鲁棒最大值选择
Robust Max Selection
AI总结：
提出不可靠比较下的鲁棒最大值选择模型，给出最小输出集及确定性查询复杂度界，并设计近最优随机两阶段算法。
摘要：
我们引入了一种新模型，用于研究不可靠信息下的算法设计，并将该模型应用于如下问题：在一个包含 n 个元素、其中有 k 个损坏元素的列表中，寻找未损坏的最大元素。在我们的模型中，算法可以在任意一对元素之间执行黑盒比较查询。然而，涉及损坏元素的查询可能产生任意输出。特别地，损坏元素不需要表现为任何一致的值，并且可能在元素的排序中引入环。这给在该设定下设计正确算法带来了新的挑战。例如，算法不能简单地输出单个元素，因为在一个包含一个损坏元素和一个未损坏元素的列表中，无法区分这两个元素。为了保证正确性，该设定下的算法必须输出一个集合，以确保未损坏的最大元素被包含在内。我们首先证明，任何算法都必须输出大小至少为 min{n, 2k+1} 的集合，才能保证输出集合包含未损坏的最大值。对于输出大小恰好为 min{n, 2k+1} 的算法，在确定性算法方面，我们证明了产生一个包含未损坏最大值的元素集合所需比较查询数的匹配上下界为 Θ(nk)。在随机化方面，我们提出了一种两阶段算法，该算法以高概率使用 O(n + k polylog k) 次比较查询找到这样的集合，几乎匹配任何随机化算法为获得恒定正确概率所必需的 Ω(n) 次查询。
英文摘要：
We introduce a new model to study algorithm design under unreliable information, and apply this model for the problem of finding the uncorrupted maximum element of a list containing $n$ elements, among which are $k$ corrupted elements. Under our model, algorithms can perform black-box comparison queries between any pair of elements. However, queries regarding corrupted elements may have arbitrary output. In particular, corrupted elements do not need to behave as any consistent values, and may introduce cycles in the elements' ordering. This imposes new challenges for designing correct algorithms under this setting. For example, one cannot simply output a single element, as it is impossible to distinguish elements of a list containing one corrupted and one uncorrupted element. To ensure correctness, algorithms under this setting must output a set to make sure the uncorrupted maximum element is included. We first show that any algorithm must output a set of size at least $\min\{n, 2k + 1\}$ to ensure that the uncorrupted maximum is contained in the output set. Restricted to algorithms whose output size is exactly $\min\{n, 2k + 1\}$, for deterministic algorithms, we show matching upper and lower bounds of $Θ(nk)$ comparison queries to produce a set of elements that contains the uncorrupted maximum. On the randomized side, we propose a 2-stage algorithm that, with high probability, uses $O(n + k \operatorname{polylog} k)$ comparison queries to find such a set, almost matching the $Ω(n)$ queries necessary for any randomized algorithm to obtain a constant probability of being correct.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
