Reporting and counting segment intersections
Journal of Computer and System Sciences
An optimal-time algorithm for slope selection
SIAM Journal on Computing
SIAM Journal on Computing
Applications of random sampling in computational geometry, II
Discrete & Computational Geometry - Selected papers from the fourth ACM symposium on computational geometry, Univ. of Illinois, Urbana-Champaign, June 6 8, 1988
Partitioning arrangements of lines, part II: applications
Discrete & Computational Geometry
Cutting hyperplane arrangements
Discrete & Computational Geometry
Randomized optimal algorithm for slope selection
Information Processing Letters
An optimal algorithm for intersecting line segments in the plane
Journal of the ACM (JACM)
Counting circular arc intersections
SIAM Journal on Computing
Optimal slope selection via expanders
Information Processing Letters
On a class of O(n2) problems in computational geometry
Computational Geometry: Theory and Applications
Davenport-Schinzel sequences and their geometric applications
Davenport-Schinzel sequences and their geometric applications
New Lower Bounds for Convex Hull Problems in Odd Dimensions
SIAM Journal on Computing
Lectures on Discrete Geometry
Reporting intersecting pairs of convex polytopes in two and three dimensions
Computational Geometry: Theory and Applications
Output-sensitive construction of the union of triangles
SODA '04 Proceedings of the fifteenth annual ACM-SIAM symposium on Discrete algorithms
SIAM Journal on Computing
Algorithms for Reporting and Counting Geometric Intersections
IEEE Transactions on Computers
Output-sensitive construction of the union of triangles
SODA '04 Proceedings of the fifteenth annual ACM-SIAM symposium on Discrete algorithms
On regular vertices on the union of planar objects
SCG '07 Proceedings of the twenty-third annual symposium on Computational geometry
On the union of fat tetrahedra in three dimensions
Journal of the ACM (JACM)
Lines avoiding balls in three dimensions revisited
Proceedings of the twenty-sixth annual symposium on Computational geometry
Cuttings for disks and axis-aligned rectangles
WADS'07 Proceedings of the 10th international conference on Algorithms and Data Structures
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We present an algorithm that efficiently counts all intersecting triples among a collection T of triangles in R^3 in nearly quadratic time. This solves a problem posed by Pellegrini [M. Pellegrini, On counting pairs of intersecting segments and off-line triangle range searching, Algorithmica 17 (1997) 380-398]. Using a variant of the technique, one can represent the set of all @k 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 planar objects of constant description complexity in R^3, 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 in the worst case.