10.1007/BF02187784
Herbert Edelsbrunner
Herbert
Edelsbrunner0000-0002-9823-6833
Guibas, Leonidas J
Leonidas
Guibas
Sharir, Micha
Micha
Sharir
The complexity and construction of many faces in arrangements of lines and of segments
Springer
1990
2018-12-11T12:06:46Z
2019-04-26T07:22:39Z
journal_article
https://research-explorer.app.ist.ac.at/record/4072
https://research-explorer.app.ist.ac.at/record/4072.json
We show that the total number of edges ofm faces of an arrangement ofn lines in the plane isO(m 2/3– n 2/3+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(m 2/3– n 2/3+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(m 2/3– n 2/3+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(m 2/3– n 2/3+2 log+n(n) log2 n logm).