On Levels in Arrangements of Curves, II: A Simple Inequality and Its Consequences

We give a surprisingly short proof that in any planar arrangement of Ò curves where each pair intersects at most a fixed number (×) of times, the -level has subquadratic (Ç´Ò¾  ½ ¾× µ) 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-× polynomials) only. When combined with existing tools (cutting curves, sampling, etc.), the new idea generates a slew of improved -level results for most of the curve families studied earlier, including a near-Ç´Ò¿ ¾µ bound for parabolas.

• Type: Conference
• Year: 2003
• Where: FOCS
• Authors: Timothy M. Chan
• Keywords: FOCS 2003, Main Open Problems, Planar Arrangement, Theoretical Computer Science, Weaker Bound