We study sorting and searching in rounds, motivated by a cake cutting problem. The search problem we consider is: we are given an array $x = (x_1, \ldots, x_n)$ and an element $z$ promised to be in the array. We have access to an oracle that answers comparison queries: "How is $x_i$ compared to $x_j$?", where the answer can be "$<$", "$=$", or "$>$". The goal is to find the location of $z$ with success probability at least $p \in [0,1]$ in at most $k$ rounds of interaction with the oracle. The problem is called ordered or unordered search, depending on whether the array $x$ is sorted or unsorted, respectively. For ordered search, we show the expected query complexity of randomized algorithms is $\Theta\bigl(k\cdot p \cdot n^{1/k}\bigr)$ in the worst case. In contrast, the expected query complexity of deterministic algorithms searching for a uniformly random element is $\Theta\bigl(k\cdot p^{1/k} \cdot n^{1/k}\bigr)$. The uniform distribution is the worst case for deterministic algorithms. For unordered search, the expected query complexity of randomized algorithms is $np\bigl(\frac{k+1}{2k}\bigr) \pm 1$ in the worst case, while the expected query complexity of deterministic algorithms searching for a uniformly random element is $np \bigl(1 - \frac{k-1}{2k}p \bigr) \pm 1$. We also discuss the connections of these search problems to the rank query model, where the array $x$ can be accessed via queries of the form "Is rank$(x_i) \leq k$?". Unordered search is equivalent to Select with rank queries (given $q$, find $x_i$ with rank $q$) and ordered search to Locate with rank queries (given $x_i$, find its rank). We show an equivalence between sorting with rank queries and proportional cake cutting with contiguous pieces for any number of rounds, as well as an improved lower bound for deterministic sorting in rounds with rank queries.

中文翻译：

---

圆形搜索，排序和切蛋糕

我们研究切蛋糕问题引起的分类和搜索。我们考虑的搜索问题是：给我们一个数组$ x =（x_1，\ ldots，x_n）$和一个元素$ z $被保证在数组中。我们可以访问一个回答比较查询的Oracle：“ $ x_i $与$ x_j $相比如何？”，其中答案可以是“ $ <$”，“ $ = $”或“ $> $”。目标是找到在与Oracle的最多$ k $次交互中，成功概率至少为$ p \ in [0,1] $的$ z $的位置。该问题称为有序搜索或无序搜索，具体取决于数组$ x $是排序还是未排序。对于有序搜索，我们显示在最坏的情况下，随机算法的预期查询复杂度为$ \ Theta \ bigl（k \ cdot p \ cdot n ^ {1 / k} \ bigr）$。相反，确定性算法搜索均匀随机元素的预期查询复杂度为$ \ Theta \ bigl（k \ cdot p ^ {1 / k} \ cdot n ^ {1 / k} \ bigr）$。对于确定性算法，均匀分布是最坏的情况。对于无序搜索，在最坏的情况下，随机算法的预期查询复杂度为$ np \ bigl（\ frac {k + 1} {2k} \ bigr）\ p​​m 1 $，而确定性算法在搜索a时的预期查询复杂度均匀随机元素是$ np \ bigl（1-\ frac {k-1} {2k} p \ bigr）\ p​​m 1 $。我们还将讨论这些搜索问题与等级查询模型的联系，在该模型中，可以通过形式为“ rank $（x_i）\ leq k $？”的查询来访问数组$ x $。无序搜索等同于使用排名查询进行选择（给定$ q $，找到具有排名$ q $的$ x_i $），而进行排序搜索则进行查找以进行排名查询（给定$ x_i $，找到其排名）。我们显示了在任意数量的回合中，使用等级查询进行排序与​​使用连续块进行比例蛋糕切割之间的等效性，以及使用等级查询进行的轮次进行确定性排序的改进下限。