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**Published in:** LIPIcs, Volume 308, 32nd Annual European Symposium on Algorithms (ESA 2024)

An outerstring graph is the intersection graph of curves lying inside a disk with one endpoint on the boundary of the disk. We show that an outerstring graph with n vertices has treewidth O(αlog n), where α denotes the arboricity of the graph, with an almost matching lower bound of Ω(α log (n/α)). As a corollary, we show that a t-biclique-free outerstring graph has treewidth O(t(log t)log n). This leads to polynomial-time algorithms for most of the central NP-complete problems such as Independent Set, Vertex Cover, Dominating Set, Feedback Vertex Set, Coloring for sparse outerstring graphs. Also, we can obtain subexponential-time (exact, parameterized, and approximation) algorithms for various NP-complete problems such as Vertex Cover, Feedback Vertex Set and Cycle Packing for (not necessarily sparse) outerstring graphs.

Shinwoo An, Eunjin Oh, and Jie Xue. Sparse Outerstring Graphs Have Logarithmic Treewidth. In 32nd Annual European Symposium on Algorithms (ESA 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 308, pp. 10:1-10:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{an_et_al:LIPIcs.ESA.2024.10, author = {An, Shinwoo and Oh, Eunjin and Xue, Jie}, title = {{Sparse Outerstring Graphs Have Logarithmic Treewidth}}, booktitle = {32nd Annual European Symposium on Algorithms (ESA 2024)}, pages = {10:1--10:18}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-338-6}, ISSN = {1868-8969}, year = {2024}, volume = {308}, editor = {Chan, Timothy and Fischer, Johannes and Iacono, John 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.2024.10}, URN = {urn:nbn:de:0030-drops-210816}, doi = {10.4230/LIPIcs.ESA.2024.10}, annote = {Keywords: Outerstring graphs, geometric intersection graphs, treewidth} }

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

In this paper, we consider the Cycle Packing problem on a unit disk graph defined as follows. Given a unit disk graph G with n vertices and an integer k, the goal is to find a set of k vertex-disjoint cycles of G if it exists. Our algorithm runs in time 2^O(√k) n^O(1). This improves the 2^O(√klog k) n^O(1)-time algorithm by Fomin et al. [SODA 2012, ICALP 2017]. Moreover, our algorithm is optimal assuming the exponential-time hypothesis.

Shinwoo An and Eunjin Oh. ETH-Tight Algorithm for Cycle Packing on Unit Disk Graphs. In 40th International Symposium on Computational Geometry (SoCG 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 293, pp. 7:1-7:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{an_et_al:LIPIcs.SoCG.2024.7, author = {An, Shinwoo and Oh, Eunjin}, title = {{ETH-Tight Algorithm for Cycle Packing on Unit Disk Graphs}}, booktitle = {40th International Symposium on Computational Geometry (SoCG 2024)}, pages = {7:1--7:15}, 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.2024.7}, URN = {urn:nbn:de:0030-drops-199522}, doi = {10.4230/LIPIcs.SoCG.2024.7}, annote = {Keywords: Unit disk graphs, cycle packing, tree decomposition, parameterized algorithm} }

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

We study a fundamental problem in Computational Geometry, the planar two-center problem. In this problem, the input is a set S of n points in the plane and the goal is to find two smallest congruent disks whose union contains all points of S. A longstanding open problem has been to obtain an O(nlog n)-time algorithm for planar two-center, matching the Ω(nlog n) lower bound given by Eppstein [SODA'97]. Towards this, researchers have made a lot of efforts over decades. The previous best algorithm, given by Wang [SoCG'20], solves the problem in O(nlog² n) time. In this paper, we present an O(nlog n)-time (deterministic) algorithm for planar two-center, which completely resolves this open problem.

Kyungjin Cho, Eunjin Oh, Haitao Wang, and Jie Xue. Optimal Algorithm for the Planar Two-Center Problem. In 40th International Symposium on Computational Geometry (SoCG 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 293, pp. 40:1-40:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{cho_et_al:LIPIcs.SoCG.2024.40, author = {Cho, Kyungjin and Oh, Eunjin and Wang, Haitao and Xue, Jie}, title = {{Optimal Algorithm for the Planar Two-Center Problem}}, booktitle = {40th International Symposium on Computational Geometry (SoCG 2024)}, pages = {40:1--40:15}, 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.2024.40}, URN = {urn:nbn:de:0030-drops-199857}, doi = {10.4230/LIPIcs.SoCG.2024.40}, annote = {Keywords: two-center, r-coverage, disk coverage, circular hulls} }

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**Published in:** LIPIcs, Volume 274, 31st Annual European Symposium on Algorithms (ESA 2023)

In this paper, we study the maximum clique problem on hyperbolic random graphs. A hyperbolic random graph is a mathematical model for analyzing scale-free networks since it effectively explains the power-law degree distribution of scale-free networks. We propose a simple algorithm for finding a maximum clique in hyperbolic random graph. We first analyze the running time of our algorithm theoretically. We can compute a maximum clique on a hyperbolic random graph G in O(m + n^{4.5(1-α)}) expected time if a geometric representation is given or in O(m + n^{6(1-α)}) expected time if a geometric representation is not given, where n and m denote the numbers of vertices and edges of G, respectively, and α denotes a parameter controlling the power-law exponent of the degree distribution of G. Also, we implemented and evaluated our algorithm empirically. Our algorithm outperforms the previous algorithm [BFK18] practically and theoretically. Beyond the hyperbolic random graphs, we have experiment on real-world networks. For most of instances, we get large cliques close to the optimum solutions efficiently.

Eunjin Oh and Seunghyeok Oh. Algorithms for Computing Maximum Cliques in Hyperbolic Random Graphs. In 31st Annual European Symposium on Algorithms (ESA 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 274, pp. 85:1-85:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{oh_et_al:LIPIcs.ESA.2023.85, author = {Oh, Eunjin and Oh, Seunghyeok}, title = {{Algorithms for Computing Maximum Cliques in Hyperbolic Random Graphs}}, booktitle = {31st Annual European Symposium on Algorithms (ESA 2023)}, pages = {85:1--85:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-295-2}, ISSN = {1868-8969}, year = {2023}, volume = {274}, editor = {G{\o}rtz, Inge Li and Farach-Colton, Martin and Puglisi, Simon J. 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.2023.85}, URN = {urn:nbn:de:0030-drops-187384}, doi = {10.4230/LIPIcs.ESA.2023.85}, annote = {Keywords: Maximum cliques, hyperbolic random graphs} }

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

In this paper, we present a linear-time approximation scheme for k-means clustering of incomplete data points in d-dimensional Euclidean space. An incomplete data point with Δ > 0 unspecified entries is represented as an axis-parallel affine subspace of dimension Δ. The distance between two incomplete data points is defined as the Euclidean distance between two closest points in the axis-parallel affine subspaces corresponding to the data points. We present an algorithm for k-means clustering of axis-parallel affine subspaces of dimension Δ that yields an (1+ε)-approximate solution in O(nd) time. The constants hidden behind O(⋅) depend only on Δ, ε and k. This improves the O(n² d)-time algorithm by Eiben et al. [SODA'21] by a factor of n.

Kyungjin Cho and Eunjin Oh. Linear-Time Approximation Scheme for k-Means Clustering of Axis-Parallel Affine Subspaces. In 32nd International Symposium on Algorithms and Computation (ISAAC 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 212, pp. 46:1-46:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{cho_et_al:LIPIcs.ISAAC.2021.46, author = {Cho, Kyungjin and Oh, Eunjin}, title = {{Linear-Time Approximation Scheme for k-Means Clustering of Axis-Parallel Affine Subspaces}}, booktitle = {32nd International Symposium on Algorithms and Computation (ISAAC 2021)}, pages = {46:1--46:13}, 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.2021.46}, URN = {urn:nbn:de:0030-drops-154794}, doi = {10.4230/LIPIcs.ISAAC.2021.46}, annote = {Keywords: k-means clustering, affine subspaces} }

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

In this paper, we present an algorithm for computing a feedback vertex set of a unit disk graph of size k, if it exists, which runs in time 2^O(√k)(n+m), where n and m denote the numbers of vertices and edges, respectively. This improves the 2^O(√klog k) n^O(1)-time algorithm for this problem on unit disk graphs by Fomin et al. [ICALP 2017]. Moreover, our algorithm is optimal assuming the exponential-time hypothesis. Also, our algorithm can be extended to handle geometric intersection graphs of similarly sized fat objects without increasing the running time.

Shinwoo An and Eunjin Oh. Feedback Vertex Set on Geometric Intersection Graphs. In 32nd International Symposium on Algorithms and Computation (ISAAC 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 212, pp. 47:1-47:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{an_et_al:LIPIcs.ISAAC.2021.47, author = {An, Shinwoo and Oh, Eunjin}, title = {{Feedback Vertex Set on Geometric Intersection Graphs}}, booktitle = {32nd International Symposium on Algorithms and Computation (ISAAC 2021)}, pages = {47:1--47:12}, 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.2021.47}, URN = {urn:nbn:de:0030-drops-154807}, doi = {10.4230/LIPIcs.ISAAC.2021.47}, annote = {Keywords: Feedback vertex set, intersection graphs, parameterized algorithm} }

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**Published in:** LIPIcs, Volume 181, 31st International Symposium on Algorithms and Computation (ISAAC 2020)

A Euclidean t-spanner for a point set V ⊂ ℝ^d is a graph such that, for any two points p and q in V, the distance between p and q in the graph is at most t times the Euclidean distance between p and q. Gudmundsson et al. [TALG 2008] presented a data structure for answering ε-approximate distance queries in a Euclidean spanner in constant time, but it seems unlikely that one can report the path itself using this data structure. In this paper, we present a data structure of size O(nlog n) that answers ε-approximate shortest-path queries in time linear in the size of the output.

Eunjin Oh. Shortest-Path Queries in Geometric Networks. In 31st International Symposium on Algorithms and Computation (ISAAC 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 181, pp. 52:1-52:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{oh:LIPIcs.ISAAC.2020.52, author = {Oh, Eunjin}, title = {{Shortest-Path Queries in Geometric Networks}}, booktitle = {31st International Symposium on Algorithms and Computation (ISAAC 2020)}, pages = {52:1--52:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-173-3}, ISSN = {1868-8969}, year = {2020}, volume = {181}, editor = {Cao, Yixin and Cheng, Siu-Wing and Li, Minming}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2020.52}, URN = {urn:nbn:de:0030-drops-133963}, doi = {10.4230/LIPIcs.ISAAC.2020.52}, annote = {Keywords: Shortest path, Euclidean spanner, data structure} }

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**Published in:** LIPIcs, Volume 123, 29th International Symposium on Algorithms and Computation (ISAAC 2018)

We study the point location problem in incremental (possibly disconnected) planar subdivisions, that is, dynamic subdivisions allowing insertions of edges and vertices only. Specifically, we present an O(n log n)-space data structure for this problem that supports queries in O(log^2 n) time and updates in O(log n log log n) amortized time. This is the first result that achieves polylogarithmic query and update times simultaneously in incremental planar subdivisions. Its update time is significantly faster than the update time of the best known data structure for fully-dynamic (possibly disconnected) planar subdivisions.

Eunjin Oh. Point Location in Incremental Planar Subdivisions. In 29th International Symposium on Algorithms and Computation (ISAAC 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 123, pp. 51:1-51:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{oh:LIPIcs.ISAAC.2018.51, author = {Oh, Eunjin}, title = {{Point Location in Incremental Planar Subdivisions}}, booktitle = {29th International Symposium on Algorithms and Computation (ISAAC 2018)}, pages = {51:1--51:12}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-094-1}, ISSN = {1868-8969}, year = {2018}, volume = {123}, editor = {Hsu, Wen-Lian and Lee, Der-Tsai and Liao, Chung-Shou}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2018.51}, URN = {urn:nbn:de:0030-drops-99991}, doi = {10.4230/LIPIcs.ISAAC.2018.51}, annote = {Keywords: Dynamic point location, general incremental planar subdivisions} }

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**Published in:** LIPIcs, Volume 123, 29th International Symposium on Algorithms and Computation (ISAAC 2018)

In this paper, we consider the quickest pair-visibility problem in polygonal domains. Given two points in a polygonal domain with h holes of total complexity n, we want to minimize the maximum distance that the two points travel in order to see each other in the polygonal domain. We present an O(n log^2 n+h^2 log^4 h)-time algorithm for this problem. We show that this running time is almost optimal unless the 3sum problem can be solved in O(n^{2-epsilon}) time for some epsilon>0.

Eunjin Oh. Minimizing Distance-to-Sight in Polygonal Domains. In 29th International Symposium on Algorithms and Computation (ISAAC 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 123, pp. 59:1-59:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{oh:LIPIcs.ISAAC.2018.59, author = {Oh, Eunjin}, title = {{Minimizing Distance-to-Sight in Polygonal Domains}}, booktitle = {29th International Symposium on Algorithms and Computation (ISAAC 2018)}, pages = {59:1--59:12}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-094-1}, ISSN = {1868-8969}, year = {2018}, volume = {123}, editor = {Hsu, Wen-Lian and Lee, Der-Tsai and Liao, Chung-Shou}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2018.59}, URN = {urn:nbn:de:0030-drops-100073}, doi = {10.4230/LIPIcs.ISAAC.2018.59}, annote = {Keywords: Visibility in polygonal domains, shortest path in polygonal domains} }

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**Published in:** LIPIcs, Volume 99, 34th International Symposium on Computational Geometry (SoCG 2018)

We study the approximate range searching for three variants of the clustering problem with a set P of n points in d-dimensional Euclidean space and axis-parallel rectangular range queries: the k-median, k-means, and k-center range-clustering query problems. We present data structures and query algorithms that compute (1+epsilon)-approximations to the optimal clusterings of P cap Q efficiently for a query consisting of an orthogonal range Q, an integer k, and a value epsilon>0.

Eunjin Oh and Hee-Kap Ahn. Approximate Range Queries for Clustering. In 34th International Symposium on Computational Geometry (SoCG 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 99, pp. 62:1-62:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{oh_et_al:LIPIcs.SoCG.2018.62, author = {Oh, Eunjin and Ahn, Hee-Kap}, title = {{Approximate Range Queries for Clustering}}, booktitle = {34th International Symposium on Computational Geometry (SoCG 2018)}, pages = {62:1--62:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-066-8}, ISSN = {1868-8969}, year = {2018}, volume = {99}, editor = {Speckmann, Bettina and T\'{o}th, Csaba D.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2018.62}, URN = {urn:nbn:de:0030-drops-87755}, doi = {10.4230/LIPIcs.SoCG.2018.62}, annote = {Keywords: Approximate clustering, orthogonal range queries} }

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**Published in:** LIPIcs, Volume 99, 34th International Symposium on Computational Geometry (SoCG 2018)

We study the point location problem on dynamic planar subdivisions that allows insertions and deletions of edges. In our problem, the underlying graph of a subdivision is not necessarily connected. We present a data structure of linear size for such a dynamic planar subdivision that supports sublinear-time update and polylogarithmic-time query. Precisely, the amortized update time is O(sqrt{n}log n(log log n)^{3/2}) and the query time is O(log n(log log n)^2), where n is the number of edges in the subdivision. This answers a question posed by Snoeyink in the Handbook of Computational Geometry. When only deletions of edges are allowed, the update time and query time are just O(alpha(n)) and O(log n), respectively.

Eunjin Oh and Hee-Kap Ahn. Point Location in Dynamic Planar Subdivisions. In 34th International Symposium on Computational Geometry (SoCG 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 99, pp. 63:1-63:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{oh_et_al:LIPIcs.SoCG.2018.63, author = {Oh, Eunjin and Ahn, Hee-Kap}, title = {{Point Location in Dynamic Planar Subdivisions}}, booktitle = {34th International Symposium on Computational Geometry (SoCG 2018)}, pages = {63:1--63:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-066-8}, ISSN = {1868-8969}, year = {2018}, volume = {99}, editor = {Speckmann, Bettina and T\'{o}th, Csaba D.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2018.63}, URN = {urn:nbn:de:0030-drops-87769}, doi = {10.4230/LIPIcs.SoCG.2018.63}, annote = {Keywords: dynamic point location, general subdivision} }

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**Published in:** LIPIcs, Volume 101, 16th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2018)

We introduce a variant of the watchman route problem, which we call the quickest pair-visibility problem. Given two persons standing at points s and t in a simple polygon P with no holes, we want to minimize the distance these persons travel in order to see each other in P. We solve two variants of this problem, one minimizing the longer distance the two persons travel (min-max) and one minimizing the total travel distance (min-sum), optimally in linear time. We also consider a query version of this problem for the min-max variant. We can preprocess a simple n-gon in linear time so that the minimum of the longer distance the two persons travel can be computed in O(log^2 n) time for any two query positions where the two persons lie.

Hee-Kap Ahn, Eunjin Oh, Lena Schlipf, Fabian Stehn, and Darren Strash. On Romeo and Juliet Problems: Minimizing Distance-to-Sight. In 16th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 101, pp. 6:1-6:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{ahn_et_al:LIPIcs.SWAT.2018.6, author = {Ahn, Hee-Kap and Oh, Eunjin and Schlipf, Lena and Stehn, Fabian and Strash, Darren}, title = {{On Romeo and Juliet Problems: Minimizing Distance-to-Sight}}, booktitle = {16th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2018)}, pages = {6:1--6:13}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-068-2}, ISSN = {1868-8969}, year = {2018}, volume = {101}, editor = {Eppstein, David}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SWAT.2018.6}, URN = {urn:nbn:de:0030-drops-88322}, doi = {10.4230/LIPIcs.SWAT.2018.6}, annote = {Keywords: Visibility polygon, shortest-path, watchman problems} }

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**Published in:** LIPIcs, Volume 92, 28th International Symposium on Algorithms and Computation (ISAAC 2017)

Motivated by map labeling, we study the problem in which we
are given a collection of n disks in the
plane that grow at possibly different speeds. Whenever two
disks meet, the one with the higher index disappears. This
problem was introduced by Funke, Krumpe, and Storandt[IWOCA 2016].
We provide the first general subquadratic algorithm for computing
the times and the order of disappearance.
Our algorithm also works for other shapes (such as rectangles)
and in any fixed dimension.
Using quadtrees, we provide an alternative
algorithm that runs in near linear time, although
this second algorithm has a logarithmic dependence
on either the ratio of the fastest speed to the slowest speed of disks
or the spread of the disk centers
(the ratio of the maximum to the minimum distance between them).
Our result improves the running times of previous algorithms by
Funke, Krumpe, and
Storandt [IWOCA 2016], Bahrdt et al. [ALENEX 2017], and
Funke and Storandt [EWCG 2017].
Finally, we give an \Omega(n\log n) lower bound on the
problem, showing that our quadtree algorithms are almost tight.

Hee-Kap Ahn, Sang Won Bae, Jongmin Choi, Matias Korman, Wolfgang Mulzer, Eunjin Oh, Ji-won Park, André van Renssen, and Antoine Vigneron. Faster Algorithms for Growing Prioritized Disks and Rectangles. In 28th International Symposium on Algorithms and Computation (ISAAC 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 92, pp. 3:1-3:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{ahn_et_al:LIPIcs.ISAAC.2017.3, author = {Ahn, Hee-Kap and Bae, Sang Won and Choi, Jongmin and Korman, Matias and Mulzer, Wolfgang and Oh, Eunjin and Park, Ji-won and van Renssen, Andr\'{e} and Vigneron, Antoine}, title = {{Faster Algorithms for Growing Prioritized Disks and Rectangles}}, booktitle = {28th International Symposium on Algorithms and Computation (ISAAC 2017)}, pages = {3:1--3:13}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-054-5}, ISSN = {1868-8969}, year = {2017}, volume = {92}, editor = {Okamoto, Yoshio and Tokuyama, Takeshi}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2017.3}, URN = {urn:nbn:de:0030-drops-82199}, doi = {10.4230/LIPIcs.ISAAC.2017.3}, annote = {Keywords: map labeling, growing disks, elimination order} }

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**Published in:** LIPIcs, Volume 92, 28th International Symposium on Algorithms and Computation (ISAAC 2017)

We consider the following problem: Preprocess a set S of n axis-parallel boxes in \mathbb{R}^d so that given a query of an axis-parallel box Q in \mathbb{R}^d, the pairs of boxes of S whose intersection intersects the query box can be reported efficiently. For the case that d=2, we present a data structure of size O(n\log n) supporting O(\log n+k) query time, where k is the size of the output. This improves the previously best known result by de Berg et al. which requires O(\log n\log^* n+ k\log n) query time using O(n\log n) space.There has been no known result for this problem for higher dimensions, except that for d=3, the best known data structure supports
O(\sqrt{n}+k\log^2\log^* n) query time using O(n\sqrt {n}\log n) space. For a constant d>2, we present a data structure supporting O(n^{1-\delta}\log^{d-1} n + k \polylog n) query time for any constant 1/d\leq\delta<1.The size of the data structure is O(n^{\delta d}\log n) if 1/d\leq\delta<1/2, or O(n^{\delta d-2\delta+1}) if 1/2\leq \delta<1.

Eunjin Oh and Hee-Kap Ahn. Finding Pairwise Intersections of Rectangles in a Query Rectangle. In 28th International Symposium on Algorithms and Computation (ISAAC 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 92, pp. 60:1-60:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{oh_et_al:LIPIcs.ISAAC.2017.60, author = {Oh, Eunjin and Ahn, Hee-Kap}, title = {{Finding Pairwise Intersections of Rectangles in a Query Rectangle}}, booktitle = {28th International Symposium on Algorithms and Computation (ISAAC 2017)}, pages = {60:1--60:12}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-054-5}, ISSN = {1868-8969}, year = {2017}, volume = {92}, editor = {Okamoto, Yoshio and Tokuyama, Takeshi}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2017.60}, URN = {urn:nbn:de:0030-drops-82417}, doi = {10.4230/LIPIcs.ISAAC.2017.60}, annote = {Keywords: Geometric data structures, axis-parallel rectangles, intersection} }

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**Published in:** LIPIcs, Volume 92, 28th International Symposium on Algorithms and Computation (ISAAC 2017)

We are given a read-only memory for input and a write-only stream for output. For a positive integer parameter s, an s-workspace algorithm is an algorithm using only O(s) words of workspace in addition to the memory for input. In this paper, we present an O(n^2/s)-time s-workspace algorithm for subdividing a simple polygon into O(\min\{n/s,s\}) subpolygons of complexity O(\max\{n/s,s\}).
As applications of the subdivision, the previously best known time-space trade-offs for the following three geometric problems are improved immediately: (1) computing the shortest path between two points inside a simple n-gon, (2) computing the shortest path tree from a point inside a simple n-gon, (3) computing a triangulation of a simple n-gon. In addition, we improve the algorithm for the second problem even further.

Eunjin Oh and Hee-Kap Ahn. A New Balanced Subdivision of a Simple Polygon for Time-Space Trade-off Algorithms. In 28th International Symposium on Algorithms and Computation (ISAAC 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 92, pp. 61:1-61:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{oh_et_al:LIPIcs.ISAAC.2017.61, author = {Oh, Eunjin and Ahn, Hee-Kap}, title = {{A New Balanced Subdivision of a Simple Polygon for Time-Space Trade-off Algorithms}}, booktitle = {28th International Symposium on Algorithms and Computation (ISAAC 2017)}, pages = {61:1--61:12}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-054-5}, ISSN = {1868-8969}, year = {2017}, volume = {92}, editor = {Okamoto, Yoshio and Tokuyama, Takeshi}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2017.61}, URN = {urn:nbn:de:0030-drops-82401}, doi = {10.4230/LIPIcs.ISAAC.2017.61}, annote = {Keywords: Time-space trade-off, simple polygon, shortest path, shortest path tree} }

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**Published in:** LIPIcs, Volume 77, 33rd International Symposium on Computational Geometry (SoCG 2017)

We consider the geodesic convex hulls of points in a simple polygonal region in the presence of non-crossing line segments (barriers) that subdivide the region into simply connected faces. We present an algorithm together with data structures for maintaining the geodesic convex hull of points in each face in a sublinear update time under the fully-dynamic setting where both input points and barriers change by insertions and deletions. The algorithm processes a mixed update sequence of insertions and deletions of points and barriers. Each update takes O(n^2/3 log^2 n) time with high probability, where n is the total number of the points and barriers at the moment. Our data structures support basic queries on the geodesic convex hull, each of which takes O(polylog n) time. In addition, we present an algorithm together with data structures for geodesic triangle counting queries under the fully-dynamic setting. With high probability, each update takes O(n^2/3 log n) time, and each query takes O(n^2/3 log n) time.

Eunjin Oh and Hee-Kap Ahn. Dynamic Geodesic Convex Hulls in Dynamic Simple Polygons. In 33rd International Symposium on Computational Geometry (SoCG 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 77, pp. 51:1-51:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{oh_et_al:LIPIcs.SoCG.2017.51, author = {Oh, Eunjin and Ahn, Hee-Kap}, title = {{Dynamic Geodesic Convex Hulls in Dynamic Simple Polygons}}, booktitle = {33rd International Symposium on Computational Geometry (SoCG 2017)}, pages = {51:1--51:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-038-5}, ISSN = {1868-8969}, year = {2017}, volume = {77}, editor = {Aronov, Boris and Katz, Matthew J.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2017.51}, URN = {urn:nbn:de:0030-drops-72198}, doi = {10.4230/LIPIcs.SoCG.2017.51}, annote = {Keywords: Dynamic geodesic convex hull, dynamic simple polygons} }

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**Published in:** LIPIcs, Volume 77, 33rd International Symposium on Computational Geometry (SoCG 2017)

Given a set of sites in a simple polygon, a geodesic Voronoi diagram partitions the polygon into regions based on distances to sites under the geodesic metric. We present algorithms for computing the geodesic nearest-point, higher-order and farthest-point Voronoi diagrams of m point sites in a simple n-gon, which improve the best known ones for m < n/polylog n. Moreover, the algorithms for the nearest-point and farthest-point Voronoi diagrams are optimal for m < n/polylog n. This partially answers a question posed by Mitchell in the Handbook of Computational Geometry.

Eunjin Oh and Hee-Kap Ahn. Voronoi Diagrams for a Moderate-Sized Point-Set in a Simple Polygon. In 33rd International Symposium on Computational Geometry (SoCG 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 77, pp. 52:1-52:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{oh_et_al:LIPIcs.SoCG.2017.52, author = {Oh, Eunjin and Ahn, Hee-Kap}, title = {{Voronoi Diagrams for a Moderate-Sized Point-Set in a Simple Polygon}}, booktitle = {33rd International Symposium on Computational Geometry (SoCG 2017)}, pages = {52:1--52:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-038-5}, ISSN = {1868-8969}, year = {2017}, volume = {77}, editor = {Aronov, Boris and Katz, Matthew J.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2017.52}, URN = {urn:nbn:de:0030-drops-72184}, doi = {10.4230/LIPIcs.SoCG.2017.52}, annote = {Keywords: Simple polygons, Voronoi diagrams, geodesic distance} }

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**Published in:** LIPIcs, Volume 64, 27th International Symposium on Algorithms and Computation (ISAAC 2016)

Given a set P of n points in the plane and a multiset W of k weights with k leq n, we assign a weight in W to a point in P to minimize the maximum weighted distance from the weighted center of P to any point in P. In this paper, we give two algorithms which take O(k^2 n^2 log^4 n) time and O(k^5 n log^4 k + kn log^3 n) time, respectively. For a constant k, the second algorithm takes only O(n log^3 n) time, which is near-linear.

Eunjin Oh and Hee-Kap Ahn. Assigning Weights to Minimize the Covering Radius in the Plane. In 27th International Symposium on Algorithms and Computation (ISAAC 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 64, pp. 58:1-58:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{oh_et_al:LIPIcs.ISAAC.2016.58, author = {Oh, Eunjin and Ahn, Hee-Kap}, title = {{Assigning Weights to Minimize the Covering Radius in the Plane}}, booktitle = {27th International Symposium on Algorithms and Computation (ISAAC 2016)}, pages = {58:1--58:12}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-026-2}, ISSN = {1868-8969}, year = {2016}, volume = {64}, editor = {Hong, Seok-Hee}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2016.58}, URN = {urn:nbn:de:0030-drops-68275}, doi = {10.4230/LIPIcs.ISAAC.2016.58}, annote = {Keywords: Weighted center, facility location, weight assignment, combinatorial op- timization, computational geometry} }

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**Published in:** LIPIcs, Volume 64, 27th International Symposium on Algorithms and Computation (ISAAC 2016)

We consider the problem of finding a shortcut connecting two vertices of a graph that minimizes the diameter of the resulting graph. We present an O(n^2 log^3 n)-time algorithm using linear space for the case that the input graph is a tree consisting of n vertices. Additionally, we present an O(n^2 log^3 n)-time algorithm using linear space for a continuous version of this problem.

Eunjin Oh and Hee-Kap Ahn. A Near-Optimal Algorithm for Finding an Optimal Shortcut of a Tree. In 27th International Symposium on Algorithms and Computation (ISAAC 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 64, pp. 59:1-59:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{oh_et_al:LIPIcs.ISAAC.2016.59, author = {Oh, Eunjin and Ahn, Hee-Kap}, title = {{A Near-Optimal Algorithm for Finding an Optimal Shortcut of a Tree}}, booktitle = {27th International Symposium on Algorithms and Computation (ISAAC 2016)}, pages = {59:1--59:12}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-026-2}, ISSN = {1868-8969}, year = {2016}, volume = {64}, editor = {Hong, Seok-Hee}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2016.59}, URN = {urn:nbn:de:0030-drops-68283}, doi = {10.4230/LIPIcs.ISAAC.2016.59}, annote = {Keywords: Network Augmentation, Shortcuts, Diameter, Trees} }

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**Published in:** LIPIcs, Volume 53, 15th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2016)

For any two points in a simple polygon P, the geodesic distance between them is the length of the shortest path contained in P that connects them. A geodesic center of a set S of sites (points) with respect to P is a point in P that minimizes the geodesic distance to its farthest site. In many realistic facility location problems, however, the facilities are constrained to lie in feasible regions. In this paper, we show how to compute the geodesic centers constrained to a set of line segments or simple polygonal regions contained in P. Our results provide substantial improvements over previous algorithms.

Eunjin Oh, Wanbin Son, and Hee-Kap Ahn. Constrained Geodesic Centers of a Simple Polygon. In 15th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 53, pp. 29:1-29:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{oh_et_al:LIPIcs.SWAT.2016.29, author = {Oh, Eunjin and Son, Wanbin and Ahn, Hee-Kap}, title = {{Constrained Geodesic Centers of a Simple Polygon}}, booktitle = {15th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2016)}, pages = {29:1--29:13}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-011-8}, ISSN = {1868-8969}, year = {2016}, volume = {53}, editor = {Pagh, Rasmus}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SWAT.2016.29}, URN = {urn:nbn:de:0030-drops-60516}, doi = {10.4230/LIPIcs.SWAT.2016.29}, annote = {Keywords: Geodesic distance, simple polygons, constrained center, facility location, farthest-point Voronoi diagram} }

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**Published in:** LIPIcs, Volume 51, 32nd International Symposium on Computational Geometry (SoCG 2016)

Given a set of sites (points) in a simple polygon, the farthest-point geodesic Voronoi diagram partitions the polygon into cells, at most one cell per site, such that every point in a cell has the same farthest site with respect to the geodesic metric. We present an O((n+m)loglogn)-time algorithm to compute the farthest-point geodesic Voronoi diagram for m sites lying on the boundary of a simple n-gon.

Eunjin Oh, Luis Barba, and Hee-Kap Ahn. The Farthest-Point Geodesic Voronoi Diagram of Points on the Boundary of a Simple Polygon. In 32nd International Symposium on Computational Geometry (SoCG 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 51, pp. 56:1-56:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{oh_et_al:LIPIcs.SoCG.2016.56, author = {Oh, Eunjin and Barba, Luis and Ahn, Hee-Kap}, title = {{The Farthest-Point Geodesic Voronoi Diagram of Points on the Boundary of a Simple Polygon}}, booktitle = {32nd International Symposium on Computational Geometry (SoCG 2016)}, pages = {56:1--56:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-009-5}, ISSN = {1868-8969}, year = {2016}, volume = {51}, editor = {Fekete, S\'{a}ndor and Lubiw, Anna}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2016.56}, URN = {urn:nbn:de:0030-drops-59481}, doi = {10.4230/LIPIcs.SoCG.2016.56}, annote = {Keywords: Geodesic distance, simple polygons, farthest-point Voronoi diagram} }

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**Published in:** LIPIcs, Volume 34, 31st International Symposium on Computational Geometry (SoCG 2015)

Let P be a closed simple polygon with n vertices. For any two points in P, the geodesic distance between them is the length of the shortest path that connects them among all paths contained in P. The geodesic center of P is the unique point in P that minimizes the largest geodesic distance to all other points of P. In 1989, Pollack, Sharir and Rote [Disc. & Comput. Geom. 89] showed an O(n log n)-time algorithm that computes the geodesic center of P. Since then, a longstanding question has been whether this running time can be improved (explicitly posed by Mitchell [Handbook of Computational Geometry, 2000]). In this paper we affirmatively answer this question and present a linear time algorithm to solve this problem.

Hee Kap Ahn, Luis Barba, Prosenjit Bose, Jean-Lou De Carufel, Matias Korman, and Eunjin Oh. A Linear-Time Algorithm for the Geodesic Center of a Simple Polygon. In 31st International Symposium on Computational Geometry (SoCG 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 34, pp. 209-223, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)

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@InProceedings{ahn_et_al:LIPIcs.SOCG.2015.209, author = {Ahn, Hee Kap and Barba, Luis and Bose, Prosenjit and De Carufel, Jean-Lou and Korman, Matias and Oh, Eunjin}, title = {{A Linear-Time Algorithm for the Geodesic Center of a Simple Polygon}}, booktitle = {31st International Symposium on Computational Geometry (SoCG 2015)}, pages = {209--223}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-83-5}, ISSN = {1868-8969}, year = {2015}, volume = {34}, editor = {Arge, Lars and Pach, J\'{a}nos}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SOCG.2015.209}, URN = {urn:nbn:de:0030-drops-51448}, doi = {10.4230/LIPIcs.SOCG.2015.209}, annote = {Keywords: Geodesic distance, facility location, 1-center problem, simple polygons} }

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