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Fast Decremental Tree Sums in Forests

Authors Benjamin Aram Berendsohn , Marek Sokołowski

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ACM Subject Classification
  • Theory of computation → Dynamic graph algorithms
Keywords
  • dynamic graphs
  • connectivity
  • group model
  • universal optimality

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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 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(OPT(F, m)). Here 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 OPT. Note that even the worst-case complexity of this algorithm remains unknown to us.
Second, we consider subtree-sum queries. Here, the forest is rooted, and a query subtree-sum(v) returns the sum of weights in the subtree rooted at v. An easy reduction from the well-known prefix sum problem shows that the general, weighted version of the problem requires Θ(n log n) time for n operations. Interestingly, we prove that the Ω(n log n) complexity lower bound still holds even if weight updates are disallowed. On the other hand, we show that the unweighted version can be solved with 𝒪((log n)/(log log n)) time per operation, and this is tight.

Cite As Get BibTex

Benjamin Aram Berendsohn and Marek Sokołowski. Fast Decremental Tree Sums in Forests. In 53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 374, pp. 26:1-26:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.ICALP.2026.26

Author Details

Benjamin Aram Berendsohn
  • Max Planck Institute for Informatics, Saarbrücken, Germany
Marek Sokołowski
  • Max Planck Institute for Informatics, Saarbrücken, Germany

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