Distributed minimum spanning tree (MST) problem is one of the most central and fundamental problems in distributed graph algorithms. Garay et al. \cite{GKP98,KP98} devised an algorithm with running time $O(D + \sqrt{n} \cdot \log^* n)$, 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 \cite{PR99} showed that the running time of the algorithm of \cite{KP98} 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. \cite{PRS16} answered this question in the affirmative, and devised a **randomized** algorithm with time $\tilde{O}(D+ \sqrt{n})$ and message complexity $\tilde{O}(m)$. They asked if such a simultaneous time- and message-optimality can be achieved by a **deterministic** algorithm.
In this paper, building upon the work of \cite{PRS16}, we answer this question in the affirmative, and devise a **deterministic** algorithm that computes MST in time $O((D + \sqrt{n}) \cdot \log n)$, using $O(m \cdot \log n + n \log n \cdot \log^* n)$ messages. The polylogarithmic factors in the time and message complexities of our algorithm are significantly smaller than the respective factors in the result of \cite{PRS16}. Also, our algorithm and its analysis are very **simple** and self-contained, as opposed to rather complicated previous sublinear-time algorithms \cite{GKP98,KP98,E04b,PRS16}.
Construction and impromptu repair of an MST in a distributed network with o(m) communication
在CONGEST模型中，通信网络是无向图，其$n$节点是处理器，其$m$边是处理器之间的通信链路。在任何给定的时间步，大小为$O(\log n)$的消息可以由每个节点发送到其每个邻居。我们展示了同步模型：如果所有节点在同一轮中开始，并且每个节点知道其ID及其邻居的ID，或者在MST的情况下，知道其入射边的不同权重并且知道$n$，然后有蒙特卡罗算法成功w. h. p.确定最小生成森林（MST）和生成森林（ST）分别使用MST的$O(n \log^2 n/\log\log n)$消息和ST的$O(n \log n )$消息。这些结果与Awerbuch等人的“民间定理”相矛盾，JACM 1990指出，广播树的分布式构造需要$Ω(m)$消息。该下限已在那里和其他论文中针对一些CONGEST模型进行了展示;我们的协议证明了这些模型的局限性。动态分布式网络是一个经历在线边插入或删除。我们还展示了如何修复一个MST或ST在一个动态的网络与异步通信。边删除可以在MST中的$O(n\log n /\log \log n)$预期消息和ST问题的$O(n)$预期消息中处理，而边插入在最坏情况下使用$O(n)$消息。我们称之为“即兴”更新，因为我们假设在边缘更新的处理之间没有额外信息的预处理或存储。以前针对这个问题的算法在每次更新时使用分摊的$o(m)$消息，需要大量的预处理和更新之间的附加本地存储。
A singularly (near) optimal distributed algorithm is one that is (near) optimal in \emph{two} criteria, namely, its time and message complexities. For \emph{synchronous} CONGEST networks, such algorithms are known for fundamental distributed computing problems such as leader election [Kutten et al., JACM 2015] and Minimum Spanning Tree (MST) construction [Pandurangan et al., STOC 2017, Elkin, PODC 2017]. However, it is open whether a singularly (near) optimal bound can be obtained for the MST construction problem in general \emph{asynchronous} CONGEST networks. We present a randomized distribu…
This paper presents a randomized Las Vegas distributed algorithm that constructs a minimum spanning tree (MST) in weighted networks with optimal (up to polylogarithmic factors) time and message complexity. This algorithm runs in $\tilde{O}(D + \sqrt{n})$ time and exchanges $\tilde{O}(m)$ messages (both with high probability), where $n$ is the number of nodes of the network, $D$ is the diameter, and $m$ is the number of edges. This is the first distributed MST algorithm that matches \emph{simultaneously} the time lower bound of $\tildeΩ(D + \sqrt{n})$ [Elkin, SIAM J. Comput. 2006] and the messa…
Distributed graph algorithms that separately optimize for either the number of rounds used or the total number of messages sent have been studied extensively. However, algorithms simultaneously efficient with respect to both measures have been elusive. For example, only very recently was it shown that for Minimum Spanning Tree (MST), an optimal message and round complexity is achievable (up to polylog terms) by a single algorithm in the CONGEST model of communication. In this paper we provide algorithms that are simultaneously round- and message-optimal for a number of well-studied distributed…
A rich line of work has been addressing the computational complexity of locally checkable labelings (LCLs), illustrating the landscape of possible complexities. In this paper, we study the landscape of LCL complexities under bandwidth restrictions. Our main results are twofold. First, we show that on trees, the CONGEST complexity of an LCL problem is asymptotically equal to its complexity in the LOCAL model. An analog statement for general (non-LCL) problems is known to be false. Second, we show that for general graphs this equivalence does not hold, by providing an LCL problem for which we sh…
The Congested Clique model proposed by Lotker et al.[SICOMP'05] was introduced in order to provide a simple abstraction for overlay networks. Congested Clique is a model of distributed (or parallel) computing, in which there are $n$ players with unique identifiers from set [n], which perform computations in synchronous rounds. Each round consists of the phase of unlimited local computation and the communication phase. While communicating, each pair of players is allowed to exchange a single message of size $O(\log n)$ bits. Since, in a single round, each player can communicate with even $Θ(n)$…
In this paper, we show that the Minimum Spanning Tree problem can be solved \emph{deterministically}, in $\mathcal{O}(1)$ rounds of the $\mathsf{Congested}$ $\mathsf{Clique}$ model. In the $\mathsf{Congested}$ $\mathsf{Clique}$ model, there are $n$ players that perform computation in synchronous rounds. Each round consist of a phase of local computation and a phase of communication, in which each pair of players is allowed to exchange $\mathcal{O}(\log n)$ bit messages. The studies of this model began with the MST problem: in the paper by Lotker et al.[SPAA'03, SICOMP'05] that defines the $\ma…
This paper focuses on showing time-message trade-offs in distributed algorithms for fundamental problems such as leader election, broadcast, spanning tree (ST), minimum spanning tree (MST), minimum cut, and many graph verification problems. We consider the synchronous CONGEST distributed computing model and assume that each node has initial knowledge of itself and the identifiers of its neighbors - the so-called KT1 model - a well-studied model that also naturally arises in many applications. Recently, it has been established that one can obtain (almost) singularly optimal algorithms, i.e., al…
The field of dynamic graph algorithms aims at achieving a thorough understanding of real-world networks whose topology evolves with time. Traditionally, the focus has been on the classic sequential, centralized setting where the main quality measure of an algorithm is its update time, i.e. the time needed to restore the solution after each update. While real-life networks are very often distributed across multiple machines, the fundamental question of finding efficient dynamic, distributed graph algorithms received little attention to date. The goal in this setting is to optimize both the roun…
This paper concerns designing distributed algorithms that are singularly optimal, i.e., algorithms that are simultaneously time and message optimal, for the fundamental leader election problem in networks. Our main result is a randomized distributed leader election algorithm for asynchronous complete networks that is essentially (up to a polylogarithmic factor) singularly optimal. Our algorithm uses $O(n)$ messages with high probability and runs in $O(\log^2 n)$ time (with high probability) to elect a unique leader. The $O(n)$ message complexity should be contrasted with the $Ω(n \log n)$ lowe…
Department of Computer Science, University of Illinois Chicago, Chicago, IL 60607, USA；Department of Computer Science \& Engineering, IIT Madras, Chennai 600036, India；Security Research Unit, Indian Statistical Institute, Kolkata, 700108, India；Department of Computer Science, Kent State University, Kent, OH 44242, USA
We show that many classical optimization problems --- such as $(1\pmε)$-approximate maximum flow, shortest path, and transshipment --- can be computed in $\newcommand{\tmix}{τ_{\text{mix}}}\tmix(G)\cdot n^{o(1)}$ rounds of distributed message passing, where $\tmix(G)$ is the mixing time of the network graph $G$. This extends the result of Ghaffari et al.\ [PODC'17], whose main result is a distributed MST algorithm in $\tmix(G)\cdot 2^{O(\sqrt{\log n \log\log n})}$ rounds in the CONGEST model, to a much wider class of optimization problems. For many practical networks of interest, e.g., peer-to…
We introduce the Adaptive Massively Parallel Computation (AMPC) model, which is an extension of the Massively Parallel Computation (MPC) model. At a high level, the AMPC model strengthens the MPC model by storing all messages sent within a round in a distributed data store. In the following round, all machines are provided with random read access to the data store, subject to the same constraints on the total amount of communication as in the MPC model. Our model is inspired by the previous empirical studies of distributed graph algorithms using MapReduce and a distributed hash table service. …
In this paper we study the problem of testing graph isomorphism (GI) in the CONGEST distributed model. In this setting we test whether the distributive network, $G_U$, is isomorphic to $G_K$ which is given as an input to all the nodes in the network, or alternatively, only to a single node. We first consider the decision variant of the problem in which the algorithm distinguishes $G_U$ and $G_K$ which are isomorphic from $G_U$ and $G_K$ which are not isomorphic. We provide a randomized algorithm with $O(n)$ rounds for the setting in which $G_K$ is given only to a single node. We prove that for…
