A Faster Algorithm for Computing Straight Skeletons

TL;DR

This paper introduces a new deterministic algorithm for computing polygon straight skeletons that improves efficiency by reducing the problem to motorcycle graph computation with better time bounds.

Contribution

The paper presents a faster, deterministic algorithm for straight skeletons that improves upon previous randomized methods by reducing complexity to $O(n (\log n)\log r)$ time.

Findings

Deterministic algorithm reduces computation time

Improved bounds for polygons with holes

Applicable to degenerate and non-degenerate polygons

Abstract

We present a new algorithm for computing the straight skeleton of a polygon. For a polygon with $n$ vertices, among which $r$ are reflex vertices, we give a deterministic algorithm that reduces the straight skeleton computation to a motorcycle graph computation in $O(n (\log n)\log r)$ time. It improves on the previously best known algorithm for this reduction, which is randomized, and runs in expected $O(n \sqrt{h+1}\log^2 n)$ time for a polygon with $h$ holes. Using known motorcycle graph algorithms, our result yields improved time bounds for computing straight skeletons. In particular, we can compute the straight skeleton of a non-degenerate polygon in $O(n (\log n) \log r + r^{4/3+\varepsilon})$ time for any $\varepsilon>0$. On degenerate input, our time bound increases to $O(n (\log n) \log r + r^{17/11+\varepsilon})$.