Deterministic Logarithmic Completeness in the Distributed Sleeping Model

Abstract

In this paper we provide a deterministic scheme for solving any decidable problem in the distributed sleeping model. The sleeping model [8,22] is a generalization of the standard message-passing model, with an additional capability of network nodes to enter a sleeping state occasionally. As long as a vertex is in the awake state, it is similar to the standard message-passing setting. However, when a vertex is asleep it cannot receive or send messages in the network nor can it perform internal computations. On the other hand, sleeping rounds do not count towards awake complexity. Awake complexity is the main complexity measurement in this setting, which is the number of awake rounds a vertex spends during an execution. In this paper we devise algorithms with worst-case guarantees on the awake complexity.We devise a deterministic scheme with awake complexity of O(log n) for solving any decidable problem in this model by constructing a structure we call Distributed Layered Tree. This structure turns out to be very powerful in the sleeping model, since it allows one to collect the entire graph information within a constant number of awake rounds. Moreover, we prove that our general technique cannot be improved in this model, by showing that the construction of distributed layered trees itself requires Ω(log n) awake rounds. This is obtained by a reduction from message-complexity lower bounds, which is of independent interest. Furthermore, our scheme also works in the CON GEST setting where we are limited to messages of size at most O(log n) bits. This result is shown for a certain class of problems, which contains problems of great interest in the research of the distributed setting. Examples for problems we can solve under this limitation are leader election, computing exact number of edges and average degree.Another result we obtain in this work is a deterministic scheme for solving any problem from a class of problems, denoted O-LOCAL, in O(log ∆ + log * n) awake rounds. This class contains various well-studied problems, such as MIS and (∆ + 1)-vertex-coloring. Our main structure in this case is a tree as well, but is sharply different from a distributed layered tree. In particular, it is constructed in the local memory of each processor, rather than distributively. Nevertheless, it provides an efficient synchronization scheme for problems of the O-LOCAL class.

Sleeping Model: Local and Dynamic Algorithms

“…Its high-probability awake complexity is O(log n), and its worst-case awake complexity is polylogarithmic. Recently Barenboim and Maimon [6] showed a completeness on the class of decidable problems with a tight bound of Θ(log n) awake time complexity. That is, they offered an algorithm for solving any decidable problem in the sleeping model in O(log n) awake time but also showed a specific decidable problem which requires at least Ω(log n) awake time.…”

“…In [6] the authors showed that given an acyclic orientation on the edge set of the input graph G, one can use the given orientation to build a binary tree internally in each vertex v and have v in the awake state exactly when one of its parents in the orientation sends information to v. This is facilitated to solve any O-LOCAL problem. In their paper, Barenboim and Maimon used a O(∆ 2 )-vertex-coloring to achieve this required acyclic orientation but we here capitalize on this idea and offer different ways to build an initial orientation such that, in terms of awake complexity, our algorithms are more efficient.…”

“…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 ).…”

“…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.…”