Maintaining the Union of Unit Discs under Insertions with Near-Optimal Overhead.

We present efficient data structures for problems on unit discs and arcs of their boundary in the plane. (i) We give an output-sensitive algorithm for the dynamic maintenance of the union of $n$ unit discs under insertions in $O(k \log^2 n)$ update time and $O(n)$ space, where $k$ is the combinatorial complexity of the structural change in the union due to the insertion of the new disc. (ii) As part of the solution of (i) we devise a fully dynamic data structure for the maintenance of lower envelopes of pseudo-lines, which we believe is of independent interest. The structure has $O(\log^2 n)$ update time and $O(\log n)$ vertical ray shooting query time. To achieve this performance, we devise a new algorithm for finding the intersection between two lower envelopes of pseudo-lines in $O(\log n)$ time, using \emph{tentative} binary search; the lower envelopes are special in that at $x=-\infty$ any pseudo-line contributing to the first envelope lies below every pseudo-line contributing to the second envelope. (iii) We also present a dynamic range searching structure for a set of circular arcs of unit radius (not necessarily on the boundary of the union of the corresponding discs), where the ranges are unit discs, with $O(n \log n)$ preprocessing time, $O(n^{1/2+\varepsilon} + \ell)$ query time and $O(\log^2 n)$ amortized update time, where $\ell$ is the size of the output and for any $\varepsilon>0$. The structure requires $O(n)$ storage space.