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 + n 3/2 ). 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.

中文翻译：

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具有接近最优时间和消息复杂性的简单确定性分布式 MST 算法

分布式最小生成树（MST）问题是分布式图算法中最核心、最基础的问题之一。Kutten 和 Peleg 设计了一种具有运行时间的算法○(D+ √n⋅ 日志* n）， 在哪里D是输入的跃点直径n-顶点米-边图，并具有消息复杂性○(米+n 3/2）。Peleg 和 Rubinovich 表明 Kutten 和 Peleg 算法的运行时间本质上是紧张的，并询问是否可以实现接近最优的运行时间和接近最优的消息复杂度。在最近的一项突破中，Pandurangan 等人。肯定地回答了这个问题，并设计了一个随时间变化的随机算法Õ(D+ √n) 和消息复杂度Õ(米）。他们询问是否可以通过确定性算法实现这种同时的时间和消息最优性。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，可以及时计算 MST○((D+ √n) ⋅ 对数n） 使用○(米⋅ 日志n+n日志n点日志* n) 消息。我们算法的时间和消息复杂度中的多对数因素明显小于 Pandurangan 等人的结果中的各个因素。此外，与之前相当复杂的亚线性时间算法相比，我们的算法及其分析非常简单且独立。最后，我们使用我们的新算法设计了一个具有运行时间的随机 MST 算法Õ(μ (G,ω) + √n) 和消息复杂度Õ(|乙|), 其中 μ 半径 μ (G,ω) ≤D是一个图参数，通常远小于D. 这改进了 Elkin 先前的界限。