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

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DOI: 10.1007/BF02187784

- Cite this article as:
- Edelsbrunner, H., Guibas, L.J. & Sharir, M. Discrete Comput Geom (1990) 5: 161. doi:10.1007/BF02187784

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## Abstract

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−δ}*n*^{2/3+2δ}+*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−δ}*n*^{2/3+2δ} 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−δ}*n*^{2/3+2δ}+*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−δ}*n*^{2/3+2δ} log+*nα*(*n*) log^{2}*n* log*m*).