Davenport-Schinzel序列的改进界和新技巧及其推广
Improved bounds and new techniques for Davenport-Schinzel sequences and their generalizations
摘要：
设λ_s（n）表示n个符号上s阶Davenport-Schinzel序列的最大长度。对于s=3，已知θ 3（n）= Theta（n alpha（n））（Hart和Sharir，1986）。对于一般的s>=4，有几乎紧的上界和下界，都是n * 2 ^poly（alpha（n））的形式（Agarwal，Sharir，and Shor，1989）。我们的第一个结果是Agarwal等人的上限技术的改进。我们得到改进的上限s>=6，这是紧的，甚至s到低阶项的指数。更重要的是，我们还提出了一种新的技术，用于推导lambda_s（n）的上界。通过这项新技术，我们：（1）重新推导了广义Davenport-Schinzel序列的上界，并得到了Adamec，Klazar，and Valtr（1992）给出的广义Davenport-Schinzel序列的改进上界。关于下界，我们证明了n = 3（n）>= 2n alpha（n）- O（n），因此系数2是紧的.我们还提出了一个更简单的版本的Agarwal，Sharir和Shor的建设，甚至s>=4，实现了已知的下限。
英文摘要：
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.
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