We study the distributed minimum spanning tree (MST) problem, a fundamental problem in distributed computing. It is well-known that distributed MST can be solved in $\tilde{O}(D+\sqrt{n})$ rounds in the standard CONGEST model (where $n$ is the network size and $D$ is the network diameter) and this is essentially the best possible round complexity (up to logarithmic factors). However, in resource-constrained networks such as ad hoc wireless and sensor networks, nodes spending so much time can lead to significant spending of resources such as energy. Motivated by the above consideration, we study distributed algorithms for MST under the \emph{sleeping model} [Chatterjee et al., PODC 2020], a model for design and analysis of resource-efficient distributed algorithms. In the sleeping model, a node can be in one of two modes in any round -- \emph{sleeping} or \emph{awake} (unlike the traditional model where nodes are always awake). Only the rounds in which a node is \emph{awake} are counted, while \emph{sleeping} rounds are ignored. A node spends resources only in the awake rounds and hence the main goal is to minimize the \emph{awake complexity} of a distributed algorithm, the worst-case number of rounds any node is awake. We present deterministic and randomized distributed MST algorithms that have an \emph{optimal} awake complexity of $O(\log n)$ time with a matching lower bound. We also show that our randomized awake-optimal algorithm has essentially the best possible round complexity by presenting a lower bound of $\tilde{\Omega}(n)$ on the product of the awake and round complexity of any distributed algorithm (including randomized) that outputs an MST. To complement our trade-off lower bound, we present a parameterized family of distributed algorithms that gives an essentially optimal trade-off between the awake complexity and the round complexity.

翻译：我们研究分布式最小生成树（MST）问题，这是分布式计算中的基本问题。众所周知，在标准CONGEST模型下，分布式MST可在$\tilde{O}(D+\sqrt{n})$轮内求解（其中$n$为网络规模，$D$为网络直径），这本质上是（对数因子范围内的）最优轮复杂度。然而，在自组织无线网络和传感器网络等资源受限网络中，节点消耗如此多时间可能导致能量等资源的显著消耗。受上述考量启发，我们研究基于"休眠模型"[Chatterjee等人，PODC 2020]的MST分布式算法，该模型专为资源高效型分布式算法的设计与分析而提出。在休眠模型中，节点每轮可处于两种模式之一——"休眠"或"清醒"（传统模型要求节点始终处于清醒状态）。仅统计节点处于"清醒"状态的轮次，而"休眠"轮次则被忽略。节点仅在清醒轮次消耗资源，因此主要目标是使分布式算法的"清醒复杂度"——即任意节点处于清醒状态的最坏情况轮次数——最小化。我们提出了确定性与随机化两类分布式MST算法，其"最优"清醒复杂度达到$O(\log n)$时间，并给出了匹配的下界。通过证明任何输出MST的分布式算法（包含随机化算法）的清醒复杂度与轮复杂度之积的下界为$\tilde{\Omega}(n)$，我们进一步表明随机化清醒最优算法具有本质最优的轮复杂度。为补充此权衡下界，我们提出参数化族分布式算法，实现了清醒复杂度与轮复杂度之间的本质最优权衡。