研究了在[1，n]中确定k个不可区分整数的集合S的问题.允许算法查询整数$q\in [1,n]$，并接收将该整数与从S中随机选择的整数进行比较的响应。该算法无法控制查询q与S的哪个元素进行比较。我们证明了这个问题的紧界。特别是，我们表明，在自然的制度，其中$k\le n$，查询的最佳数量，以达到$n^{-Ω(1)}$错误概率是$Θ(k^3 \log n)$。在$k>n$的情况下，查询的最佳数量是$Θ(n^2 k \log n)$。我们的主要技术工具包括使用信息论来推导下界，以及在Feige，Raghavan，Peleg和Upfal（1994）的精神下应用噪声二分搜索。特别是，我们的下限技术可能适用于其他情况下，涉及搜索不确定性。
英文摘要：
We investigate the problem of determining a set S of k indistinguishable integers in the range [1,n]. The algorithm is allowed to query an integer $q\in [1,n]$, and receive a response comparing this integer to an integer randomly chosen from S. The algorithm has no control over which element of S the query q is compared to. We show tight bounds for this problem. In particular, we show that in the natural regime where $k\le n$, the optimal number of queries to attain $n^{-Ω(1)}$ error probability is $Θ(k^3 \log n)$. In the regime where $k>n$, the optimal number of queries is $Θ(n^2 k \log n)$.
Our main technical tools include the use of information theory to derive the lower bounds, and the application of noisy binary search in the spirit of Feige, Raghavan, Peleg, and Upfal (1994). In particular, our lower bound technique is likely to be applicable in other situations that involve search under uncertainty.
Parameterized Quantum Query Complexity of Graph Collision
我们提出了三个新的量子算法的量子查询模型中的\textsc{graph-collision}问题：\begin{itemize} \item an algorithm based on tree decomposition that uses $O\left(\sqrt{n}t^{\sfrac{1}{6}}\right)$ queries where $t$ is the treewidth of the graph; \item an algorithm constructed on a span program that improves a result by Gavinsky and Ito. The algorithm uses $O(\sqrt{n}+\sqrt{α^{**}})$ queries, where $α^{**}(G)$ is a graph parameter defined by \[α^{**}(G):=\min_{VC\text{-- vertex cover of}G}{\max_{\substack{I\subseteq VC\\I\text{-- independent set}}}{\sum_{v\in I}{°{v}}}};\] \item an algorithm for a subclass of circulant graphs t…
