Improved bounds and new techniques for Davenport-Schinzel sequences and their generalizations

TL;DR

This paper advances the understanding of Davenport-Schinzel sequences by improving upper bounds for higher orders, introducing new techniques for bounds derivation, and confirming tightness of the lower bounds for s=3.

Contribution

It presents a novel technique for deriving upper bounds of Davenport-Schinzel sequences and improves existing bounds for s>=6, also confirming tight bounds for s=3.

Findings

Improved upper bounds for s>=6 sequences.

Re-derived bounds for s=3 sequences matching known tight bounds.

Presented simpler constructions achieving known lower bounds for even s>=4.

Abstract

Let lambda_s(n) denote the maximum length of a Davenport-Schinzel sequence of order s on n symbols. For s=3 it is known that lambda_3(n) = Theta(n alpha(n)) (Hart and Sharir, 1986). For general s>=4 there are almost-tight upper and lower bounds, both of the form n * 2^poly(alpha(n)) (Agarwal, Sharir, and Shor, 1989). Our first result is an improvement of the upper-bound technique of Agarwal et al. We obtain improved upper bounds for s>=6, which are tight for even s up to lower-order terms in the exponent. More importantly, we also present a new technique for deriving upper bounds for lambda_s(n). With this new technique we: (1) re-derive the upper bound of lambda_3(n) <= 2n alpha(n) + O(n sqrt alpha(n)) (first shown by Klazar, 1999); (2) re-derive our own new upper bounds for general s; and (3) obtain improved upper bounds for the generalized Davenport-Schinzel sequences considered by Adamec, Klazar, and Valtr (1992). Regarding lower bounds, we show that lambda_3(n) >= 2n alpha(n) - O(n), and therefore, the coefficient 2 is tight. We also present a simpler version of the construction of Agarwal, Sharir, and Shor that achieves the known lower bounds for even s>=4.