## 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)≤2n 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)≥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 λ_{s}(n) ≥n·^{2}(1/t!) α(n)^{t} - O(α(n) ^{t-1}) for s≥4 even.

Original language | English |
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Article number | 17 |

Number of pages | 44 |

Journal | Journal of the ACM |

Volume | 57 |

Issue number | 3 |

DOIs | |

State | Published - 1 Mar 2010 |

Externally published | Yes |

## Keywords

- Davenport-Schinzel sequence
- Inverse Ackermann function
- Lower envelope