succinctly: it is supposed, but not yet proved, that P != NP. where P is the class of problems with optimal algorithmic solution that is polynomial time complexity and NP is the class of problems where proposed solutions can be checked in polynomial time complexity but...
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Most programmers who use Big-O do not fully understand it, at least by this definition. There's also some deepness to "complexity classes" that isn't fully communicated via asymptotic complexity. You need to understand algorithmic reductions too.
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