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

Published: 01 December 1990 Publication History

Abstract

We show that the total number of edges ofm faces of an arrangement ofn lines in the plane isO(m2/3 n2/3+2 +n) for any >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 n2/3+2 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 n2/3+2 +n (n) logm) for any >0, where (n) is the functional inverse of Ackermann's function. We give a (randomized) algorithm that produces these faces and takes expected timeO(m2/3 n2/3+2 log+n (n) log2n logm).

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      cover image Discrete & Computational Geometry
      Discrete & Computational Geometry  Volume 5, Issue 2
      March 1990
      119 pages

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      Springer-Verlag

      Berlin, Heidelberg

      Publication History

      Published: 01 December 1990

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