# O Time A

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 number of elementary operations performed by the algorithm, supposing that each 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 are taken to be related by a constant factor.

## O time A

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Algorithmic complexities are classified according to the type of function appearing in the big O notation. For example, an algorithm with time complexity O ( n ) \displaystyle O(n) is a linear time algorithm and an algorithm with time complexity O ( n α ) \displaystyle O(n^\alpha ) for some constant α > 1 \displaystyle \alpha >1 is a polynomial time algorithm.

An algorithm is said to be constant time (also written as O ( 1 ) \textstyle O(1) time) if the value of T ( n ) \textstyle T(n) (the complexity of the algorithm) is bounded by a value that does not depend on the size of the input. For example, accessing any single element in an array takes constant time as only one 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 in the array is needed in order to determine the minimal value. Hence it is a linear time operation, taking O ( n ) \textstyle 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.

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.[9] They are however allowed to be 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 1 , . . . , b k \displaystyle b_1,...,b_k it is assumed that the algorithm can in time O ( 1 ) \displaystyle O(1) request and obtain the value of b i \displaystyle b_i for any i.

Sub-linear time algorithms are typically randomized, and provide only 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.

An algorithm is said to take linear time, or O ( n ) \displaystyle O(n) time, if its time complexity is O ( n ) \displaystyle 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 c such that the running time is at most c n \displaystyle cn for every input of size 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.

For example, simple, comparison-based sorting algorithms 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.

An algorithm is said to be of polynomial time if its running time is upper bounded by a polynomial expression in the size of the input for the algorithm, that is, T(n) = O(nk) for some positive constant k.[1][12] Problems for which a deterministic polynomial-time algorithm exists belong to the complexity class 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".[13]

In some contexts, especially in 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:[14]

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. The second condition is strictly necessary: given the integer 2 n \displaystyle 2^n (which takes up space proportional to n in the Turing machine model), it is possible to compute 2 2 n \displaystyle 2^2^n with n multiplications using repeated squaring. However, the space used to represent 2 2 n \displaystyle 2^2^n is proportional to 2 n \displaystyle 2^n , 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.

An algorithm that runs in polynomial time but that is not strongly polynomial is said to run in weakly polynomial time.[15]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, which depends linearly on the magnitude of values in the problem and is not truly polynomial time.

P is the smallest time-complexity class on a deterministic machine which is 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.

An algorithm is defined to take superpolynomial time if T(n) is not bounded above by any polynomial. Using little omega notation, it is ω(nc) time for all constants c, where n is the input parameter, typically the number of bits in the input.

An algorithm that requires superpolynomial time lies outside the complexity class P. Cobham's thesis posits that these algorithms are impractical, and in many cases they are. Since the P versus NP problem is unresolved, it is unknown whether NP-complete problems require superpolynomial time.

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 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.[16]

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 the square of the number of vertices.) This conjecture (for the k-SAT problem) is known as the exponential time hypothesis.[18] 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.

The term 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,[19] and we list the two most widely used ones below.

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(2nε). The set of all such problems is the complexity class SUBEXP which can be defined in terms of DTIME as follows.[6][20][21][22]

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 ( L , k ) \displaystyle (L,k) of decision problems 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:[26] 041b061a72