A Simple Dynamization of Trapezoidal Point Location in Planar Subdivisions

TL;DR

This paper introduces a method to dynamically update the Trapezoidal Search Tree for planar subdivisions, enabling efficient online point location with practical implementation and experimental validation.

Contribution

It extends the Trapezoidal Search Tree to a dynamic setting, maintaining linear size and logarithmic query time, with an open-source implementation and experimental results.

Findings

Expected update time is O(log^2|S|) per operation.

The method maintains linear size of the data structure.

Experimental results demonstrate practical efficiency.

Abstract

We study how to dynamize the Trapezoidal Search Tree - a well known randomized point location structure for planar subdivisions of kinetic line segments. Our approach naturally extends incremental leaf-level insertions to recursive methods and allows adaptation for the online setting. Moreover, the dynamization carries over to the Trapezoidal Search DAG, offering a linear sized data structure with logarithmic point location costs as a by-product. On a set $S$ of non-crossing segments, each update performs expected ${\mathcal O}(\log^2|S|)$ operations. We demonstrate the practicality of our method with an open-source implementation, based on the Computational Geometry Algorithms Library, and experiments on the update performance.