Permutation pattern-avoidance is a central concept of both enumerative and extremal combinatorics. In this paper we study the effect of permutation pattern-avoidance on the complexity of optimization problems. In the context of the dynamic optimality conjecture (Sleator, Tarjan, STOC 1983), Chalermsook, Goswami, Kozma, Mehlhorn, and Saranurak (FOCS 2015) conjectured that the amortized access cost of an optimal binary search tree (BST) is $O(1)$ whenever the access sequence avoids some fixed pattern. They showed a bound of $2^{\alpha{(n)}^{O(1)}}$, which was recently improved to $2^{\alpha{(n)}(1+o(1))}$ by Chalermsook, Pettie, and Yingchareonthawornchai (2023); here $n$ is the BST size and $\alpha(\cdot)$ the inverse-Ackermann function. In this paper we resolve the conjecture, showing a tight $O(1)$ bound. This indicates a barrier to dynamic optimality: any candidate online BST (e.g., splay trees or greedy trees) must match this optimum, but current analysis techniques only give superconstant bounds. More broadly, we argue that the easiness of pattern-avoiding input is a general phenomenon, not limited to BSTs or even to data structures. To illustrate this, we show that when the input avoids an arbitrary, fixed, a priori unknown pattern, one can efficiently compute a $k$-server solution of $n$ requests from a unit interval, with total cost $n^{O(1/\log k)}$, in contrast to the worst-case $\Theta(n/k)$ bound; and a traveling salesman tour of $n$ points from a unit box, of length $O(\log{n})$, in contrast to the worst-case $\Theta(\sqrt{n})$ bound; similar results hold for the euclidean minimum spanning tree, Steiner tree, and nearest-neighbor graphs. We show both results to be tight. Our techniques build on the Marcus-Tardos proof of the Stanley-Wilf conjecture, and on the recently emerging concept of twin-width; we believe our techniques to be more generally applicable.

翻译：排列模式避免是计数组合学与极值组合学中的核心概念。本文研究排列模式避免对优化问题复杂度的影响。在动态最优性猜想（Sleator, Tarjan, STOC 1983）的背景下，Chalermsook、Goswami、Kozma、Mehlhorn 与 Saranurak（FOCS 2015）推测：当访问序列避免某个固定模式时，最优二叉搜索树（BST）的均摊访问成本为 $O(1)$。他们证明了 $2^{\alpha{(n)}^{O(1)}}$ 的上界，该结果近期被 Chalermsook、Pettie 与 Yingchareonthawornchai（2023）改进为 $2^{\alpha{(n)}(1+o(1))}$；其中 $n$ 为 BST 规模，$\alpha(\cdot)$ 为反阿克曼函数。本文解决了该猜想，证明了紧的 $O(1)$ 上界。这表明动态最优性存在一个障碍：任何候选在线 BST（如伸展树或贪心树）必须匹配该最优值，但现有分析技术仅能给出超常数界。更广泛地，我们论证模式避免输入的易处理性是一种普遍现象，不仅限于 BST 甚至数据结构领域。为阐明这一点，我们证明：当输入避免任意固定且先验未知的模式时，可高效计算单位区间上 $n$ 个请求的 $k$-服务器解，总代价为 $n^{O(1/\log k)}$（相较最坏情况下的 $\Theta(n/k)$ 界）；以及单位盒子中 $n$ 个点的旅行商回路，长度为 $O(\log{n})$（相较最坏情况下的 $\Theta(\sqrt{n})$ 界）；类似结果对欧几里得最小生成树、斯坦纳树及近邻图同样成立。我们证明这两个结果均为紧界。我们的技术建立在 Marcus-Tardos 对 Stanley-Wilf 猜想的证明以及新兴的孪宽概念之上；我们相信这些技术具有更广泛的适用性。