The complexity and construction of many faces in arrangements of lines and of segments
 Herbert Edelsbrunner,
 Leonidas J. Guibas,
 Micha Sharir
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Abstract
We show that the total number of edges ofm faces of an arrangement ofn lines in the plane isO(m ^{2/3−δ } n ^{2/3+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(m ^{2/3−δ } n ^{2/3+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(m ^{2/3−δ } n ^{2/3+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(m ^{2/3−δ } n ^{2/3+2δ } log+nα(n) log^{2} n logm).
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 Title
 The complexity and construction of many faces in arrangements of lines and of segments
 Journal

Discrete & Computational Geometry
Volume 5, Issue 1 , pp 161196
 Cover Date
 19901201
 DOI
 10.1007/BF02187784
 Print ISSN
 01795376
 Online ISSN
 14320444
 Publisher
 SpringerVerlag
 Additional Links
 Topics
 Industry Sectors
 Authors

 Herbert Edelsbrunner ^{(1)}
 Leonidas J. Guibas ^{(2)} ^{(3)}
 Micha Sharir ^{(4)} ^{(5)}
 Author Affiliations

 1. Department of Computer Science, University of Illinois at UrbanaChampaign, 61801, Urbana, Ill, USA
 2. DEC Systems Research Center, 130 Lytton Avenue Palo Alto, 94301, Ca, USA
 3. Department of Computer Science, Stanford University, 94305, Ca, USA
 4. Courant Institute of Mathematical Sciences, New York University, 10012, New York, NY, USA
 5. School of Mathematical Science, Tel Aviv University, 69978, Tel Aviv, Israel