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**Published in:** LIPIcs, Volume 293, 40th International Symposium on Computational Geometry (SoCG 2024)

Given a set of n points in the Euclidean plane, the k-MinSumRadius problem asks to cover this point set using k disks with the objective of minimizing the sum of the radii of the disks. After a long line of research on related problems, it was finally discovered that this problem admits a polynomial time algorithm [GKKPV '12]; however, the running time of this algorithm is 𝒪(n^881), and its relevance is thereby mostly of theoretical nature. A practically and structurally interesting special case of the k-MinSumRadius problem is that of small k. For the 2-MinSumRadius problem, a near-quadratic time algorithm with expected running time 𝒪(n² log² n log² log n) was given over 30 years ago [Eppstein '92].
We present the first improvement of this result, namely, a near-linear time algorithm to compute the 2-MinSumRadius that runs in expected 𝒪(n log² n log² log n) time. We generalize this result to any constant dimension d, for which we give an 𝒪(n^{2-1/(⌈d/2⌉ + 1) + ε}) time algorithm. Additionally, we give a near-quadratic time algorithm for 3-MinSumRadius in the plane that runs in expected 𝒪(n² log² n log² log n) time. All of these algorithms rely on insights that uncover a surprisingly simple structure of optimal solutions: we can specify a linear number of lines out of which one separates one of the clusters from the remaining clusters in an optimal solution.

Mikkel Abrahamsen, Sarita de Berg, Lucas Meijer, André Nusser, and Leonidas Theocharous. Clustering with Few Disks to Minimize the Sum of Radii. In 40th International Symposium on Computational Geometry (SoCG 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 293, pp. 2:1-2:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{abrahamsen_et_al:LIPIcs.SoCG.2024.2, author = {Abrahamsen, Mikkel and de Berg, Sarita and Meijer, Lucas and Nusser, Andr\'{e} and Theocharous, Leonidas}, title = {{Clustering with Few Disks to Minimize the Sum of Radii}}, booktitle = {40th International Symposium on Computational Geometry (SoCG 2024)}, pages = {2:1--2:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-316-4}, ISSN = {1868-8969}, year = {2024}, volume = {293}, editor = {Mulzer, Wolfgang and Phillips, Jeff M.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2024.2}, URN = {urn:nbn:de:0030-drops-199472}, doi = {10.4230/LIPIcs.SoCG.2024.2}, annote = {Keywords: geometric clustering, minimize sum of radii, covering points with disks} }

Document

**Published in:** LIPIcs, Volume 293, 40th International Symposium on Computational Geometry (SoCG 2024)

We devise a data structure that can answer shortest path queries for two query points in a polygonal domain P on n vertices. For any ε > 0, the space complexity of the data structure is O(n^{10+ε}) and queries can be answered in O(log n) time. Alternatively, we can achieve a space complexity of O(n^{9+ε}) by relaxing the query time to O(log² n). This is the first improvement upon a conference paper by Chiang and Mitchell from 1999. They presented a data structure with O(n^{11}) space complexity and O(log n) query time. Our main result can be extended to include a space-time trade-off. Specifically, we devise data structures with O(n^{9+ε}/𝓁^{4+O(ε)}) space complexity and O(𝓁 log² n) query time, for any integer 1 ≤ 𝓁 ≤ n.
Furthermore, we present improved data structures for the special case where we restrict one (or both) of the query points to lie on the boundary of P. When one of the query points is restricted to lie on the boundary, and the other query point is unrestricted, the space complexity becomes O(n^{6+ε}) and the query time O(log²n). When both query points are on the boundary, the space complexity is decreased further to O(n^{4+ε}) and the query time to O(log n), thereby improving an earlier result of Bae and Okamoto.

Sarita de Berg, Tillmann Miltzow, and Frank Staals. Towards Space Efficient Two-Point Shortest Path Queries in a Polygonal Domain. In 40th International Symposium on Computational Geometry (SoCG 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 293, pp. 17:1-17:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{deberg_et_al:LIPIcs.SoCG.2024.17, author = {de Berg, Sarita and Miltzow, Tillmann and Staals, Frank}, title = {{Towards Space Efficient Two-Point Shortest Path Queries in a Polygonal Domain}}, booktitle = {40th International Symposium on Computational Geometry (SoCG 2024)}, pages = {17:1--17:16}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-316-4}, ISSN = {1868-8969}, year = {2024}, volume = {293}, editor = {Mulzer, Wolfgang and Phillips, Jeff M.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2024.17}, URN = {urn:nbn:de:0030-drops-199628}, doi = {10.4230/LIPIcs.SoCG.2024.17}, annote = {Keywords: data structure, polygonal domain, geodesic distance} }

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

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).

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.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} }

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

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.

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.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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