TY - CONF

T1 - Counting and representing intersections among triangles in three dimensions

AU - Ezra, Esther

AU - Sharir, Micha

N1 - Funding Information:
Keywords: Triangles in three dimensions; Curve-sensitive cuttings; Counting intersections; Arrangements; 3SUM-hard problems ✩ Work on this paper has been supported by NSF Grants CCR-97-32101 and CCR-00-98246, by a grant from the US–Israeli Binational Science Foundation, by a grant from the Israel Science Fund, Israeli Academy of Sciences, for a Center of Excellence in Geometric Computing at Tel Aviv University, and by the Hermann Minkowski–MINERVA Center for Geometry at Tel Aviv University. A preliminary version of this paper has appeared in Proc. 20th Annu. ACM Sympos. Comput. Geom., 2004, pp. 210– 219. * Corresponding author. E-mail addresses: estere@post.tau.ac.il (E. Ezra), michas@post.tau.ac.il (M. Sharir).

PY - 2004

Y1 - 2004

N2 - We present an algorithm that efficiently counts all intersecting triples among a collection T of triangles in ℝ3 in nearly-quadratic time. This solves a problem posed by Pellegrini. Using a variant of the technique, one can represent the set of all κ triple intersections, in compact form, as the disjoint union of complete tripartite hypergraphs, which requires nearly-quadratic construction time and storage. Our approach also applies to any collection of convex planar objects of constant description complexity in ℝ3, with the same performance bounds. We also prove that this counting problem belongs to the 3SUM-hard family, and thus our algorithm is likely to be nearly optimal (since it is believed that 3SUM-hard problems cannot be solved in subquadratic time).

AB - We present an algorithm that efficiently counts all intersecting triples among a collection T of triangles in ℝ3 in nearly-quadratic time. This solves a problem posed by Pellegrini. Using a variant of the technique, one can represent the set of all κ triple intersections, in compact form, as the disjoint union of complete tripartite hypergraphs, which requires nearly-quadratic construction time and storage. Our approach also applies to any collection of convex planar objects of constant description complexity in ℝ3, with the same performance bounds. We also prove that this counting problem belongs to the 3SUM-hard family, and thus our algorithm is likely to be nearly optimal (since it is believed that 3SUM-hard problems cannot be solved in subquadratic time).

KW - 3SUM-hard problems

KW - Arrangements

KW - Counting intersections

KW - Curve-sensitive cuttings

KW - Triangles in three dimensions

UR - http://www.scopus.com/inward/record.url?scp=4544313323&partnerID=8YFLogxK

U2 - 10.1145/997817.997851

DO - 10.1145/997817.997851

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AN - SCOPUS:4544313323

SP - 210

EP - 219

Y2 - 9 June 2004 through 11 June 2004

ER -