On the Locality of Nash-Williams Forest Decomposition and Star-Forest Decomposition

Harris, David G.; Su, Hsin-Hao; Vu, Hoa T.

doi:10.1145/3465084.3467908

Cited by 7 publications

(2 citation statements)

References 56 publications

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“…Barenboim & Elkin gave a (2 + ε)-approximation in [7]. This has since then been improved to (1 + ε)-approximations [22,23,40,21].…”

Section: Arboricity Decompositionsmentioning

confidence: 99%

Fully-dynamic $α+ 2$ Arboricity Decomposition and Implicit Colouring

Christiansen¹,

Rotenberg²

2022

Preprint

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In the implicit dynamic colouring problem, the task is to maintain a representation of a proper colouring as a dynamic graph is subject to insertions and deletions of edges, while facilitating interspersed queries to the colours of vertices. The goal is to use few colours, while still efficiently handling edge-updates and responding to colour-queries. For an n-vertex dynamic graph of arboricity α, we present an algorithm that maintains an implicit vertex colouring with 4•2 α colours, in amortised poly-log n update time, and with O(α log n) worst-case query time. The previous best implicit dynamic colouring algorithm uses 2 40α colours, and has a more efficient update time of O(log 3 n) and the same query time of O(α log n) [25].For graphs undergoing arboricity α preserving updates, we give a fully-dynamic α + 2 arboricity decomposition in poly(log n, α) time, which matches the number of forests in the best near-linear static algorithm by Blumenstock and Fischer [12] who obtain α + 2 forests in near-linear time.Our construction goes via dynamic bounded out-degree orientations, where we present a fullydynamic explicit, deterministic, worst-case algorithm for (1+ε)α +2 bounded out-degree orientation with update time O(ε −6 α 2 log 3 n). The state-of-the-art explicit, deterministic, worst-case algorithm for bounded out-degree orientations maintains a β • α + log β n out-orientation in O(β 2 α 2 + βα log β n) time [28].

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“…Barenboim & Elkin gave a (2 + ε)-approximation in [7]. This has since then been improved to (1 + ε)-approximations [22,23,40,21].…”

Section: Arboricity Decompositionsmentioning

confidence: 99%

Fully-dynamic $α+ 2$ Arboricity Decomposition and Implicit Colouring

Christiansen¹,

Rotenberg²

2022

Preprint

View full text Add to dashboard Cite

show abstract

“…Hence, the logarithmic barrier for graphs of bounded arboricity seems to be inevitable by using existing techniques. There is a progress in providing a decomposition with as fewest forests as possible [14,15]. Breaking the logarithmic round complexity barrier for a constant number of forests contradicts the known lower bound of Linial [20] for coloring unoriented trees.…”

Section: Bottlenecks Of Designing Algorithms For Bounded Arboricity G...mentioning

confidence: 99%