Optimal Window Queries on Line Segments using the Trapezoidal Search DAG
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We propose new query applications of the well-known Trapezoidal Search DAG (TSD) on a set of n line segments in the plane, where queries are allowed to be vertical line segments. We show that our algorithm 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 segment set and the query. This improves on the query time 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 report the k segments which intersect an axis aligned query window in O(k + log n) expected time. Our publicly available implementation handles degeneracies exactly, including segments with overlap and multi-intersections. Experiments on real and synthetic data sets show that the method is practical and provides more reliable query times in comparison to R-trees and the segment tree based data structure.
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