Tree-Packing Revisited: Faster Fully Dynamic Min-Cut and Arboricity
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arXiv
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2024
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| _version_ | 1866913596115517440 |
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| author | de Vos, Tijn Christiansen, Aleksander B. G. |
| author_facet | de Vos, Tijn Christiansen, Aleksander B. G. |
| contents | A tree-packing is a collection of spanning trees of a graph. It has been a useful tool for computing the minimum cut in static, dynamic, and distributed settings. In particular, [Thorup, Comb. 2007] used them to obtain his dynamic min-cut algorithm with $\tilde O(λ^{14.5}\sqrt{n})$ worst-case update time. We reexamine this relationship, showing that we need to maintain fewer spanning trees for such a result; we show that we only need to pack $Θ(λ^3 \log m)$ greedy trees to guarantee a 1-respecting cut or a trivial cut in some contracted graph.
Based on this structural result, we then provide a deterministic algorithm for fully dynamic exact min-cut, that has $\tilde O(λ^{5.5}\sqrt{n})$ worst-case update time, for min-cut value bounded by $λ$. In particular, this also leads to an algorithm for general fully dynamic exact min-cut with $\tilde O(m^{1-1/12})$ amortized update time, improving upon $\tilde O(m^{1-1/31})$ [Goranci et al., SODA 2023].
We also give the first fully dynamic algorithm that maintains a $(1+\varepsilon)$-approximation of the fractional arboricity -- which is strictly harder than the integral arboricity. Our algorithm is deterministic and has $O(α\log^6m/\varepsilon^4)$ amortized update time, for arboricity at most $α$. We extend these results to a Monte Carlo algorithm with $O(\text{poly}(\log m,\varepsilon^{-1}))$ amortized update time against an adaptive adversary. Our algorithms work on multi-graphs as well.
Both result are obtained by exploring the connection between the min-cut/arboricity and (greedy) tree-packing. We investigate tree-packing in a broader sense; including a lower bound for greedy tree-packing, which - to the best of our knowledge - is the first progress on this topic since [Thorup, Comb. 2007]. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_09141 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Tree-Packing Revisited: Faster Fully Dynamic Min-Cut and Arboricity de Vos, Tijn Christiansen, Aleksander B. G. Data Structures and Algorithms A tree-packing is a collection of spanning trees of a graph. It has been a useful tool for computing the minimum cut in static, dynamic, and distributed settings. In particular, [Thorup, Comb. 2007] used them to obtain his dynamic min-cut algorithm with $\tilde O(λ^{14.5}\sqrt{n})$ worst-case update time. We reexamine this relationship, showing that we need to maintain fewer spanning trees for such a result; we show that we only need to pack $Θ(λ^3 \log m)$ greedy trees to guarantee a 1-respecting cut or a trivial cut in some contracted graph. Based on this structural result, we then provide a deterministic algorithm for fully dynamic exact min-cut, that has $\tilde O(λ^{5.5}\sqrt{n})$ worst-case update time, for min-cut value bounded by $λ$. In particular, this also leads to an algorithm for general fully dynamic exact min-cut with $\tilde O(m^{1-1/12})$ amortized update time, improving upon $\tilde O(m^{1-1/31})$ [Goranci et al., SODA 2023]. We also give the first fully dynamic algorithm that maintains a $(1+\varepsilon)$-approximation of the fractional arboricity -- which is strictly harder than the integral arboricity. Our algorithm is deterministic and has $O(α\log^6m/\varepsilon^4)$ amortized update time, for arboricity at most $α$. We extend these results to a Monte Carlo algorithm with $O(\text{poly}(\log m,\varepsilon^{-1}))$ amortized update time against an adaptive adversary. Our algorithms work on multi-graphs as well. Both result are obtained by exploring the connection between the min-cut/arboricity and (greedy) tree-packing. We investigate tree-packing in a broader sense; including a lower bound for greedy tree-packing, which - to the best of our knowledge - is the first progress on this topic since [Thorup, Comb. 2007]. |
| title | Tree-Packing Revisited: Faster Fully Dynamic Min-Cut and Arboricity |
| topic | Data Structures and Algorithms |
| url | https://arxiv.org/abs/2405.09141 |