Undirected $(1+\varepsilon)$-Shortest Paths via Minor-Aggregates: Near-Optimal Deterministic Parallel & Distributed Algorithms

TL;DR

This paper introduces near-optimal deterministic parallel and distributed algorithms for approximate shortest paths in undirected graphs, utilizing novel tools like minor-aggregates and low-diameter decompositions to achieve efficiency and simplicity.

Contribution

The paper develops a deterministic reduction of shortest path problems to Minor-Aggregations, leading to optimal parallel and distributed algorithms with new techniques for graph decomposition and routing.

Findings

Deterministic parallel algorithms with $ ilde{O}(1)$ depth and near-linear work.

Universal deterministic distributed algorithms with optimal round complexity.

New tools for graph decomposition, routing, and flow rounding.

Abstract

This paper presents near-optimal deterministic parallel and distributed algorithms for computing $(1+\varepsilon)$-approximate single-source shortest paths in any undirected weighted graph. On a high level, we deterministically reduce this and other shortest-path problems to $\tilde{O}(1)$ 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 $\tilde{O}(1)$ depth and near-linear work. Universally-optimal deterministic distributed (CONGEST) algorithms, whenever deterministic Minor-Aggregate algorithms exist. For example, an optimal $\tilde{O}(HopDiameter(G))$-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 $\frac{D}{2}$". 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 $\tilde{O}(1)$-competitive $\ell_1$-oblivious routing based on low-diameter decompositions that can be evaluated in near-linear work. The previous such routing [ZGY+20] was $n^{o(1)}$-competitive and required $n^{o(1)}$ 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.