Motorcycle graphs and straight skeletons

We present a new algorithm to compute a motorcycle graph. It runs in O(nrootn log n) time when n is the size of the input. We give a new characterization of the straight skeleton of a polygon possibly with holes. For a simple polygon, we show that it yields a randomized algorithm that reduces the straight skeleton computation to a motorcycle graph computation in 0(n log(2) n) time. Combining these results, we can compute the straight skeleton of a non-degenerate simple polygon with n vertices, among which r are reflex vertices, in O(n log(2) n + rrootr log r) expected time. For a degenerate simple polygon, our expected time bound becomes O(n log(2) n + r (17/11+epsilon))