Universally-Optimal Distributed Shortest Paths and Transshipment via Graph-Based L1-Oblivious Routing
(2110.15944)Abstract
We provide universally-optimal distributed graph algorithms for $(1+\varepsilon)$-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 $T \cdot n{o(1)}$ rounds whenever a $T$-round algorithm exists for $G$. This includes $D \cdot n{o(1)}$-round algorithms for any planar or excluded-minor network. Our algorithms never require more than $(\sqrt{n} + D) \cdot n{o(1)}$ rounds, resulting in the first sub-linear-round distributed algorithm for transshipment. The key technical contribution leading to these results is the first efficient $n{o(1)}$-competitive linear $\ell1$-oblivious routing operator that does not require the use of $\ell1$-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.
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