The complexity and construction of many faces in arrangements of lines and of segments

We show that the total number of edges ofm faces of an arrangement ofn lines in the plane isO(m2/3−δn2/3+2δ+n) for anyδ>0. The proof takes an algorithmic approach, that is, we describe an algorithm for the calculation of thesem faces and derive the upper bound from the analysis of the algorithm. The algorithm uses randomization and its expected time complexity isO(m2/3−δn2/3+2δ logn+n logn logm). If instead of lines we have an arrangement ofn line segments, then the maximum number of edges ofm faces isO(m2/3−δn2/3+2δ+nα (n) logm) for anyδ>0, whereα(n) is the functional inverse of Ackermann's function. We give a (randomized) algorithm that produces these faces and takes expected timeO(m2/3−δn2/3+2δ log+nα(n) log2n logm).

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