A low out-degree orientation directs each edge of an undirected graph with the goal of minimizing the maximum out-degree of a vertex. In the parallel batch-dynamic setting, one can insert or delete batches of edges, and the goal is to process the entire batch in parallel with work per edge similar to that of a single sequential update and with span (or depth) for the entire batch that is polylogarithmic. In this paper we present faster parallel batch-dynamic algorithms for maintaining a low out-degree orientation of an undirected graph. All results herein achieve polylogarithmic depth, with high probability (whp); the focus of this paper is on minimizing the work, which varies across results. Our first result is the first parallel batch-dynamic algorithm to maintain an asymptotically optimal orientation with asymptotically optimal expected work bounds, in an amortized sense, improving over the prior best work bounds of Liu et al.~[SPAA~'22] by a logarithmic factor. Our second result is a $O(c \log n)$ orientation algorithm with expected worst-case $O(\sqrt{\log n})$ work per edge update, where $c$ is a known upper-bound on the arboricity of the graph. This matches the best-known sequential worst-case $O(c \log n)$ orientation algorithm given by Berglin and Brodal ~[Algorithmica~'18], albeit in expectation. Our final result is a $O(c + \log n)$-orientation algorithm with $O(\log^2 n)$ expected worst-case work per edge update. This algorithm significantly improves upon the recent result of Ghaffari and Koo~[SPAA~'25], which maintains a $O(c)$-orientation with $O(\log^9 n)$ worst-case work per edge whp.

翻译：低出度定向通过为无向图的每条边指定方向，以最小化顶点的最大出度为目标。在并行批量动态场景中，可以插入或删除批量边，目标是以接近单次顺序更新的每边工作量并行处理整个批量，同时使整个批量的跨度（或深度）为多对数级别。本文提出了更快的并行批量动态算法，用于维护无向图的低出度定向。本文所有结果均以高概率实现多对数深度；本文重点在于最小化工作量，不同结果的工作量有所不同。我们的第一个结果是首个在摊销意义上以渐近最优的期望工作量界限维持渐近最优定向的并行批量动态算法，将Liu等人[SPAA'22]先前的最佳工作量界限改进了一个对数因子。我们的第二个结果是$O(c \log n)$定向算法，其每边更新的期望最坏情况工作量为$O(\sqrt{\log n})$，其中$c$是图树宽度的已知上界。这与Berglin和Brodal[Algorithmica'18]提出的最佳已知顺序最坏情况$O(c \log n)$定向算法相匹配，尽管是在期望意义上。我们的最终结果是$O(c + \log n)$定向算法，其每边更新的期望最坏情况工作量为$O(\log^2 n)$。该算法显著改进了Ghaffari和Koo[SPAA'25]的最新结果，后者以高概率维持$O(c)$定向，每边最坏情况工作量为$O(\log^9 n)$。