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

An eight-partition of a finite set of points (respectively, of a continuous mass distribution) in ℝ³ consists of three planes that divide the space into 8 octants, such that each open octant contains at most 1/8 of the points (respectively, of the mass). In 1966, Hadwiger showed that any mass distribution in ℝ³ admits an eight-partition; moreover, one can prescribe the normal direction of one of the three planes. The analogous result for finite point sets follows by a standard limit argument.
We prove the following variant of this result: Any mass distribution (or point set) in ℝ³ admits an eight-partition for which the intersection of two of the planes is a line with a prescribed direction.
Moreover, we present an efficient algorithm for calculating an eight-partition of a set of n points in ℝ³ (with prescribed normal direction of one of the planes) in time O^*(n^{5/2}).

Boris Aronov, Abdul Basit, Indu Ramesh, Gianluca Tasinato, and Uli Wagner. Eight-Partitioning Points in 3D, and Efficiently Too. In 40th International Symposium on Computational Geometry (SoCG 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 293, pp. 8:1-8:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{aronov_et_al:LIPIcs.SoCG.2024.8, author = {Aronov, Boris and Basit, Abdul and Ramesh, Indu and Tasinato, Gianluca and Wagner, Uli}, title = {{Eight-Partitioning Points in 3D, and Efficiently Too}}, booktitle = {40th International Symposium on Computational Geometry (SoCG 2024)}, pages = {8:1--8:15}, 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.2024.8}, URN = {urn:nbn:de:0030-drops-199538}, doi = {10.4230/LIPIcs.SoCG.2024.8}, annote = {Keywords: Mass partitions, partitions of points in three dimensions, Borsuk-Ulam Theorem, Ham-Sandwich Theorem} }

Document

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

It is unlikely that the discrete Fréchet distance between two curves of length n can be computed in strictly subquadratic time. We thus consider the setting where one of the curves, P, is known in advance. In particular, we wish to construct data structures (distance oracles) of near-linear size that support efficient distance queries with respect to P in sublinear time. Since there is evidence that this is impossible for query curves of length Θ(n^α), for any α > 0, we focus on query curves of (small) constant length, for which we are able to devise distance oracles with the desired bounds.
We extend our tools to handle subcurves of the given curve, and even arbitrary vertex-to-vertex subcurves of a given geometric tree. That is, we construct an oracle that can quickly compute the distance between a short polygonal path (the query) and a path in the preprocessed tree between two query-specified vertices. Moreover, we define a new family of geometric graphs, t-local graphs (which strictly contains the family of geometric spanners with constant stretch), for which a similar oracle exists: we can preprocess a graph G in the family, so that, given a query segment and a pair u,v of vertices in G, one can quickly compute the smallest discrete Fréchet distance between the segment and any (u,v)-path in G. The answer is exact, if t = 1, and approximate if t > 1.

Boris Aronov, Tsuri Farhana, Matthew J. Katz, and Indu Ramesh. Discrete Fréchet Distance Oracles. In 40th International Symposium on Computational Geometry (SoCG 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 293, pp. 10:1-10:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{aronov_et_al:LIPIcs.SoCG.2024.10, author = {Aronov, Boris and Farhana, Tsuri and Katz, Matthew J. and Ramesh, Indu}, title = {{Discrete Fr\'{e}chet Distance Oracles}}, booktitle = {40th International Symposium on Computational Geometry (SoCG 2024)}, pages = {10:1--10:14}, 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.2024.10}, URN = {urn:nbn:de:0030-drops-199554}, doi = {10.4230/LIPIcs.SoCG.2024.10}, annote = {Keywords: discrete Fr\'{e}chet distance, distance oracle, heavy-path decomposition, t-local graphs} }

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