The number of edges of many faces in a line segment arrangement

We show that the maximum number of edges boundingm faces in an arrangement ofn line segments in the plane isO(m2/3n2/3+nα(n)+nlogm). This improves a previous upper bound of Edelsbrunner et al. [5] and almost matches the best known lower bound which is Ω(m2/3n2/3+nα(n)). In addition, we show that the number of edges bounding anym faces in an arrangement ofn line segments with a total oft intersecting pairs isO(m2/3t1/3+nα(t/n)+nmin{logm,logt/n}), almost matching the lower bound of Ω(m2/3t1/3+nα(t/n)) demonstrated in this paper.

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