The complexity and construction of many faces in arrangements of lines and of segments
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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) log2 n logm).
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- The complexity and construction of many faces in arrangements of lines and of segments
Discrete & Computational Geometry
Volume 5, Issue 1 , pp 161-196
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- 1. Department of Computer Science, University of Illinois at Urbana-Champaign, 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