Deterministic Dynamic Edge Colouring
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866908634460454912 |
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| author | Christiansen, Aleksander B. G. |
| author_facet | Christiansen, Aleksander B. G. |
| contents | Given a dynamic graph $G$ with $n$ vertices and $m$ edges subject to insertion an deletions of edges, we show how to maintain a $(1+\varepsilon)Δ$-edge-colouring of $G$ without the use of randomisation.
More specifically, we show a deterministic dynamic algorithm with an amortised update time of $2^{\tilde{O}_{\log \varepsilon^{-1}}(\sqrt{\log n})}$ using $(1+\varepsilon)Δ$ colours. If $\varepsilon^{-1} \in 2^{O(\log^{0.49} n)}$, then our update time is sub-polynomial in $n$.
While there exists randomised algorithms maintaining colourings with the same number of colours [Christiansen STOC'23, Duan, He, Zhang SODA'19, Bhattacarya, Costa, Panski, Solomon SODA'24] in polylogarithmic and even constant update time, this is the first deterministic algorithm to go below the greedy threshold of $2Δ-1$ colours for all input graphs.
On the way to our main result, we show how to dynamically maintain a shallow hierarchy of degree-splitters with both recourse and update time in $n^{o(1)}$. We believe that this algorithm might be of independent interest. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_13139 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Deterministic Dynamic Edge Colouring Christiansen, Aleksander B. G. Data Structures and Algorithms Given a dynamic graph $G$ with $n$ vertices and $m$ edges subject to insertion an deletions of edges, we show how to maintain a $(1+\varepsilon)Δ$-edge-colouring of $G$ without the use of randomisation. More specifically, we show a deterministic dynamic algorithm with an amortised update time of $2^{\tilde{O}_{\log \varepsilon^{-1}}(\sqrt{\log n})}$ using $(1+\varepsilon)Δ$ colours. If $\varepsilon^{-1} \in 2^{O(\log^{0.49} n)}$, then our update time is sub-polynomial in $n$. While there exists randomised algorithms maintaining colourings with the same number of colours [Christiansen STOC'23, Duan, He, Zhang SODA'19, Bhattacarya, Costa, Panski, Solomon SODA'24] in polylogarithmic and even constant update time, this is the first deterministic algorithm to go below the greedy threshold of $2Δ-1$ colours for all input graphs. On the way to our main result, we show how to dynamically maintain a shallow hierarchy of degree-splitters with both recourse and update time in $n^{o(1)}$. We believe that this algorithm might be of independent interest. |
| title | Deterministic Dynamic Edge Colouring |
| topic | Data Structures and Algorithms |
| url | https://arxiv.org/abs/2402.13139 |