2 Search Results for "Huang, Zhengcheng"

Document
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.
Cite as

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.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}
}
Document
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.
Cite as

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.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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