  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].
