Redondo Beach, California
Nov. 12, 2000 to Nov. 14, 2000
T.M. Chan , Dept. of Comput. Sci., Waterloo Univ., Ont., Canada
Analyzing the worst-case complexity of the k-level in a planar arrangement of n curves is a fundamental problem in combinatorial geometry. We give the first subquadratic upper bound (roughly O(nk/sup 1-2/3/*)) for curves that are graphs of polynomial functions of an arbitrary fixed degree s. Previously, nontrivial results were known only for the case s=1 and s=2. We also improve the earlier bound for pseudo-parabolas (curves that pairwise intersect at most twice) to O(nk/sup 7/9/log/sup 2/3/ k). The proofs are simple and rely on a theorem of Tamaki and Tokuyama on cutting pseudo-parabolas into pseudo-segments, as well as a new observation for cutting pseudo-segments into pieces that can be extended to pseudo-lines. We mention applications to parametric and kinetic minimum spanning trees.
computational complexity; computational geometry; polynomials; worst-case complexity; planar arrangement; combinatorial geometry; subquadratic upper bound; polynomial functions; pseudo-parabolas; pseudo-segments; kinetic minimum spanning trees
T.M. Chan, "On levels in arrangements of curves", FOCS, 2000, 2013 IEEE 54th Annual Symposium on Foundations of Computer Science, 2013 IEEE 54th Annual Symposium on Foundations of Computer Science 2000, pp. 219, doi:10.1109/SFCS.2000.892109