Undirected (1+)-shortest paths via minor-aggregates: near-optimal deterministic parallel and distributed algorithms

This paper presents near-optimal deterministic parallel and distributed algorithms for computing (1+eps)-approximate single-source shortest paths in any undirected weighted graph. On a high level, we deterministically reduce this and other shortest-path problems to Õ(1) Minor-Aggregations. A Minor-Aggregation computes an aggregate (e.g., max or sum) of node-values for every connected component of some subgraph. Our reduction immediately implies: Optimal deterministic parallel (PRAM) algorithms with Õ(1) depth and near-linear work. Universally-optimal deterministic distributed (CONGEST) algorithms, whenever deterministic Minor-Aggregate algorithms exist. For example, an optimal Õ(hopDiameterG)-round deterministic CONGEST algorithm for excluded-minor networks. Several novel tools developed for the above results are interesting in their own right: A local iterative approach for reducing shortest path computations "up to distance D"to computing low-diameter decompositions "up to distance D/2". Compared to the recursive vertex-reduction approach of [Li20], our approach is simpler, suitable for distributed algorithms, and eliminates many derandomization barriers. A simple graph-based Õ(1)-competitive g.,"1-oblivious routing based on low-diameter decompositions that can be evaluated in near-linear work. The previous such routing [ZGY+20] was no(1)-competitive and required no(1) more work. A deterministic algorithm to round any fractional single-source transshipment flow into an integral tree solution. The first distributed algorithms for computing Eulerian orientations