The 2-Center Problem in Three Dimensions

TL;DR

This paper introduces two randomized algorithms for solving the 2-center problem in three-dimensional space, aiming to find two congruent balls covering a set of points with minimal radius, with varying efficiency depending on the configuration.

Contribution

The paper presents novel randomized algorithms for the 2-center problem in 3D, improving computational efficiency based on the relative positions of the covering balls.

Findings

First algorithm runs in O(n^3 log^5 n) expected time.

Second algorithm runs faster when the 2-centers are not too close to each other.

The second algorithm's efficiency depends on the ratio of the radii and the proximity of the centers.

Abstract

Let P be a set of n points in R^3. The 2-center problem for P is to find two congruent balls of minimum radius whose union covers P. We present two randomized algorithms for computing a 2-center of P. The first algorithm runs in O(n^3 log^5 n) expected time, and the second algorithm runs in O((n^2 log^5 n) /(1-r*/r_0)^3) expected time, where r* is the radius of the 2-center balls of P and r_0 is the radius of the smallest enclosing ball of P. The second algorithm is faster than the first one as long as r* is not too close to r_0, which is equivalent to the condition that the centers of the two covering balls be not too close to each other.