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 search cost of an optimal binary search tree (BST) is constant whenever the search sequence is pattern-avoiding. The best known bound to date is $2^{\alpha{(n)}(1+o(1))}$ recently obtained by Chalermsook, Pettie, and Yingchareonthawornchai (SODA 2024); 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.

翻译：排列模式规避是枚举组合数学与极值组合数学的核心概念之一。本文研究排列模式规避对优化问题复杂性的影响。在动态最优性猜想（Sleator, Tarjan, STOC 1983）的背景下，Chalermsook, Goswami, Kozma, Mehlhorn 和 Saranurak（FOCS 2015）猜想：当搜索序列具有模式规避特性时，最优二叉搜索树（BST）的摊销搜索成本为常数。目前已知的最佳上界是 Chalermsook, Pettie 和 Yingchareonthawornchai（SODA 2024）近期获得的 $2^{\alpha{(n)}(1+o(1))}$，其中 $n$ 为 BST 的规模，$\alpha(\cdot)$ 为反阿克曼函数。本文证实了该猜想，给出了紧的 $O(1)$ 上界。这一结果表明了动态最优性面临的一个障碍：任何候选的在线 BST（例如伸展树或贪心树）都必须达到这一最优值，但现有的分析技术只能给出超常数上界。更广泛地说，我们认为模式规避输入带来的易处理性是一种普遍现象，并不局限于 BST 乃至数据结构领域。为说明这一点，我们证明了当输入规避一个任意的、固定的、先验未知的模式时，可以高效地计算单位区间上 $n$ 个请求的 $k$ 服务器问题解，其总成本为 $n^{O(1/\log k)}$，而最坏情况下的成本为 $\Theta(n/k)$；同时可以计算单位正方形内 $n$ 个点的旅行商回路，其长度为 $O(\log{n})$，而最坏情况下的长度为 $\Theta(\sqrt{n})$；类似的结果也适用于欧几里得最小生成树、斯坦纳树和最近邻图。我们证明了这些结果均是紧的。我们的技术建立在 Marcus-Tardos 对 Stanley-Wilf 猜想的证明以及近期兴起的双宽度概念之上。