We give improved algorithms for maintaining edge-orientations of a fully-dynamic graph, such that the out-degree of each vertex is bounded. On one hand, we show how to orient the edges such that the out-degree of each vertex 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 either $O(\log n \log \alpha)$, amortised, or $O(\log ^2 n \log \alpha)$, worst-case time, for the problem of maintaining an edge-orientation with at most $O(\alpha + \log n)$ out-edges per vertex. Since our algorithms have update times with worst-case guarantees, the number of changes to the solution (i.e. the recourse) is naturally limited. Our algorithms adapt to the current arboricity of the graph, and yield improvements over previous work: Firstly, we obtain an $O(\varepsilon^{-6}\log^3 n \log \rho)$ worst-case update time algorithm for maintaining a $(1+\varepsilon)$ approximation of the maximum subgraph density, $\rho$. Secondly, we obtain an $O(\varepsilon^{-6}\log^3 n \log \alpha)$ worst-case update time algorithm for maintaining a $(1 + \varepsilon) \cdot OPT + 2$ approximation of the optimal out-orientation of a graph with adaptive arboricity $\alpha$. This yields the first worst-case polylogarithmic dynamic algorithm for decomposing into $O(\alpha)$ forests.Thirdly, we obtain arboricity-adaptive fully-dynamic deterministic algorithms for a variety, of problems including maximal matching, $\Delta+1$ coloring, and matrix vector multiplication. All update times are worst-case $O(\alpha+\log^2n \log \alpha)$, where $\alpha$ is the current arboricity of the graph.

翻译：我们针对全动态图的边定向维护问题提出了改进算法，使得每个顶点的出度有界。一方面，我们展示如何在摊销更新时间为$O(\log^2 n \log \alpha)$或最坏情况更新时间为$O(\log^3 n \log \alpha)$的条件下，将边定向为每个顶点的出度与图的树性$\alpha$成比例。另一方面，受包括动态最大匹配在内的应用驱动，我们获得了另一种权衡方案：在维护每个顶点出度最多为$O(\alpha + \log n)$的边定向问题时，可实现摊销时间$O(\log n \log \alpha)$或最坏情况时间$O(\log^2 n \log \alpha)$。由于算法具有最坏情况保证的更新时间，解的变化次数（即回溯）自然受限。算法自适应于图的当前树性，并较先前工作取得以下改进：首先，我们针对最大子图密度$\rho$的$(1+\varepsilon)$近似维护问题，获得了最坏情况更新时间为$O(\varepsilon^{-6}\log^3 n \log \rho)$的算法；其次，针对自适应树性$\alpha$的图，我们实现了最坏情况更新时间为$O(\varepsilon^{-6}\log^3 n \log \alpha)$的算法，用于维护最优外向定向的$(1 + \varepsilon) \cdot OPT + 2$近似，由此得到首个最坏情况多项式对数时间的动态分解为$O(\alpha)$个森林的算法；第三，我们为包括最大匹配、$\Delta+1$着色和矩阵向量乘法在内的多种问题，设计了树性自适应的全动态确定性算法。所有更新时间在最坏情况下均为$O(\alpha+\log^2n \log \alpha)$，其中$\alpha$为图的当前树性。