Sparsity-Parameterised Dynamic Edge Colouring
arxiv(2023)
摘要
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.
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