Distributed exact weighted all-pairs shortest paths in near-linear time

In the distributed all-pairs shortest paths problem (APSP), every node in the weighted undirected distributed network (the CONGEST model) needs to know the distance from every other node using least number of communication rounds (typically called time complexity). The problem admits (1 + o(1))-approximation Θ (n)time algorithm and a nearly-tight Ω (n) lower bound [Nanongkai, STOC’14; Lenzen and Patt-Shamir PODC’15]. For the exact case, Elkin [STOC’17] presented an O(n⁵/³ log²/³ n) time bound, which was later improved to Õ(n⁵/⁴) in [Huang, Nanongkai, Saranurak FOCS’17]. It was shown that any super-linear lower bound (in n) requires a new technique [Censor-Hillel, Khoury, Paz, DISC’17], but otherwise it remained widely open whether there exists a Õ(n)-time algorithm for the exact case, which would match the best possible approximation algorithm. This paper resolves this question positively: we present a randomized (Las Vegas) Õ(n)-time algorithm, matching the lower bound up to polylogarithmic factors. Like the previous Õ(n⁵/⁴) bound, our result works for directed graphs with zero (and even negative) edge weights. In addition to the improved running time, our algorithm works in a more general setting than that required by the previous Õ(n⁵/⁴) bound; in our setting (i) the communication is only along edge directions (as opposed to bidirectional), and (ii) edge weights are arbitrary (as opposed to integers in (1, 2, . . ., poly(n))). The previously best algorithm for this more difficult setting required Õ(n³/²) time [Agarwal and Ramachandran, ArXiv’18] (this can be improved to Õ(n⁴/³) if one allows bidirectional communication). Our algorithm is extremely simple and relies on a new technique called Random Filtered Broadcast. Given any sets of nodes A, B ⊆ V and assuming that every b ∈ B knows all distances from nodes in A, and every node v ∈ V knows all distances from nodes in B, we want every v ∈ V to know DistThrough_B(a,v) = min_b∈_B dist(a,b) + dist(b,v) for every a ∈ A. Previous works typically solve this problem by broadcasting all knowledge of every b ∈ B, causing super-linear edge congestion and time. We show a randomized algorithm that can reduce edge congestions and thus solve this problem in Õ(n) expected time.