论文标题
改善稀疏图的动态着色
Improved Dynamic Colouring of Sparse Graphs
论文作者
论文摘要
给定一个动态图受到边缘插入和删除的约束，我们将展示如何更新适当的顶点着色的隐式表示，以便在查询时间时可以计算顶点的颜色。我们给出了一种确定性算法，该算法使用$ O（α^2）$颜色用于动态图$α$的动态图，以及使用$ o（\ min \ {α\logα，α\ log log \ log \ log \ log \ log \ n \}）的随机算法中的颜色。我们的确定性算法在$α$和$ \ log n $中具有更新和查询时间的多项式，并且我们的随机算法摊销了更新和查询的时间，高概率是$ \ log n $中的多项式，而不依赖于这种支撑性。 因此，与最新的隐性着色相比，我们将颜色的数量呈指数增长，即从$ o（2^α）$颜色，并且我们接近这种ARBORICITY-PARAMITY-PAREMATICY-PARMETION-PARMETION-PRAMETICE-PLAIN BOUST BOUDEN BOUDEND BOUSTING BOUNTING BOUNTING $（α）$的理论下限。同时，我们的随机算法改善了更新和查询时间，以及时运行$ \ log n $，而无需依赖于$α$。我们的算法完全适应了查询或更新时间时动态树皮性的当前价值。
Given a dynamic graph subject to edge insertions and deletions, we show how to update an implicit representation of a proper vertex colouring, such that colours of vertices are computable upon query time. We give a deterministic algorithm that uses $O(α^2)$ colours for a dynamic graph of arboricity $α$, and a randomised algorithm that uses $O(\min\{α\log α, α\log \log \log n\})$ colours in the oblivious adversary model. Our deterministic algorithm has update- and query times polynomial in $α$ and $\log n$, and our randomised algorithm has amortised update- and query time that with high probability is polynomial in $\log n$ with no dependency on the arboricity. Thus, we improve the number of colours exponentially compared to the state-of-the art for implicit colouring, namely from $O(2^α)$ colours, and we approach the theoretical lower bound of $Ω(α)$ for this arboricity-parameterised approach. Simultaneously, our randomised algorithm improves the update- and query time to run in time solely polynomial in $\log n$ with no dependency on $α$. Our algorithms are fully adaptive to the current value of the dynamic arboricity at query or update time.
