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

We show that the total number of edges of m faces of an arrangement of n lines in the plane is O(m^2/3-δn^{2/3+2 δ}+n) for any δ>0. The proof takes an algorithmic approach, that is, we describe an algorithm for the calculation of these m faces and derive the upper bound from the analysis of the algorithm. The algorithm uses randomization and its expected time complexity is O(m^2/3-δn^{2/3+2 δ} log n+n log n log m). If instead of lines we have an arrangement of n line segments, then the maximum number of edges of m faces is O(m^2/3-δn^{2/3+2 δ}+nα (n) log m) for any δ>0, where α(n) is the functional inverse of Ackermann's function. We give a (randomized) algorithm that produces these faces and takes expected time O(m^2/3-δn^{2/3+2 δ} log+nα(n) log²n log m).