New incremental (1+ε)-approximate APSP data structures for sparse digraphs reach O~(m^{3/2}n^{3/4}) deterministic and O~(m^{4/3}n^{5/6}) randomized total update time against adaptive adversaries, plus a near-linear offline SSSP algorithm.
More asymmetry yields faster matrix multiplication
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On Incremental Approximate Shortest Paths in Directed Graphs
New incremental (1+ε)-approximate APSP data structures for sparse digraphs reach O~(m^{3/2}n^{3/4}) deterministic and O~(m^{4/3}n^{5/6}) randomized total update time against adaptive adversaries, plus a near-linear offline SSSP algorithm.