--- _id: '4065' abstract: - lang: eng text: We prove that given n⩾3 convex, compact, and pairwise disjoint sets in the plane, they may be covered with n non-overlapping convex polygons with a total of not more than 6n−9 sides, and with not more than 3n−6 distinct slopes. Furthermore, we construct sets that require 6n−9 sides and 3n−6 slopes for n⩾3. The upper bound on the number of slopes implies a new bound on a recently studied transversal problem. acknowledgement: 'The first author acknowledges the support by Amoco Fnd. Fat. Dev. Comput. Sci. l-6-44862. Work on this paper by the second author was supported by a Shell Fellowship in Computer Science. The third author as supported by the office of Naval Research under grant NOOO14-86K-0416. ' article_processing_charge: No article_type: original author: - first_name: Herbert full_name: Edelsbrunner, Herbert id: 3FB178DA-F248-11E8-B48F-1D18A9856A87 last_name: Edelsbrunner orcid: 0000-0002-9823-6833 - first_name: Arch full_name: Robison, Arch last_name: Robison - first_name: Xiao full_name: Shen, Xiao last_name: Shen citation: ama: Edelsbrunner H, Robison A, Shen X. Covering convex sets with non-overlapping polygons. Discrete Mathematics. 1990;81(2):153-164. doi:10.1016/0012-365X(90)90147-A apa: Edelsbrunner, H., Robison, A., & Shen, X. (1990). Covering convex sets with non-overlapping polygons. Discrete Mathematics. Elsevier. https://doi.org/10.1016/0012-365X(90)90147-A chicago: Edelsbrunner, Herbert, Arch Robison, and Xiao Shen. “Covering Convex Sets with Non-Overlapping Polygons.” Discrete Mathematics. Elsevier, 1990. https://doi.org/10.1016/0012-365X(90)90147-A. ieee: H. Edelsbrunner, A. Robison, and X. Shen, “Covering convex sets with non-overlapping polygons,” Discrete Mathematics, vol. 81, no. 2. Elsevier, pp. 153–164, 1990. ista: Edelsbrunner H, Robison A, Shen X. 1990. Covering convex sets with non-overlapping polygons. Discrete Mathematics. 81(2), 153–164. mla: Edelsbrunner, Herbert, et al. “Covering Convex Sets with Non-Overlapping Polygons.” Discrete Mathematics, vol. 81, no. 2, Elsevier, 1990, pp. 153–64, doi:10.1016/0012-365X(90)90147-A. short: H. Edelsbrunner, A. Robison, X. Shen, Discrete Mathematics 81 (1990) 153–164. date_created: 2018-12-11T12:06:44Z date_published: 1990-04-15T00:00:00Z date_updated: 2022-02-22T15:45:55Z day: '15' doi: 10.1016/0012-365X(90)90147-A extern: '1' intvolume: ' 81' issue: '2' language: - iso: eng main_file_link: - url: https://www.sciencedirect.com/science/article/pii/0012365X9090147A?via%3Dihub month: '04' oa_version: None page: 153 - 164 publication: Discrete Mathematics publication_identifier: eissn: - 1872-681X issn: - 0012-365X publication_status: published publisher: Elsevier publist_id: '2060' quality_controlled: '1' scopus_import: '1' status: public title: Covering convex sets with non-overlapping polygons type: journal_article user_id: ea97e931-d5af-11eb-85d4-e6957dddbf17 volume: 81 year: '1990' ... --- _id: '4074' abstract: - lang: eng text: We present upper and lower bounds for extremal problems defined for arrangements of lines, circles, spheres, and alike. For example, we prove that the maximum number of edges boundingm cells in an arrangement ofn lines is Θ(m 2/3 n 2/3 +n), and that it isO(m 2/3 n 2/3 β(n) +n) forn unit-circles, whereβ(n) (and laterβ(m, n)) is a function that depends on the inverse of Ackermann's function and grows extremely slowly. If we replace unit-circles by circles of arbitrary radii the upper bound goes up toO(m 3/5 n 4/5 β(n) +n). The same bounds (without theβ(n)-terms) hold for the maximum sum of degrees ofm vertices. In the case of vertex degrees in arrangements of lines and of unit-circles our bounds match previous results, but our proofs are considerably simpler than the previous ones. The maximum sum of degrees ofm vertices in an arrangement ofn spheres in three dimensions isO(m 4/7 n 9/7 β(m, n) +n 2), in general, andO(m 3/4 n 3/4 β(m, n) +n) if no three spheres intersect in a common circle. The latter bound implies that the maximum number of unit-distances amongm points in three dimensions isO(m 3/2 β(m)) which improves the best previous upper bound on this problem. Applications of our results to other distance problems are also given. acknowledgement: The research of the second author was supported by the National Science Foundation under Grant CCR-8714565. Work by the fourth author has been supported by Office of Naval Research Grant N00014-87-K-0129, by National Science Foundation Grant No. NSF-DCR-83-20085, by grants from the Digital Equipment Corporation and the IBM Corporation, and by a research grant from the NCRD, the Israeli National Council for Research and Development. A preliminary version of this paper has appeared in theProceedings of the 29th IEEE Symposium on Foundations of Computer Science, 1988. article_processing_charge: No article_type: original author: - first_name: Kenneth full_name: Clarkson, Kenneth last_name: Clarkson - first_name: Herbert full_name: Edelsbrunner, Herbert id: 3FB178DA-F248-11E8-B48F-1D18A9856A87 last_name: Edelsbrunner orcid: 0000-0002-9823-6833 - first_name: Leonidas full_name: Guibas, Leonidas last_name: Guibas - first_name: Micha full_name: Sharir, Micha last_name: Sharir - first_name: Emo full_name: Welzl, Emo last_name: Welzl citation: ama: Clarkson K, Edelsbrunner H, Guibas L, Sharir M, Welzl E. Combinatorial complexity bounds for arrangements of curves and spheres. Discrete & Computational Geometry. 1990;5(1):99-160. doi:10.1007/BF02187783 apa: Clarkson, K., Edelsbrunner, H., Guibas, L., Sharir, M., & Welzl, E. (1990). Combinatorial complexity bounds for arrangements of curves and spheres. Discrete & Computational Geometry. Springer. https://doi.org/10.1007/BF02187783 chicago: Clarkson, Kenneth, Herbert Edelsbrunner, Leonidas Guibas, Micha Sharir, and Emo Welzl. “Combinatorial Complexity Bounds for Arrangements of Curves and Spheres.” Discrete & Computational Geometry. Springer, 1990. https://doi.org/10.1007/BF02187783. ieee: K. Clarkson, H. Edelsbrunner, L. Guibas, M. Sharir, and E. Welzl, “Combinatorial complexity bounds for arrangements of curves and spheres,” Discrete & Computational Geometry, vol. 5, no. 1. Springer, pp. 99–160, 1990. ista: Clarkson K, Edelsbrunner H, Guibas L, Sharir M, Welzl E. 1990. Combinatorial complexity bounds for arrangements of curves and spheres. Discrete & Computational Geometry. 5(1), 99–160. mla: Clarkson, Kenneth, et al. “Combinatorial Complexity Bounds for Arrangements of Curves and Spheres.” Discrete & Computational Geometry, vol. 5, no. 1, Springer, 1990, pp. 99–160, doi:10.1007/BF02187783. short: K. Clarkson, H. Edelsbrunner, L. Guibas, M. Sharir, E. Welzl, Discrete & Computational Geometry 5 (1990) 99–160. date_created: 2018-12-11T12:06:47Z date_published: 1990-03-01T00:00:00Z date_updated: 2022-02-17T15:41:04Z day: '01' doi: 10.1007/BF02187783 extern: '1' intvolume: ' 5' issue: '1' language: - iso: eng main_file_link: - url: https://link.springer.com/article/10.1007/BF02187783 month: '03' oa_version: None page: 99 - 160 publication: Discrete & Computational Geometry publication_identifier: eissn: - 1432-0444 issn: - 0179-5376 publication_status: published publisher: Springer publist_id: '2048' quality_controlled: '1' status: public title: Combinatorial complexity bounds for arrangements of curves and spheres type: journal_article user_id: ea97e931-d5af-11eb-85d4-e6957dddbf17 volume: 5 year: '1990' ... --- _id: '4078' abstract: - lang: eng text: In this paper we derived combinatorial point selection results for geometric objects defined by pairs of points. In a nutshell, the results say that if many pairs of a set of n points in some fixed dimension each define a geometric object of some type, then there is a point covered by many of these objects. Based on such a result for three-dimensional spheres we show that the combinatorial size of the Delaunay triangulation of a point set in space can be reduced by adding new points. We believe that from a practical point of view this is the most important result of this paper. article_processing_charge: No author: - first_name: Bernard full_name: Chazelle, Bernard last_name: Chazelle - first_name: Herbert full_name: Edelsbrunner, Herbert id: 3FB178DA-F248-11E8-B48F-1D18A9856A87 last_name: Edelsbrunner orcid: 0000-0002-9823-6833 - first_name: Leonidas full_name: Guibas, Leonidas last_name: Guibas - first_name: John full_name: Hershberger, John last_name: Hershberger - first_name: Raimund full_name: Seidel, Raimund last_name: Seidel - first_name: Micha full_name: Sharir, Micha last_name: Sharir citation: ama: 'Chazelle B, Edelsbrunner H, Guibas L, Hershberger J, Seidel R, Sharir M. Slimming down by adding; selecting heavily covered points. In: Proceedings of the 6th Annual Symposium on Computational Geometry. ACM; 1990:116-127. doi:10.1145/98524.98551' apa: 'Chazelle, B., Edelsbrunner, H., Guibas, L., Hershberger, J., Seidel, R., & Sharir, M. (1990). Slimming down by adding; selecting heavily covered points. In Proceedings of the 6th annual symposium on computational geometry (pp. 116–127). Berkley, CA, United States: ACM. https://doi.org/10.1145/98524.98551' chicago: Chazelle, Bernard, Herbert Edelsbrunner, Leonidas Guibas, John Hershberger, Raimund Seidel, and Micha Sharir. “Slimming down by Adding; Selecting Heavily Covered Points.” In Proceedings of the 6th Annual Symposium on Computational Geometry, 116–27. ACM, 1990. https://doi.org/10.1145/98524.98551. ieee: B. Chazelle, H. Edelsbrunner, L. Guibas, J. Hershberger, R. Seidel, and M. Sharir, “Slimming down by adding; selecting heavily covered points,” in Proceedings of the 6th annual symposium on computational geometry, Berkley, CA, United States, 1990, pp. 116–127. ista: 'Chazelle B, Edelsbrunner H, Guibas L, Hershberger J, Seidel R, Sharir M. 1990. Slimming down by adding; selecting heavily covered points. Proceedings of the 6th annual symposium on computational geometry. SCG: Symposium on Computational Geometry, 116–127.' mla: Chazelle, Bernard, et al. “Slimming down by Adding; Selecting Heavily Covered Points.” Proceedings of the 6th Annual Symposium on Computational Geometry, ACM, 1990, pp. 116–27, doi:10.1145/98524.98551. short: B. Chazelle, H. Edelsbrunner, L. Guibas, J. Hershberger, R. Seidel, M. Sharir, in:, Proceedings of the 6th Annual Symposium on Computational Geometry, ACM, 1990, pp. 116–127. conference: end_date: 1990-06-09 location: Berkley, CA, United States name: 'SCG: Symposium on Computational Geometry' start_date: 1990-06-07 date_created: 2018-12-11T12:06:48Z date_published: 1990-01-01T00:00:00Z date_updated: 2022-02-17T10:09:54Z day: '01' doi: 10.1145/98524.98551 extern: '1' language: - iso: eng main_file_link: - url: https://dl.acm.org/doi/10.1145/98524.98551 month: '01' oa_version: None page: 116 - 127 publication: Proceedings of the 6th annual symposium on computational geometry publication_identifier: isbn: - 978-0-89791-362-1 publication_status: published publisher: ACM publist_id: '2046' quality_controlled: '1' scopus_import: '1' status: public title: Slimming down by adding; selecting heavily covered points type: conference user_id: ea97e931-d5af-11eb-85d4-e6957dddbf17 year: '1990' ... --- _id: '4076' abstract: - lang: eng text: We present an algorithm to compute a Euclidean minimum spanning tree of a given set S of n points in Ed in time O(Td(N, N) logd N), where Td(n, m) is the time required to compute a bichromatic closest pair among n red and m blue points in Ed. If Td(N, N) = Ω(N1+ε), for some fixed ε > 0, then the running time improves to O(Td(N, N)). Furthermore, we describe a randomized algorithm to compute a bichromatic closets pair in expected time O((nm log n log m)2/3+m log2 n + n log2 m) in E3, which yields an O(N4/3log4/3 N) expected time algorithm for computing a Euclidean minimum spanning tree of N points in E3. article_processing_charge: No author: - first_name: Pankaj full_name: Agarwal, Pankaj last_name: Agarwal - first_name: Herbert full_name: Edelsbrunner, Herbert id: 3FB178DA-F248-11E8-B48F-1D18A9856A87 last_name: Edelsbrunner orcid: 0000-0002-9823-6833 - first_name: Otfried full_name: Schwarzkopf, Otfried last_name: Schwarzkopf - first_name: Emo full_name: Welzl, Emo last_name: Welzl citation: ama: 'Agarwal P, Edelsbrunner H, Schwarzkopf O, Welzl E. Euclidean minimum spanning trees and bichromatic closest pairs. In: Proceedings of the 6th Annual Symposium on Computational Geometry. ACM; 1990:203-210. doi:10.1145/98524.98567' apa: 'Agarwal, P., Edelsbrunner, H., Schwarzkopf, O., & Welzl, E. (1990). Euclidean minimum spanning trees and bichromatic closest pairs. In Proceedings of the 6th annual symposium on Computational geometry (pp. 203–210). Berkeley, CA, United States: ACM. https://doi.org/10.1145/98524.98567' chicago: Agarwal, Pankaj, Herbert Edelsbrunner, Otfried Schwarzkopf, and Emo Welzl. “ Euclidean Minimum Spanning Trees and Bichromatic Closest Pairs.” In Proceedings of the 6th Annual Symposium on Computational Geometry, 203–10. ACM, 1990. https://doi.org/10.1145/98524.98567. ieee: P. Agarwal, H. Edelsbrunner, O. Schwarzkopf, and E. Welzl, “ Euclidean minimum spanning trees and bichromatic closest pairs,” in Proceedings of the 6th annual symposium on Computational geometry, Berkeley, CA, United States, 1990, pp. 203–210. ista: 'Agarwal P, Edelsbrunner H, Schwarzkopf O, Welzl E. 1990. Euclidean minimum spanning trees and bichromatic closest pairs. Proceedings of the 6th annual symposium on Computational geometry. SCG: Symposium on Computational Geometry, 203–210.' mla: Agarwal, Pankaj, et al. “ Euclidean Minimum Spanning Trees and Bichromatic Closest Pairs.” Proceedings of the 6th Annual Symposium on Computational Geometry, ACM, 1990, pp. 203–10, doi:10.1145/98524.98567. short: P. Agarwal, H. Edelsbrunner, O. Schwarzkopf, E. Welzl, in:, Proceedings of the 6th Annual Symposium on Computational Geometry, ACM, 1990, pp. 203–210. conference: end_date: 1990-06-09 location: Berkeley, CA, United States name: 'SCG: Symposium on Computational Geometry' start_date: 1990-06-07 date_created: 2018-12-11T12:06:48Z date_published: 1990-01-01T00:00:00Z date_updated: 2022-02-16T15:30:22Z day: '01' doi: 10.1145/98524.98567 extern: '1' language: - iso: eng main_file_link: - url: https://dl.acm.org/doi/10.1145/98524.98567 month: '01' oa_version: None page: 203 - 210 publication: Proceedings of the 6th annual symposium on Computational geometry publication_identifier: isbn: - 978-0-89791-362-1 publication_status: published publisher: ACM publist_id: '2044' quality_controlled: '1' scopus_import: '1' status: public title: ' Euclidean minimum spanning trees and bichromatic closest pairs' type: conference user_id: ea97e931-d5af-11eb-85d4-e6957dddbf17 year: '1990' ... --- _id: '4077' abstract: - lang: eng text: We prove that for any set S of n points in the plane and n3-α triangles spanned by the points of S there exists a point (not necessarily of S) contained in at least n3-3α/(512 log25 n) of the triangles. This implies that any set of n points in three - dimensional space defines at most 6.4n8/3 log5/3 n halving planes. article_processing_charge: No author: - first_name: Boris full_name: Aronov, Boris last_name: Aronov - first_name: Bernard full_name: Chazelle, Bernard last_name: Chazelle - first_name: Herbert full_name: Edelsbrunner, Herbert id: 3FB178DA-F248-11E8-B48F-1D18A9856A87 last_name: Edelsbrunner orcid: 0000-0002-9823-6833 - first_name: Leonidas full_name: Guibas, Leonidas last_name: Guibas - first_name: Micha full_name: Sharir, Micha last_name: Sharir - first_name: Rephael full_name: Wenger, Rephael last_name: Wenger citation: ama: 'Aronov B, Chazelle B, Edelsbrunner H, Guibas L, Sharir M, Wenger R. Points and triangles in the plane and halving planes in space. In: Proceedings of the 6th Annual Symposium on Computational Geometry. ACM; 1990:112-115. doi:10.1145/98524.98548' apa: 'Aronov, B., Chazelle, B., Edelsbrunner, H., Guibas, L., Sharir, M., & Wenger, R. (1990). Points and triangles in the plane and halving planes in space. In Proceedings of the 6th annual symposium on Computational geometry (pp. 112–115). Berkley, CA, United States: ACM. https://doi.org/10.1145/98524.98548' chicago: Aronov, Boris, Bernard Chazelle, Herbert Edelsbrunner, Leonidas Guibas, Micha Sharir, and Rephael Wenger. “Points and Triangles in the Plane and Halving Planes in Space.” In Proceedings of the 6th Annual Symposium on Computational Geometry, 112–15. ACM, 1990. https://doi.org/10.1145/98524.98548. ieee: B. Aronov, B. Chazelle, H. Edelsbrunner, L. Guibas, M. Sharir, and R. Wenger, “Points and triangles in the plane and halving planes in space,” in Proceedings of the 6th annual symposium on Computational geometry, Berkley, CA, United States, 1990, pp. 112–115. ista: 'Aronov B, Chazelle B, Edelsbrunner H, Guibas L, Sharir M, Wenger R. 1990. Points and triangles in the plane and halving planes in space. Proceedings of the 6th annual symposium on Computational geometry. SCG: Symposium on Computational Geometry, 112–115.' mla: Aronov, Boris, et al. “Points and Triangles in the Plane and Halving Planes in Space.” Proceedings of the 6th Annual Symposium on Computational Geometry, ACM, 1990, pp. 112–15, doi:10.1145/98524.98548. short: B. Aronov, B. Chazelle, H. Edelsbrunner, L. Guibas, M. Sharir, R. Wenger, in:, Proceedings of the 6th Annual Symposium on Computational Geometry, ACM, 1990, pp. 112–115. conference: end_date: 1990-06-09 location: Berkley, CA, United States name: 'SCG: Symposium on Computational Geometry' start_date: 1990-06-07 date_created: 2018-12-11T12:06:48Z date_published: 1990-01-01T00:00:00Z date_updated: 2022-02-17T09:42:27Z day: '01' doi: 10.1145/98524.98548 extern: '1' language: - iso: eng main_file_link: - url: https://dl.acm.org/doi/10.1145/98524.98548 month: '01' oa_version: None page: 112 - 115 publication: Proceedings of the 6th annual symposium on Computational geometry publication_identifier: isbn: - 978-0-89791-362-1 publication_status: published publisher: ACM publist_id: '2045' quality_controlled: '1' scopus_import: '1' status: public title: Points and triangles in the plane and halving planes in space type: conference user_id: ea97e931-d5af-11eb-85d4-e6957dddbf17 year: '1990' ...