Editing Open Problems:82
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Equivalence up to a permutation would then be the variant where one must test whether $p$ and $q$ are equal ''up to relabeling of the elements'': | Equivalence up to a permutation would then be the variant where one must test whether $p$ and $q$ are equal ''up to relabeling of the elements'': | ||
− | (i) $\exists \pi\in\mathcal{S}_n$ s.t. $p=q\circ\pi$ vs. (ii) $\forall \pi\in\mathcal{S}_n$, $\operatorname{d}_{\rm TV}(p,q\circ \pi)>\varepsilon$. Results of Valiant and Valiant {{cite|ValiantV-11}} imply that this question has sample complexity $\Theta | + | (i) $\exists \pi\in\mathcal{S}_n$ s.t. $p=q\circ\pi$ vs. (ii) $\forall \pi\in\mathcal{S}_n$, $\operatorname{d}_{\rm TV}(p,q\circ \pi)>\varepsilon$. Results of Valiant and Valiant {{cite|ValiantV-11}} imply that this question has sample complexity $\Theta(\frac{n}{\varepsilon^2\log n})$. |
The most general question then is: | The most general question then is: |