We give simple 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 $O(\log^2 n \log \alpha)$. On the other hand, motivated by applications in dynamic maximal matching, we obtain a different trade-off, namely the improved worst case update time of $O(\log n \log \alpha)$ 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 make choices based entirely on local information, which makes them automatically adaptive to the current arboricity of the graph. In other words, they are arboricity-oblivious, while they are arboricity-sensitive. This both simplifies and improves upon previous work, by having fewer assumptions or better asymptotic guarantees. As a consequence, one obtains an algorithm with improved efficiency for maintaining a $(1+\varepsilon)$ approximation of the maximum subgraph density, and an algorithm for dynamic maximal matching whose worst-case update time is guaranteed to be upper bounded by $O(\alpha + \log n\log \alpha)$, where $\alpha$ is the arboricity at the time of the update.
翻译:我们用简单的算法来维持一个完全动态的图形的边缘方向, 这样每个顶点的偏差度是相近的。 一方面, 我们展示如何调整边缘的偏差度, 这样每个顶点的偏差度与图表的偏差值成正比值 $\ ALpha$, 最差的更新时间是 $O (\ log2, n\ log\ ALpha) 。 另一方面, 在动态最大匹配应用程序的驱动下, 我们得到了不同的取舍, 即美元( log n ) 的最大偏差更新时间是 $O( log n log ) 的最大偏差时间值。 一方面, 我们的算法将最差的偏差值更新时间值更新到 $( alphalpha) 。 一方面, 最差的直径差的直径比值更新时间值是更短的。 一方面, 最差的直径直径比值是比值更新的, 其直径更近的直值是更短的。, 其直径直到直径更近的直的直径比值更新, 。