Intersection of Unit-balls and Diameter of a Point Set in 3

We describe an algorithm for computing the intersection of n balls of equal radius in IR 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 IR 3 (the maximum distance between any pair) which runs in time O(n lg 5 n). The algorithms are deterministic and elementary; this is in contrast with the running time O(n logn) in both cases that can be achieved using randomization 3], and the running times O(n lg n) and O(n lg 3 n) using deterministic geometric sampling 2, 1].