Lower Bounds for Geometric Diameter Problems
نویسندگان
چکیده
The diameter of a set P of n points in R 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 R 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 R to the diameter problem for a point set in R.
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