Optimal Window Queries on Line Segments using the Trapezoidal Search DAG

We propose new query applications of the well known randomized incremental construction of the Trapezoidal Search DAG (TSD) on a set of $n$ line segments in the plane, where queries are allowed to be any axis aligned window. We show that our algorithm reports the $m$ trapezoids that are intersected by the query in $\mathcal{O}(m+\log n)$ expected time, regardless of the spatial location of the segment set and the query. In case the query is a {\em vertical segment}, the query time bound reduces to $\mathcal{O}(k +\log n)$ where $k$ is the number of segments that are intersected. This improves on the query and space bound of the well known Segment Tree based approach, which is to date the theoretical bottleneck for optimal query time. In the case where the set of segments is a connected planar subdivision, this method can easily be extended to an algorithm which reports the $k$ segments which intersect an axis aligned query window in $\mathcal{O}(k + \log n)$ expected time. Our publicly available implementation handles degeneracies exactly, including segments with overlap and multi-intersections. Experiments show that the method is practical and provides more reliable query times in comparison to R-trees and the segment tree based data structure on real-world and synthetic data sets.