Applications of random sampling in computational geometry, II
Discrete & Computational Geometry - Selected papers from the fourth ACM symposium on computational geometry, Univ. of Illinois, Urbana-Champaign, June 6 8, 1988
An upper bound on the number of planar K-sets
Discrete & Computational Geometry
On levels in arrangements of lines, segments, planes, and triangles
SCG '97 Proceedings of the thirteenth annual symposium on Computational geometry
A Characterization of Planar Graphs by Pseudo-Line Arrangements
ISAAC '97 Proceedings of the 8th International Symposium on Algorithms and Computation
Topological graphs with no self-intersecting cycle of length 4
Proceedings of the nineteenth annual symposium on Computational geometry
Lenses in arrangements of pseudo-circles and their applications
Journal of the ACM (JACM)
On levels in arrangements of surfaces in three dimensions
SODA '05 Proceedings of the sixteenth annual ACM-SIAM symposium on Discrete algorithms
On Levels in Arrangements of Curves, II: A Simple Inequality and Its Consequences
Discrete & Computational Geometry
Intersection reverse sequences and geometric applications
Journal of Combinatorial Theory Series A
On the bichromatic k-set problem
Proceedings of the nineteenth annual ACM-SIAM symposium on Discrete algorithms
On the bichromatic k-set problem
ACM Transactions on Algorithms (TALG)
Kinetic Euclidean minimum spanning tree in the plane
Journal of Discrete Algorithms
| Hi-index | 0.00 |
We revisit the problem of bounding the combinatorial complexity of the k-level in a two-dimensional arrangement of n curves. We give a number of small improvements over the results from the author's previous paper (FOCS'03). For example: For pseudo-parabolas, we obtain an upper bound of O(n3/2 log n), which improves the previous bound of O(n3/2 log2 n). For 3-intersecting curves, we obtain an upper bound of O(n2-1/(3+√7))=O(n1.823), the first improvement over the previous bound of O(n11/6)=O(n1.834). For s-intersecting curves or curve segments with s ≥ 3, we obtain an upper bound of O(n2--1/2s--δs) if s is odd, and O(n2--1/2(s--1)--δs) if s is even, for some constant δs 0. For pseudo-segments, we obtain an upper bound of O(n4/3 log1/3--δ n) for some constant δ 0. the previous bound was O(n4/3 log 2/3 n). For s-intersecting curve segments such that all but B pairs intersect at most once, we obtain an upper bound of O((n4/3 + n1+δB1/3--δ) log1/3--δ n + B) for some constant δ 0. We also observe that better concrete bounds for k-levels for constant values of n could in principle lead to better asymptotic bounds for arbitrary n.