Distributed symmetry-breaking with improved vertex-averaged complexity

Barenboim, Leonid; Tzur, Yaniv

doi:10.1145/3288599.3288601

Cited by 10 publications

(14 citation statements)

References 30 publications

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“…In the current paper we investigate yet another distributed setting, namely, the Sleeping Setting. Several variants of this setting have attracted the attention of researchers recently [5,8,10,13,18,22]. The particular setting and complexity measure we consider in this paper were introduced by Chatterjee, Gmyr and Pandurangan [8] in PODC '20. This sleeping setting is similar to the standard distributed LOCAL setting [24], but has an additional capability, as follows.…”

Section: Our Resultsmentioning

confidence: 99%

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Deterministic Logarithmic Completeness in the Distributed Sleeping Model

Barenboim,

Maimon

2021

Preprint

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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.

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Section: Our Resultsmentioning

confidence: 99%

“…An important efficiency measurement in this setting is the vertex-averaged awake complexity. This setting was further studied by Barenboim and Tzur [5], who obtained various symmetry-breaking algorithms with improved vertex-averaged complexity.…”

Section: Related Workmentioning

confidence: 99%

Deterministic Logarithmic Completeness in the Distributed Sleeping Model

Barenboim,

Maimon

2021

Preprint

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“…Several variants of the sleeping setting were studied recently [7,8,9,11,12,13]. The particular setting and complexity measurement we consider in this paper were introduced by Chatterjee, Gmyr and Pandurangan [8] in PODC'20.…”

Section: Previous Workmentioning

confidence: 99%

“…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.…”

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Sleeping Model: Local and Dynamic Algorithms

Maimon¹

2021

Preprint

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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.

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“…Recently Barenboim and Tzur [6] showed that MIS can be solved in O(a + log * n) rounds under node-averaged complexity deterministically, where a is the arboricity of the graph. It is a open question whether one can design an algorithm with O(1) (or even o(log n)) node-averaged round complexity in the traditional model for general graphs (which can have arboricity as high as Θ(n)).…”

Section: Mis In O(1)-rounds Node Average Complexity?mentioning

confidence: 99%

Sleeping is Efficient: MIS in $O(1)$-rounds Node-averaged Awake Complexity

Chatterjee¹,

Gmyr²,

Pandurangan³

2020

Preprint

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Maximal Independent Set (MIS) is one of the fundamental problems in distributed computing. The round (time) complexity of distributed MIS has traditionally focused on the worstcase time for all nodes to finish. The best-known (randomized) MIS algorithms take O(log n) worst-case rounds on general graphs (where n is the number of nodes). Breaking the O(log n) worst-case bound has been a longstanding open problem, while currently the best-known lower bound is Ω( log n log log n ) rounds. Motivated by the goal to reduce total energy consumption in energy-constrained networks such as sensor and ad hoc wireless networks, we take an alternative approach to measuring performance. We focus on minimizing the total (or equivalently, the average) time for all nodes to finish. It is not clear whether the currently best-known algorithms yield constant-round (or even o(log n)) node-averaged round complexity for MIS in general graphs. We posit the sleeping model, a generalization of the traditional model, that allows nodes to enter either "sleep" or "waking" states at any round. While waking state corresponds to the default state in the traditional model, in sleeping state a node is "offline", i.e., it does not send or receive messages (and messages sent to it are dropped as well) and does not incur any time, communication, or local computation cost. Hence, in this model, only rounds in which a node is awake are counted and we are interested in minimizing the average as well as the worst-case number of rounds a node spends in the awake state, besides the traditional worst-case round complexity (i.e., the rounds for all nodes to finish including both the awake and sleeping rounds).Our main result is that we show that MIS can be solved in (expected) O(1) rounds under node-averaged awake complexity measure in the sleeping model. In particular, we present a randomized distributed algorithm for MIS that has expected O(1)-rounds node-averaged awake complexity and, with high probability has O(log n)-rounds worst-case awake complexity and O(log 3.41 n)-rounds worst-case complexity.Our work is a step towards understanding the node-averaged complexity of MIS both in the traditional and sleeping models. It is also the first step in designing energy-efficient distributed algorithms for energy-constrained networks.

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Distributed symmetry-breaking with improved vertex-averaged complexity

Cited by 10 publications

References 30 publications

Deterministic Logarithmic Completeness in the Distributed Sleeping Model

Deterministic Logarithmic Completeness in the Distributed Sleeping Model

Sleeping Model: Local and Dynamic Algorithms

Sleeping is Efficient: MIS in $O(1)$-rounds Node-averaged Awake Complexity

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