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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 n ε). The set of all such problems is the complexity class SUBEXP which can be defined in terms of DTIME as follows. [6] [19] [20] [21]
The graph shows the running time vs. problem size for a knapsack problem of a state-of-the-art, specialized algorithm. The quadratic fit suggests that the algorithmic complexity of the problem is O((log(n)) 2). [24] All of the above discussion has assumed that P means "easy" and "not in P" means "difficult", an assumption known as Cobham's ...
A complexity class is a set of problems of related complexity. Simpler complexity classes are defined by the following factors: The type of computational problem: The most commonly used problems are decision problems. However, complexity classes can be defined based on function problems, counting problems, optimization problems, promise ...
In computational complexity theory, NP (nondeterministic polynomial time) is a complexity class used to classify decision problems. NP is the set of decision problems for which the problem instances , where the answer is "yes", have proofs verifiable in polynomial time by a deterministic Turing machine , or alternatively the set of problems ...
Note that the study of complexity classes is intended primarily to understand the inherent complexity required to solve computational problems. Complexity theorists are thus generally concerned with finding the smallest complexity class that a problem falls into and are therefore concerned with identifying which class a computational problem ...
NC = P problem The P vs NP problem is a major unsolved question in computer science that asks whether every problem whose solution can be quickly verified by a computer (NP) can also be quickly solved by a computer (P). This question has profound implications for fields such as cryptography, algorithm design, and computational theory.
Thus, such problems have a complexity that is at least linear, that is, using big omega notation, a complexity (). The solution of some problems, typically in computer algebra and computational algebraic geometry, may be very large. In such a case, the complexity is lower bounded by the maximal size of the output, since the output must be written.
P can also be defined as an algorithmic complexity class for problems that are not decision problems [11] (even though, for example, finding the solution to a 2-satisfiability instance in polynomial time automatically gives a polynomial algorithm for the corresponding decision problem). In that case P is not a subset of NP, but P∩DEC is ...