Halving Balls in Deterministic Linear Time

Let D be a set of n pairwise disjoint unit balls in R and P the set of their center points. A hyperplane H is an m-separator for D if each closed halfspace bounded by H contains at leastm points from P. This generalizes the notion of halving hyperplanes, which correspond to n/2-separators. The analogous notion for point sets has been well studied. Separators have various applications, for instance, in divide-and-conquer schemes. In such a scheme any ball that is intersected by the separating hyperplane may still interact with both sides of the partition. Therefore it is desirable that the separating hyperplane intersects a small number of balls only. We present three deterministic algorithms to bisect or approximately bisect a given set of disjoint unit balls by a hyperplane: Firstly, we present a simple linear-time algorithm to construct an αn-separator for balls in R, for any 0 < α < 1/2, that intersects at most cn balls, for some constant c that depends on d and α. The number of intersected balls is best possible up to the constant c. Secondly, we present a near-linear time algorithm to construct an (n/2 − o(n))-separator in R that intersects o(n) balls. Finally, we give a linear-time algorithm to construct a halving line in R that intersects O(n) disks. Our results improve the runtime of a disk sliding algorithm by Bereg, Dumitrescu and Pach. In addition, our results improve and derandomize an algorithm to construct a space decomposition used by Löffler and Mulzer to construct an onion (convex layer) decomposition for imprecise points (any point resides at an unknown location within a given disk). Figure 1: A set of 18 disks in R and three separators. The dashed line forms a 6-separator. Both the solid line and the dotted line are halving lines. The solid line is preferable to the other two lines because it separates perfectly and intersects no disks. ar X iv :1 40 5. 18 94 v1 [ cs .C G ] 8 M ay 2 01 4