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 levels in arrangements of curves, iii: further improvements
Proceedings of the twenty-fourth annual symposium on Computational geometry
Tangencies between families of disjoint regions in the plane
Proceedings of the twenty-sixth annual symposium on Computational geometry
On the bichromatic k-set problem
ACM Transactions on Algorithms (TALG)
Tangencies between families of disjoint regions in the plane
Computational Geometry: Theory and Applications
Crossings between curves with many tangencies
WALCOM'10 Proceedings of the 4th international conference on Algorithms and Computation
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We give a surprisingly short proof that in any planar arrangement of n curves where each pair intersects at most a fixed number (s) of times, the k-level has subquadratic (O(n2-1/2s) complexity. This answers one of the main open problems from the author’s previous paper [DCG 29, 375-393 (2003)], which provided a weaker upper bound for a restricted class of curves only (graphs of degree-s polynomials). When combined with existing tools (cutting curves, sampling, etc.), the new idea generates a slew of improved k-level results for most of the curve families studied earlier, including a near-O(n3/2 bound for parabolas.