
Amount of resources to perform an algorithm
arithmetic complexity. For example, the arithmetic complexity of the computation of the determinant of a n×n integer matrix is O ( n 3 ) {\displaystyle O(n^{3})}
Computational_complexity
Topics referred to by the same term
Computational complexity, of algorithms Computational complexity theory Game complexity, in combinatorial game theory Integer complexity, in number theory
Complexity_(disambiguation)
Unsolved problem in computer science
axioms for integer arithmetic, then nearly polynomial-time algorithms exist for all NP problems. Therefore, assuming (as most complexity theorists do)
P_versus_NP_problem
Method to solve optimization problems
H. Freeman. ISBN 978-0-7167-1045-5. A6: MP1: INTEGER PROGRAMMING, pg.245. (computer science, complexity theory) Gärtner, Bernd; Matoušek, Jiří (2006)
Linear_programming
Unproven computational hardness assumption
three-party set disjointness problem in communication complexity, three subsets of the integers in some range [1,m] are specified, and three communicating
Exponential_time_hypothesis
Estimate of number of possible chess games
efficiently computable bijection between integers and chess positions. Allis also estimated the game-tree complexity to be at least 10123, "based on an average
Shannon_number
Measure of algorithmic complexity
a complexity formula K(s) ≥ L. The strings s, and the integer L in turn, are computable by procedure: def string_nth_proof(n: int) def complexity
Kolmogorov_complexity
Typographic symbol
|a|:‖A‖ reads "propositional truncation of A") Integer complexity: ‖n‖; reads "the complexity of the integer n". In LaTeX mathematical mode, the ASCII vertical
Vertical_bar
Algorithmic runtime requirements for matrix multiplication
complexity bounds on algorithms for computing the canonical structure of finite abelian groups and the Hermite and Smith normal forms of an integer matrix"
Computational complexity of matrix multiplication
Computational_complexity_of_matrix_multiplication
polyhedron with facet complexity at most f, and v is a rational vector with distance from P of at most 2-6nf, and q and w are integer vectors satisfying
N-dimensional_polyhedron
Computation modulo a fixed integer
mathematics, modular arithmetic is a system of arithmetic operations for integers, differing from the usual ones in that numbers "wrap around" when reaching
Modular_arithmetic
Greatest integer less than or equal to square root
number theory, the integer square root (isqrt) of a non-negative integer n is the non-negative integer m which is the greatest integer less than or equal
Integer_square_root
Problem a computer might be able to solve
a positive integer n, find a nontrivial prime factor of n." is a computational problem that has a solution, as there are many known integer factorization
Computational_problem
Largest integer that divides given integers
of two or more integers, which are not all zero, is the largest positive integer that divides each of the integers. For two integers x, y, the greatest
Greatest_common_divisor
Branch of computational complexity theory
In computer science, parameterized complexity is a branch of computational complexity theory that focuses on classifying computational problems according
Parameterized_complexity
Self-referential paradox
self-referential paradox arising from an expression like "The smallest positive integer not definable in under sixty letters" (a phrase with fifty-seven letters)
Berry_paradox
Calculations where numbers' precision is only limited by computer memory
for bignums, and others have libraries available for arbitrary-precision integer and floating-point math. Rather than storing values as a fixed number of
Arbitrary-precision arithmetic
Arbitrary-precision_arithmetic
Open problem on 3x+1 and x/2 functions
"Mortality of iterated piecewise affine functions over the integers: Decidability and complexity". Computability. 1 (1): 19–56. doi:10.3233/COM-150032. Michel
Collatz_conjecture
Model of computation
problem. If the input is an integer circuit, however, it is unknown whether this problem is decidable. Circuit complexity attempts to classify Boolean
Circuit_(computer_science)
Algorithm that arranges lists in order
better than O(n log n) time complexity assuming certain constraints, including: Thorup's algorithm, a randomized integer sorting algorithm, taking O(n
Sorting_algorithm
Function that counts distinct factors of a string
of a positive integer n to be the number of different words of length n in L The complexity function of a word is thus the complexity function of the
Complexity_function
Quantum algorithm for integer factorization
Harvey and van der Hoeven, thus demonstrating that the integer factorization problem is in complexity class BQP. Shor's algorithm is asymptotically faster
Shor's_algorithm
Online database of integer sequences
The On-Line Encyclopedia of Integer Sequences (OEIS) is an online database of integer sequences. It was created and maintained by Neil Sloane while researching
On-Line Encyclopedia of Integer Sequences
On-Line_Encyclopedia_of_Integer_Sequences
Decidable first-order theory of the natural numbers with addition
structure of non-negative integers with constants 0 {\displaystyle 0} , 1 {\displaystyle 1} , and the addition of non-negative integers. Presburger arithmetic
Presburger_arithmetic
Class of problems in computer science
In complexity theory, PP, or PPT is the class of decision problems solvable by a probabilistic Turing machine in polynomial time with an error probability
PP_(complexity)
Attribute of machine learning models
there exists a positive integer N, such that, for all sample sizes n≥N, we have Prρn[ℰ(hn)−ℰℋ∗≥ε]<δ.The sample complexity of 𝒜 is then the minimum
Sample_complexity
2002 documentary film by Jeffrey Blitz
a PhD in Mathematics from the University of Michigan, focusing on integer complexity. As of 2025, Altman is involved in the New York rationalist community
Spellbound_(2002_film)
Venezuelan computer scientist
recognition of his contributions to the foundations of computational complexity theory and its application to cryptography and program checking". Blum
Manuel_Blum
Approach to the study of finite semigroups and automata
semigroups have complexity 0, while non-trivial finite groups have complexity 1. In fact, there are semigroups of every non-negative integer complexity. For example
Krohn–Rhodes_theory
Model of computational complexity
In theoretical computer science, circuit complexity is a branch of computational complexity theory in which Boolean functions are classified according
Circuit_complexity
Notion in combinatorial game theory
Combinatorial game theory measures game complexity in several ways: State-space complexity (the number of legal game positions from the initial position)
Game_complexity
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
Integer_circuit
Number without repeated prime factors
In mathematics, a square-free integer (or squarefree integer) is an integer that is divisible by no square number other than 1. That is, its prime factorization
Square-free_integer
Graver bases enable iterative solutions of linear and various nonlinear integer programming problems in polynomial time. They were introduced by Jack E
Graver_basis
Study of resources used by an algorithm
the analysis of algorithms is the process of finding the computational complexity of algorithms—the amount of time, storage, or other resources needed to
Analysis_of_algorithms
Type of mathematical expression
addition, subtraction, multiplication and exponentiation to nonnegative integer powers, and has a finite number of terms. An example of a polynomial of
Polynomial
Complexity class
In computational complexity theory, a computational problem H is called NP-hard if, for every problem L which can be solved in non-deterministic polynomial-time
NP-hardness
Hypothesis in computational complexity theory
In computational complexity theory, a computational hardness assumption is the hypothesis that a particular problem cannot be solved efficiently (where
Computational hardness assumption
Computational_hardness_assumption
Subfield of computational complexity theory
Fine-grained complexity is a subfield of computational complexity theory that studies the time required to solve computational problems, especially those
Fine_grained_complexity
Measure of complexity of real-valued functions
learning theory (machine learning and theory of computation), Rademacher complexity, named after Hans Rademacher, measures richness of a class of sets with
Rademacher_complexity
Algorithm to compute the maximum flow in a network
found in O ( E ) {\displaystyle O(E)} time and increases the flow by an integer amount of at least 1 {\displaystyle 1} , with the upper bound f {\displaystyle
Ford–Fulkerson_algorithm
Combinational digital circuit
combinational digital circuit that performs arithmetic and bitwise operations on integer binary numbers. This is in contrast to a floating-point unit (FPU), which
Arithmetic_logic_unit
Ancient algorithm for generating prime numbers
numbers less than or equal to a given integer n by Eratosthenes's method: Create a list of consecutive integers from 2 through n: (2, 3, 4, ..., n). Initially
Sieve_of_Eratosthenes
Type of computer science algorithm
that space complexity also has varied choices in whether or not to count the index lengths as part of the space used. Often, the space complexity is given
In-place_algorithm
Product of numbers from 1 to n
factorial of a non-negative integer n {\displaystyle n} , denoted by n ! {\displaystyle n!} , is the product of all positive integers less than or equal to
Factorial
Used to count, measure, and label
Hindu–Arabic numeral system, a decimal system which can display any non-negative integer using a combination of ten Arabic numeral symbols called digits. Numerals
Number
Algorithm for computing greatest common divisors
of the smaller integer. This was proven by Gabriel Lamé in 1844 (Lamé's Theorem), and marks the beginning of computational complexity theory. Additional
Euclidean_algorithm
Class in computational complexity theory
}{=}}{\mathsf {P}}} More unsolved problems in computer science In computational complexity theory, the class NC (for Nick's class) is the set of decision problems
NC_(complexity)
Discrete Fourier transform algorithm
of sparse (mostly zero) factors. As a result, it manages to reduce the complexity of computing the DFT from O(n2), which arises if one simply applies the
Fast_Fourier_transform
Overview of and topical guide to algorithms
includes their design, proof of correctness, efficiency, computational complexity, and implementation in computer programs. Algorithm — finite sequence
Outline_of_algorithms
Complexity class
computational complexity theory, co-NP is a complexity class. A decision problem X is a member of co-NP if and only if its complement X is in the complexity class
Co-NP
Divide and conquer sorting algorithm
"Algorithms and Complexity". Proceedings of the 3rd Italian Conference on Algorithms and Complexity. Italian Conference on Algorithms and Complexity. Lecture
Merge_sort
Complexity class
In computational complexity theory, the complexity class PPP (polynomial pigeonhole principle) is a subclass of TFNP. It is the class of search problems
PPP_(complexity)
Search tree data structure
trie, which uses individual bits from fixed-length binary data (such as integers or memory addresses) as keys. The idea of a trie for representing a set
Trie
Methodic assignment of colors to elements of a graph
representations, it is typical to use the first few positive or non-negative integers as the "colors". In general, one can use any finite set as the "color set"
Graph_coloring
Algorithm that employs a degree of randomness as part of its logic or procedure
Carlo algorithms are considered, and several complexity classes are studied. The most basic randomized complexity class is RP, which is the class of decision
Randomized_algorithm
PR is the complexity class of all primitive recursive functions—or, equivalently, the set of all formal languages that can be decided in time bounded by
PR_(complexity)
Exponent of a power of two
The number of digits (bits) in the binary representation of a positive integer n is the integral part of 1 + log2 n, i.e. ⌊ log 2 n ⌋ + 1. {\displaystyle
Binary_logarithm
Product of the prime factors of an integer
Encyclopedia of Integer Sequences. OEIS Foundation. Adleman, Leonard M.; McCurley, Kevin S. (1994). "Open Problems in Number Theoretic Complexity, II". Algorithmic
Radical_of_an_integer
Problem of inverting exponentiation in groups
be defined for all integers k {\displaystyle k} , and the discrete logarithm log b ( a ) {\displaystyle \log _{b}(a)} is an integer k {\displaystyle k}
Discrete_logarithm
This is a list of computability and complexity topics, by Wikipedia page. Computability theory is the part of the theory of computation that deals with
List of computability and complexity topics
List_of_computability_and_complexity_topics
problem and the magnitudes of the data involved (provided these are given as integers), rather than the base-two logarithms of their magnitudes. Such algorithms
Weak_NP-completeness
Mathematical theorem
identifies the integration formulas for integrands with and without consecutive integer exponents and for single and double integrals. The integration formula
Ramanujan's_master_theorem
Problem in combinatorial optimization
{\displaystyle x_{i}} of copies of each kind of item to a maximum non-negative integer value c {\displaystyle c} : maximize ∑ i = 1 n v i x i {\displaystyle \sum
Knapsack_problem
Notation describing limiting behavior in computational number theory
computational number theory, to express the complexity of algorithms for difficult number theory problems, e.g. sieves for integer factorization and methods for solving
L-notation
Complexity class of bounded-depth circuits
representations of integers). Since it is a circuit class, like P/poly, AC0 also contains every unary language. From a descriptive complexity viewpoint, DLOGTIME-uniform
AC0
Computational complexity of quantum algorithms
Quantum complexity theory is the subfield of computational complexity theory that deals with complexity classes defined using quantum computers, a computational
Quantum_complexity_theory
Algorithm for determinants of integers
Bareiss algorithm is not commonly used for integer matrices, because multi-modular arithmetic allows a complexity similar to that of the Bareiss algorithm
Bareiss_algorithm
Function used in computer cryptography
"easy" and "hard" are to be understood in the sense of computational complexity theory, specifically the theory of polynomial time problems. This has
One-way_function
Problem of determining if a Boolean formula could be made true
NP-complete – this is the Cook–Levin theorem. This means that all problems in the complexity class NP, which includes a wide range of natural decision and optimization
Boolean satisfiability problem
Boolean_satisfiability_problem
NP-hard problem in combinatorial optimization
optimization: algorithms and complexity, Mineola, NY: Dover, pp.308-309. Tucker, A. W. (1960), "On Directed Graphs and Integer Programs", IBM Mathematical
Travelling_salesman_problem
American computer scientist (born 1969)
"Time-Space Tradeoffs for Counting NP Solutions Modulo Integers", Computational Complexity, 17 (2): 179–219, doi:10.1007/s00037-008-0248-y, S2CID 8815358
Ryan Williams (computer scientist)
Ryan_Williams_(computer_scientist)
Arithmetical operation
Joris van der Hoeven submitted a paper presenting an integer multiplication algorithm with a complexity of O(nlogn). The algorithm, also based on the fast
Multiplication
Implementation of arithmetic operations
implementations of the arithmetic operations. It includes: Fixed-size arithmetic "Integer arithmetic", which in practice is modular arithmetic by a power of 2. Fixed-point
Computer_arithmetic
Subfield of mathematical optimization
is related to operations research, algorithm theory, and computational complexity theory. It has important applications in several fields, including artificial
Combinatorial_optimization
Complexity class used in circuit complexity
numbers, integer division or recognizing the Dyck language with multiple types of parentheses. It is commonly used to model the computational complexity of
TC0
Scheduling algorithm for the network scheduler
to 0. Variables and Constants const integer N // Nb of queues const integer Q[1..N] // Per queue quantum integer DC[1..N] // Per queue deficit counter
Deficit_round_robin
Class of binary number
On-Line Encyclopedia of Integer Sequences, OEIS Foundation Charlier, Émilie; Cisternino, Célia; Massuir, Adeline (2019), "State complexity of the multiples of
Evil_number
of complexity classes in computational complexity theory. For other computational and complexity subjects, see list of computability and complexity topics
List_of_complexity_classes
Set of computational problems stated by Richard Karp (1973)
In computational complexity theory, Karp's 21 NP-complete problems are a set of computational problems which are NP-complete. In his 1972 paper, "Reducibility
Karp's 21 NP-complete problems
Karp's_21_NP-complete_problems
Decision problem in computer science
multiple subset sum problem. The time complexity of SSP depends on two parameters: n - the number of input integers. If n is a small fixed number, then
Subset_sum_problem
Complexity class
reduction from the problem of integer factorization to problems complete for PPA. Christos Papadimitriou (1994). "On the complexity of the parity argument and
PPA_(complexity)
Central computer component that executes instructions
encoded integer) that the CPU can process in one operation, which is commonly called word size, bit width, data path width, integer precision, or integer size
Central_processing_unit
Computational task of sorting whole numbers
science, integer sorting is the algorithmic problem of sorting a collection of data values by integer keys. Algorithms designed for integer sorting may
Integer_sorting
List of unsolved computational problems
cryptography possible? Log-rank conjecture Hartmanis–Stearns conjecture Can integer factorization be done in polynomial time on a classical (non-quantum) computer
List of unsolved problems in computer science
List_of_unsolved_problems_in_computer_science
Randomized polynomial time class of computational complexity theory
In computational complexity theory, randomized polynomial time (RP) is the complexity class of decision problems for which a probabilistic Turing machine
RP_(complexity)
Natural sorting algorithm
representing the positive integer 3 (because they each contain three beads) while the top row represents the positive integer 2 (as it only contains two
Bead_sort
Date and time representation system widely used in computing
referred to as the Unix epoch. Unix time is typically encoded as a signed integer. The Unix time 0 is exactly midnight UTC on 1 January 1970, with Unix time
Unix_time
Number whose square is a given number
roots of positive integers that are not perfect squares are always irrational numbers: numbers not expressible as a ratio of two integers (that is, they
Square_root
Problem in computer science
Unsolved problem in computer science What is the Turing run-time complexity of the square-root sum problem? More unsolved problems in computer science
Square-root_sum_problem
Data structure for integer priorities
minimum (or maximum) priority. In the bucket queue, the priorities must be integers, and it is particularly suited to applications in which the priorities
Bucket_queue
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