Three Approaches to Building Time-Windowed Geometric Data Structures
University of Waterloo

Given a set of geometric objects (points or line segments) each associated with a time value, we wish to determine whether a given property is true for a subset of those objects whose time values fall within a query time window. We call such problems time-windowed decision problems. We present algorithms to preprocess for the time-windowed closest pair decision problem in $O(n)$ expected time, for the time-windowed 2D diameter decision problem in $O(n \log n)$ time, the time-windowed 2D convex hull area decision problem in $O(n\, \alpha(n) \log n)$ time (where $\alpha$ is the inverse Ackermann function), and the time-windowed 3D diameter decision and orthogonal segment intersection detection problems in $O(n \operatorname{polylog} n)$ time.

Our first approach is to reduce the closest pair decision problem to 2D dominance range emptiness using grids to compute candidate satisfying pairs. We extend this approach to find the closest pair of points by reducing the problem to 2D dominance range minimum, which we further reduce to 2D point location.

Our second approach is to reduce time-windowed decision problems to a generalized range successor problem, which we solve using a novel way to search range trees.

Our third approach is to use dynamic data structures directly, taking advantage of a new observation that the total number of combinatorial changes to a planar convex hull is near linear for any FIFO update sequence, in which deletions occur in the same order as insertions.