Sleeping Model: Local and Dynamic Algorithms

Abstract

The distributed setting represents a communication network where each node is a processor with its own memory, and each edge is a direct communication line between two nodes. Algorithms are usually measured by the number of synchronized communication rounds required until the algorithm terminates in all the nodes of the input graph. In this work we refer to this as the clock round complexity. The setting is studied under several models, more known among them are the LOCAL model and the CON GEST model. In recent years the sleeping model (or some variation thereof) came to the focus of researchers [7,8,9,11,12,13,6]. In this model nodes can go into a sleep state in which they spend no energy but at the same time cannot receive or send messages, nor can they perform internal computations. This model captures energy considerations of a problem. In [6] Barenboim and Maimon defined the class of O-LOCAL 1 problems and showed that for this class of problems there is a deterministic algorithm that runs in O(log ∆) awake time. The clock round complexity of that algorithm is O(∆ 2 ). Well-studied O-LOCAL problems include coloring and Maximal Independence Set (MIS).In this work we present several deterministic results in the sleeping model: 1. We offer three algorithms for the O-LOCAL class of problems with a trade off between awake complexity and clock round complexity. One of these algorithms requires only O(∆ 1+ ) clock rounds for some constant > 0 but still only O(log ∆) awake time which improves on the algorithm in [6]. We add to this two other algorithms that trade a higher awake complexity for lower clock round complexity. We note that the awake time incurred is not that significant. 2. We offer dynamic algorithms in the sleeping model. We show three algorithms for solving dynamic problems in the O-LOCAL class as well as an algorithm for solving any dynamic decidable problem. 3. We show that one can solve any O-LOCAL problem in constant awake time in graphs with constant neighborhood independence. Specifically, our algorithm requires O(K) awake time where K is the neighborhood independence of the input graph. Graphs with bounded neighborhood independence are well studied with several results in recent years for several core problem in the distributed setting [1,16,3,4,5]. ACM Subject ClassificationComputing methodologies → Distributed algorithms Keywords and phrases Distributed Computing, Sleeping model, Bounded Neighborhood Independence Digital Object Identifier 10.4230/LIPIcs.OPODIS.2021. Acknowledgements Anonymous acknowledgements 11 A problem P is an O-LOCAL problem if, given an acyclic orientation on the edges of the input graph, one can solve the problem as follows. Each vertex awaits the decisions of its parents according to the given orientation and can make its own decision in regard to P using only the information about its parents decisions.

Distributed MIS in O(log log n) Awake Complexity

“…They showed that problems in this class admit a deterministic algorithm that runs in 𝑂 (log Δ) awake time and 𝑂 (Δ 2 ) round complexity. Maimon [34] presents trade-offs between awake and round complexity for O-LOCAL problems.…”

Sleeping is Superefficient: MIS in Exponentially Better Awake Complexity

“…They also define the class of O-LOCAL problems (that includes MIS and coloring) and showed that problems in this class admit a deterministic algorithm that runs in O(log ∆) awake time and O(∆ 2 ) round complexity. Maimon [28] presents trade-offs between awake and round complexity for O-LOCAL problems.…”