Optimal Window Queries on Line Segments using the Trapezoidal Search DAG

Abstract

We propose a new query application for the well-known Trapezoidal Search DAG (TSD) of a set of $n$~line segments in the plane, where queries are allowed to be {\em vertical line segments}. We show that a simple Depth-First Search reports the $k$ trapezoids that are intersected by the query segment in $O(k+\log n)$ expected time, regardless of the spatial location of the query. This bound is optimal and matches known data structures with $O(n)$ size. In the important case of edges from a connected, planar graph, our simplistic approach yields an expected $O(n \log^*\!n)$ construction time, which improves on the construction time of known structures for vertical segment-queries. Also for connected input, a simple extension allows the TSD approach to directly answer axis-aligned window-queries in $O(k + \log n)$ expected time, where $k$ is the result size.