A (fully) dynamic graph algorithm is a data structure that supports edge insertions, edge deletions, and answers specific queries pertinent to the problem at hand. In this work, we address the fully dynamic edge orientation problem, also known as the fully dynamic $\Delta$-orientation problem. The objective is to maintain an orientation of the edges in an undirected graph such that the out-degree of any vertex remains low. When edges are inserted or deleted, it may be necessary to reorient some edges to prevent vertices from having excessively high out-degrees. In this paper, we introduce the first algorithm that maintains an optimal edge orientation during both insertions and deletions. In experiments comparing with recent nearly exact algorithms, we achieve a 32% lower running time. The update time of our algorithm is up to 6 orders of magnitude faster than static exact algorithms.

翻译：全动态图算法是一种支持边插入、边删除并回答与特定问题相关查询的数据结构。本研究针对全动态边定向问题（亦称全动态Δ-定向问题），其目标是在无向图中维持边的定向，使得任意顶点的出度保持较低水平。当边被插入或删除时，可能需要重新定向部分边以防止顶点出度过高。本文提出了首个在插入和删除操作中均能维持最优边定向的算法。在与近期近似精确算法的对比实验中，本算法实现了运行时间降低32%的效果，其更新速度比静态精确算法快达6个数量级。