A Faster Algorithm for Computing Motorcycle Graphs

Abstract: We present a new algorithm for computing motorcycle graphs that runs in O(n^4/3+e) time for any e>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(n.sqrt(h+1)log^2(n)+n^4/3+e) expected time. If all input coordinates are O(log n)-bit rational numbers, we can compute the straight skeleton of a (possibly degenerate) polygon with h holes in O(n.sqrt(h+1)log^3(n)) expected time. In particular, it means that we can compute the straight skeleton of a simple polygon in O(n.log^3(n)) expected time if all input coordinates are O(\log n)-bit rationals, while all previously known algorithms have worst-case running time larger than n^3/2.