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'''Kandala (or similar)''' may refer to:
{{Redirect|Running time|the film|Running Time (film)}}
{{comparison computational complexity.svg}}
In [[computer science]], the '''time complexity''' is the [[computational complexity]] that measures or estimates the time taken for running an [[algorithm]]. Time complexity is commonly estimated by counting the number of elementary operations performed by the algorithm, supposing that an elementary operation takes a fixed amount of time to perform. Thus, the amount of time taken and the number of elementary operations performed by the algorithm differ by at most a [[constant factor]].
*[[Kandala (pogost)]], a Sami ''pogost'' (populated place) in [[Kolsky Uyezd]] of the Tsardom of Russia
Since an algorithm's running time may vary with different inputs of the same size, one commonly considers the [[Worst-case complexity|worst-case time complexity]], which is the maximum amount of time taken on inputs of a given size. Less common, and usually specified explicitly, is the [[average-case complexity]], which is the average of the time taken on inputs of a given size (this makes sense, as there are only a finite number of possible inputs of a given size).
*[[Kandala Subrahmanyam]] (born 1920), Indian lawyer, socialist leader, freedom activist, and parliamentarian
*[[Kevin Kandala]] (born 1992), Zimbabwean first-class cricketer; see [[2013–14 Pro50 Championship]]
In both cases, the time complexity is generally expressed as a [[Function (mathematics)|function]] of the size of the input.<ref name=Sipser />{{RP|226}} Since this function is generally difficult to compute exactly, and the running time is usually not critical for small input, one focuses commonly on the behavior of the complexity when the input size increases; that is, on the [[asymptotic analysis|asymptotic behavior]] of the complexity. Therefore, the time complexity is commonly expressed using [[big O notation]], typically <math>O(n),</math>
<math>O(n\log n),</math> <math>O(n^\alpha),</math> <math>O(2^n),</math> etc., where {{mvar|n}} is the input size measured by the number of [[bit]]s needed for representing it.
Algorithm complexities are classified by the function appearing in the big O notation. For example, an algorithm with time complexity <math>O(n)</math> is a ''linear time algorithm'', an algorithm with time complexity <math>O(n^\alpha)</math> for some constant <math>\alpha \ge 1</math> is a ''polynomial time algorithm''.
==Table of common time complexities==
{{Further information|Computational complexity of mathematical operations}}
The following table summarizes some classes of commonly encountered time complexities. In the table, poly(''x'') = ''x''<sup>''O''(1)</sup>, i.e., polynomial in ''x''.
{| class="wikitable sortable"
|-
! Name !! [[Complexity class]] !! Running time (''T''(''n'')) !! Examples of running times !! Example algorithms
|-
| constant time || || ''O''(1) || 10 || Determining if an integer (represented in binary) is even or odd
|-
| [[inverse Ackermann function|inverse Ackermann]] time || || ''O''(''α''(''n'')) || || [[Amortized time]] per operation using a [[disjoint set data structure|disjoint set]]
|-
| [[iterated logarithm]]ic time || || ''O''({{log-star}} ''n'') || || [[Cole-Vishkin algorithm|Distributed coloring of cycles]]
|-
| log-logarithmic || || ''O''(log log ''n'') || || Amortized time per operation using a bounded [[priority queue]]<ref>{{Cite journal|first1=Kurt |last1=Mehlhorn |first2=Stefan |last2=Naher|year=1990|title=Bounded ordered dictionaries in O(log log N) time and O(n) space|journal=Information Processing Letters|doi=10.1016/0020-0190(90)90022-P|volume=35|issue=4|pages=183–189}}</ref>
|-
| logarithmic time || [[DLOGTIME]] || ''O''(log ''n'') || log ''n'', log(''n''<sup>2</sup>) || [[Binary search]]
|-
| polylogarithmic time || || poly(log ''n'') || (log ''n'')<sup>2</sup> ||
|-
|fractional power || || {{math|''O''(''n''<sup>c</sup>)}} where {{Math|0 < c < 1}} || ''n''<sup>1/2</sup>, ''n''<sup>2/3</sup> || Searching in a [[kd-tree]]
|-
| linear time || || ''O''(''n'') || ''n'' || Finding the smallest or largest item in an unsorted [[Array data structure|array]]
|-
| "n log star n" time || || ''O''(''n'' {{log-star}} ''n'') || || [[Raimund Seidel|Seidel]]'s [[polygon triangulation]] algorithm.
|-
| quasilinear time || || ''O''(''n'' log ''n'') || ''n'' log ''n'', log ''n''! || Fastest possible [[comparison sort]]; [[Fast Fourier transform]].
|-
| quadratic time || || ''O''(''n''<sup>2</sup>) || ''n''<sup>2</sup> || [[Bubble sort]]; [[Insertion sort]]; [[Convolution theorem|Direct convolution]]
|-
| cubic time || || ''O''(''n''<sup>3</sup>) || ''n''<sup>3</sup> || Naive multiplication of two ''n''×''n'' matrices. Calculating [[partial correlation]].
|-
| polynomial time || [[P (complexity)|P]] || 2<sup>''O''(log ''n'')</sup> = poly(''n'') || ''n'', ''n'' log ''n'', ''n''<sup>10</sup> || [[Karmarkar's algorithm]] for [[linear programming]]; [[AKS primality test]]
|-
| quasi-polynomial time || QP || 2<sup>poly(log ''n'')</sup> || ''n''<sup>log log ''n''</sup>, ''n''<sup>log ''n''</sup> || Best-known O(log<sup>2</sup> ''n'')-[[approximation algorithm]] for the directed [[Steiner tree problem]].
|-
| sub-exponential time<br/>(first definition) || SUBEXP || ''O''(2<sup>''n''<sup>''ε''</sup></sup>) for all ''ε'' > 0 || ''O''(2<sup>log ''n''<sup>log log ''n''</sup></sup>) || Assuming complexity theoretic conjectures, [[Bounded-error probabilistic polynomial|BPP]] is contained in SUBEXP.<ref name="bpp" />
|-
| sub-exponential time<br/>(second definition) || || 2<sup>''o''(''n'')</sup> || 2<sup>''n''<sup>1/3</sup></sup> || Best-known algorithm for [[integer factorization]] and [[graph isomorphism problem|graph isomorphism]]
|-
| exponential time<br/>(with linear exponent) || [[E (complexity)|E]] || 2<sup>''O''(''n'')</sup> || 1.1<sup>''n''</sup>, 10<sup>''n''</sup> || Solving the [[traveling salesman problem]] using [[dynamic programming]]
|-
| exponential time || [[EXPTIME]] || 2<sup>poly(''n'')</sup> || 2<sup>''n''</sup>, 2<sup>''n''<sup>2</sup></sup> || Solving [[matrix chain multiplication]] via [[brute-force search]]
|-
| factorial time || || ''O''(''n''!) || ''n''! || Solving the [[Travelling salesman problem|traveling salesman problem]] via [[brute-force search]]
|-
| double exponential time || [[2-EXPTIME]] || 2<sup>2<sup>poly(''n'')</sup></sup> || 2<sup>2<sup>''n''</sup></sup> || Deciding the truth of a given statement in [[Presburger arithmetic]]
|}
==Constant time==
An algorithm is said to be '''constant time''' (also written as '''O(1)''' time) if the value of ''T''(''n'') is bounded by a value that does not depend on the size of the input. For example, accessing any single element in an [[array data structure|array]] takes constant time as only one [[Instruction (computer science)|operation]] has to be performed to locate it. In a similar manner, finding the minimal value in an array sorted in ascending order; it is the first element. However, finding the minimal value in an unordered array is not a constant time operation as scanning over each [[element (math)|element]] in the array is needed in order to determine the minimal value. Hence it is a linear time operation, taking O(n) time. If the number of elements is known in advance and does not change, however, such an algorithm can still be said to run in constant time.
Despite the name "constant time", the running time does not have to be independent of the problem size, but an upper bound for the running time has to be bounded independently of the problem size. For example, the task "exchange the values of ''a'' and ''b'' if necessary so that ''a''≤''b''" is called constant time even though the time may depend on whether or not it is already true that ''a'' ≤ ''b''. However, there is some constant ''t'' such that the time required is always ''at most'' ''t''.
Here are some examples of code fragments that run in constant time :
int index = 5;
int item = list[index];
'''if''' (condition true) '''then'''
perform some operation that runs in constant time
'''else'''
perform some other operation that runs in constant time
'''for''' i = 1 '''to''' 100
'''for''' j = 1 '''to''' 200
perform some operation that runs in constant time
If ''T''(''n'') is O(''any constant value''), this is equivalent to and stated in standard notation as ''T''(''n'') being O(1).
==Logarithmic time==
{{Further information|Logarithmic growth}}
An algorithm is said to take '''logarithmic time''' if ''T''(''n'') = '''O(log ''n'')'''. Due to the use of the [[binary numeral system]] by computers, the [[logarithm]] is frequently base 2 (that is, log<sub>2</sub> ''n'', sometimes written lg ''n''). However, by the [[Logarithmic identities#Changing the base|change of base]] for logarithms, log<sub>a</sub> ''n'' and log<sub>b</sub> ''n'' differ only by a constant multiplier, which in big-O notation is discarded; thus O(log ''n'') is the standard notation for logarithmic time algorithms regardless of the base of the logarithm.
Algorithms taking logarithmic time are commonly found in operations on [[binary tree]]s or when using [[binary search]].
An O(log n) algorithm is considered highly efficient, as the ratio of the number of operations to the size of the input decreases and tends to zero when ''n'' increases. An algorithm that must access all elements of its input cannot take logarithmic time, as the time taken for reading an input of size ''n'' is of the order of ''n''.
An example of logarithmic time is given by dictionary search. Let us consider a [[dictionary]] which contains ''n'' entries, sorted by [[alphabetical order]]. We suppose that, for {{math|1 ≤ ''k'' ≤ ''n''}}, one may access to the {{mvar|k}}th entry of the dictionary in a constant time. Let {{math|''D''[''k'']}} denote this {{mvar|k}}th entry. Under these hypotheses, the test if a word {{mvar|D}} is in the dictionary may be done in logarithmic time: consider <math>D(\lfloor n/2 \rfloor),</math> where <math>\lfloor\;\rfloor</math> denotes the [[floor function]]. If <math>D=D(\lfloor n/2 \rfloor),</math> then we are done. Else, if <math>D<D(\lfloor n/2 \rfloor),</math> continue the search in the same way in the left half of the dictionary, otherwise continue similarly with the right half of the dictionary. This algorithm is similar to the method often used to find an entry in a paper dictionary.
==Polylogarithmic time==
An algorithm is said to run in '''[[polylogarithmic function|polylogarithmic]] time''' if ''T''(''n'') = O((log ''n'')<sup>''k''</sup>), for some constant ''k''. For example, [[Matrix chain multiplication|matrix chain ordering]] can be solved in polylogarithmic time on a [[parallel random-access machine]].<ref>{{Cite journal| last1=Bradford | first1=Phillip G. | last2=Rawlins | first2=Gregory J. E. | last3=Shannon | first3=Gregory E. | title=Efficient Matrix Chain Ordering in Polylog Time | publisher=[[Society for Industrial and Applied Mathematics]] | location=Philadelphia | year=1998 | journal=SIAM Journal on Computing | issn=1095-7111 | volume=27 | issue=2 | pages=466–490 | doi=10.1137/S0097539794270698}}</ref>
==Sub-linear time==
An algorithm is said to run in '''sub-linear time''' (often spelled '''sublinear time''') if ''T''(''n'') = o(''n''). In particular this includes algorithms with the time complexities defined above, as well as others such as the O(''n''<sup>½</sup>) [[Grover's algorithm|Grover's search]] algorithm.
Typical algorithms that are exact and yet run in sub-linear time use [[parallel algorithm|parallel processing]] (as the NC<sub>1</sub> matrix determinant calculation does), [[quantum algorithm|non-classical processing]] (as Grover's search does), or alternatively have guaranteed assumptions on the input structure (as the logarithmic time [[binary search algorithm|binary search]] and many tree maintenance algorithms do). However, [[formal language]]s such as the set of all strings that have a 1-bit in the position indicated by the first log(n) bits of the string may depend on every bit of the input and yet be computable in sub-linear time.
The specific term ''sublinear time algorithm'' is usually reserved to algorithms that are unlike the above in that they are run over classical serial machine models and are not allowed prior assumptions on the input.<ref>{{Cite journal
| last1 = Kumar | first1 = Ravi
| last2 = Rubinfeld | first2 = Ronitt
| title = Sublinear time algorithms
| journal = SIGACT News
| volume = 34
| issue = 4
| pages = 57–67
| url = http://www.cs.princeton.edu/courses/archive/spr04/cos598B/bib/kumarR-survey.pdf
| year = 2003
| doi=10.1145/954092.954103}}</ref> They are however allowed to be [[randomized algorithm|randomized]], and indeed must be randomized for all but the most trivial of tasks.
As such an algorithm must provide an answer without reading the entire input, its particulars heavily depend on the access allowed to the input. Usually for an input that is represented as a binary string ''b''<sub>1</sub>,...,''b<sub>k</sub>'' it is assumed that the algorithm can in time O(1) request and obtain the value of ''b<sub>i</sub>'' for any ''i''.
Sub-linear time algorithms are typically randomized, and provide only [[approximation algorithm|approximate]] solutions. In fact, the property of a binary string having only zeros (and no ones) can be easily proved not to be decidable by a (non-approximate) sub-linear time algorithm. Sub-linear time algorithms arise naturally in the investigation of [[property testing]].
==Linear time==
An algorithm is said to take '''linear time''', or {{math|''O''(''n'')}} time, if its time complexity is {{math|''O''(''n'')}}. Informally, this means that the running time increases at most linearly with the size of the input. More precisely, this means that there is a constant {{mvar|c}} such that the running time is at most {{math|''cn''}} for every input of size {{mvar|n}}. For example, a procedure that adds up all elements of a list requires time proportional to the length of the list, if the adding time is constant, or, at least, bounded by a constant.
Linear time is the best possible time complexity in situations where the algorithm has to sequentially read its entire input. Therefore, much research has been invested into discovering algorithms exhibiting linear time or, at least, nearly linear time. This research includes both software and hardware methods. There are several hardware technologies which exploit [[Parallel computing|parallelism]] to provide this. An example is [[content-addressable memory]]. This concept of linear time is used in string matching algorithms such as the [[Boyer–Moore string search algorithm|Boyer–Moore algorithm]] and [[Ukkonen's algorithm]].
==Quasilinear time==
An algorithm is said to run in quasilinear time (also referred to as log-linear time) if ''T''(''n'') = '''O(''n'' log<sup>''k''</sup> ''n'')''' for some positive constant ''k''; linearithmic time is the case ''k'' = 1.<ref>{{cite journal|last1=Naik|first1=Ashish V.|last2=Regan|first2=Kenneth W.|last3=Sivakumar|first3=D.|title=On Quasilinear Time Complexity Theory|journal=Theoretical Computer Science|date=1995|volume=148|pages=325–349|url=http://www.cse.buffalo.edu/~regan/papers/pdf/NRS95.pdf|accessdate=23 February 2015|doi=10.1016/0304-3975(95)00031-q}}</ref><ref>Sedgewick, R. and Wayne K (2011). [http://algs4.cs.princeton.edu/home/ Algorithms, 4th Ed]. p. 186. Pearson Education, Inc.</ref> Using [[soft O notation]] these algorithms are Õ(''n''). Quasilinear time algorithms are also O(''n''<sup>1+ε</sup>) for every constant ε > 0, and thus run faster than any polynomial time algorithm whose time bound includes a term ''n''<sup>''c''</sup> for any ''c'' > 1.
Algorithms which run in quasilinear time include:
* [[In-place merge sort]], O(''n'' log<sup>''2''</sup> ''n'')
* [[Quicksort]], O(''n'' log ''n''), in its randomized version, has a running time that is O(''n'' log ''n'') in expectation on the worst-case input. Its non-randomized version has a O(''n'' log ''n'') running time only when considering average case complexity.
* [[Heapsort]], O(''n'' log ''n''), [[merge sort]], [[introsort]], binary tree sort, [[smoothsort]], [[patience sorting]], etc. in the worst case
* [[Fast Fourier transform]]s, O(''n'' log ''n'')
* [[Monge array]] calculation, O(''n'' log ''n'')
In many cases, the ''n'' · log ''n'' running time is simply the result of performing a Θ(log ''n'') operation ''n'' times (for the notation, see {{slink|Big O notation|Family of Bachmann–Landau notations}}). For example, [[binary tree sort]] creates a [[binary tree]] by inserting each element of the ''n''-sized array one by one. Since the insert operation on a [[self-balancing binary search tree]] takes ''O''(log ''n'') time, the entire algorithm takes ''O''(''n'' log ''n'') time.
[[Comparison sort]]s require at least ''O''(''n'' log ''n'') number of comparisons in the worst case because log(''n''!) = Θ(''n'' log ''n''), by [[Stirling's approximation]]. They also frequently arise from the [[recurrence relation]] ''T''(''n'') = 2''T''(''n''/2) + ''O''(''n'').
==Sub-quadratic time==
An [[algorithm]] is said to be '''subquadratic time''' if ''T''(''n'') = o(''n''<sup>2</sup>).
For example, simple, comparison-based [[sorting algorithm]]s are quadratic (e.g. [[insertion sort]]), but more advanced algorithms can be found that are subquadratic (e.g. [[Shell sort]]). No general-purpose sorts run in linear time, but the change from quadratic to sub-quadratic is of great practical importance.
==Polynomial time==
An algorithm is said to be of '''polynomial time''' if its running time is [[upper bound]]ed by a [[polynomial expression]] in the size of the input for the algorithm, i.e., ''T''(''n'') = O(''n''<sup>''k''</sup>) for some non-negative, non-zero constant ''k''.<ref name=Sipser>{{Cite book| last=Sipser | first=Michael | authorlink=Michael Sipser | title=Introduction to the Theory of Computation | year=2006 | publisher=Course Technology Inc | location= | isbn=0-619-21764-2 | pages=}}</ref><ref>{{Cite book| last=Papadimitriou | first=Christos H. | authorlink=Christos H. Papadimitriou | title=Computational complexity | year=1994 | publisher=Addison-Wesley | location=Reading, Mass. | isbn=0-201-53082-1 | pages=}}</ref> [[Decision problem|Problems]] for which a deterministic polynomial time algorithm exists belong to the [[complexity class]] '''[[P (complexity)|P]]''', which is central in the field of [[computational complexity theory]]. [[Cobham's thesis]] states that polynomial time is a synonym for "tractable", "feasible", "efficient", or "fast".<ref>{{Cite book| last=Cobham | first=Alan | authorlink=Alan Cobham | year = 1965 | chapter = The intrinsic computational difficulty of functions | title = Proc. Logic, Methodology, and Philosophy of Science II | publisher = North Holland}}</ref>
Some examples of polynomial time algorithms:
* The [[selection sort]] sorting algorithm on ''n'' integers performs <math>An^2</math> operations for some constant ''A''. Thus it runs in time <math>O(n^2)</math> and is a polynomial time algorithm.
* All the basic arithmetic operations (addition, subtraction, multiplication, division, and comparison) can be done in polynomial time.
* [[Maximum matching]]s in [[Graph (discrete mathematics)|graphs]] can be found in polynomial time.
===Strongly and weakly polynomial time===
<!--[[Strongly polynomial]] redirects here.-->
In some contexts, especially in [[Optimization (mathematics)|optimization]], one differentiates between '''strongly polynomial time''' and '''weakly polynomial time''' algorithms. These two concepts are only relevant if the inputs to the algorithms consist of integers.
Strongly polynomial time is defined in the arithmetic model of computation. In this model of computation the basic arithmetic operations (addition, subtraction, multiplication, division, and comparison) take a unit time step to perform, regardless of the sizes of the operands. The algorithm runs in strongly polynomial time if <ref>{{Cite book| last=Grötschel | first=Martin | author1-link= Martin Grötschel |author2=[[László Lovász]] |author3=[[Alexander Schrijver]] | year = 1988 | chapter = Complexity, Oracles, and Numerical Computation| title = Geometric Algorithms and Combinatorial Optimization | publisher = Springer | isbn=0-387-13624-X}}</ref>
# the number of operations in the arithmetic model of computation is bounded by a polynomial in the number of integers in the input instance; and
# the space used by the algorithm is bounded by a polynomial in the size of the input.
Any algorithm with these two properties can be converted to a polynomial time algorithm by replacing the arithmetic operations by suitable algorithms for performing the arithmetic operations on a [[Turing machine]]. If the second of the above requirements is not met, then this is not true anymore. Given the integer <math>2^n</math> (which takes up space proportional to n in the Turing machine model), it is possible to compute <math>2^{2^n}</math>
with n multiplications using [[repeated squaring]]. However, the space used to represent <math>2^{2^n}</math> is proportional to <math>2^n</math>, and thus exponential rather than polynomial in the space used to represent the input. Hence, it is not possible to carry out this computation in polynomial time on a Turing machine, but it is possible to compute it by polynomially many arithmetic operations.
Conversely, there are algorithms which run in a number of Turing machine steps bounded by a polynomial in the length of binary-encoded input, but do not take a number of arithmetic operations bounded by a polynomial in the number of input numbers. The [[Euclidean algorithm]] for computing the [[greatest common divisor]] of two integers is one example. Given two integers <math>a</math> and <math>b</math> the running time of the algorithm is bounded by a number of Turing machine steps that is polynomial in the size of a binary representation of <math>a</math> and <math>b</math>. At the same time, the number of arithmetic operations cannot be bounded by the number of integers in the input (which is constant in this case, there are always only two integers in the input). Due to the latter observation, the algorithm does not run in strongly polynomial time. Its real running time depends on the magnitudes of <math>a</math> and <math>b</math> and not only on the number of integers in the input.
An algorithm which runs in polynomial time but which is not strongly polynomial is said to run in '''weakly polynomial time'''.<ref>{{Cite book| last=Schrijver | first=Alexander | authorlink = Alexander Schrijver| year = 2003 | chapter = Preliminaries on algorithms and Complexity | title = Combinatorial Optimization: Polyhedra and Efficiency | volume = 1 | publisher = Springer | isbn=3-540-44389-4}}</ref>
A well-known example of a problem for which a weakly polynomial-time algorithm is known, but is not known to admit a strongly polynomial-time algorithm,
is [[linear programming]]. Weakly polynomial-time should not be confused with [[pseudo-polynomial time]].
===Complexity classes===
The concept of polynomial time leads to several complexity classes in computational complexity theory. Some important classes defined using polynomial time are the following.
* '''[[P (complexity)|P]]''': The [[complexity class]] of [[decision problem]]s that can be solved on a [[deterministic Turing machine]] in polynomial time.
* '''[[NP (complexity)|NP]]''': The complexity class of decision problems that can be solved on a [[non-deterministic Turing machine]] in polynomial time.
* '''[[ZPP (complexity)|ZPP]]''': The complexity class of decision problems that can be solved with zero error on a [[probabilistic Turing machine]] in polynomial time.
* '''[[RP (complexity)|RP]]''': The complexity class of decision problems that can be solved with 1-sided error on a probabilistic Turing machine in polynomial time.
* '''[[BPP (complexity)|BPP]]''': The complexity class of decision problems that can be solved with 2-sided error on a probabilistic Turing machine in polynomial time.
* '''[[BQP]]''': The complexity class of decision problems that can be solved with 2-sided error on a [[quantum Turing machine]] in polynomial time.
P is the smallest time-complexity class on a deterministic machine which is [[Robustness (computer science)|robust]] in terms of machine model changes. (For example, a change from a single-tape Turing machine to a multi-tape machine can lead to a quadratic speedup, but any algorithm that runs in polynomial time under one model also does so on the other.) Any given [[abstract machine]] will have a complexity class corresponding to the problems which can be solved in polynomial time on that machine.
==Superpolynomial time==
An algorithm is said to take '''superpolynomial time''' if ''T''(''n'') is not bounded above by any polynomial. Using [[big O notation#Family of Bachmann–Landau notations|little omega notation]], it is ω(''n''<sup>''c''</sup>) time for all constants ''c'', where ''n'' is the input parameter, typically the number of bits in the input.
For example, an algorithm that runs for 2<sup>''n''</sup> steps on an input of size ''n'' requires superpolynomial time (more specifically, exponential time).
An algorithm that uses exponential resources is clearly superpolynomial, but some algorithms are only very weakly superpolynomial. For example, the [[Adleman–Pomerance–Rumely primality test]] runs for ''n''<sup>O(log log ''n'')</sup> time on ''n''-bit inputs; this grows faster than any polynomial for large enough ''n'', but the input size must become impractically large before it cannot be dominated by a polynomial with small degree.
An algorithm that requires superpolynomial time lies outside the [[complexity class]] '''[[P (complexity)|P]]'''. [[Cobham's thesis]] posits that these algorithms are impractical, and in many cases they are. Since the [[P versus NP problem]] is unresolved, no algorithm for an [[NP-complete]] problem is currently known to run in polynomial time.
==Quasi-polynomial time==
'''Quasi-polynomial time''' algorithms are algorithms that run slower than polynomial time, yet not so slow as to be exponential time. The worst case running time of a quasi-polynomial time algorithm is <math>2^{O((\log n)^c)}</math> for some fixed <math>c > 0</math>. If the constant "c" in the definition of quasi-polynomial time algorithms is equal to 1, we get a polynomial time algorithm, and if it is less than 1, we get a sub-linear time algorithm.
Quasi-polynomial time algorithms typically arise in [[reduction (complexity)|reductions]] from an [[NP-hard]] problem to another problem. For example, one can take an instance of an NP hard problem, say [[Boolean satisfiability problem|3SAT]], and convert it to an instance of another problem B, but the size of the instance becomes <math>2^{O((\log n)^c)}</math>. In that case, this reduction does not prove that problem B is NP-hard; this reduction only shows that there is no polynomial time algorithm for B unless there is a quasi-polynomial time algorithm for 3SAT (and thus all of [[NP (complexity)|NP]]). Similarly, there are some problems for which we know quasi-polynomial time algorithms, but no polynomial time algorithm is known. Such problems arise in approximation algorithms; a famous example is the directed [[Steiner tree problem]], for which there is a quasi-polynomial time approximation algorithm achieving an approximation factor of <math>O(\log^3 n)</math> (n being the number of vertices), but showing the existence of such a polynomial time algorithm is an open problem.
Other computational problems with quasi-polynomial time solutions but no known polynomial time solution include the [[planted clique]] problem in which the goal is to [[clique problem|find a large clique]] in the union of a clique and a [[random graph]]. Although quasi-polynomially solvable, it has been conjectured that the planted clique problem has no polynomial time solution; this planted clique conjecture has been used as a [[computational hardness assumption]] to prove the difficulty of several other problems in computational [[game theory]], [[property testing]], and [[machine learning]].<ref>{{citation
| last1 = Braverman | first1 = Mark
| last2 = Ko | first2 = Young Kun
| last3 = Rubinstein | first3 = Aviad
| last4 = Weinstein | first4 = Omri
| arxiv = 1504.08352
| title = ETH hardness for densest-{{mvar|k}}-subgraph with perfect completeness
| year = 2015}}.</ref>
The complexity class '''QP''' consists of all problems that have quasi-polynomial time algorithms. It can be defined in terms of [[DTIME]] as follows.<ref>{{ComplexityZoo|Class QP: Quasipolynomial-Time|Q#qp}}</ref>
:<math>\mbox{QP} = \bigcup_{c \in \mathbb{N}} \mbox{DTIME}(2^{(\log n)^c})</math>
===Relation to NP-complete problems===
In complexity theory, the unsolved [[P versus NP]] problem asks if all problems in NP have polynomial-time algorithms. All the best-known algorithms for [[NP-complete]] problems like 3SAT etc. take exponential time. Indeed, it is conjectured for many natural NP-complete problems that they do not have sub-exponential time algorithms. Here "sub-exponential time" is taken to mean the second definition presented below. (On the other hand, many graph problems represented in the natural way by adjacency matrices are solvable in subexponential time simply because the size of the input is square of the number of vertices.) This conjecture (for the k-SAT problem) is known as the [[exponential time hypothesis]].<ref name="ETH">{{Cite journal| last1=Impagliazzo | first1=R. | last2=Paturi | first2=R. | title=On the complexity of k-SAT | publisher=[[Elsevier]] | year=2001 | journal=Journal of Computer and System Sciences | issn=1090-2724 | volume=62 | issue=2 | pages=367–375 | doi=10.1006/jcss.2000.1727}}</ref> Since it is conjectured that NP-complete problems do not have quasi-polynomial time algorithms, some inapproximability results in the field of [[approximation algorithms]] make the assumption that NP-complete problems do not have quasi-polynomial time algorithms. For example, see the known inapproximability results for the [[set cover]] problem.
=={{anchor|subexponential time}}Sub-exponential time==
The term '''[[Infra-exponential|sub-exponential]] time''' is used to express that the running time of some algorithm may grow faster than any polynomial but is still significantly smaller than an exponential. In this sense, problems that have sub-exponential time algorithms are somewhat more tractable than those that only have exponential algorithms. The precise definition of "sub-exponential" is not generally agreed upon,<ref>{{Cite web|url=http://scottaaronson.com/blog/?p=394 |title=A not-quite-exponential dilemma |author=Aaronson, Scott |date=5 April 2009 |work=Shtetl-Optimized |accessdate=2 December 2009}}</ref> and we list the two most widely used ones below.
===First definition===<!-- [[SUBEXP]] redirects here -->
A problem is said to be sub-exponential time solvable if it can be solved in running times whose logarithms grow smaller than any given polynomial. More precisely, a problem is in sub-exponential time if for every ε > 0 there exists an algorithm which solves the problem in time O(2<sup>n<sup>ε</sup></sup>). The set of all such problems is the complexity class '''SUBEXP''' which can be defined in terms of [[DTIME]] as follows.<ref name="bpp">{{Cite journal| last1=Babai | first1=László | author1-link = László Babai | last2=Fortnow | first2=Lance | author2-link = Lance Fortnow | last3=Nisan | first3=N. | author3-link = Noam Nisan | last4=Wigderson | first4=Avi | author4-link = Avi Wigderson | title=BPP has subexponential time simulations unless EXPTIME has publishable proofs | publisher=[[Springer-Verlag]] | location=Berlin, New York | year=1993 | journal=Computational Complexity | volume=3 | issue=4 | pages=307–318 | doi=10.1007/BF01275486}}</ref><ref>{{ComplexityZoo|Class SUBEXP: Deterministic Subexponential-Time|S#subexp}}</ref><ref>{{Cite journal| last1=Moser | first1=P. | title=Baire's Categories on Small Complexity Classes | publisher=Springer-Verlag | location=Berlin, New York | year=2003 | journal=[[Lecture Notes in Computer Science]] | issn=0302-9743 | pages=333–342}}
</ref><ref>{{Cite journal| last1=Miltersen | first1=P.B. | title=DERANDOMIZING COMPLEXITY CLASSES | publisher=Kluwer Academic Pub | year=2001 | journal=Handbook of Randomized Computing | page=843}}</ref>
:<math>\text{SUBEXP}=\bigcap_{\varepsilon>0} \text{DTIME}\left(2^{n^\varepsilon}\right)</math>
Note that this notion of sub-exponential is non-uniform in terms of ε in the sense that ε is not part of the input and each ε may have its own algorithm for the problem.
===Second definition===
Some authors define sub-exponential time as running times in 2<sup>o(''n'')</sup>.<ref name="ETH" /><ref>{{Cite journal| last1=Kuperberg | first1=Greg | title=A Subexponential-Time Quantum Algorithm for the Dihedral Hidden Subgroup Problem | publisher=[[Society for Industrial and Applied Mathematics]] | location=Philadelphia | year=2005 | journal=SIAM Journal on Computing | issn=1095-7111 | volume=35 | issue=1 | page=188 | doi=10.1137/s0097539703436345}}</ref><ref>{{cite arXiv|eprint=quant-ph/0406151v1|author1=Oded Regev|title=A Subexponential Time Algorithm for the Dihedral Hidden Subgroup Problem with Polynomial Space|class=quant-ph|year=2004}}</ref> This definition allows larger running times than the first definition of sub-exponential time. An example of such a sub-exponential time algorithm is the best-known classical algorithm for integer factorization, the [[general number field sieve]], which runs in time about <math>2^{\tilde{O}(n^{1/3})}</math>, where the length of the input is ''n''. Another example is the best-known algorithm for the [[graph isomorphism problem]], which runs in time <math>2^{O(\sqrt{n \log n})}</math>.
Note that it makes a difference whether the algorithm is allowed to be sub-exponential in the size of the instance, the number of vertices, or the number of edges. In [[parameterized complexity]], this difference is made explicit by considering pairs <math>(L,k)</math> of [[decision problem]]s and parameters ''k''. '''SUBEPT''' is the class of all parameterized problems that run in time sub-exponential in ''k'' and polynomial in the input size ''n'':<ref>{{Cite book
| last=Flum | first=Jörg
| last2=Grohe | first2=Martin
| title = Parameterized Complexity Theory | year = 2006 | publisher = Springer
| url = https://www.springer.com/east/home/generic/search/results?SGWID=5-40109-22-141358322-0
| isbn = 978-3-540-29952-3 |ref=harv |accessdate=2010-03-05| page=417}}</ref>
:<math>\text{SUBEPT}=\text{DTIME}\left(2^{o(k)} \cdot \text{poly}(n)\right).</math>
More precisely, SUBEPT is the class of all parameterized problems <math>(L,k)</math> for which there is a [[computable function]] <math>f : \mathbb N\to\mathbb N</math> with <math>f \in o(k)</math> and an algorithm that decides ''L'' in time <math>2^{f(k)} \cdot \text{poly}(n)</math>.
====Exponential time hypothesis====
{{Main article|Exponential time hypothesis}}
The '''exponential time hypothesis''' ('''ETH''') is that [[3SAT]], the satisfiability problem of Boolean formulas in [[conjunctive normal form]] with, at most, three literals per clause and with ''n'' variables, cannot be solved in time 2<sup>''o''(''n'')</sup>. More precisely, the hypothesis is that there is some absolute constant {{math|''c''>0}} such that 3SAT cannot be decided in time 2<sup>''cn''</sup> by any deterministic Turing machine. With ''m'' denoting the number of clauses, ETH is equivalent to the hypothesis that ''k''SAT cannot be solved in time 2<sup>''o''(''m'')</sup> for any integer {{math|''k'' ≥ 3}}.<ref>{{Cite journal|first1=R.|last1=Impagliazzo|author1-link=Russell Impagliazzo|first2=R.|last2=Paturi|first3=F.|last3=Zane|title=Which problems have strongly exponential complexity?|journal=[[Journal of Computer and System Sciences]]|volume=63|issue=4|year=2001|pages=512–530|doi=10.1006/jcss.2001.1774}}</ref> The exponential time hypothesis implies [[P ≠ NP]].
==Exponential time==
An algorithm is said to be '''exponential time''', if ''T''(''n'') is upper bounded by 2<sup>poly(''n'')</sup>, where poly(''n'') is some polynomial in ''n''. More formally, an algorithm is exponential time if ''T''(''n'') is bounded by O(2<sup>''n''<sup>''k''</sup></sup>) for some constant ''k''. Problems which admit exponential time algorithms on a deterministic Turing machine form the complexity class known as '''[[EXP]]'''.
:<math>\text{EXP} = \bigcup_{c \in \mathbb{N}} \text{DTIME}\left(2^{n^c}\right)</math>
Sometimes, exponential time is used to refer to algorithms that have ''T''(''n'') = 2<sup>O(''n'')</sup>, where the exponent is at most a linear function of ''n''. This gives rise to the complexity class '''[[E (complexity)|E]]'''.
:<math>\text{E} = \bigcup_{c \in \mathbb{N}} \text{DTIME}\left(2^{cn}\right)</math>
==Double exponential time==
An algorithm is said to be [[double exponential function|double exponential]] time if ''T''(''n'') is upper bounded by 2<sup>2<sup>poly(''n'')</sup></sup>, where poly(''n'') is some polynomial in ''n''. Such algorithms belong to the complexity class [[2-EXPTIME]].
:<math>\mbox{2-EXPTIME} = \bigcup_{c \in \mathbb{N}} \mbox{DTIME}\left( 2^{2^{n^c}}\right)</math>
Well-known double exponential time algorithms include:
* Decision procedures for [[Presburger arithmetic]]
* Computing a [[Gröbner basis]] (in the worst case <ref>Mayr,E. & Mayer,A.: The Complexity of the Word Problem for Commutative Semi-groups and
Polynomial Ideals. ''Adv. in Math.'' 46(1982) pp. 305–329</ref>)
* [[Quantifier elimination]] on [[real closed field]]s takes at least double exponential time,<ref>J.H. Davenport & J. Heintz: Real Quantifier Elimination is Doubly Exponential.
''J. Symbolic Comp.'' 5(1988) pp. 29–35.</ref> and can be done in this time.<ref>G.E. Collins: Quantifier Elimination for Real Closed Fields by Cylindrical Algebraic
Decomposition. Proc. 2nd. GI Conference Automata Theory & Formal Languages (Springer Lecture
Notes in Computer Science 33) pp. 134–183</ref>
==See also==
*[[Kandla]], a port in Gujarat state, India
* [[L-notation]]
*[[Kandhla]], town in Uttar Pradesh, India
* [[Space complexity]]
*[[Khandala]], hill station in Maharashtra, India
*[[Qandala]], an ancient port city in Somalia
==References==
*[[Kangaroo]], kangaroo kandala
{{Reflist|colwidth=33em}}
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[[Category:Analysis of algorithms]]
[[Category:Computational complexity theory]]
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