Sharp Bounds on Davenport-Schinzel Sequences of Every Order

Abstract

One of the longest-standing open problems in computational geometry is bounding the complexity of 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 bound the running times of geometric algorithms, data structures, and the combinatorial complexity of geometric arrangements. Let λ s ( n ) be the maximum length of an order- s DS sequence over n letters. What is λ s asymptotically? This question has been answered satisfactorily [Hart and Sharir 1986; Agarwal et al. 1989; Klazar 1999; Nivasch 2010], when s is even or s ≤ 3. However, since the work of Agarwal et al. 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, λ s ( n ) behaves essentially like λ s -1 ( n ) when s is odd. This refutes conjectures by Alon et al. [2008] and Nivasch [2010].