A Simple Deterministic Distributed MST Algorithm with Near-Optimal Time and Message Complexities

Michael Elkin
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引用次数: 2

Abstract

The distributed minimum spanning tree (MST) problem is one of the most central and fundamental problems in distributed graph algorithms. Kutten and Peleg devised an algorithm with running time O(D + √n ⋅ log* n), where D is the hop diameter of the input n-vertex m-edge graph, and with message complexity O(m + n3/2). Peleg and Rubinovich showed that the running time of the algorithm of Kutten and Peleg is essentially tight and asked if one can achieve near-optimal running time together with near-optimal message complexity. In a recent breakthrough, Pandurangan et al. answered this question in the affirmative and devised a randomized algorithm with time Õ(D+ √ n) and message complexity Õ(m). They asked if such a simultaneous time- and message optimality can be achieved by a deterministic algorithm. In this article, building on the work of Pandurangan et al., we answer this question in the affirmative and devise a deterministic algorithm that computes MST in time O((D + √ n) ⋅ log n) using O(m ⋅ log n + n log n cdot log* n) messages. The polylogarithmic factors in the time and message complexities of our algorithm are significantly smaller than the respective factors in the result of Pandurangan et al. In addition, our algorithm and its analysis are very simple and self-contained as opposed to rather complicated previous sublinear-time algorithms. Finally, we use our new algorithm to devise a randomized MST algorithm with running time Õ(μ (G,ω) + √ n) and message complexity Õ(|E|), where μ-radius μ (G,ω) ≤ D is a graph parameter, which is typically much smaller than D. This improves a previous bound from Elkin.
具有近最优时间复杂度和消息复杂度的简单确定性分布式MST算法
分布式最小生成树问题是分布式图算法中最核心、最基本的问题之一。Kutten和Peleg设计了一种运行时间为O(D +√n⋅log* n)的算法,其中D为输入n顶点m边图的跳直径,消息复杂度为O(m + n3/2)。Peleg和Rubinovich表明Kutten和Peleg算法的运行时间本质上是紧的,并询问是否可以在接近最优的消息复杂度下实现接近最优的运行时间。在最近的一项突破中,Pandurangan等人肯定地回答了这个问题,并设计了一个随机算法,时间为Õ(D+√n),消息复杂度为Õ(m)。他们问,这种同时的时间和消息最优性是否可以通过确定性算法实现。在本文中,基于Pandurangan等人的工作,我们肯定地回答了这个问题,并设计了一种确定性算法,该算法使用O(m⋅log n + n log n cdot log* n)条消息计算时间为O((D +√n)⋅log n)的MST。我们算法的时间复杂度和消息复杂度的多对数因子明显小于Pandurangan等人的结果中各自的因子。此外,我们的算法及其分析相对于以往复杂的亚线性时间算法来说非常简单和独立。最后,我们利用新算法设计了一个随机化的MST算法,其运行时间为Õ(μ (G,ω) +√n),消息复杂度为Õ(|E|),其中μ-半径μ (G,ω)≤D是一个图参数,通常比D小得多。
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