Approximating the lambda-low-density value

TL;DR

This paper presents an efficient method to approximate the lambda-low-density value for line segment sets, enabling better algorithm selection in geometric applications, with practical validation on real-world data.

Contribution

It introduces a 3-approximation algorithm for lambda-low-density value computation and a dynamic maintenance method with efficient updates.

Findings

The approximation algorithm runs in O(n log n + λ n) time.

The dynamic maintenance algorithm operates in O(log n + λ^2) amortized time per update.

Many real-world datasets have small lambda-low-density values, justifying specialized algorithms.

Abstract

The use of realistic input models has gained popularity in the theory community. Assuming a realistic input model often precludes complicated hypothetical inputs, and the analysis yields bounds that better reflect the behaviour of algorithms in practice. One of the most popular models for polygonal curves and line segments is $\lambda$-low-density. To select the most efficient algorithm for a certain input, one often needs to compute the $\lambda$-low-density value, or at least an approximate value. In this paper, we show that given a set of $n$ line segments in $\mathbb{R}^2$ one can compute a $3$-approximation of the $\lambda$-low density value in $O(n \log n + \lambda n)$ time. We also show how to maintain a $3$-approximation of the $\lambda$-low density value while allowing insertions of new segments in $O(\log n + \lambda^2)$ amortized time per update. Finally, we argue that many real-world data sets have a small $\lambda$-low density value, warranting the recent development of specialised algorithms. This is done by computing approximate $\lambda$-low density values for $12$ real-world data sets.