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P, NP, and NP-Complete Problems - Coggle Diagram
P, NP, and NP-Complete Problems
We say that an algorithm solves a problem in polynomial time if its worst-case time efficiency belongs to O(p(n)) where p(n) is a polynomial of the problem’s input size n.
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P and NP Problems
Class P is a class of decision problems that can be solved in polynomial time by (deterministic) algorithms. This class of problems is called polynomial.
problems that can be
solved in polynomial time as the set that computer science theoreticians call P.
decision problems, which are problems with yes/no answers.
Such problems are called undecidable, as opposed to decidable problems that can be solved by an algorithm.
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There are many important problems, however, for which no polynomial-time algorithm
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A nondeterministic algorithm is a two-stage procedure that takes as its input an instance I of a decision problem and does the following.
a nondeterministic algorithm is said to be nondeterministic polynomial if the time efficiency of its verification stage is polynomial.
Class NP is the class of decision problems that can be solved by nondeterministic polynomial algorithms. This class of problems is called nondeterministic polynomial.
Most decision problems are in NP. First of all, this class includes all the
problems in P: P ⊆ NP.
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A decision problem D1 is said to be polynomially reducible to a decision problem D2, if there exists a function t that transforms instances of D1 to instances of D2 such that:
t maps all yes instances of D1 to yes instances of D2 and all no instances of D1 to no instances of D2
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Their algorithm does not solve, however, the related problem of factoring large composite integers, which lies at the heart of the widely used encryption method called the RSA algorithm