A Faster Algorithm for Computing Motorcycle Graphs

Abstract

We present a new algorithm for computing motorcycle graphs that runs in $$O(n^{4/3+\varepsilon })$$O(n4/3+?) time for any $$\varepsilon >0$$?>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 $$h$$h holes in $$O(n \sqrt{h+1} \log ^2 n+n^{4/3+\varepsilon })$$O(nh+1log2n+n4/3+?) expected time. If all input coordinates are $$O(\log n)$$O(logn)-bit rational numbers, we can compute the straight skeleton of a (possibly degenerate) polygon with $$h$$h holes in $$O(n \sqrt{h+1}\log ^3 n)$$O(nh+1log3n) expected time. In particular, it means that we can compute the straight skeleton of a simple polygon in $$O(n\log ^3n)$$O(nlog3n) expected time if all input coordinates are $$O(\log n)$$O(logn)-bit rationals, while all previously known algorithms have worst-case running time $$\omega (n^{3/2})$$?(n3/2).