The Preisach extremum stack $Π_n$ is the minimal sufficient statistic for the class $\mathcal{R}$ of computable rate-independent functionals in the Kolmogorov complexity sense [1]. Its standard update algorithm runs in amortised $O(1)$ time, but adversarial inputs can force $Θ(k)$ operations per step (where $k$ is the current depth). We establish a three-level complexity picture: (i) any compact exact $\mathcal{R}$-minimal representation incurs $Θ(k)$ output changes per step in the worst case (in a model-independent output-change metric); (ii) the monotone ordering of the Preisach wiping property enables binary search, reducing boundary detection to $O(log k)$, though physical deletion remains $Θ(d)$; (iii) a finger-tree implementation achieves $O(log k)$ worst-case time per step for both search and deletion, at the cost of a more complex data structure, while maintaining exact $\mathcal{R}$-minimality with no approximation error. These results settle the worst-case complexity of the Preisach extremum stack across all three levels.

翻译：Preisach极值栈 $Π_n$ 是 Kolmogorov 复杂度意义下 [1] 可计算率无关泛函类 $\mathcal{R}$ 的最小充分统计量。其标准更新算法具有均摊 $O(1)$ 时间，但对抗性输入可能强制每步 $Θ(k)$ 次操作（其中 $k$ 为当前深度）。我们建立了三层次复杂度图景：(i) 任何紧致精确 $\mathcal{R}$-最小表示在最坏情况下每步会引发 $Θ(k)$ 次输出变化（基于模型无关的输出变化度量）；(ii) Preisach 擦除性质的单调序结构支持二分查找，将边界检测降至 $O(\log k)$，而物理删除仍为 $Θ(d)$；(iii) 基于手指树的实现可在搜索和删除上达到每步 $O(\log k)$ 最坏情况时间，代价是更复杂的数据结构，同时保持无近似误差的精确 $\mathcal{R}$-最小性。这些结果在所有三个层次上解决了 Preisach 极值栈的最坏情况复杂度。