轮次中的搜索、排序与蛋糕分割
Searching, Sorting, and Cake Cutting in Rounds
AI总结：
本研究将轮次中的比例蛋糕分割归约为带排名查询的排序，并分析无序与有序搜索的查询复杂度，给出随机与确定性算法的精确期望界。
摘要：
我们研究轮次中的搜索与排序问题，其动机源于一个公平分配问题：给定一个包含 $n$ 个参与者的蛋糕分割问题，在与参与者的至多 $k$ 轮交互中计算一个公平分配方案。轮次设置介于同步设置与完全自适应设置之间，同时也捕捉了并行复杂度。我们发现轮次中的比例蛋糕分割等价于轮次中带排名查询的排序问题。我们为轮次中的比例蛋糕分割设计了一个协议，而 Alon 和 Azar 给出了轮次中带排名查询的排序问题的下界。受排名查询模型的启发，我们随后考虑了两个基本搜索问题：有序搜索与无序搜索。\n在无序搜索中，我们给定一个数组 $\vec{x}=(x_1, \ldots, x_n)$ 和一个元素 $z$，并保证 $z$ 在数组中。我们可以访问一个预言机，它接收形如“$z$ 是否在位置 $i$？”的查询，并回答“是”或“否”。目标是在与预言机至多 $k$ 轮交互中，以至少 $p$ 的成功概率找到 $z$ 的位置。\n我们证明，在最坏情况输入下，随机化算法的期望查询复杂度为 $np\bigl(\frac{k+1}{2k}\bigr) \pm O(1)$，而在最坏情况输入分布下，确定性算法的期望查询复杂度为 $np \bigl(1 - \frac{k-1}{2k}p \bigr) \pm O(1)$。这些界甚至适用于完全自适应的无序搜索，其中两种复杂度之间的比值随着数组规模的增长收敛于 $2-p$。\n在有序搜索中，我们给定排序数组 $\vec{x}=(x_1, \ldots, x_n)$ 和元素 $z$，并保证 $z$ 在数组中。我们可以访问一个接收比较查询的预言机。这里我们发现，在最坏情况输入下随机化算法的期望查询复杂度与在最坏情况输入分布下确定性算法的期望查询复杂度本质上相同：$p k \cdot n^{\frac{1}{k}} \pm O(1+pk)$。
英文摘要：
We study searching and sorting in rounds motivated by a fair division question: given a cake cutting problem with $n$ players, compute a fair allocation in at most $k$ rounds of interaction with the players. Rounds interpolate between the simultaneous and the fully adaptive settings, also capturing parallel complexity. We find that proportional cake cutting in rounds is equivalent to sorting with rank queries in rounds. We design a protocol for proportional cake cutting in rounds, while lower bounds for sorting in rounds with rank queries were given by Alon and Azar. Inspired by the rank query model, we then consider two basic search problems: ordered and unordered search. In unordered search, we get an array $\vec{x}=(x_1, \ldots, x_n)$ and an element $z$ promised to be in $\vec{x}$. We have access to an oracle that receives queries of the form "Is $z$ at location $i$?" and answers "Yes" or "No". The goal is to find the location of $z$ with success probability at least $p$ in at most $k$ rounds of interaction with the oracle. We show the expected query complexity of randomized algorithms on a worst case input is $np\bigl(\frac{k+1}{2k}\bigr) \pm O(1)$, while that of deterministic algorithms on a worst case input distribution is $np \bigl(1 - \frac{k-1}{2k}p \bigr) \pm O(1)$. These bounds apply even to fully adaptive unordered search, where the ratio between the two complexities converges to $2-p$ as the size of the array grows. In ordered search, we get sorted array $\vec{x}=(x_1, \ldots, x_n)$ and element $z$ promised to be in $\vec{x}$. We have access to an oracle that gets comparison queries. Here we find that the expected query complexity of randomized algorithms on a worst case input and deterministic algorithms on a worst case input distribution is essentially the same: $p k \cdot n^{\frac{1}{k}} \pm O(1+pk)$.
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。
