Construction of 1-d lower envelopes and applications
SCG '97 Proceedings of the thirteenth annual symposium on Computational geometry
Deterministic algorithms for 3-D diameter and some 2-D lower envelopes
Proceedings of the sixteenth annual symposium on Computational geometry
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We describe an algorithm for computing the intersection of $n$ balls of equal radius in $\RR^3$ which runs in time $O(n\lg^2 n)$. The algorithm can be parallelized so that the comparisons that involve the radius of the balls are performed in $O(\lg^3 n)$ batches. Using parametric search, these algorithms are used to obtain an algorithm for computing the diameter of a set of $n$ points in $\RR^3$ (the maximum distance between any pair) which runs in time $O(n\lg^5 n)$. The algorithms are deterministic and elementary. For both problems there are $\Omega(n\lg n)$ lower bounds which are matched by randomized algorithms.