Ailon et al. [SICOMP'11] proposed self-improving algorithms for sorting and Delaunay triangulation (DT) when the input instances $x_1,\cdots,x_n$ follow some unknown \emph{product distribution}. That is, $x_i$ comes from a fixed unknown distribution $\mathsf{D}_i$, and the $x_i$'s are drawn independently. After spending $O(n^{1+\varepsilon})$ time in a learning phase, the subsequent expected running time is $O((n+ H)/\varepsilon)$, where $H \in \{H_\mathrm{S},H_\mathrm{DT}\}$, and $H_\mathrm{S}$ and $H_\mathrm{DT}$ are the entropies of the distributions of the sorting and DT output, respectively. In this paper, we allow dependence among the $x_i$'s under the \emph{group product distribution}. There is a hidden partition of $[1,n]$ into groups; the $x_i$'s in the $k$-th group are fixed unknown functions of the same hidden variable $u_k$; and the $u_k$'s are drawn from an unknown product distribution. We describe self-improving algorithms for sorting and DT under this model when the functions that map $u_k$ to $x_i$'s are well-behaved. After an $O(\mathrm{poly}(n))$-time training phase, we achieve $O(n + H_\mathrm{S})$ and $O(n\alpha(n) + H_\mathrm{DT})$ expected running times for sorting and DT, respectively, where $\alpha(\cdot)$ is the inverse Ackermann function.

中文翻译：

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自我改进算法的概括

艾隆等人。[SICOMP'11] 当输入实例 $x_1,\cdots,x_n$ 遵循一些未知的 \emph{product distribution} 时，提出了用于排序和 Delaunay 三角剖分 (DT) 的自我改进算法。也就是说，$x_i$ 来自固定的未知分布 $\mathsf{D}_i$，并且 $x_i$ 是独立绘制的。在学习阶段花费$O(n^{1+\varepsilon})$时间后，后续的预期运行时间为$O((n+H)/\varepsilon)$，其中$H\in\{H_\mathrm {S},H_\mathrm{DT}\}$ 和 $H_\mathrm{S}$ 和 $H_\mathrm{DT}$ 分别是排序和 DT 输出的分布的熵。在本文中，我们允许在 \emph {group product distribution} 下 $x_i$ 之间存在依赖关系。有 $[1,n]$ 的隐藏分区成组；$x_i$' $k$-th组中的s是同一个隐藏变量$u_k$的固定未知函数；并且 $u_k$ 来自未知的产品分布。当将 $u_k$ 映射到 $x_i$ 的函数表现良好时，我们描述了在此模型下用于排序和 DT 的自我改进算法。在 $O(\mathrm{poly}(n))$ 时间训练阶段之后，我们实现了 $O(n + H_\mathrm{S})$ 和 $O(n\alpha(n) + H_\mathrm{ DT})$ 分别为排序和 DT 的预期运行时间，其中 $\alpha(\cdot)$ 是逆阿克曼函数。