Distributed MST Computation in the Sleeping Model: Awake-Optimal Algorithms and Lower Bounds

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 ˜ O ( D + √ 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 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 — sleeping or awake (unlike the traditional model where nodes are always awake). Only the rounds in which a node is awake are counted, while sleeping rounds are ignored. A node spends resources only in the awake rounds and hence the main goal is to minimize the awake complexity of a distributed algorithm, the worst-case number of rounds any node is awake. We present distributed MST algorithms that have optimal awake complexity with a matching lower bound. We also show that our awake-optimal algorithms have essentially the best possible round complexity by presenting a lower bound on the product of the awake and round complexity of any distributed algorithm (including randomized). Specifically, we show the following results: We show that both the above algorithms have optimal awake complexity by proving that Ω(log n ) is a lower bound on the awake complexity for computing an MST even for randomized algorithms. To better understand the relationship between awake and round complexities, we prove a lower bound 1 of ˜Ω( n ) on the product of round complexity and awake complexity for any distributed algorithm (even randomized) that outputs an MST. This lower bound shows that our randomized algorithm that has the optimal awake complexity of O (log n ) also has essentially the best possible round complexity of O ( n log n ).