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 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 worst case update time of $O(\log ^2 n \log \alpha)$ 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：我们给出了改进的算法，用于维护全动态图的边定向，使得每个顶点的出度有界。一方面，我们展示了如何在最坏情况更新时间为$O(\log^3 n \log \alpha)$的条件下，对边进行定向，使得每个顶点的出度与图的树性$\alpha$成正比。另一方面，受动态最大匹配等应用的启发，我们获得了另一种权衡，即在每个顶点出度最多为$O(\alpha + \log n)$的边定向维护问题中，实现了改进的最坏情况更新时间$O(\log^2 n \log \alpha$)。由于我们的算法具有最坏情况保证的更新时间，解的变更次数（即追索）自然受到限制。我们的算法能自适应图当前的树性，并在此前工作基础上取得改进：首先，我们获得了维护最大子图密度$\rho$的$(1+\varepsilon)$近似算法的$O(\varepsilon^{-6}\log^3 n \log \rho)$最坏情况更新时间算法。其次，对于具有自适应树性$\alpha$的图，我们获得了维护最优外向定向的$(1 + \varepsilon) \cdot OPT + 2$近似算法的$O(\varepsilon^{-6}\log^3 n \log \alpha)$最坏情况更新时间算法，这首次实现了将图分解为$O(\alpha)$个森林的最坏情况多对数动态算法。第三，我们针对多种问题（包括最大匹配、$\Delta+1$着色和矩阵向量乘法）获得了树性自适应的全动态确定性算法。所有更新时间均为最坏情况$O(\alpha+\log^2n \log \alpha)$，其中$\alpha$是图的当前树性。