The distributed single-source shortest paths problem is one of the most fundamental and central problems in the message-passing distributed computing. Classical Bellman-Ford algorithm solves it in O ( n ) time, where n is the number of vertices in the input graph G . Peleg and Rubinovich [49] showed a lower bound of ˜Ω( D + √ n ) for this problem, where D is the hop-diameter of G . Whether or not this problem can be solved in O ( n ) time when D is relatively small is a major open question. Despite intensive research [10, 17, 33, 41, 45] that yielded near-optimal algorithms for the approximate variant of this problem, no progress was reported for the original problem. In this article, we answer this question in the affirmative. We devise an algorithm that requires O (( n log n ) 5/6 ) time, for D = O (√ n log n ), and O ( D 1/3 ⋅ ( n log n ) 2/3 ) time, for larger D . This running time is sublinear in n in almost the entire range of parameters, specifically, for D = o ( n / log 2 n ). We also generalize our result in two directions. One is when edges have bandwidth b ≥ 1, and the other is the s -sources shortest paths problem. For both problems, our algorithm provides bounds that improve upon the previous state-of-the-art in almost the entire range of parameters. In particular, we provide an all-pairs shortest paths algorithm that requires O ( n 5/3 ⋅ log 2/3 n ) time, even for b = 1, for all values of D . We also devise the first algorithm with non-trivial complexity guarantees for computing exact shortest paths in the multipass semi-streaming model of computation. From the technical viewpoint, our distributed algorithm computes a hopset G ′′ of a skeleton graph G ′ of G without first computing G ′ itself. We then conduct a Bellman-Ford exploration in G ′ ∪ G ′′ , while computing the required edges of G ′ on the fly . As a result, our distributed algorithm computes exactly those edges of G ′ that it really needs, rather than computing approximately the entire G ′ .

中文翻译：

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次线性时间内分布的精确最短路径

分布式单源最短路径问题是消息传递分布式计算中最基本、最核心的问题之一。经典的 Bellman-Ford 算法解决了它○(n) 时间，在哪里n是输入图中的顶点数G. Peleg 和 Rubinovich [49] 显示了 ∼Ω(D+ √n) 对于这个问题，其中D是跳跃直径G. 这个问题能不能解决○(n) 时间D相对较小是一个主要的悬而未决的问题。尽管深入研究 [10, 17, 33, 41, 45] 产生了近乎最优的算法近似此问题的变体，原始问题未报告任何进展。在这篇文章中，我们肯定地回答了这个问题。我们设计了一个算法，需要○((n日志n)5/6） 的时间D=○(√n日志n）， 和○(D 1/3⋅ (n日志n)2/3) 时间，对于更大的D. 这个运行时间是次线性的n在几乎整个参数范围内，特别是对于D=○(n/ 日志2 n）。我们还将我们的结果推广到两个方向。一种是当边缘有带宽时b≥1，另一个是s-sources 最短路径问题。对于这两个问题，我们的算法提供了在几乎整个参数范围内改进先前最先进技术的界限。特别是，我们提供了一个全对最短路径算法，它需要○(n 5/3⋅ 日志2/3 n) 时间，即使对于b= 1，对于所有值D. 我们还设计了第一个具有非平凡复杂性保证的算法，用于计算精确的最短路径多通道半流计算模型。从技术角度来看，我们的分布式算法计算了一个 hopsetG ''骨架图的G '的G 无需先计算 G '本身。然后我们在G '∪G ''，同时计算所需的边G ' 在飞行中. 因此，我们的分布式算法计算确切地那些边缘G '它确实需要，而不是大约计算整个G '.