2 Search Results for "de Berg, Sarita"

Document
DOI: 10.4230/LIPIcs.SoCG.2023.16
The Complexity of Geodesic Spanners

Authors: Sarita de Berg, Marc van Kreveld, and Frank Staals

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

Abstract
A geometric t-spanner for a set S of n point sites is an edge-weighted graph for which the (weighted) distance between any two sites p,q ∈ S is at most t times the original distance between p and q. We study geometric t-spanners for point sets in a constrained two-dimensional environment P. In such cases, the edges of the spanner may have non-constant complexity. Hence, we introduce a novel spanner property: the spanner complexity, that is, the total complexity of all edges in the spanner. Let S be a set of n point sites in a simple polygon P with m vertices. We present an algorithm to construct, for any constant ε > 0 and fixed integer k ≥ 1, a (2k + ε)-spanner with complexity O(mn^{1/k} + nlog² n) in O(nlog²n + mlog n + K) time, where K denotes the output complexity. When we consider sites in a polygonal domain P with holes, we can construct such a (2k+ε)-spanner of similar complexity in O(n² log m + nmlog m + K) time. Additionally, for any constant ε ∈ (0,1) and integer constant t ≥ 2, we show a lower bound for the complexity of any (t-ε)-spanner of Ω(mn^{1/(t-1)} + n).
Cite as

Sarita de Berg, Marc van Kreveld, and Frank Staals. The Complexity of Geodesic Spanners. In 39th International Symposium on Computational Geometry (SoCG 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 258, pp. 16:1-16:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{deberg_et_al:LIPIcs.SoCG.2023.16,
  author =	{de Berg, Sarita and van Kreveld, Marc and Staals, Frank},
  title =	{{The Complexity of Geodesic Spanners}},
  booktitle =	{39th International Symposium on Computational Geometry (SoCG 2023)},
  pages =	{16:1--16:16},
  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.2023.16},
  URN =		{urn:nbn:de:0030-drops-178669},
  doi =		{10.4230/LIPIcs.SoCG.2023.16},
  annote =	{Keywords: spanner, simple polygon, polygonal domain, geodesic distance, complexity}
}
Document
DOI: 10.4230/LIPIcs.ISAAC.2021.14
Dynamic Data Structures for k-Nearest Neighbor Queries

Authors: Sarita de Berg and Frank Staals

Published in: LIPIcs, Volume 212, 32nd International Symposium on Algorithms and Computation (ISAAC 2021)

Abstract
Our aim is to develop dynamic data structures that support k-nearest neighbors (k-NN) queries for a set of n point sites in O(f(n) + k) time, where f(n) is some polylogarithmic function of n. The key component is a general query algorithm that allows us to find the k-NN spread over t substructures simultaneously, thus reducing a O(tk) term in the query time to O(k). Combining this technique with the logarithmic method allows us to turn any static k-NN data structure into a data structure supporting both efficient insertions and queries. For the fully dynamic case, this technique allows us to recover the deterministic, worst-case, O(log²n/log log n +k) query time for the Euclidean distance claimed before, while preserving the polylogarithmic update times. We adapt this data structure to also support fully dynamic geodesic k-NN queries among a set of sites in a simple polygon. For this purpose, we design a shallow cutting based, deletion-only k-NN data structure. More generally, we obtain a dynamic k-NN data structure for any type of distance functions for which we can build vertical shallow cuttings. We apply all of our methods in the plane for the Euclidean distance, the geodesic distance, and general, constant-complexity, algebraic distance functions.
Cite as

Sarita de Berg and Frank Staals. Dynamic Data Structures for k-Nearest Neighbor Queries. In 32nd International Symposium on Algorithms and Computation (ISAAC 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 212, pp. 14:1-14:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{deberg_et_al:LIPIcs.ISAAC.2021.14,
  author =	{de Berg, Sarita and Staals, Frank},
  title =	{{Dynamic Data Structures for k-Nearest Neighbor Queries}},
  booktitle =	{32nd International Symposium on Algorithms and Computation (ISAAC 2021)},
  pages =	{14:1--14:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-214-3},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{212},
  editor =	{Ahn, Hee-Kap and Sadakane, Kunihiko},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2021.14},
  URN =		{urn:nbn:de:0030-drops-154473},
  doi =		{10.4230/LIPIcs.ISAAC.2021.14},
  annote =	{Keywords: data structure, simple polygon, geodesic distance, nearest neighbor searching}
}
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