Computational geometry: an introduction
Computational geometry: an introduction
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
Farthest neighbors, maximum spanning trees and related problems in higher dimensions
Computational Geometry: Theory and Applications
A practical approach for computing the diameter of a point set
SCG '01 Proceedings of the seventeenth annual symposium on Computational geometry
Computing farthest neighbors on a convex polytope
Theoretical Computer Science - Computing and combinatorics
Lower bounds for algebraic computation trees
STOC '83 Proceedings of the fifteenth annual ACM symposium on Theory of computing
Algebraic Complexity Theory
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The diameter of a set P of n points in ${\mathbb R}^d$ is the maximum Euclidean distance between any two points in P. If P is the vertex set of a 3–dimensional convex polytope, and if the combinatorial structure of this polytope is given, we prove that, in the worst case, deciding whether the diameter of P is smaller than 1 requires Ω(n log n) time in the algebraic computation tree model. It shows that the O(n log n) time algorithm of Ramos for computing the diameter of a point set in ${\mathbb R}^3$ is optimal for computing the diameter of a 3–polytope. We also give a linear time reduction from Hopcroft's problem of finding an incidence between points and lines in ${\mathbb R}^2$ to the diameter problem for a point set in ${\mathbb R}^7$.