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：我们提出了改进的算法，用于维护完全动态图的边定向问题，使得每个顶点的出度有界。一方面，我们展示了如何定向边使得每个顶点的出度与图的可达性树数α成正比，在均摊更新时间为O(log² n log α)或最坏情况更新时间为O(log³ n log α)的条件下实现。另一方面，受动态最大匹配等应用启发，我们获得了另一种权衡：对于维护每个顶点出度不超过O(α + log n)的边定向问题，均摊时间O(log n log α)或最坏情况时间O(log² n log α)。由于我们的算法具有最坏情况保证的更新时间，解的变化次数（即重排成本）自然受限。算法自适应于图的当前可达性树数，并在先前工作基础上取得了改进：首先，我们获得了维护最大子图密度ρ的(1+ε)近似的最坏情况更新时间为O(ε^{-6} log³ n log ρ)的算法。其次，我们获得了维护具有自适应可达性树数α的图的最优定向的(1+ε)·OPT+2近似的最坏情况更新时间为O(ε^{-6} log³ n log α)的算法。这首次实现了将图分解为O(α)个森林的最坏情况多对数动态算法。第三，我们针对一系列问题（包括最大匹配、Δ+1着色和矩阵向量乘法）获得了可达性树数自适应的完全动态确定性算法。所有更新时间均为最坏情况O(α+log² n log α)，其中α是图的当前可达性树数。