Is the time complexity of the empty algorithm O(0)?

Question

So given the following program:

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Is the time complexity of this program O(0)? In other words, is 0 O(0)?

I thought answering this in a separate question

Answer 1

I would say it's O(1) by definition, but O(0) if you want to get technical about it: since O(k₁g(n)) is equivalent to O(k₂g(n)) for any constants k₁ and k₂, it follows that O(1 * 1) is equivalent to O(0 * 1), and therefore O(0) is equivalent to O(1).

However, the empty algorithm is not like, for example, the identity function, whose definition is something like "return your input". The empty algorithm is more like an empty statement, or whatever happens between two statements. Its definition is "do absolutely nothing with your input", presumably without even the implied overhead of simply having input.

Consequently, the complexity of the empty algorithm is unique in that O(0) has a complexity of zero times whatever function strikes your fancy, or simply zero. It follows that since the whole business is so wacky, and since O(0) doesn't already mean something useful, and since it's slightly ridiculous to even discuss such things, a reasonable special case for O(0) is something like this:

The complexity of the empty algorithm is O(0) in time and space. An algorithm with time complexity O(0) is equivalent to the empty algorithm.

So there you go.