Negative-Weight Single-Source Shortest Paths in Near-linear Time
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We present a randomized algorithm that computes single-source shortest paths (SSSP) in O(mlog^8(n)log W) time when edge weights are integral and can be negative. This essentially resolves the classic negative-weight SSSP problem. The previous bounds are Õ((m+n^1.5)log W) [BLNPSSSW FOCS'20] and m^4/3+o(1)log W [AMV FOCS'20]. Near-linear time algorithms were known previously only for the special case of planar directed graphs [Fakcharoenphol and Rao FOCS'01]. In contrast to all recent developments that rely on sophisticated continuous optimization methods and dynamic algorithms, our algorithm is simple: it requires only a simple graph decomposition and elementary combinatorial tools. In fact, ours is the first combinatorial algorithm for negative-weight SSSP to break through the classic Õ(m√(n)log W) bound from over three decades ago [Gabow and Tarjan SICOMP'89].
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