Online Hitting Sets for Disks of Bounded Radii

We present algorithms for the online minimum hitting set problem in geometric range spaces: given a set $P$ of $n$ points in the plane and a sequence of geometric objects that arrive one-by-one, we need to maintain a hitting set at all times by making irrevocable decisions. For disks of radii in the interval $[1,M]$, we present an $O(\log M \log n)$-competitive algorithm. This result generalizes from disks to positive homothets of any convex body in the plane with scaling factors in the interval $[1,M]$. As a main technical tool, we reduce the problem to the online hitting set problem for a finite subset of integer points and geometric objects with the lowest point property, introduced in this paper, which behave similarly to bottomless rectangles. Specifically, for a given $N>1$, we present an $O(\log N)$-competitive algorithm for the variant where $P$ is a subset of an $N\times N$ section of the integer lattice, and the geometric objects have the lowest point property.