We study two fundamental decremental dynamic graph problems. In both problems, we need to maintain a vertex-weighted forest of size $n$ under edge deletions, weight updates, and a certain information-retrieval query. Both problems can be solved in $O(\log n)$ time per update/query using standard dynamic forest data structures like top trees, even if additionally edge insertions are allowed. We investigate whether the deletion-only problem can be solved faster. First, we consider $\texttt{tree-sum}$ queries, where we ask for the sum of vertex weights in one of the connected components (i.e., trees) in the forest. We give a data structure with $O(n)$ preprocessing time and $O(\log^* n)$ time per operation, based on a micro-macro tree decomposition (Alstrup et al., 1997). If the forest is unweighted (i.e., all weights are 1 and cannot be changed), then the operation time can be improved to $O(1)$. Additionally, we give an asymptotically universally optimal algorithm. More specifically, our algorithm works in the group model, and processes $m$ operations on an initial forest $F$ in running time $O( \mathrm{OPT}(F, m) )$. Here $\mathrm{OPT}(F, m)$ is the number of weight additions and subtractions that a best possible algorithm performs to handle a worst-case instance for a fixed initial forest $F$ and a fixed number $m$ of operations. We achieve this with a combination of the aforementioned decomposition technique, precomputation of optimal data structures for very small instances, and some insights into the behavior of $\mathrm{OPT}$. Note that even the worst-case complexity of this algorithm remains unknown to us. Second, we consider $\texttt{subtree-sum}$ queries. Here, the forest is rooted, and a query $\texttt{subtree-sum}(v)$ returns the sum of weights in the subtree rooted at $v$. We show tight bounds for several variants of this problem. [...]

翻译：我们研究两个基础的递减式动态图问题。在这两个问题中，我们需要在边删除、权重更新以及特定信息检索查询操作下，维护一个大小为 $n$ 的顶点加权森林。使用标准的动态森林数据结构（如 top trees），即使允许边插入，这两个问题也能在每次更新/查询操作中实现 $O(\log n)$ 时间。我们探讨仅限删除操作的问题能否更快解决。首先，我们考虑 $\texttt{tree-sum}$ 查询，即询问森林中某个连通分量（即树）中顶点权重的和。基于微-宏观树分解方法（Alstrup 等，1997），我们给出一个数据结构，其预处理时间为 $O(n)$，每次操作时间为 $O(\log^* n)$。若森林无权重（即所有权重为 1 且不可更改），则操作时间可优化至 $O(1)$。此外，我们提出一个渐近全局最优算法。具体而言，该算法在群模型下工作，对初始森林 $F$ 处理 $m$ 次操作，运行时间为 $O( \mathrm{OPT}(F, m) )$，其中 $\mathrm{OPT}(F, m)$ 表示针对固定初始森林 $F$ 和固定操作次数 $m$ 的最坏情况实例时，最优算法执行的权重加减操作次数。我们通过结合上述分解技术、对极小实例预计算最优数据结构、以及对 $\mathrm{OPT}$ 行为的深入理解来实现这一目标。注意，即使该算法的最坏情况复杂度对我们而言仍是未知的。其次，我们考虑 $\texttt{subtree-sum}$ 查询。这里森林是有根的，查询 $\texttt{subtree-sum}(v)$ 返回以 $v$ 为根的子树中权重的和。我们给出该问题多个变体的紧界。[……]