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Assume for some algorithmic problem it holds that, for each $\epsilon>0$, there is some algorithm that needs space at most $O(n^\epsilon)$.

Is there an established name for this kind of bound?

I'd like to stress that it is not the same as having an algorithm with space complexity $n^{o(1)}$, since in the above, the algorithm can be different, for each $\epsilon$.

I am actually not interested in such a bound with respect to space, but with respect to work for parallel algorithms.

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Have you seen the concept of PTAS (Polynomial-time approximation scheme)? I think is kind of what you are looking for. See link

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    $\begingroup$ Thank you. I know PTAS and I can see the analogy you see (for every $\epsilon$ there is an algorithm...). But I am specifically interested in settings, where the bound is of the form $n^\epsilon$. $\endgroup$
    – Thomas S
    Commented Mar 28, 2023 at 11:51

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