Smoothed analysis of dynamic networks

Dinitz, Michael; Fineman, Jeremy T.; Gilbert, Seth; Newport, Calvin

doi:10.1007/s00446-017-0300-8

Cited by 10 publications

(15 citation statements)

References 17 publications

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“…They studied the mixing times, extended our bound, and proved an

O ({(1 - λ)}^{- 1} \log n)

upper bound on the mixing time where λ is the upper bound on the second largest (in absolute value) eigenvalue of all transition matrices of graphs in

scriptG

. A recent paper took a smoothed analysis perspective on dynamic networks and studied our examples, among other things . Giakkoupis et al used evolving graphs in a different context, they were constructed to delay a rumor spreading process.…”

Section: Introductionmentioning

confidence: 96%

Cover time and mixing time of random walks on dynamic graphs

Avin

Koucký

Lotker

2017

Random Struct Algorithms

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The application of simple random walks on graphs is a powerful tool that is useful in many algorithmic settings such as network exploration, sampling, information spreading, and distributed computing. This is due to the reliance of a simple random walk on only local data, its negligible memory requirements, and its distributed nature. It is well known that for static graphs the cover time, that is, the expected time to visit every node of the graph, and the mixing time, that is, the time to sample a node according to the stationary distribution, are at most polynomial relative to the size of the graph. Motivated by real world networks, such as peer-to-peer and wireless networks, the conference version of this paper was the first to study random walks on arbitrary dynamic networks. We study the most general model in which an oblivious adversary is permitted to change the graph after every step of the random walk. In contrast to static graphs, and somewhat counter-intuitively, we show that there are adversary strategies that force the expected cover time and the mixing time of the simple random walk on dynamic graphs to be exponentially long, even when at each time step the network is well connected and rapidly mixing. To resolve this, we propose a simple strategy, the lazy random walk, which guarantees, under minor conditions, polynomial cover time and polynomial mixing time regardless of the changes made by the adversary. K E Y W O R D Scover time, dynamic graphs, mixing time, random walks, stationary distribution * The preliminary version of this paper appeared as How to explore a fast-changing world (cover time of a simple random walk on evolving graphs), in the How to cite this article: Avin C., Koucký M., Lotker Z. Cover time and mixing time of random walks on dynamic graphs. Random Struct Alg. 2018;52:576-596.

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“…They studied the mixing times, extended our bound, and proved an

O ({(1 - λ)}^{- 1} \log n)

upper bound on the mixing time where λ is the upper bound on the second largest (in absolute value) eigenvalue of all transition matrices of graphs in

scriptG

Section: Introductionmentioning

confidence: 96%

Cover time and mixing time of random walks on dynamic graphs

Avin

Koucký

Lotker

2017

Random Struct Algorithms

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“…The only work that we are aware of in the context of smoothed analysis of distributed algorithms is that of Dinitz et al [5] who study smoothed analysis of distributed algorithms for dynamic networks. Their dynamic network model is a dynamic graph H = G 1 , G 2 , .…”

Section: Related Workmentioning

confidence: 99%

“…In this paper, we initiate the study of smoothed analysis of distributed graph algorithms. Our paper is motivated by the work of Dinitz et al [5] who initiated the study of smoothed analysis for dynamic networks (we refer to Section 1.1 for more details). A main contribution of our paper is positing smoothing models in the context of distributed graph algorithms and performing analyses of the models.…”

Section: Introductionmentioning

confidence: 99%

“…Or more precisely, is it specific to the choice of the graph topology or the edge weights or both? If small perturbations do not change the worst-case bound by too much then we can say that the lower bound is robust and, if they do we can say that the bounds are fragile [5]. Thus smoothed analysis can lead to a better understanding of the complexity of distributed MST by studying perturbations of the worst-case input.…”

Section: Introductionmentioning

confidence: 99%

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Distributed MST: A Smoothed Analysis

Chatterjee,

Pandurangan,

Pham

2019

Preprint

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We study smoothed analysis of distributed graph algorithms, focusing on the fundamental minimum spanning tree (MST) problem. With the goal of studying the time complexity of distributed MST as a function of the "perturbation" of the input graph, we posit a smoothing model that is parameterized by a smoothing parameter 0 ≤ ǫ(n) ≤ 1 which controls the amount of random edges that can be added to an input graph G per round. Informally, ǫ(n) is the probability (typically a small function of n, e.g., n − 1 4 ) that a random edge can be added to a node per round. The added random edges, once they are added, can be used (only) for communication.We show upper and lower bounds on the time complexity of distributed MST in the above smoothing model. We present a distributed algorithm that, with high probability, 1 computes an MST and runs in Õ(min{rounds 2 where ǫ is the smoothing parameter, D is the network diameter and n is the network size. To complement our upper bound, we also show a lower bound of Ω(min{ 1 √ ǫ(n), D + √ n}). We note that the upper and lower bounds essentially match except for a multiplicative 2 O( √ log n) polylog(n) factor. Our work can be considered as a first step in understanding the smoothed complexity of distributed graph algorithms.

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“…It elected one leader in a complete graph, running O(1) round, and the message complexity is O √ n log 3 2 n . In [25], Dinitz et al generalized smoothed analysis to distributed algorithms in dynamic networks. They applied this technique to solve three problems of standard dynamic network with the worst-case lower bounds, these problems are random walks, flooding, and aggregation.…”

Section: Related Workmentioning

confidence: 99%

Multi-leader Election in Dynamic Sensor Networks

Meng¹,

Yu²,

Yu³

2016

2016 International Conference on Identification, Information and Knowledge in the Internet of Things (IIKI)

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The leader election problem is one of the fundamental problems in distributed computing. Different from most of the existing results studying the multi-leader election in static networks or one leader election in dynamic networks, in this paper, we focus on the multi-leader election in dynamic sensor networks where nodes are deployed randomly. A centralized simple leader election algorithm (VLE), a distributed leader election algorithm (NMDLE), and a multi-leader election algorithm (PSMLE) are proposed so as to elect multi-leaders for the purpose of saving energy and prolonging the network lifetime, respectively. Specifically, the proposed algorithms aim at using less leaders to control the whole network, which is controlled by at least k opt leaders, here k opt denotes the optimal number of network partitions. Then we analyze the impacts of the sleep scheme of nodes and node moving on energy consumption and establish a theoretical model for energy cost. Finally, we provide extensive simulation results valuating the correctness of theoretical analysis.

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Smoothed analysis of dynamic networks

Cited by 10 publications

References 17 publications

Cover time and mixing time of random walks on dynamic graphs

Cover time and mixing time of random walks on dynamic graphs

Distributed MST: A Smoothed Analysis

Multi-leader Election in Dynamic Sensor Networks

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