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Sparsity-Parameterised Dynamic Edge Colouring

Authors Aleksander B. G. Christiansen , Eva Rotenberg , Juliette Vlieghe

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  • This paper is based on the master’s thesis of Juliette Vlieghe, supervised by Eva Rotenberg and Aleksander B. G. Christiansen and defended on 30th June 2023.
  • Master's Thesis https://doi.org/10.13140/RG.2.2.18471.52648

Subject Classification

ACM Subject Classification
  • Theory of computation → Dynamic graph algorithms
Keywords
  • edge colouring
  • arboricity
  • hierarchical partition
  • dynamic algorithms
  • amortized analysis

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Abstract

We study the edge-colouring problem, and give efficient algorithms where the number of colours is parameterised by the graph’s arboricity, α. In a dynamic graph, subject to insertions and deletions, we give a deterministic algorithm that updates a proper Δ + O(α) edge colouring in poly(log n) amortized time. Our algorithm is fully adaptive to the current value of the maximum degree and arboricity.
In this fully-dynamic setting, the state-of-the-art edge-colouring algorithms are either a randomised algorithm using (1 + ε)Δ colours in poly(log n, ε^{-1}) time per update, or the naive greedy algorithm which is a deterministic 2Δ -1 edge colouring with log(Δ) update time. 
Compared to the (1+ε)Δ algorithm, our algorithm is deterministic and asymptotically faster, and when α is sufficiently small compared to Δ, it even uses fewer colours. In particular, ours is the first Δ+O(1) edge-colouring algorithm for dynamic forests, and dynamic planar graphs, with polylogarithmic update time. 
Additionally, in the static setting, we show that we can find a proper edge colouring with Δ + 2α colours in O(mlog n) time. Moreover, the colouring returned by our algorithm has the following local property: every edge uv is coloured with a colour in {1, max{deg(u), deg(v)} + 2α}. The time bound matches that of the greedy algorithm that computes a 2Δ-1 colouring of the graph’s edges, and improves the number of colours when α is sufficiently small compared to Δ.

Cite As Get BibTex

Aleksander B. G. Christiansen, Eva Rotenberg, and Juliette Vlieghe. Sparsity-Parameterised Dynamic Edge Colouring. In 19th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 294, pp. 20:1-20:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024) https://doi.org/10.4230/LIPIcs.SWAT.2024.20

Author Details

Aleksander B. G. Christiansen
  • Technical University of Denmark, Lyngby, Denmark
Eva Rotenberg
  • Technical University of Denmark, Lyngby, Denmark
Juliette Vlieghe
  • Technical University of Denmark, Lyngby, Denmark

Funding

This work was supported by the VILLUM Foundation grant VIL37507 "Efficient Recomputations for Changeful Problems".

Acknowledgements

We thank Jacob Holm for his interest in this work, and for comments and improvements to an earlier version of this manuscript. We also thank an anonymous reviewer for their highly detailed feedback and suggestions for improvement.

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