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Here, complexity refers to the time complexity of performing computations on a multitape Turing machine. [1] See big O notation for an explanation of the notation used. Note: Due to the variety of multiplication algorithms, M ( n ) {\displaystyle M(n)} below stands in for the complexity of the chosen multiplication algorithm.
In theoretical 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 ...
In computational complexity theory, the complexity class NTIME(f(n)) is the set of decision problems that can be solved by a non-deterministic Turing machine which runs in time O(f(n)). Here O is the big O notation , f is some function, and n is the size of the input (for which the problem is to be decided).
Therefore, the time complexity, generally called bit complexity in this context, may be much larger than the 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})} for the usual algorithms ( Gaussian elimination ).
In computational complexity theory, DTIME (or TIME) is the computational resource of computation time for a deterministic Turing machine. It represents the amount of time (or number of computation steps) that a "normal" physical computer would take to solve a certain computational problem using a certain algorithm .
In computational complexity theory, the time hierarchy theorems are important statements about time-bounded computation on Turing machines. Informally, these theorems say that given more time, a Turing machine can solve more problems. For example, there are problems that can be solved with n 2 time but not n time, where n is the input length.
Hasse diagram of complexity classes including P, NP, co-NP, BPP, P/poly, PH, and PSPACE. The polynomial hierarchy is an analogue (at much lower complexity) of the exponential hierarchy and arithmetical hierarchy. It is known that PH is contained within PSPACE, but it is not known whether the two classes are equal.
In computational complexity theory, although it would be a non-formal usage of the term, the time/space complexity of a particular problem in terms of all algorithms that solve it with computational resources (i.e., time or space) bounded by a function of the input's size.