Abstract
This paper presents parallel and distributed algorithms for single-source
shortest paths when edges can have negative weights (negative-weight SSSP). We
show a framework that reduces negative-weight SSSP in either setting to
$n^{o(1)}$ calls to any SSSP algorithm that works with a virtual source. More
specifically, for a graph with $m$ edges, $n$ vertices, undirected hop-diameter
$D$, and polynomially bounded integer edge weights, we show randomized
algorithms for negative-weight SSSP with (i) $W_{SSSP}(m,n)n^{o(1)}$ work and
$S_{SSSP}(m,n)n^{o(1)}$ span, given access to an SSSP algorithm with
$W_{SSSP}(m,n)$ work and $S_{SSSP}(m,n)$ span in the parallel model, (ii)
$T_{SSSP}(n,D)n^{o(1)}$, given access to an SSSP algorithm that takes
$T_{SSSP}(n,D)$ rounds in $\mathsf{CONGEST}$. This work builds off the recent
result of [Bernstein, Nanongkai, Wulff-Nilsen, FOCS'22], which gives a
near-linear time algorithm for negative-weight SSSP in the sequential setting.
Using current state-of-the-art SSSP algorithms yields randomized algorithms
for negative-weight SSSP with (i) $m^{1+o(1)}$ work and $n^{1/2+o(1)}$ span in
the parallel model, (ii) $(n^{2/5}D^{2/5} + \sqrt{n} + D)n^{o(1)}$ rounds in
$\mathsf{CONGEST}$.
Our main technical contribution is an efficient reduction for computing a
low-diameter decomposition (LDD) of directed graphs to computations of SSSP
with a virtual source. Efficiently computing an LDD has heretofore only been
known for undirected graphs in both the parallel and distributed models. The
LDD is a crucial step of the algorithm in [Bernstein, Nanongkai, Wulff-Nilsen,
FOCS'22], and we think that its applications to other problems in parallel and
distributed models are far from being exhausted.
