[Paper Review] Improved bounds and new techniques for Davenport--Schinzel sequences and their generalizations

Gabriel Nivasch|arXiv (Cornell University)|Jan 4, 2009

Algorithms and Data Compression15 references5 citations

TL;DR

This paper improves upper and lower bounds for Davenport–Schinzel sequences of order s, introducing a novel recurrence-based technique that re-derives and strengthens prior results. It establishes tighter bounds for λ₃(n) and general s, showing the coefficient 2 in the upper bound is tight and improving the leading constant in the exponent for even s via a refined analysis of prior methods.

ABSTRACT

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[EQUATION]where t = [(s - 2)/2], and α(n) denotes the inverse Ackermann function. The previous upper bounds, by Agarwal, Sharir, and Shor (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 λs(n). This new technique is based on some recurrences very similar to those used by the author, together with Alon, Kaplan, Sharir, and Smorodinsky (SODA 2008), for the problem of stabbing interval chains with j-tuples. With this new technique we: (1) re-derive the upper bound of λ3(n) ≤ 2nα(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, Klazar, and Valtr (1992).Regarding lower bounds, we show that λ3(n) ≥ 2nα(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, Sharir, and Shor that achieves the known lower bounds of λs(n) ≥ n·2(1/t!)α(n)t - O (α(n)t-1) for s ≥ 4 even.

Motivation & Objective

• To improve the upper and lower bounds for the maximum length λₛ(n) of Davenport–Schinzel sequences of order s on n symbols.
• To develop a new analytical technique based on recurrences inspired by interval chain stabbing problems to derive tighter bounds.
• To re-derive and strengthen existing bounds for λ₃(n) and generalized Davenport–Schinzel sequences.
• To establish the tightness of the coefficient 2 in the λ₃(n) upper bound and improve the leading constant in the exponent for even s.
• To simplify and re-verify known lower bound constructions for λₛ(n), particularly for s ≥ 4 even.

Proposed method

• Introduce a new recurrence-based method for upper bounding λₛ(n), modeled on techniques from interval chain stabbing problems.
• Apply the recurrence framework to re-derive the λ₃(n) ≤ 2nα(n) + O(n√α(n)) bound, confirming its tightness.
• Use the new technique to derive improved upper bounds for general s, with a leading coefficient of 1/t! in the exponent where t = ⌊(s−2)/2⌋.
• Re-analyze and simplify the construction of Agarwal, Sharir, and Shor to achieve known lower bounds for s ≥ 4 even.
• Leverage the inverse Ackermann function α(n) to express bounds with improved asymptotic precision.
• Combine recurrence analysis with structural decomposition of sequences to control forbidden alternations in Davenport–Schinzel sequences.

Experimental results

Research questions

• RQ1 What is the tightest possible upper bound for λ₃(n), and is the coefficient 2 in the 2nα(n) term optimal?
• RQ2 Can a new recurrence-based method be developed to unify and improve existing upper bounds for λₛ(n) across all s?
• RQ3 How do the new bounds compare to known lower bounds, particularly for even s ≥ 4?
• RQ4 Can the leading coefficient in the exponent of the upper bound for even s be improved from 1 to 1/t!?
• RQ5 Is there a simpler construction that achieves the known lower bounds for λₛ(n) when s ≥ 4 even?

Key findings

• The upper bound for λ₃(n) is tightened to 2nα(n) + O(n√α(n)), and the coefficient 2 is proven tight via a matching lower bound.
• For general s, the upper bound is improved to λₛ(n) ≤ n · 2(1/t!)α(n)t − O(α(n)t−1) with t = ⌊(s−2)/2⌋, improving the leading coefficient in the exponent.
• The new recurrence-based technique successfully re-derives the λ₃(n) bound and generalizes to yield improved bounds for all s.
• The lower bound for λ₃(n) is improved to 2nα(n) − O(n), confirming that the coefficient 2 in the upper bound is optimal.
• A simplified variant of the Agarwal–Sharir–Shor construction achieves the known lower bounds for s ≥ 4 even, validating the tightness of the exponent in the bound.
• The bounds for even s are now tight up to lower-order terms in the exponent, resolving a long-standing open problem in sequence extremal combinatorics.