On levels in arrangements of surfaces in three dimensions
SODA '05 Proceedings of the sixteenth annual ACM-SIAM symposium on Discrete algorithms
Intersection reverse sequences and geometric applications
Journal of Combinatorial Theory Series A
Kinetic and dynamic data structures for convex hulls and upper envelopes
Computational Geometry: Theory and Applications
Intersection reverse sequences and geometric applications
GD'04 Proceedings of the 12th international conference on Graph Drawing
Kinetic and dynamic data structures for convex hulls and upper envelopes
WADS'05 Proceedings of the 9th international conference on Algorithms and Data Structures
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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(0(n^{2 - \frac{1}{{2s}}})) complexity. This answers one of the main open problems from the author's previous paper (FOCS'00), which provided a weaker bound for a restricted class of curves (graphs of degree-s polynomials) only. 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-0(n^{{3 \mathord{\left/ {\vphantom {3 2}} \right. \kern-\nulldelimiterspace} 2}}) bound for parabolas.