Abstract
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.
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Cheng, SW., Vigneron, A. Motorcycle Graphs and Straight Skeletons. Algorithmica 47, 159–182 (2007). https://doi.org/10.1007/s00453-006-1229-7
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DOI: https://doi.org/10.1007/s00453-006-1229-7