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**Published in:** LIPIcs, Volume 129, 35th International Symposium on Computational Geometry (SoCG 2019)

We consider a range-search variant of the closest-pair problem. Let Gamma be a fixed shape in the plane. We are interested in storing a given set of n points in the plane in some data structure such that for any specified translate of Gamma, the closest pair of points contained in the translate can be reported efficiently. We present results on this problem for two important settings: when Gamma is a polygon (possibly with holes) and when Gamma is a general convex body whose boundary is smooth. When Gamma is a polygon, we present a data structure using O(n) space and O(log n) query time, which is asymptotically optimal. When Gamma is a general convex body with a smooth boundary, we give a near-optimal data structure using O(n log n) space and O(log^2 n) query time. Our results settle some open questions posed by Xue et al. at SoCG 2018.

Jie Xue, Yuan Li, Saladi Rahul, and Ravi Janardan. Searching for the Closest-Pair in a Query Translate. In 35th International Symposium on Computational Geometry (SoCG 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 129, pp. 61:1-61:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{xue_et_al:LIPIcs.SoCG.2019.61, author = {Xue, Jie and Li, Yuan and Rahul, Saladi and Janardan, Ravi}, title = {{Searching for the Closest-Pair in a Query Translate}}, booktitle = {35th International Symposium on Computational Geometry (SoCG 2019)}, pages = {61:1--61:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-104-7}, ISSN = {1868-8969}, year = {2019}, volume = {129}, editor = {Barequet, Gill and Wang, Yusu}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2019.61}, URN = {urn:nbn:de:0030-drops-104659}, doi = {10.4230/LIPIcs.SoCG.2019.61}, annote = {Keywords: Closest pair, Range search, Geometric data structures, Translation query} }

Document

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

Given a dataset S of points in R^2, the range closest-pair (RCP) problem aims to preprocess S into a data structure such that when a query range X is specified, the closest-pair in S cap X can be reported efficiently. The RCP problem can be viewed as a range-search version of the classical closest-pair problem, and finds applications in many areas. Due to its non-decomposability, the RCP problem is much more challenging than many traditional range-search problems. This paper revisits the RCP problem, and proposes new data structures for various query types including quadrants, strips, rectangles, and halfplanes. Both worst-case and average-case analyses (in the sense that the data points are drawn uniformly and independently from the unit square) are applied to these new data structures, which result in new bounds for the RCP problem. Some of the new bounds significantly improve the previous results, while the others are entirely new.

Jie Xue, Yuan Li, Saladi Rahul, and Ravi Janardan. New Bounds for Range Closest-Pair Problems. In 34th International Symposium on Computational Geometry (SoCG 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 99, pp. 73:1-73:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{xue_et_al:LIPIcs.SoCG.2018.73, author = {Xue, Jie and Li, Yuan and Rahul, Saladi and Janardan, Ravi}, title = {{New Bounds for Range Closest-Pair Problems}}, booktitle = {34th International Symposium on Computational Geometry (SoCG 2018)}, pages = {73:1--73:14}, 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-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2018.73}, URN = {urn:nbn:de:0030-drops-87865}, doi = {10.4230/LIPIcs.SoCG.2018.73}, annote = {Keywords: Closest-pair, Range search, Candidate pairs, Average-case analysis} }

Document

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

In this paper, we study the linear separability problem for stochastic geometric objects under the well-known unipoint/multipoint uncertainty models. Let S=S_R U S_B be a given set of stochastic bichromatic points, and define n = min{|S_R|, |S_B|} and N = max{|S_R|, |S_B|}. We show that the separable-probability (SP) of S can be computed in O(nN^{d-1}) time for d >= 3 and O(min{nN log N, N^2}) time for d=2, while the expected separation-margin (ESM) of S can be computed in O(nN^d) time for d >= 2. In addition, we give an Omega(nN^{d-1}) witness-based lower bound for computing SP, which implies the optimality of our algorithm among all those in this category. Also, a hardness result for computing ESM is given to show the difficulty of further improving our algorithm. As an extension, we generalize the same problems from points to general geometric objects, i.e., polytopes and/or balls, and extend our algorithms to solve the generalized SP and ESM problems in O(nN^d) and O(nN^{d+1}) time, respectively. Finally, we present some applications of our algorithms to stochastic convex-hull related problems.

Jie Xue, Yuan Li, and Ravi Janardan. On the Separability of Stochastic Geometric Objects, with Applications. In 32nd International Symposium on Computational Geometry (SoCG 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 51, pp. 62:1-62:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{xue_et_al:LIPIcs.SoCG.2016.62, author = {Xue, Jie and Li, Yuan and Janardan, Ravi}, title = {{On the Separability of Stochastic Geometric Objects, with Applications}}, booktitle = {32nd International Symposium on Computational Geometry (SoCG 2016)}, pages = {62:1--62:16}, 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-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2016.62}, URN = {urn:nbn:de:0030-drops-59544}, doi = {10.4230/LIPIcs.SoCG.2016.62}, annote = {Keywords: Stochastic objects, linear separability, separable-probability, expected separation-margin, convex hull} }

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