Optimal Window Queries on Line Segments using the Trapezoidal Search DAG

TL;DR

This paper introduces an efficient method for performing vertical and window segment queries on line segments using the Trapezoidal Search DAG, achieving optimal or improved expected query and construction times.

Contribution

It presents a simple DFS-based approach for trapezoid intersection queries with optimal expected time and improved construction time for certain graph classes.

Findings

Query time is O(k + log n) expected, matching the best known bounds.

Construction time for certain graphs is improved to O(n log* n).

The method extends to axis-aligned window queries with efficient expected performance.

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.