Universally-Optimal Distributed Shortest Paths and Transshipment via Graph-Based ℓ1-Oblivious Routing

We provide universally-optimal distributed graph algorithms for (1+∊)-approximate shortest path problems including shortest-path-tree and transshipment. The universal optimality of our algorithms guarantees that, on any n-node network G, our algorithm completes in T · no(1) rounds whenever a T-round algorithm exists for G. This includes D · no(1)-round algorithms for any planar or excluded-minor network. Our algorithms never require more than rounds, resulting in the first sub-linear-round distributed algorithm for transshipment. The key technical contribution leading to these results is the first efficient no(1)-competitive linear ℓ1-oblivious routing operator that does not require the use of ℓ1-embeddings. Our construction is simple, solely based on low-diameter decompositions, and—in contrast to all known constructions—directly produces an oblivious flow instead of just an approximation of the optimal flow cost. This also has the benefit of simplifying the interaction with Sherman's multiplicative weight framework [SODA'17] in the distributed setting and its subsequent rounding procedures.