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DOI: 10.4230/LIPIcs.SoCG.2023.23

Constant-Hop Spanners for More Geometric Intersection Graphs, with Even Smaller Size

Authors: Timothy M. Chan and Zhengcheng Huang

Published in: LIPIcs, Volume 258, 39th International Symposium on Computational Geometry (SoCG 2023)

Abstract

In SoCG 2022, Conroy and Tóth presented several constructions of sparse, low-hop spanners in geometric intersection graphs, including an O(nlog n)-size 3-hop spanner for n disks (or fat convex objects) in the plane, and an O(nlog² n)-size 3-hop spanner for n axis-aligned rectangles in the plane. Their work left open two major questions: (i) can the size be made closer to linear by allowing larger constant stretch? and (ii) can near-linear size be achieved for more general classes of intersection graphs? We address both questions simultaneously, by presenting new constructions of constant-hop spanners that have almost linear size and that hold for a much larger class of intersection graphs. More precisely, we prove the existence of an O(1)-hop spanner for arbitrary string graphs with O(nα_k(n)) size for any constant k, where α_k(n) denotes the k-th function in the inverse Ackermann hierarchy. We similarly prove the existence of an O(1)-hop spanner for intersection graphs of d-dimensional fat objects with O(nα_k(n)) size for any constant k and d. We also improve on some of Conroy and Tóth’s specific previous results, in either the number of hops or the size: we describe an O(nlog n)-size 2-hop spanner for disks (or more generally objects with linear union complexity) in the plane, and an O(nlog n)-size 3-hop spanner for axis-aligned rectangles in the plane. Our proofs are all simple, using separator theorems, recursion, shifted quadtrees, and shallow cuttings.

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Timothy M. Chan and Zhengcheng Huang. Constant-Hop Spanners for More Geometric Intersection Graphs, with Even Smaller Size. In 39th International Symposium on Computational Geometry (SoCG 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 258, pp. 23:1-23:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{chan_et_al:LIPIcs.SoCG.2023.23,
  author =	{Chan, Timothy M. and Huang, Zhengcheng},
  title =	{{Constant-Hop Spanners for More Geometric Intersection Graphs, with Even Smaller Size}},
  booktitle =	{39th International Symposium on Computational Geometry (SoCG 2023)},
  pages =	{23:1--23:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-273-0},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{258},
  editor =	{Chambers, Erin W. and Gudmundsson, Joachim},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2023.23},
  URN =		{urn:nbn:de:0030-drops-178738},
  doi =		{10.4230/LIPIcs.SoCG.2023.23},
  annote =	{Keywords: Hop spanners, geometric intersection graphs, string graphs, fat objects, separators, shallow cuttings}
}

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DOI: 10.4230/LIPIcs.ESA.2021.28

Dynamic Colored Orthogonal Range Searching

Authors: Timothy M. Chan and Zhengcheng Huang

Published in: LIPIcs, Volume 204, 29th Annual European Symposium on Algorithms (ESA 2021)

Abstract

In the colored orthogonal range reporting problem, we want a data structure for storing n colored points so that given a query axis-aligned rectangle, we can report the distinct colors among the points inside the rectangle. This natural problem has been studied in a series of papers, but most prior work focused on the static case. In this paper, we give a dynamic data structure in the 2D case which can answer queries in O(log^{1+o(1)} n + klog^{1/2+o(1)}n) time, where k denotes the output size (the number of distinct colors in the query range), and which can support insertions and deletions in O(log^{2+o(1)}n) time (amortized) in the standard RAM model. This is the first fully dynamic structure with polylogarithmic update time whose query cost per color reported is sublogarithmic (near √{log n}). We also give an alternative data structure with O(log^{1+o(1)} n + klog^{3/4+o(1)}n) query time and O(log^{3/2+o(1)}n) update time (amortized). We also mention extensions to higher constant dimensions.

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Timothy M. Chan and Zhengcheng Huang. Dynamic Colored Orthogonal Range Searching. In 29th Annual European Symposium on Algorithms (ESA 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 204, pp. 28:1-28:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{chan_et_al:LIPIcs.ESA.2021.28,
  author =	{Chan, Timothy M. and Huang, Zhengcheng},
  title =	{{Dynamic Colored Orthogonal Range Searching}},
  booktitle =	{29th Annual European Symposium on Algorithms (ESA 2021)},
  pages =	{28:1--28:13},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-204-4},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{204},
  editor =	{Mutzel, Petra and Pagh, Rasmus and Herman, Grzegorz},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2021.28},
  URN =		{urn:nbn:de:0030-drops-146090},
  doi =		{10.4230/LIPIcs.ESA.2021.28},
  annote =	{Keywords: Range searching, dynamic data structures, word RAM}
}

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Documents authored by Huang, Zhengcheng

Constant-Hop Spanners for More Geometric Intersection Graphs, with Even Smaller Size

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Dynamic Colored Orthogonal Range Searching

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