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DOI: 10.4230/LIPIcs.SoCG.2026.89

Approximating Convex Hulls via Range Queries

Authors: Thomas Schibler, Jie Xue, and Jiumu Zhu

Published in: LIPIcs, Volume 367, 42nd International Symposium on Computational Geometry (SoCG 2026)

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 ⊆ [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 ⊕ CH(P) = (C∖CH(P)) ∪ (CH(P)∖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/√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/√q) if the queries are non-adaptive, and Θ̃(1/q²) if the queries are adaptive. Here Θ̃(⋅) hides logarithmic factors.

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Thomas Schibler, Jie Xue, and Jiumu Zhu. Approximating Convex Hulls via Range Queries. In 42nd International Symposium on Computational Geometry (SoCG 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 367, pp. 89:1-89:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{schibler_et_al:LIPIcs.SoCG.2026.89,
  author =	{Schibler, Thomas and Xue, Jie and Zhu, Jiumu},
  title =	{{Approximating Convex Hulls via Range Queries}},
  booktitle =	{42nd International Symposium on Computational Geometry (SoCG 2026)},
  pages =	{89:1--89:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-418-5},
  ISSN =	{1868-8969},
  year =	{2026},
  volume =	{367},
  editor =	{Ahn, Hee-Kap and Hoffmann, Michael and Nayyeri, Amir},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2026.89},
  URN =		{urn:nbn:de:0030-drops-258956},
  doi =		{10.4230/LIPIcs.SoCG.2026.89},
  annote =	{Keywords: convex hull, range searching}
}

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DOI: 10.4230/LIPIcs.SoCG.2025.73

Embedding Graphs as Euclidean kNN-Graphs

Authors: Thomas Schibler, Subhash Suri, and Jie Xue

Published in: LIPIcs, Volume 332, 41st International Symposium on Computational Geometry (SoCG 2025)

Abstract

Let G = (V,E) be a directed graph on n vertices where each vertex has out-degree k. We say that G is kNN-realizable in d-dimensional Euclidean space if there exists a point set P = {p_1, p_2, …, p_n} in ℝ^d along with a one-to-one mapping ϕ: V → P such that for any u,v ∈ V, u is an out-neighbor of v in G if and only if ϕ(u) is one of the k nearest neighbors of ϕ(v); we call the map ϕ a kNN-realization of G in ℝ^d. The kNN-realization problem, which aims to compute a kNN-realization of an input graph in ℝ^d, is known to be NP-hard already for d = 2 and k = 1 [Eades and Whitesides, Theoretical Computer Science, 1996], and to the best of our knowledge has not been studied in dimension d = 1. The main results of this paper are the following: - For any fixed dimension d ≥ 2, we can efficiently compute an embedding realizing at least a 1 - ε fraction of G’s edges, or conclude that G is not kNN-realizable in ℝ^d. - For d = 1, we can decide in O(kn) time whether G is kNN-realizable and, if so, compute a realization in O(n^{2.5} poly(log n)) time.

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Thomas Schibler, Subhash Suri, and Jie Xue. Embedding Graphs as Euclidean kNN-Graphs. In 41st International Symposium on Computational Geometry (SoCG 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 332, pp. 73:1-73:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{schibler_et_al:LIPIcs.SoCG.2025.73,
  author =	{Schibler, Thomas and Suri, Subhash and Xue, Jie},
  title =	{{Embedding Graphs as Euclidean kNN-Graphs}},
  booktitle =	{41st International Symposium on Computational Geometry (SoCG 2025)},
  pages =	{73:1--73:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-370-6},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{332},
  editor =	{Aichholzer, Oswin and Wang, Haitao},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2025.73},
  URN =		{urn:nbn:de:0030-drops-232253},
  doi =		{10.4230/LIPIcs.SoCG.2025.73},
  annote =	{Keywords: Geometric graphs, k-nearest neighbors, graph embedding, approximation algorithms}
}

Document

DOI: 10.4230/LIPIcs.ESA.2017.65

K-Dominance in Multidimensional Data: Theory and Applications

Authors: Thomas Schibler and Subhash Suri

Published in: LIPIcs, Volume 87, 25th Annual European Symposium on Algorithms (ESA 2017)

Abstract

We study the problem of k-dominance in a set of d-dimensional vectors, prove bounds on the number of maxima (skyline vectors), under both worst-case and average-case models, perform experimental evaluation using synthetic and real-world data, and explore an application of k-dominant skyline for extracting a small set of top-ranked vectors in high dimensions where the full skylines can be unmanageably large.

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Thomas Schibler and Subhash Suri. K-Dominance in Multidimensional Data: Theory and Applications. In 25th Annual European Symposium on Algorithms (ESA 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 87, pp. 65:1-65:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{schibler_et_al:LIPIcs.ESA.2017.65,
  author =	{Schibler, Thomas and Suri, Subhash},
  title =	{{K-Dominance in Multidimensional Data: Theory and Applications}},
  booktitle =	{25th Annual European Symposium on Algorithms (ESA 2017)},
  pages =	{65:1--65:13},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-049-1},
  ISSN =	{1868-8969},
  year =	{2017},
  volume =	{87},
  editor =	{Pruhs, Kirk and Sohler, Christian},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2017.65},
  URN =		{urn:nbn:de:0030-drops-78402},
  doi =		{10.4230/LIPIcs.ESA.2017.65},
  annote =	{Keywords: Dominance, skyline, database search, average case analysis, random vectors}
}

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Documents authored by Schibler, Thomas

Approximating Convex Hulls via Range Queries

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Embedding Graphs as Euclidean kNN-Graphs

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K-Dominance in Multidimensional Data: Theory and Applications

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