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

Abstract

We present several new results regarding λ s ( n ), the maximum length of a Davenport--Schinzel sequence of order s on n distinct symbols. First, we prove that λ s ( n ) ≤ n · 2 (1/ t !)α( n ) t + O (α( n ) t -1 ) for s ≥ 4 even, and λ s ( n ) ≤ n · 2 (1/t!)α( n ) t log 2 α( n ) + O (α( n ) t ) for s ≥ 3 odd, where t = ⌊( s -2)/2⌋, and α( n ) denotes the inverse Ackermann function. The previous upper bounds, by Agarwal et al. [1989], had a leading coefficient of 1 instead of 1/ t ! in the exponent. The bounds for even s are now tight up to lower-order terms in the exponent. These new bounds result from a small improvement on the technique of Agarwal et al. More importantly, we also present a new technique for deriving upper bounds for λ s ( n ). This new technique is very similar to the one we applied to the problem of stabbing interval chains [Alon et al. 2008]. With this new technique we: (1) re-derive the upper bound of λ 3 ( n ) ≤ 2 n α( n ) + O ( n √α( 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 et al. [1992]. Regarding lower bounds, we show that λ 3 ( n ) ≥ 2 n α( n ) - O ( n ) (the previous lower bound (Sharir and Agarwal, 1995) had a coefficient of 1/2), so the coefficient 2 is tight. We also present a simpler variant of the construction of Agarwal et al. [1989] that achieves the known lower bounds of λ s ( n ) ≥ n · 2 (1/ t !) α( n ) t - O (α( n ) t -1 ) for s ≥ 4 even.