Recently, motivated by the rapid increase of the data size in various applications, Monemizadeh [APPROX'23] and Driemel, Monemizadeh, Oh, Staals, and Woodruff [SoCG'25] studied geometric problems in the setting where the only access to the input point set is via querying a range-search oracle. Algorithms in this setting are evaluated on two criteria: (i) the number of queries to the oracle and (ii) the error of the output. In this paper, we continue this line of research and investigate one of the most fundamental geometric problems in the oracle setting, i.e., the convex hull problem. Let $P$ be an unknown set of points in $[0,1]^d$ equipped with a range-emptiness oracle. Via querying the oracle, the algorithm is supposed to output a convex polygon $C \subseteq [0,1]^d$ as an estimation of the convex hull $CH(P)$ of $P$. The error of the output is defined as the volume of the symmetric difference $C \oplus CH(P) = (C \backslash CH(P)) \cup (CH(P) \backslash C)$. We prove tight and near-tight tradeoffs between the number of queries and the error of the output for different variants of the problem, depending on the type of the range-emptiness queries and whether the queries are non-adaptive or adaptive. - Orthogonal emptiness queries in $d$-dimensional space: We show that the minimum error a deterministic algorithm can achieve with $q$ queries is $Θ(q^{-1/d})$ if the queries are non-adaptive, and $Θ(q^{-1/(d-1)})$ if the queries are adaptive. In particular, in 2D, the bounds are $Θ(1/\sqrt{q})$ and $Θ(1/q)$ for non-adaptive and adaptive queries, respectively. - Halfplane emptiness queries in 2D: We show that the minimum error a deterministic algorithm can achieve with $q$ queries is $Θ(1/\sqrt{q})$ if the queries are non-adaptive, and $\widetildeΘ(1/q^2)$ if the queries are adaptive. Here $\widetildeΘ(\cdot)$ hides logarithmic factors.

翻译：摘要：近年来，受各类应用中数据量急剧增长的驱动，Monemizadeh [APPROX'23] 以及 Driemel、Monemizadeh、Oh、Staals 和 Woodruff [SoCG'25] 研究了在仅能通过查询范围搜索预言机来访问输入点集这一设定下的几何问题。在此设定下，算法的评估标准有两个：(i) 对预言机的查询次数，以及 (ii) 输出的误差。本文延续这一研究方向，探讨了预言机设定下最基础的几何问题之一——凸包问题。设 $P$ 是 $[0,1]^d$ 中的一个未知点集，配置有一个范围空性预言机。通过查询该预言机，算法应输出一个凸多边形 $C \subseteq [0,1]^d$ 作为 $P$ 的凸包 $CH(P)$ 的估计。输出的误差定义为对称差 $C \oplus CH(P) = (C \backslash CH(P)) \cup (CH(P) \backslash C)$ 的体积。针对问题的不同变体，我们证明了查询次数与输出误差之间紧致或近乎紧致的权衡关系，这些变体取决于范围空性查询的类型以及查询是非自适应还是自适应的。 - $d$ 维空间中的正交空性查询：我们证明，对于非自适应查询，确定性算法通过 $q$ 次查询所能实现的最小误差为 $Θ(q^{-1/d})$；对于自适应查询，则为 $Θ(q^{-1/(d-1)})$。特别地，在二维空间中，非自适应查询和自适应查询的界分别为 $Θ(1/\sqrt{q})$ 和 $Θ(1/q)$。 - 二维空间中的半平面空性查询：我们证明，对于非自适应查询，确定性算法通过 $q$ 次查询所能实现的最小误差为 $Θ(1/\sqrt{q})$；对于自适应查询，则为 $\widetildeΘ(1/q^2)$。其中 $\widetildeΘ(\cdot)$ 隐藏了对数因子。