Dynamic Maintenance of the Lower Envelope of Pseudo-Lines

TL;DR

This paper introduces a dynamic data structure for efficiently maintaining the lower envelope of pseudo-lines, supporting fast updates and queries, with a novel intersection-finding algorithm using tentative binary search.

Contribution

It presents a new fully dynamic data structure with optimal update and query times for lower envelopes of pseudo-lines, including a novel intersection algorithm.

Findings

O(log^2 n) update time

O(log n) query time

O(n) space complexity

Abstract

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 $x=-\infty$ 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.