Faster Reductions for Straight Skeletons to Motorcycle Graphs

TL;DR

This paper presents an improved algorithm that reduces the straight skeleton computation to the motorcycle graph problem in near-linear time, significantly speeding up the process for polygons and planar straight line graphs.

Contribution

It introduces a faster reduction algorithm for straight skeletons to motorcycle graphs, improving previous complexity bounds for polygons and PSLGs.

Findings

Achieves $O(n ext{log}n)$ time for simple polygons.

Achieves $O(n( ext{log}n) ext{log}m)$ time for PSLGs.

Provides substantial speedups over previous algorithms.

Abstract

We give an algorithm that reduces the straight skeleton to the motorcycle graph in $O(n\log n)$ time for simple polygons and $O(n(\log n)\log m)$ time for a planar straight line graph (PSLG) with $m$ connected components. This improves on the previous best of $O(n(\log n)\log r)$ for polygons with $r$ reflex vertices (possibly with holes) and $O(n^2\log n)$ for general planar straight line graphs. This allows us to speed up the straight skeleton algorithm for polygons and PSLGs. For a polygon with $h$ holes and $r$ reflex vertices we achieve a speedup from $O(n(\log n)\log r + r^{4/3+\epsilon})$ time to $O(n(\log n)\log h + r^{4/3 + \epsilon})$ time in the non-degenerate case and from $O(n(\log n)\log r + r^{17/11 + \epsilon})$ to $O(n(\log n)\log h + r^{17/11 + \epsilon})$ in degenerate cases. For a PSLG with $m$ connected components and $r$ reflex vertices, we gain a speed up from $O(n^{1 + \epsilon} + n^{8/11 + \epsilon}r^{9/11+\epsilon})$ to $O(n(\log n)\log m + r^{4/3 + \epsilon})$ in the non-degenerate case and from $O(n^{1 + \epsilon} + n^{8/11 + \epsilon}r^{9/11+\epsilon})$ to $O(n(\log n)\log m + r^{17/11 + \epsilon})$ in the degenerate case.