Dynamic Maintenance of the Lower Envelope of Pseudo-Lines

We present a fully dynamic data structure for the maintenance of lower envelopes of pseudo-lines. 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 $x=-\mathrm{\infty }$ any pseudo-line contributing to the first envelope lies below every pseudo-line contributing to the second envelope. The structure requires $O(n)$ storage space.