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Are there any O(1/n) algorithms?
This just popped in my head for no particular reason, and I suppose it’s a strange question. Are there any known algorithms or problems which actually get easier or faster to solve with larger input? I’m guessing that if there are, it wouldn’t be for things like mutations or sorting, it would be for decision problems. Perhaps there’s some problem where having a ton of input makes it easy to decide something, but I can’t imagine what.
If there is no such thing as negative complexity, is there a proof that there cannot be? Or is it just that no one has found it yet?
No that is not possible. Since Big-Oh is suppose to be an approximation of the number of operations an algorithm performs related to its domain size then it would not make sense to describe an algorithm as using a negative number of operations.
The formal definition section of the wikipedia article actually defines the Big-Oh notation in terms of using positive real numbers. So there actually is not even a proof because the whole concept of Big-Oh has no meaning on the negative real numbers per the formal definition.
Short answer: Its not possible because the definition says so.