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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 also
Integer_sorting
Algorithm that arranges lists in order
sorted lists. Sorting is also often useful for canonicalizing data and for producing human-readable output. Formally, the output of any sorting algorithm
Sorting_algorithm
Sorting algorithm
counting sort is an algorithm for sorting a collection of objects according to keys that are small positive integers; that is, it is an integer sorting algorithm
Counting_sort
Non-comparative lexicographical sorting algorithm
most suitable for sorting strings or fixed-length integer representations. A sequence like [b, c, e, d, f, g, ba] would be sorted as [b, ba, c, d, e
Radix_sort
This is a list of notable integer sequences with links to their entries in the On-Line Encyclopedia of Integer Sequences. OEIS core sequences Index to
List_of_integer_sequences
Type of sorting algorithm that works by comparing pairs of elements
log n) bound for comparison sorting by using non-comparison sorts; an example is integer sorting, where all keys are integers. When the keys form a small
Comparison_sort
Problem of sorting pairs of numbers by their sum
unsolved problems in computer science In computer science, X + Y sorting is the problem of sorting pairs of numbers by their sums. Applications of the problem
X_+_Y_sorting
Number used for counting
are 0 (if included), 1, 2, 3, and so on. The terms positive integers, non-negative integers, whole numbers, and counting numbers are also used. The set
Natural_number
Model of computation in computer science
word RAM model, integer sorting can be done fairly efficiently. Yijie Han and Mikkel Thorup created a randomized algorithm to sort integers in expected time
Word_RAM
Computer memory concept
required, or at least valuable, in many algorithms such as binary search, integer sorting, or certain versions of sieve of Eratosthenes. Other data structures
Random_access
Divide and conquer sorting algorithm
1961. It is still a commonly used algorithm for sorting. Overall, it is slightly faster than merge sort and heapsort for randomized data, particularly
Quicksort
Query of largest element in a set less than an element
problems like integer sorting. The problem consists of maintaining a set S, which contains a subset of U integers. Each of these integers can be stored
Predecessor_problem
Fast sort algorithm for items with limited-size integer keys
Kirkpatrick–Reisch sorting is a fast sorting algorithm for items with limited-size integer keys. It is notable for having an asymptotic time complexity
Kirkpatrick–Reisch_sort
Theoretical model of computation
a problem such as integer sorting in which there are n integers to be sorted, the transdichotomous model assumes that each integer may be stored in a
Transdichotomous_model
Inefficient recursive sorting algorithm
Stooge sort is a recursive sorting algorithm. It is notable for its exceptionally poor time complexity of O ( n log 3 / log 1.5 ) {\displaystyle O(n^{\log
Stooge_sort
Type of computer science algorithm
not-in-place sorting algorithms The bit space requirement of a pointer is O(log n), but pointer size can be considered a constant in most sorting applications
In-place_algorithm
Divide and conquer sorting algorithm
Merge sort (also commonly spelled as mergesort or merge-sort) is an efficient, general-purpose, comparison-based sorting algorithm. Most implementations
Merge_sort
Simple sorting algorithm using comparisons
Bubble sort, sometimes referred to as sinking sort, is a simple sorting algorithm that repeatedly steps through the input list element by element, comparing
Bubble_sort
Natural sorting algorithm
Bead sort, also called gravity sort, is a natural sorting algorithm, developed by Joshua J. Arulanandham, Cristian S. Calude and Michael J. Dinneen in
Bead_sort
Minimum spanning forest algorithm that greedily adds edges
where they have small enough integer weight to allow integer sorting algorithms such as counting sort or radix sort to sort them in linear time, the disjoint
Kruskal's_algorithm
Sorting algorithm
In computer science, bogosort (also known as permutation sort and stupid sort) is a sorting algorithm based on the generate and test paradigm. The function
Bogosort
Mathematical model of computer
instance, it allows fast integer sorting algorithms, while sorting on the real RAM must be done with slower comparison sorting algorithms. However, some
Real_RAM
Efficient sorting algorithm that combines insert and merge operations
Block sort, or block merge sort, is a sorting algorithm combining at least two merge operations with an insertion sort to arrive at O(n log n) (see Big
Block_sort
American computer scientist (died 2023)
{\displaystyle O(n(1+{\tfrac {\log N}{\log n}}))} by radix sorting. However, it was assumed that integer sorting algorithms would necessarily have a time bound depending
Dan_Willard
Generalized alphabetical order
CE. This date ordering makes computerized sorting of dates easier by avoiding the need for a separate sorting algorithm. The monoid of words over an alphabet
Lexicographic_order
assumed to be integers, this can be sped up using integer sorting algorithms, to have the same asymptotic running time as the sorting algorithms. For
Maxima_of_a_point_set
Ordering of strings in alphabetical order with numeric value consideration
natural sorting by default to display file lists. Collation Lexicographical order ISO 8601 Semantic versioning "Sorting for Humans : Natural Sort Order"
Natural_sort_order
Assembly of written information into a standard order
long as only integers are to be sorted, although it can slow down sorting significantly. For example, Microsoft Windows does this when sorting file names
Collation
Sorting algorithm which uses multiple comparison intervals
Shell sort or Shell's method, is an in-place comparison sort. It can be understood as either a generalization of sorting by exchange (bubble sort) or sorting
Shellsort
Sorting algorithm using the heap data structure
git/tree/lib/sort.c#n205 Linux kernel source Maus, Arne [in Norwegian] (14 May 2014). "Sorting by generating the sorting permutation, and the
Heapsort
Mathematics problem
Pancake sorting is the mathematical problem of sorting a disordered stack of pancakes in order of size when a spatula can be inserted at any point in the
Pancake_sorting
Method for finding kth smallest value
by the sorting step, which requires Θ ( n log n ) {\displaystyle \Theta (n\log n)} time using a comparison sort. Even when integer sorting algorithms
Selection_algorithm
Abstract devices built up of a fixed number of "wires"
perform sorting on fixed numbers of values, in which case they are called sorting networks. Sorting networks differ from general comparison sorts in that
Sorting_network
Sorting algorithm
different sorting algorithm, or by recursively applying the bucket sorting algorithm. It is a distribution sort, a generalization of pigeonhole sort that allows
Bucket_sort
Class of algorithms in computational geometry
allow numbers to be sorted more quickly than O ( n log n ) {\displaystyle O(n\log n)} time, for instance by using integer sorting algorithms, planar
Convex_hull_algorithms
Sequence in computer science
for integer keys that are smaller than the number of items, and is frequently used as part of radix sort, a fast algorithm for sorting integers that
Prefix_sum
Algorithm for shuffling a finite sequence
sorts the set according to the assigned numbers. The sorting method has the same asymptotic time complexity as Fisher–Yates: although general sorting
Fisher–Yates_shuffle
Optimization problem
Algorithms. Albers, Susanne; Torben Hagerup (1992). "Improved parallel integer sorting without concurrent writing". Proceedings of the third annual ACM-SIAM
Job-shop_scheduling
Use of functions that call themselves
initial conditions. * @param data an array of integers SORTED in ASCENDING order * @param target the integer to search for * @param count the total number
Recursion_(computer_science)
Parallel sorting algorithm
for sorting. It is also used as a construction method for building a sorting network. The algorithm was devised by Ken Batcher. The resulting sorting networks
Bitonic_sorter
IEEE standard for floating-point arithmetic
encoding allows one to easily sort them without using floating-point hardware, as if the bits represented sign-magnitude integers, although it is unclear whether
IEEE_754
Natural number
(Number of primitive sorting networks on n elements; also number of rhombic tilings of a 2n-gon)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation
900_(number)
Tree data structure
are proved. Efficient imperative Standard ML code can be generated. Integer sorting Tango tree Peter van Emde Boas: Preserving order in a forest in less
Van_Emde_Boas_tree
Relaxed variant of the sorting problem
In computer science, partial sorting is a relaxed variant of the sorting problem. Total sorting is the problem of returning a list of items such that its
Partial_sorting
Integer associated with a civil day
ahargaṇa (Kali ahargaṇa number or Kalidina) is an integer associated with a civil day. The integer represents the number of civil days in a collection
Kali_ahargana
Property of being an even or odd number
In mathematics, parity is the property of an integer of whether it is even or odd. An integer is even if it is divisible by 2, and odd if it is not. For
Parity_(mathematics)
Function for sorting in C++ standard library
merge sort for in-place stable sorting and regular merge sort for stable sorting with additional memory. Partial sorting is implemented by partial_sort, which
Sort_(C++)
Worst-case number of comparisons used by sorting algorithms
computer science, the sorting numbers are a sequence of numbers introduced in 1950 by Hugo Steinhaus for the analysis of comparison sort algorithms. These
Sorting_number
Triangle with integer side lengths
An integer triangle or integral triangle is a triangle all of whose side lengths are integers. A rational triangle is one whose side lengths are rational
Integer_triangle
Search algorithm used in sorted arrays
than linear search for sorted arrays except if the array is short, although the array needs to be sorted beforehand. All sorting algorithms based on comparing
Binary_search
Sorting algorithm
Strand sort is a recursive sorting algorithm that sorts items of a list into increasing order. It has O(n2) worst-case time complexity, which occurs when
Strand_sort
Path-finding using high-weight graph edges
if the edge weights are represented as integers, then the time bounds for integer sorting a list of m integers would apply also to this problem. Berman
Widest_path_problem
Estimate of time taken for running an algorithm
Some examples of polynomial-time algorithms: The selection sort sorting algorithm on n integers performs An2 operations for some constant A. Thus it runs
Time_complexity
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
Rational Approximation of the Binary Thue-Morse-Mahler Number". Journal of Integer Sequences. 16 (13.2.3). Cuyt et al. 2008, p. 185. Cuyt et al. 2008, p. 186
List of mathematical constants
List_of_mathematical_constants
Product of two distinct primes ≡ 3 (mod 4)
form 4t + 3, for some integer t. Integers of this form are referred to as Blum primes. This means that the factors of a Blum integer are Gaussian primes
Blum_integer
Arithmetic operation
involving two numbers: the base, b, and the exponent, n. When n is a positive integer, exponentiation corresponds to repeated multiplication of the base: that
Exponentiation
Decision problem in computer science
{\displaystyle S} of integers and a target-sum T {\displaystyle T} , and the question is to decide whether any subset of the integers sum to precisely T
Subset_sum_problem
Order of bytes in a computer word
particularly of manipulating integer data by computers. In pure form, this is valid for moderately sized non-negative integers, e.g., of C data type unsigned
Endianness
Number divisible only by 1 and itself
number n, called trial division, tests whether n is a multiple of any integer between 2 and n. Faster algorithms include the Miller–Rabin primality
Prime_number
Ten raised to an integer power
of the integer powers of the number ten; in other words, ten multiplied by itself a certain number of times (when the power is a positive integer). By definition
Power_of_10
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
Mathematical proportionality to a square
real-valued function of an integer or natural number variable). Examples of quadratic growth include: Any quadratic polynomial. Certain integer sequences such as
Quadratic_growth
Product of an integer with itself
number or perfect square is an integer that is the square of an integer; in other words, it is the product of some integer with itself. For example, 9 is
Square_number
Cuboid whose edges and face diagonals have integer lengths
Euler, is a rectangular cuboid whose edges and face diagonals all have integer lengths. A primitive Euler brick is an Euler brick whose edge lengths are
Euler_brick
Two raised to an integer power
number of the form 2n where n is an integer, that is, the result of exponentiation with the number 2 as the base and integer n as the exponent. In the fast-growing
Power_of_two
Israeli-American computer scientist
triconnected components, ear decomposition, max-flow, 2-3 trees, hashing, integer sorting, near constant-time algorithms and data structures, exact string matching
Uzi_Vishkin
(PDE) using the fast Fourier transform (FFT) IS Integer Sort Sort small integers using the bucket sort EP Embarrassingly Parallel Generate independent
NAS_Parallel_Benchmarks
divisors of the numbers 1 to 1000. A divisor of an integer n is an integer m, for which n/m is again an integer (which is necessarily also a divisor of n).
Table_of_divisors
Quotient of two integers
integers, a numerator p and a nonzero denominator q. For example, 3 7 {\displaystyle {\tfrac {3}{7}}} is a rational number, as is every integer (for
Rational_number
Variant of radix sort
flag sort is an efficient, in-place variant of radix sort that distributes items into buckets. Non-comparative sorting algorithms such as radix sort and
American_flag_sort
Mapping arbitrary data to fixed-size values
32-bit integer. Thus the 32-bit integer Integer and 32-bit floating-point Float objects can simply use the value directly, whereas the 64-bit integer Long
Hash_function
Integer having a non-trivial divisor
number is a positive integer that can be formed by multiplying two smaller positive integers. Accordingly, it is a positive integer that has at least one
Composite_number
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
Mathematical table
Gaussian integer is either the zero, one of the four units (±1, ±i), a Gaussian prime or composite. The article is a table of Gaussian Integers x + iy followed
Table of Gaussian integer factorizations
Table_of_Gaussian_integer_factorizations
Encyclopedia of Integer Sequences. OEIS Foundation. Sloane, N. J. A. (ed.). "Sequence A000628". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation
List of straight-chain alkanes
List_of_straight-chain_alkanes
Number that permute or shift cyclically when multiplied by another number
mathematics, the transposable integers are integers that permute or shift cyclically when they are multiplied by another integer n {\displaystyle n} . Examples
Transposable_integer
Programming language
Printing(output); var i : integer; procedure PrintAnInteger(j : integer); begin ... end; function triple(x: integer): integer; begin triple := x * 3 end;
Pascal_(programming_language)
Theoretical programming language for describing concurrent computations
fixed by flipping k manually. Program bubblesort declare n: integer, A: array [0..n-1] of integer initially n = 20 # <|| i : 0 <= i and i < n :: A[i] = rand() %
UNITY_(programming_language)
positive integer, n! is the product of the first n positive integers, and is read as "n factorial". 2. Double factorial: if n is a positive integer, n!!
Glossary of mathematical symbols
Glossary_of_mathematical_symbols
Branch of pure mathematics
rational numbers), or defined as generalizations of the integers (for example, algebraic integers). Integers can be considered either in themselves or as solutions
Number_theory
Number representing a continuous quantity
negative numbers. The real numbers include the rational numbers, such as the integer − 5 {\displaystyle -5} and the fraction 4 / 3 {\displaystyle 4/3} . Real
Real_number
Data serialization format
data structures in a simple and compact format. Integers are encoded as i<base10 integer>e. The integer is encoded in base 10 and may be negative (indicated
Bencode
Associative array for storing key–value pairs
function in which an unsigned integer that is initially zero is repeatedly left shifted one bit and then xor'ed with the integer value of the next character
Hash_table
Standard library function in the C programming language
implements a sorting algorithm for arrays of arbitrary objects according to a user-provided comparison function. It is named after the "quicker sort" algorithm
Qsort
Comparison-based sorting algorithm
In computer science, smoothsort is a comparison-based sorting algorithm. A variant of heapsort, it was invented and published by Edsger Dijkstra in 1981
Smoothsort
Integer having only small prime factors
In number theory, an n-smooth (or n-friable) number is an integer whose prime factors are all less than or equal to n. For example, a 7-smooth number
Smooth_number
Numbers significantly larger than those used regularly
redirect targets Power of 10 – Ten raised to an integer power Power of two – Two raised to an integer power Tetration – Arithmetic operation Rayo's number
Large_numbers
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
Natural number
Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-06-15. "Sloane's A081357 : Sublime numbers". The On-Line Encyclopedia of Integer Sequences
12_(number)
Problem in computational complexity theory
in the real RAM model. The authors state that these results refute the integer and real-valued versions of the 3SUM conjecture (and refute other conjectures)
3SUM
Algorithm that combines multiple sorted lists into one
various sorting algorithms, most famously merge sort. The merge algorithm plays a critical role in the merge sort algorithm, a comparison-based sorting algorithm
Merge_algorithm
Numbers with many divisors
a positive integer that has more divisors than all smaller positive integers. If d(n) denotes the number of divisors of a positive integer n, then a positive
Highly_composite_number
Instructions for the x86 microprocessors
coding scheme. AVX2 (also known as Haswell New Instructions) expands most integer commands to 256 bits and introduces new instructions. They were first supported
Advanced_Vector_Extensions
Search tree data structure
corresponds to one call of the radix sorting routine, as the trie structure reflects the execution pattern of the top-down radix sort. If null links are encountered
Trie
Number that is less than the sum of its proper divisors
excessive number is a positive integer for which the sum of its proper divisors is greater than the number. The integer 12 is the first abundant number:
Abundant_number
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
Topics referred to by the same term
sieve, an integer factorization algorithm Quality Score, a variable used by search engines to set the rank and cost of ads Quicksort, a sorting algorithm
QS
Mathematical sequence involving arithmetic progressions
In mathematics, a Stanley sequence is an integer sequence generated by a greedy algorithm that chooses the sequence members to avoid arithmetic progressions
Stanley_sequence
Result of multiplying four instances of a number together
fourth power is always 1. Every positive integer can be expressed as the sum of at most 19 fourth powers; every integer larger than 13792 can be expressed as
Fourth_power
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INTEGER SORTING
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INTEGER SORTING
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INTEGER SORTING
INTEGER SORTING
INTEGER SORTING
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