We revisit the problem of computing with noisy information considered in Feige et al. 1994, which includes computing the OR function from noisy queries, and computing the MAX, SEARCH and SORT functions from noisy pairwise comparisons. For $K$ given elements, the goal is to correctly recover the desired function with probability at least $1-\delta$ when the outcome of each query is flipped with probability $p$. We consider both the adaptive sampling setting where each query can be adaptively designed based on past outcomes, and the non-adaptive sampling setting where the query cannot depend on past outcomes. The prior work provides tight bounds on the worst-case query complexity in terms of the dependence on $K$. However, the upper and lower bounds do not match in terms of the dependence on $\delta$ and $p$. We improve the lower bounds for all the four functions under both adaptive and non-adaptive query models. Most of our lower bounds match the upper bounds up to constant factors when either $p$ or $\delta$ is bounded away from $0$, while the ratio between the best prior upper and lower bounds goes to infinity when $p\rightarrow 0$ or $p\rightarrow 1/2$. On the other hand, we also provide matching upper and lower bounds for the number of queries in expectation, improving both the upper and lower bounds for the variable-length query model.

翻译：我们重新审视了Feige等人（1994）中考虑的计算带噪声信息的问题，包括从噪声查询中计算OR函数，以及从噪声成对比较中计算MAX、SEARCH和SORT函数。对于给定的K个元素，目标是在每次查询结果以概率p翻转的情况下，以至少1-δ的概率正确恢复所需函数。我们考虑自适应采样设置（每次查询可根据先前结果动态设计）和非自适应采样设置（查询不能依赖于先前结果）。先前工作在关于K的依赖关系上给出了最坏情况查询复杂度的紧界。然而，上下界在δ和p的依赖关系上并不匹配。我们改进了所有四个函数在自适应和非自适应查询模型下的下界。当p或δ远离0时，我们的大多数下界与上界在常数因子内匹配，而先前的上下界之比在p→0或p→1/2时趋于无穷。另一方面，我们还提供了期望查询次数的匹配上下界，改进了变长查询模型中的上界和下界。