--- _id: '2012' abstract: - lang: eng text: The classical sphere packing problem asks for the best (infinite) arrangement of non-overlapping unit balls which cover as much space as possible. We define a generalized version of the problem, where we allow each ball a limited amount of overlap with other balls. We study two natural choices of overlap measures and obtain the optimal lattice packings in a parameterized family of lattices which contains the FCC, BCC, and integer lattice. acknowledgement: We thank Herbert Edelsbrunner for his valuable discussions and ideas on the topic of this paper. The second author has been supported by the Max Planck Center for Visual Computing and Communication article_number: '1401.0468' article_processing_charge: No author: - first_name: Mabel full_name: Iglesias Ham, Mabel id: 41B58C0C-F248-11E8-B48F-1D18A9856A87 last_name: Iglesias Ham - first_name: Michael full_name: Kerber, Michael last_name: Kerber orcid: 0000-0002-8030-9299 - first_name: Caroline full_name: Uhler, Caroline id: 49ADD78E-F248-11E8-B48F-1D18A9856A87 last_name: Uhler orcid: 0000-0002-7008-0216 citation: ama: Iglesias Ham M, Kerber M, Uhler C. Sphere packing with limited overlap. arXiv. doi:10.48550/arXiv.1401.0468 apa: Iglesias Ham, M., Kerber, M., & Uhler, C. (n.d.). Sphere packing with limited overlap. arXiv. https://doi.org/10.48550/arXiv.1401.0468 chicago: Iglesias Ham, Mabel, Michael Kerber, and Caroline Uhler. “Sphere Packing with Limited Overlap.” ArXiv, n.d. https://doi.org/10.48550/arXiv.1401.0468. ieee: M. Iglesias Ham, M. Kerber, and C. Uhler, “Sphere packing with limited overlap,” arXiv. . ista: Iglesias Ham M, Kerber M, Uhler C. Sphere packing with limited overlap. arXiv, 1401.0468. mla: Iglesias Ham, Mabel, et al. “Sphere Packing with Limited Overlap.” ArXiv, 1401.0468, doi:10.48550/arXiv.1401.0468. short: M. Iglesias Ham, M. Kerber, C. Uhler, ArXiv (n.d.). date_created: 2018-12-11T11:55:12Z date_published: 2014-01-01T00:00:00Z date_updated: 2023-10-18T08:06:45Z day: '01' department: - _id: HeEd - _id: CaUh doi: 10.48550/arXiv.1401.0468 external_id: arxiv: - '1401.0468' language: - iso: eng main_file_link: - open_access: '1' url: http://cccg.ca/proceedings/2014/papers/paper23.pdf month: '01' oa: 1 oa_version: Submitted Version publication: arXiv publication_status: submitted publist_id: '5064' status: public title: Sphere packing with limited overlap type: preprint user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87 year: '2014' ... --- _id: '2209' abstract: - lang: eng text: "A straight skeleton is a well-known geometric structure, and several algorithms exist to construct the straight skeleton for a given polygon or planar straight-line graph. In this paper, we ask the reverse question: Given the straight skeleton (in form of a planar straight-line graph, with some rays to infinity), can we reconstruct a planar straight-line graph for which this was the straight skeleton? We show how to reduce this problem to the problem of finding a line that intersects a set of convex polygons. We can find these convex polygons and all such lines in $O(nlog n)$ time in the Real RAM computer model, where $n$ denotes the number of edges of the input graph. We also explain how our approach can be used for recognizing Voronoi diagrams of points, thereby completing a partial solution provided by Ash and Bolker in 1985.\r\n" alternative_title: - '2013 10th International Symposium on Voronoi Diagrams in Science and Engineering (ISVD 2013) ' author: - first_name: Therese full_name: Biedl, Therese last_name: Biedl - first_name: Martin full_name: Held, Martin last_name: Held - first_name: Stefan full_name: Huber, Stefan id: 4700A070-F248-11E8-B48F-1D18A9856A87 last_name: Huber orcid: 0000-0002-8871-5814 citation: ama: 'Biedl T, Held M, Huber S. Recognizing straight skeletons and Voronoi diagrams and reconstructing their input. In: IEEE; 2013:37-46. doi:10.1109/ISVD.2013.11' apa: 'Biedl, T., Held, M., & Huber, S. (2013). Recognizing straight skeletons and Voronoi diagrams and reconstructing their input (pp. 37–46). Presented at the ISVD: Voronoi Diagrams in Science and Engineering, St. Petersburg, Russia: IEEE. https://doi.org/10.1109/ISVD.2013.11' chicago: Biedl, Therese, Martin Held, and Stefan Huber. “Recognizing Straight Skeletons and Voronoi Diagrams and Reconstructing Their Input,” 37–46. IEEE, 2013. https://doi.org/10.1109/ISVD.2013.11. ieee: 'T. Biedl, M. Held, and S. Huber, “Recognizing straight skeletons and Voronoi diagrams and reconstructing their input,” presented at the ISVD: Voronoi Diagrams in Science and Engineering, St. Petersburg, Russia, 2013, pp. 37–46.' ista: 'Biedl T, Held M, Huber S. 2013. Recognizing straight skeletons and Voronoi diagrams and reconstructing their input. ISVD: Voronoi Diagrams in Science and Engineering, 2013 10th International Symposium on Voronoi Diagrams in Science and Engineering (ISVD 2013) , , 37–46.' mla: Biedl, Therese, et al. Recognizing Straight Skeletons and Voronoi Diagrams and Reconstructing Their Input. IEEE, 2013, pp. 37–46, doi:10.1109/ISVD.2013.11. short: T. Biedl, M. Held, S. Huber, in:, IEEE, 2013, pp. 37–46. conference: end_date: 2013-07-10 location: St. Petersburg, Russia name: 'ISVD: Voronoi Diagrams in Science and Engineering' start_date: 2013-07-08 date_created: 2018-12-11T11:56:20Z date_published: 2013-12-01T00:00:00Z date_updated: 2021-01-12T06:56:00Z day: '01' department: - _id: HeEd doi: 10.1109/ISVD.2013.11 language: - iso: eng month: '12' oa_version: None page: 37 - 46 publication_identifier: eisbn: - '978-0-7695-5037-4 ' publication_status: published publisher: IEEE publist_id: '4763' quality_controlled: '1' scopus_import: 1 status: public title: Recognizing straight skeletons and Voronoi diagrams and reconstructing their input type: conference user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87 year: '2013' ... --- _id: '2210' abstract: - lang: eng text: 'A straight skeleton is a well-known geometric structure, and several algorithms exist to construct the straight skeleton for a given polygon. In this paper, we ask the reverse question: Given the straight skeleton (in form of a tree with a drawing in the plane, but with the exact position of the leaves unspecified), can we reconstruct the polygon? We show that in most cases there exists at most one polygon; in the remaining case there is an infinite number of polygons determined by one angle that can range in an interval. We can find this (set of) polygon(s) in linear time in the Real RAM computer model.' author: - first_name: Therese full_name: Biedl, Therese last_name: Biedl - first_name: Martin full_name: Held, Martin last_name: Held - first_name: Stefan full_name: Huber, Stefan id: 4700A070-F248-11E8-B48F-1D18A9856A87 last_name: Huber orcid: 0000-0002-8871-5814 citation: ama: 'Biedl T, Held M, Huber S. Reconstructing polygons from embedded straight skeletons. In: 29th European Workshop on Computational Geometry. TU Braunschweig; 2013:95-98.' apa: 'Biedl, T., Held, M., & Huber, S. (2013). Reconstructing polygons from embedded straight skeletons. In 29th European Workshop on Computational Geometry (pp. 95–98). Braunschweig, Germany: TU Braunschweig.' chicago: Biedl, Therese, Martin Held, and Stefan Huber. “Reconstructing Polygons from Embedded Straight Skeletons.” In 29th European Workshop on Computational Geometry, 95–98. TU Braunschweig, 2013. ieee: T. Biedl, M. Held, and S. Huber, “Reconstructing polygons from embedded straight skeletons,” in 29th European Workshop on Computational Geometry, Braunschweig, Germany, 2013, pp. 95–98. ista: 'Biedl T, Held M, Huber S. 2013. Reconstructing polygons from embedded straight skeletons. 29th European Workshop on Computational Geometry. EuroCG: European Workshop on Computational Geometry, 95–98.' mla: Biedl, Therese, et al. “Reconstructing Polygons from Embedded Straight Skeletons.” 29th European Workshop on Computational Geometry, TU Braunschweig, 2013, pp. 95–98. short: T. Biedl, M. Held, S. Huber, in:, 29th European Workshop on Computational Geometry, TU Braunschweig, 2013, pp. 95–98. conference: end_date: 2013-03-20 location: Braunschweig, Germany name: 'EuroCG: European Workshop on Computational Geometry' start_date: 2013-03-17 date_created: 2018-12-11T11:56:21Z date_published: 2013-03-01T00:00:00Z date_updated: 2021-01-12T06:56:00Z day: '01' department: - _id: HeEd language: - iso: eng main_file_link: - open_access: '1' url: http://www.ibr.cs.tu-bs.de/alg/eurocg13/booklet_eurocg13.pdf month: '03' oa: 1 oa_version: Submitted Version page: 95 - 98 publication: 29th European Workshop on Computational Geometry publication_status: published publisher: TU Braunschweig publist_id: '4762' status: public title: Reconstructing polygons from embedded straight skeletons type: conference user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87 year: '2013' ... --- _id: '2304' abstract: - lang: eng text: This extended abstract is concerned with the irregularities of distribution of one-dimensional permuted van der Corput sequences that are generated from linear permutations. We show how to obtain upper bounds for the discrepancy and diaphony of these sequences, by relating them to Kronecker sequences and applying earlier results of Faure and Niederreiter. acknowledgement: This research is supported by the Graduate school of IST Austria (Institute of Science and Technology Austria). author: - first_name: Florian full_name: Pausinger, Florian id: 2A77D7A2-F248-11E8-B48F-1D18A9856A87 last_name: Pausinger orcid: 0000-0002-8379-3768 citation: ama: Pausinger F. Van der Corput sequences and linear permutations. Electronic Notes in Discrete Mathematics. 2013;43:43-50. doi:10.1016/j.endm.2013.07.008 apa: Pausinger, F. (2013). Van der Corput sequences and linear permutations. Electronic Notes in Discrete Mathematics. Elsevier. https://doi.org/10.1016/j.endm.2013.07.008 chicago: Pausinger, Florian. “Van Der Corput Sequences and Linear Permutations.” Electronic Notes in Discrete Mathematics. Elsevier, 2013. https://doi.org/10.1016/j.endm.2013.07.008. ieee: F. Pausinger, “Van der Corput sequences and linear permutations,” Electronic Notes in Discrete Mathematics, vol. 43. Elsevier, pp. 43–50, 2013. ista: Pausinger F. 2013. Van der Corput sequences and linear permutations. Electronic Notes in Discrete Mathematics. 43, 43–50. mla: Pausinger, Florian. “Van Der Corput Sequences and Linear Permutations.” Electronic Notes in Discrete Mathematics, vol. 43, Elsevier, 2013, pp. 43–50, doi:10.1016/j.endm.2013.07.008. short: F. Pausinger, Electronic Notes in Discrete Mathematics 43 (2013) 43–50. date_created: 2018-12-11T11:56:53Z date_published: 2013-09-05T00:00:00Z date_updated: 2021-01-12T06:56:39Z day: '05' department: - _id: HeEd doi: 10.1016/j.endm.2013.07.008 intvolume: ' 43' language: - iso: eng month: '09' oa_version: None page: 43 - 50 publication: Electronic Notes in Discrete Mathematics publication_status: published publisher: Elsevier publist_id: '4623' quality_controlled: '1' scopus_import: 1 status: public title: Van der Corput sequences and linear permutations type: journal_article user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87 volume: 43 year: '2013' ... --- _id: '2807' abstract: - lang: eng text: 'We consider several basic problems of algebraic topology, with connections to combinatorial and geometric questions, from the point of view of computational complexity. The extension problem asks, given topological spaces X; Y , a subspace A ⊆ X, and a (continuous) map f : A → Y , whether f can be extended to a map X → Y . For computational purposes, we assume that X and Y are represented as finite simplicial complexes, A is a subcomplex of X, and f is given as a simplicial map. In this generality the problem is undecidable, as follows from Novikov''s result from the 1950s on uncomputability of the fundamental group π1(Y ). We thus study the problem under the assumption that, for some k ≥ 2, Y is (k - 1)-connected; informally, this means that Y has \no holes up to dimension k-1" (a basic example of such a Y is the sphere Sk). We prove that, on the one hand, this problem is still undecidable for dimX = 2k. On the other hand, for every fixed k ≥ 2, we obtain an algorithm that solves the extension problem in polynomial time assuming Y (k - 1)-connected and dimX ≤ 2k - 1. For dimX ≤ 2k - 2, the algorithm also provides a classification of all extensions up to homotopy (continuous deformation). This relies on results of our SODA 2012 paper, and the main new ingredient is a machinery of objects with polynomial-time homology, which is a polynomial-time analog of objects with effective homology developed earlier by Sergeraert et al. We also consider the computation of the higher homotopy groups πk(Y ), k ≥ 2, for a 1-connected Y . Their computability was established by Brown in 1957; we show that πk(Y ) can be computed in polynomial time for every fixed k ≥ 2. On the other hand, Anick proved in 1989 that computing πk(Y ) is #P-hard if k is a part of input, where Y is a cell complex with certain rather compact encoding. We strengthen his result to #P-hardness for Y given as a simplicial complex. ' author: - first_name: Martin full_name: Čadek, Martin last_name: Čadek - first_name: Marek full_name: Krcál, Marek id: 33E21118-F248-11E8-B48F-1D18A9856A87 last_name: Krcál - first_name: Jiří full_name: Matoušek, Jiří last_name: Matoušek - first_name: Lukáš full_name: Vokřínek, Lukáš last_name: Vokřínek - first_name: Uli full_name: Wagner, Uli id: 36690CA2-F248-11E8-B48F-1D18A9856A87 last_name: Wagner orcid: 0000-0002-1494-0568 citation: ama: 'Čadek M, Krcál M, Matoušek J, Vokřínek L, Wagner U. Extending continuous maps: Polynomiality and undecidability. In: 45th Annual ACM Symposium on Theory of Computing. ACM; 2013:595-604. doi:10.1145/2488608.2488683' apa: 'Čadek, M., Krcál, M., Matoušek, J., Vokřínek, L., & Wagner, U. (2013). Extending continuous maps: Polynomiality and undecidability. In 45th Annual ACM Symposium on theory of computing (pp. 595–604). Palo Alto, CA, United States: ACM. https://doi.org/10.1145/2488608.2488683' chicago: 'Čadek, Martin, Marek Krcál, Jiří Matoušek, Lukáš Vokřínek, and Uli Wagner. “Extending Continuous Maps: Polynomiality and Undecidability.” In 45th Annual ACM Symposium on Theory of Computing, 595–604. ACM, 2013. https://doi.org/10.1145/2488608.2488683.' ieee: 'M. Čadek, M. Krcál, J. Matoušek, L. Vokřínek, and U. Wagner, “Extending continuous maps: Polynomiality and undecidability,” in 45th Annual ACM Symposium on theory of computing, Palo Alto, CA, United States, 2013, pp. 595–604.' ista: 'Čadek M, Krcál M, Matoušek J, Vokřínek L, Wagner U. 2013. Extending continuous maps: Polynomiality and undecidability. 45th Annual ACM Symposium on theory of computing. STOC: Symposium on the Theory of Computing, 595–604.' mla: 'Čadek, Martin, et al. “Extending Continuous Maps: Polynomiality and Undecidability.” 45th Annual ACM Symposium on Theory of Computing, ACM, 2013, pp. 595–604, doi:10.1145/2488608.2488683.' short: M. Čadek, M. Krcál, J. Matoušek, L. Vokřínek, U. Wagner, in:, 45th Annual ACM Symposium on Theory of Computing, ACM, 2013, pp. 595–604. conference: end_date: 2013-06-04 location: Palo Alto, CA, United States name: 'STOC: Symposium on the Theory of Computing' start_date: 2013-06-01 date_created: 2018-12-11T11:59:42Z date_published: 2013-06-01T00:00:00Z date_updated: 2021-01-12T06:59:51Z day: '01' ddc: - '510' department: - _id: UlWa - _id: HeEd doi: 10.1145/2488608.2488683 file: - access_level: open_access checksum: 06c2ce5c1135fbc1f71ca15eeb242dcf content_type: application/pdf creator: system date_created: 2018-12-12T10:14:29Z date_updated: 2020-07-14T12:45:48Z file_id: '5081' file_name: IST-2016-533-v1+1_Extending_continuous_maps_polynomiality_and_undecidability.pdf file_size: 447945 relation: main_file file_date_updated: 2020-07-14T12:45:48Z has_accepted_license: '1' language: - iso: eng month: '06' oa: 1 oa_version: Submitted Version page: 595 - 604 publication: 45th Annual ACM Symposium on theory of computing publication_status: published publisher: ACM publist_id: '4078' pubrep_id: '533' quality_controlled: '1' scopus_import: 1 status: public title: 'Extending continuous maps: Polynomiality and undecidability' type: conference user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87 year: '2013' ...