Fast decremental tree sums in forests

Abstract

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. [...]