We consider the problem of maintaining a $(1-\epsilon)$-approximation to the densest subgraph (DSG) in an undirected multigraph as it undergoes edge insertions and deletions (the fully dynamic setting). Sawlani and Wang [SW20] developed a data structure that, for any given $\epsilon > 0$, maintains a $(1-\epsilon)$-approximation with $O(\log^4 n/\epsilon^6)$ worst-case update time for edge operations, and $O(1)$ query time for reporting the density value. Their data structure was the first to achieve near-optimal approximation, and improved previous work that maintained a $(1/4 - \epsilon)$ approximation in amortized polylogarithmic update time [BHNT15]. In this paper we develop a data structure for $(1-\epsilon)$-approximate DSG that improves the one from [SW20] in two aspects. First, the data structure uses linear space improving the space bound in [SW20] by a logarithmic factor. Second, the data structure maintains a $(1-\epsilon)$-approximation in amortized $O(\log^2 n/\epsilon^4)$ time per update while simultaneously guaranteeing that the worst case update time is $O(\log^3 n \log \log n/\epsilon^6)$. We believe that the space and update time improvements are valuable for current large scale graph data sets. The data structure extends in a natural fashion to hypergraphs and yields improvements in space and update times over recent work [BBCG22] that builds upon [SW20].

翻译：我们考虑在未定向的多面图中将美元(1-\ epsilon) 美元($1-\ epsilon) 和美元($1-\ epsilon) 与最稠密的子谱(DSG) 同步化的问题。 Sawlani 和 Wang [SW20] 开发了一个数据结构,对于任何给定的 $( epsilon > 0),将美元( 1-\\ epsilon) 和美元( epsilon) 与美元( 最坏的) 最坏的情况更新时间( DSG), 和美元(1美元) 来报告密度值。它们的数据结构是第一个实现接近最优化的插入和删除( 完全动态) 。 数据结构将保持美元( 1/4\\ \ \ \ eepsialsonlon) 的缩放速度( mexcial) 数据更新到 meximal 数据。