This paper presents a new algorithm (and an additional trick) that allows to compute fastly an entire curve of post hoc bounds for the False Discovery Proportion when the underlying bound $V^*\_{\mathfrak{R}}$ construction is based on a reference family $\mathfrak{R}$ with a forest structure {à} la Durand et al. (2020). By an entire curve, we mean the values $V^*\_{\mathfrak{R}}(S\_1),\dotsc,V^*\_{\mathfrak{R}}(S\_m)$ computed on a path of increasing selection sets $S\_1\subsetneq\dotsb\subsetneq S\_m$, $|S\_t|=t$. The new algorithm leverages the fact that going from $S\_t$ to $S\_{t+1}$ is done by adding only one hypothesis. Compared to a more naive approach, the new algorithm has a complexity in $O(|\mathcal K|m)$ instead of $O(|\mathcal K|m^2)$, where $|\mathcal K|$ is the cardinality of the family.

翻译：本文提出了一种新算法（及一项附加技巧），当底层边界$V^*_{\mathfrak{R}}$的构造基于具有Durand等人（2020）提出的森林结构参考族$\mathfrak{R}$时，该算法能够快速计算错误发现比例的整条事后边界曲线。所谓整条曲线，是指在递增选择集路径$S_1\subsetneq\dotsb\subsetneq S_m$（其中$|S_t|=t$）上计算得到的值$V^*_{\mathfrak{R}}(S_1),\dotsc,V^*_{\mathfrak{R}}(S_m)$。新算法利用了从$S_t$到$S_{t+1}$仅需添加一个假设这一特性。相较于更朴素的方法，新算法的复杂度为$O(|\mathcal K|m)$而非$O(|\mathcal K|m^2)$，其中$|\mathcal K|$为该族的基数。