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

2026-02-19

AI Summary

This work addresses the problem of efficiently maintaining a low out-degree orientation in dynamic graphs under parallel batch updates. It presents the first randomized parallel algorithm achieving asymptotically optimal out-degree and work bounds. Leveraging the graph’s arboricity parameter and combining amortized with worst-case analysis, the algorithm ensures polylogarithmic depth while significantly reducing the expected work per edge update. The main contributions are three novel algorithms: the first achieves asymptotically optimal amortized work; the second guarantees out-degree $O(c \log n)$ with $O(\sqrt{\log n})$ expected worst-case work per edge; and the third attains out-degree $O(c + \log n)$ with $O(\log^2 n)$ expected worst-case work per edge—substantially improving upon the previous $O(\log^9 n)$ bound and matching the expected performance of the best sequential algorithm.

Technology Category

• Reasoning under Uncertainty: Stochastic Optimization
• Data Mining & Knowledge Management: Scalability, Parallel & Distributed Systems
• Machine Learning: Graph-based Machine Learning

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

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.

Problem

• low out-degree orientation
• parallel batch-dynamic
• graph orientation
• arboricity
• dynamic graph algorithms

Innovation

• parallel batch-dynamic algorithms
• low out-degree orientation
• graph arboricity
• polylogarithmic depth
• amortized work