Fast decremental tree sums in forests
Benjamin Aram Berendsohn, Marek Soko{\l}owski

TL;DR
This paper introduces efficient decremental algorithms for maintaining sums in forests, achieving near-optimal and constant-time operations under certain conditions, improving over standard methods.
Contribution
The authors develop a new data structure with sublogarithmic update time and an asymptotically optimal algorithm for decremental forest sum problems, including unweighted cases.
Findings
O(1) query time for unweighted forests
An algorithm with running time matching the theoretical OPT
Tight bounds established for subtree-sum variants
Abstract
We study two fundamental decremental dynamic graph problems. In both problems, we need to maintain a vertex-weighted forest of size under edge deletions, weight updates, and a certain information-retrieval query. Both problems can be solved in 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 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 preprocessing time and 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…
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