Faster Parallel Batch-Dynamic Algorithms for Low Out-Degree Orientation

arXiv:2602.17811v1 Announce Type: new
Abstract: 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.