Adaptive Out-Orientations with Applications

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.