Node-Weighted APSP is solvable in Õ(n^{2.686}) time (n^{2.5} if omega=2), and APSP with at most n^{3-omega-delta} distinct weights per node is solvable in subcubic time.
Subcubic equivalences between graph centrality problems, apsp, and diameter
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All-Pairs Shortest Paths with Few Weights per Node
Node-Weighted APSP is solvable in Õ(n^{2.686}) time (n^{2.5} if omega=2), and APSP with at most n^{3-omega-delta} distinct weights per node is solvable in subcubic time.