Approximating Convex Hulls via Range Queries

TL;DR

This paper investigates the tradeoffs between the number of range-queries and the accuracy of convex hull approximations in high-dimensional spaces, providing tight bounds for different query types and adaptivity settings.

Contribution

It establishes tight bounds on the error versus query complexity for convex hull approximation using range-queries, extending understanding in the oracle query model.

Findings

Deterministic algorithms have bounds of Θ(q^{-1/d}) and Θ(q^{-1/(d-1)}) for non-adaptive and adaptive orthogonal queries.

In 2D, non-adaptive orthogonal queries achieve error Θ(1/√q), adaptive ones Θ(1/q).

Halfplane queries in 2D have error bounds of Θ(1/√q) non-adaptive and ~Θ(1/q^2) adaptive.

Abstract

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 $\Theta(q^{-1/d})$ if the queries are non-adaptive, and $\Theta(q^{-1/(d-1)})$ if the queries are adaptive. In particular, in 2D, the bounds are $\Theta(1/\sqrt{q})$ and $\Theta(1/q)$ 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 $\Theta(1/\sqrt{q})$ if the queries are non-adaptive, and $\widetilde{\Theta}(1/q^2)$ if the queries are adaptive. Here $\widetilde{\Theta}(\cdot)$ hides logarithmic factors.