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In this work we revisit the fundamental Single-Source Shortest Paths (SSSP)problem with possibly negative edge weights. A recent breakthrough result byBernstein, 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 boundon the magnitude of the smallest negative-weight edge. In this work we improvethe running time to $O(m \log^2(n) \log(nW) \log\log n)$, which is animprovement by nearly six log-factors. Some of these log-factors are easy toshave (e.g. replacing the priority queue used in Dijkstra's algorithm), whileothers are significantly more involved (e.g. to find negative cycles we designan algorithm reminiscent of noisy binary search and analyze it with driftanalysis).  As side results, we obtain an algorithm to compute the minimum cycle mean inthe same running time as well as a new construction for computing Low-DiameterDecompositions in directed graphs.

Negative-Weight Single-Source Shortest Paths in Near-Linear Time: Now Faster!