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

• Authors: Gabriel Nivasch
• Venue: Journal of the ACM (JACM)
• Year: 2010

We present several new results regarding λs(n), the maximum length of a Davenport--Schinzel sequence of order s on n distinct symbols. First, we prove that λs(n) ≤ n ⋅ 2(1/t&excel;)α(n)t + O(α(n)t−1) for s ≥ 4 even, and λs(n) ≤ n ⋅ 2(1/t&excel;)α(n)t log2 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&excel; 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 λs(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 λ3(n) ≤ 2n α(n) + O(n &sqrt;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 λ3(n) ≥ 2n 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 λs(n) ≥ n ⋅ 2(1/t&excel;) α(n)t − O(α(n)t−1) for s ≥ 4 even.