  Let $\D$ be a set of $n$ pairwise disjoint unit balls in $\R^d$ and $P$ the
set of their center points. A hyperplane $\Hy$ is an \emph{$m$-separator} for
$\D$ if each closed halfspace bounded by $\Hy$ contains at least $m$ 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 $\alpha n$-separator for balls
in $\R^d$, for any $0<\alpha<1/2$, that intersects at most $cn^{(d-1)/d}$
balls, for some constant $c$ that depends on $d$ and $\alpha$. 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 $\R^d$
that intersects $o(n)$ balls. Finally, we give a linear-time algorithm to
construct a halving line in $\R^2$ that intersects $O(n^{(5/6)+\epsilon})$
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 L{\"o}ffler and Mulzer to
construct an onion (convex layer) decomposition for imprecise points (any point
resides at an unknown location within a given disk).
