Distributed Exact Shortest Paths in Sublinear Time

Michael Elkin
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引用次数: 4

Abstract

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(D1/3 ⋅ (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/ log2 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(n5/3 ⋅ 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′.
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亚线性时间下的精确分布最短路径
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Bellman-Ford算法在O(n)时间内求解该问题,其中n为输入图g的顶点数。Peleg和Rubinovich[49]给出了该问题的下界为~ Ω(D +√n),其中D为g的跳径。当D相对较小时,该问题能否在O(n)时间内求解是一个主要的开放问题。尽管进行了深入的研究[10,17,33,41,45],为该问题的近似变体提供了近乎最优的算法,但对于原始问题没有任何进展。在本文中,我们肯定地回答这个问题。我们设计了一种算法,对于D = O(√n log n)需要O((n log n)5/6)时间,对于较大的D需要O(d3 /3·(n log n)2/3)时间。在几乎整个参数范围内,特别是对于D = O(n/ log2n),运行时间在n上是次线性的。我们还将结果推广到两个方向。一类是边带宽b≥1,另一类是s源最短路径问题。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边界。特别是,我们提供了一种全对最短路径算法,即使b = 1,对于所有d值也需要O(n5/3⋅log 2/3 n)时间。我们还设计了第一种具有非平凡复杂性保证的算法,用于计算多通道半流模型中精确的最短路径。从技术角度来看,我们的分布式算法不需要先计算G '本身,就可以计算G的骨架图G '的hopset G "。然后我们在G '∪G '中进行Bellman-Ford探索,同时动态计算G '的所需边。因此,我们的分布式算法精确地计算它真正需要的G '的那些边,而不是近似地计算整个G '。
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