  In this paper, we consider the problem of approximating the densest subgraph
in the dynamic graph stream model. In this model of computation, the input
graph is defined by an arbitrary sequence of edge insertions and deletions and
the goal is to analyze properties of the resulting graph given memory that is
sub-linear in the size of the stream. We present a single-pass algorithm that
returns a $(1+\epsilon)$ approximation of the maximum density with high
probability; the algorithm uses $O(\epsilon^{-2} n \polylog n)$ space,
processes each stream update in $\polylog (n)$ time, and uses $\poly(n)$
post-processing time where $n$ is the number of nodes. The space used by our
algorithm matches the lower bound of Bahmani et al.~(PVLDB 2012) up to a
poly-logarithmic factor for constant $\epsilon$. The best existing results for
this problem were established recently by Bhattacharya et al.~(STOC 2015). They
presented a $(2+\epsilon)$ approximation algorithm using similar space and
another algorithm that both processed each update and maintained a
$(4+\epsilon)$ approximation of the current maximum density in $\polylog (n)$
time per-update.
