One of the longest-standing open problems in computational geometry is to
bound the lower envelope of n univariate functions, each pair of which
crosses at most s times, for some fixed s. This problem is known to be
equivalent to bounding the length of an order-s Davenport-Schinzel sequence,
namely a sequence over an n-letter alphabet that avoids alternating
subsequences of the form a⋯b⋯a⋯b⋯ with length
s+2. These sequences were introduced by Davenport and Schinzel in 1965 to
model a certain problem in differential equations and have since been applied
to bounding the running times of geometric algorithms, data structures, and the
combinatorial complexity of geometric arrangements.
Let λs​(n) be the maximum length of an order-s DS sequence over n
letters. What is λs​ asymptotically? This question has been answered
satisfactorily (by Hart and Sharir, Agarwal, Sharir, and Shor, Klazar, and
Nivasch) when s is even or s≤3. However, since the work of Agarwal,
Sharir, and Shor in the mid-1980s there has been a persistent gap in our
understanding of the odd orders.
In this work we effectively close the problem by establishing sharp bounds on
Davenport-Schinzel sequences of every order s. Our results reveal that,
contrary to one's intuition, λs​(n) behaves essentially like
λs−1​(n) when s is odd. This refutes conjectures due to Alon et al.
(2008) and Nivasch (2010).Comment: A 10-page extended abstract will appear in the Proceedings of the
Symposium on Computational Geometry, 201