Abstract
The Euclidean Steiner tree problem asks to find a min-cost metric graph that connects a given set of terminal points X in \(\mathbb {R}^d\), possibly using points not in X which are called Steiner points. Even though near-linear time \((1 + \epsilon )\)-approximation was obtained in the offline setting in seminal works of Arora and Mitchell, efficient dynamic algorithms for Steiner tree is still open. We give the first algorithm that (implicitly) maintains a \((1 + \epsilon )\)-approximate solution which is accessed via a set of tree traversal queries, subject to point insertion and deletions, with amortized update and query time \(O(\textrm{poly}\log n)\) with high probability. Our approach is based on an Arora-style geometric dynamic programming, and our main technical contribution is to maintain the DP subproblems in the dynamic setting efficiently. We also need to augment the DP subproblems to support the tree traversal queries.
Full version of this paper is available at [5]
T-H. Hubert Chan was partially supported by the Hong Kong RGC grant 17203122.
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Notes
- 1.
In this informal statement we consider d and \(\epsilon \) as constants and their dependence are hidden in the big-O.
- 2.
We state the exact dependence of d which was not accurately calculated in [1], see the appendix for the details.
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Chan, TH.H., Goranci, G., Jiang, S.HC., Wang, B., Xue, Q. (2024). Fully Dynamic Algorithms for Euclidean Steiner Tree. In: Uehara, R., Yamanaka, K., Yen, HC. (eds) WALCOM: Algorithms and Computation. WALCOM 2024. Lecture Notes in Computer Science, vol 14549. Springer, Singapore. https://doi.org/10.1007/978-981-97-0566-5_6
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