Approximating Convex Hulls via Range Queries

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 $Θ(q^{-1/d})$ 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.

Figures (3)

• Figure 1: The lower bound construction in $d = 2$ dimensions for non-adaptive orthogonal queries with underlying points $P$ (red) along the main diagonal $H_0$. The depicted query $\Box$ is $1$-good since its lower left corner lies between $H_0$ and $H_1$. The length of $H_1 \cap \Box$ (blue) is $O(\delta^{d-1})$, so $\Omega(1/\delta^{d-1})$$1$-good queries are needed to cover $H_1$ in order to determine whether $P$ contains a point on $H_1$. In total, $\Omega(n\delta^{d-1})$$i$-good queries are needed to cover all $H_1, \cdots, H_n$.
• Figure 2: One iteration of Algorithm \ref{['alg-AdaptiveRectangle']} for $d=2, \vec{v}=(1,1)$. The four queries in $\mathcal{R'}$ are filled in gray. The upper right corners of these queries are used to compute the set $U$ with boundary shown in blue, while the lower left corners produce $L$ in red. The boundary of $\mathcal{CH}_{\vec{v}}(P)$ lies in $U\setminus L$. The smallest dashed queries form the set $\mathcal{R}$ for the next iteration; their union covers $U \setminus L$.
• Figure 3: Lower bound construction for non-adaptive halfplane queries. The point $p^*$ (red) is offset from the midpoint $(x/2, y/2)$ of $l_{x,y}$ to lie sufficiently far above $l_{x,y}$ (blue) but below both $l_{x+\delta,y}$ and $l_{x,y+\delta}$. Only a segment with both $x$-intercept in the interval $[x, x+\delta]$ and $y$-intercept in the interval $[y, y+\delta]$ can separate both $(x, 0), (0, y)$ from $p^*$. Doing so for all $\Theta(1/\delta^2) > q$ paired choices of $x, y$ (gray lines) necessitates as many queries. Range [1/3, 2/3] enlarged for clarity.

Theorems & Definitions (33)

• Lemma 1
• Theorem 2
• Lemma 3
• Lemma 4
• Lemma 5
• Lemma 6
• Lemma 7
• Theorem 8
• Lemma 9
• Lemma 10