  In this work we revisit the fundamental Single-Source Shortest Paths (SSSP)
problem with possibly negative edge weights. A recent breakthrough result by
Bernstein, Nanongkai and Wulff-Nilsen established a near-linear $O(m \log^8(n)
\log(W))$-time algorithm for negative-weight SSSP, where $W$ is an upper bound
on the magnitude of the smallest negative-weight edge. In this work we improve
the running time to $O(m \log^2(n) \log(nW) \log\log n)$, which is an
improvement by nearly six log-factors. Some of these log-factors are easy to
shave (e.g. replacing the priority queue used in Dijkstra's algorithm), while
others are significantly more involved (e.g. to find negative cycles we design
an algorithm reminiscent of noisy binary search and analyze it with drift
analysis).
  As side results, we obtain an algorithm to compute the minimum cycle mean in
the same running time as well as a new construction for computing Low-Diameter
Decompositions in directed graphs.
