We give improved algorithms for maintaining edge-orientations of a fully-dynamic graph, such that the maximum out-degree is bounded. On one hand, we show how to orient the edges such that maximum out-degree is proportional to the arboricity $\alpha$ of the graph, in, either, an amortised update time of $O(\log^2 n \log \alpha)$, or a worst-case update time of $O(\log^3 n \log \alpha)$. On the other hand, motivated by applications including dynamic maximal matching, we obtain a different trade-off. Namely, the improved update time of either $O(\log n \log \alpha)$, amortised, or $O(\log ^2 n \log \alpha)$, worst-case, for the problem of maintaining an edge-orientation with at most $O(\alpha + \log n)$ out-edges per vertex. Finally, all of our algorithms naturally limit the recourse to be polylogarithmic in $n$ and $\alpha$. Our algorithms adapt to the current arboricity of the graph. Moreover, further analysis shows that they can yield a $(1 + \varepsilon)$-approximation of the arboricity or the subgraph density at the cost of increased update time.

翻译：我们提出了改进的全动态图边定向维护算法，使得最大出度有界。一方面，我们展示了如何在摊销更新时间复杂度为 $O(\log^2 n \log \alpha)$ 或最坏情况更新时间复杂度为 $O(\log^3 n \log \alpha)$ 的情况下，将边定向为最大出度与图的树度 $\alpha$ 成比例。另一方面，受动态最大匹配等应用启发，我们获得了不同的权衡。具体而言，对于每个顶点最多具有 $O(\alpha + \log n)$ 条外向边的边定向维护问题，我们实现了改进的摊销更新时间复杂度 $O(\log n \log \alpha)$ 或最坏情况更新时间复杂度 $O(\log^2 n \log \alpha)$。最后，我们所有算法均自然地将回溯次数限制为 $n$ 和 $\alpha$ 的多对数函数。我们的算法能够自适应调整至图的当前树度。进一步分析表明，通过增加更新时间复杂度，这些算法能够实现对树度或子图密度的 $(1 + \varepsilon)$-近似。