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Track A: Algorithms, Complexity and Games

DOI: 10.4230/LIPIcs.ICALP.2026.34

Near-Optimal Dynamic Data Structures for Maximum Depth and Klee’s Measure of Boxes

Authors: Sujoy Bhore, Subhash Suri, Jie Xue, Xiongxin Yang, and Jiumu Zhu

Published in: LIPIcs, Volume 374, 53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026)

Abstract

We study two fundamental geometric problems on a dynamic set of n axis-parallel boxes in d-dimensional space. The maximum depth problem asks for the largest number of boxes that contain a common point, whereas Klee’s measure problem asks for the volume of the union of the boxes. We present fully dynamic exact data structures for both problems achieving Õ(n^{(d-1)/2}) amortized update time. This update time is optimal for an exact dynamic algorithm, up to logarithmic factors, assuming the Combinatorial k-Clique Hypothesis. Previously, matching bounds were established only for d = 1 [Imai and Asano, J. Algo.'83], and for d = 2 [Suri, Xue, Yang, and Zhu, SoCG'25]. Our approach integrates a classic grid-based partition framework with a novel charging analysis that controls the cost of structure-sensitive offline routines within each cell. This argument allows us to perform a global aggregation of the update time, by circumventing the worst-case costs associated with individual cell updates. We believe this technique may be of independent interest for other dynamic geometric problems.

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Sujoy Bhore, Subhash Suri, Jie Xue, Xiongxin Yang, and Jiumu Zhu. Near-Optimal Dynamic Data Structures for Maximum Depth and Klee’s Measure of Boxes. In 53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 374, pp. 34:1-34:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{bhore_et_al:LIPIcs.ICALP.2026.34,
  author =	{Bhore, Sujoy and Suri, Subhash and Xue, Jie and Yang, Xiongxin and Zhu, Jiumu},
  title =	{{Near-Optimal Dynamic Data Structures for Maximum Depth and Klee’s Measure of Boxes}},
  booktitle =	{53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026)},
  pages =	{34:1--34:21},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-428-4},
  ISSN =	{1868-8969},
  year =	{2026},
  volume =	{374},
  editor =	{Bhattacharya, Sayan and Nanongkai, Danupon and Benedikt, Michael and Puppis, Gabriele},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2026.34},
  URN =		{urn:nbn:de:0030-drops-264230},
  doi =		{10.4230/LIPIcs.ICALP.2026.34},
  annote =	{Keywords: dynamic algorithms, maximum depth}
}

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DOI: 10.4230/LIPIcs.SoCG.2025.77

Dynamic Maximum Depth of Geometric Objects

Authors: Subhash Suri, Jie Xue, Xiongxin Yang, and Jiumu Zhu

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

Abstract

Given a set of geometric objects in the plane (rectangles, squares, disks etc.), its maximum depth (or geometric clique) is the largest number of objects with a common intersection. In this paper, we present data structures for dynamically maintaining the maximum depth under insertions and deletions of geometric objects, with sublinear update time. We achieve the following results: - a 1/k-approximate dynamic maximum-depth data structure for (axis-parallel) rectangles with O(n^{1/(k+1)} log n) amortized update time, for any fixed k ∈ ℤ^+. In particular, when k = 1, this gives an exact data structure for rectangles with O(√n log n) amortized update time, almost matching the best known bound for the offline version of the problem. - a (1/2-ε)-approximate dynamic maximum-depth data structure for disks with n^{2/3} log^{O(1)}n amortized update time, for any constant ε > 0. Having exact data structures for disks with sublinear update time is unlikely, since the static maximum-depth problem for disks is 3SUM-hard and thus does not admit subquadratic-time algorithms.

Cite as

Subhash Suri, Jie Xue, Xiongxin Yang, and Jiumu Zhu. Dynamic Maximum Depth of Geometric Objects. In 41st International Symposium on Computational Geometry (SoCG 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 332, pp. 77:1-77:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{suri_et_al:LIPIcs.SoCG.2025.77,
  author =	{Suri, Subhash and Xue, Jie and Yang, Xiongxin and Zhu, Jiumu},
  title =	{{Dynamic Maximum Depth of Geometric Objects}},
  booktitle =	{41st International Symposium on Computational Geometry (SoCG 2025)},
  pages =	{77:1--77:13},
  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.2025.77},
  URN =		{urn:nbn:de:0030-drops-232295},
  doi =		{10.4230/LIPIcs.SoCG.2025.77},
  annote =	{Keywords: dynamic algorithms, maximum depth}
}

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Near-Optimal Dynamic Data Structures for Maximum Depth and Klee’s Measure of Boxes

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Dynamic Maximum Depth of Geometric Objects

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