# A faster algorithm for computing motorcycle graphs

A. Vigneron, L. Yan

Discrete & Computational Geometry, 52(3), 492-514, (2014)

## Keywords

Algorithms design and analysis, Motorcycle graph, Straight skeleton ,Medial axis, Polygon

## Abstract

We present a new algorithm for computing motorcycle graphs that runs in O(n4/3+ε) time for any ε>0, improving on all previously known algorithms. The main application of this result is to computing the straight skeleton of a polygon. It allows us to compute the straight skeleton of a non-degenerate polygon with h holes in O(nh+1−−−−−√log2n+n4/3+ε) expected time. If all input coordinates are O(logn)-bit rational numbers, we can compute the straight skeleton of a (possibly degenerate) polygon with h holes in O(nh+1−−−−−√log3n) expected time. In particular, it means that we can compute the straight skeleton of a simple polygon in O(nlog3n) expected time if all input coordinates are O(logn)-bit rationals, while all previously known algorithms have worst-case running time ω(n3/2).

## Code

DOI: 10.1007/s00454-014-9625-2