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.
翻译:本文展示了接近最佳的确定性平行和分布式算法, 用于计算 $( 1 ⁇ varepsilon) 和 $( 1 ⁇ varepsilon) 的近似单一来源的最短路径。 在高水平上, 我们确定性地将这个和其他最短路径的问题降低到 $( tilde{ O} $ 美元 ) 的最小分类法。 一个小分类为某些子图的每个连接部分计算一个( 例如, 最大或总) 共( ) 无偏差值 ) 。 我们的削减性( PRAM) 的优化确定性平行算法( PRAM) 以 $( tilldede) =O} 深度和近线性工作。 通用性最佳确定性确定性算法( CONEONEGEST) 将一个最优的 $( $( G) 20 美元 ) 的确定性 CONEST 值算法( ) 用于排除性网络 。 为上述结果而开发的一些新工具本身很有意思 : 更清洁的 降低性 路径计算法 至 至 最短的计算法 。