Motorcycle Graphs and Straight Skeletons

We present a new algorithm to compute motorcycle graphs. It runs in $O(n \sqrt{n}\log n)$ time when n is the number of motorcycles. We give a new characterization of the straight skeleton of a nondegenerate polygon. For a polygon with n vertices and h holes, we show that it yields a randomized algorithm that reduces the straight skeleton computation to a motorcycle graph computation in expected $O(n\sqrt{h+1}\log^2 n)$ time. Combining these results, we can compute the straight skeleton of a nondegenerate polygon with h holes and with n vertices, among which r are reflex vertices, in $O(n\sqrt{h+1}\log^2 n+r \sqrt{r} \log r)$ expected time. In particular, we cancompute the straight skeleton of a nondegenerate polygon with n vertices in $O(n\sqrt{n}\log^2n)$ expected time.