Tight Bounds for Coalescing-Branching Random Walks on Regular Graphs

Berenbrink, Petra; Giakkoupis, George; Kling, Peter

doi:10.1137/1.9781611975031.112

Cited by 6 publications

(4 citation statements)

References 24 publications

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“…Token-based processes have also been used to implement efficient, randomised rumour spreading protocols. For example, Perenbrink, Giakkoupis and Kling [19] analysed the cover time of a synchronous coalescing-branching random walk on regular graphs: Initially there is a single token located at some node. In each round, (1) every node v that has a token, splits its token into k parts and sends them k randomly chosen neighbours, and (2) at the end of the round, all tokens at a single node coalesce into a single token.…”

Section: Dynamics Of Interacting Particles and Token Shuffling Protocolsmentioning

confidence: 99%

Fast Graphical Population Protocols

Alistarh,

Gelashvili,

Rybicki

2021

Preprint

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Let G be a graph on n nodes. In the stochastic population protocol model, a collection of n indistinguishable, resource-limited nodes collectively solve tasks via pairwise interactions. In each interaction, two randomly chosen neighbors first read each other's states, and then update their local states. A rich line of research has established tight upper and lower bounds on the complexity of fundamental tasks, such as majority and leader election, in this model, when G is a clique. Specifically, in the clique, these tasks can be solved fast, i.e., in n polylog n pairwise interactions, with high probability, using at most polylog n states per node.In this work, we consider the more general setting where G is an arbitrary graph, and present a technique for simulating protocols designed for fully-connected networks in any connected regular graph. Our main result is a simulation that is efficient on many interesting graph families: roughly, the simulation overhead is polylogarithmic in the number of nodes, and quadratic in the conductance of the graph. As a sample application, we show that, in any regular graph with conductance φ, both leader election and exact majority can be solved in φ −2 • n polylog n pairwise interactions, with high probability, using at most φ −2 • polylog n states per node. This shows that there are fast and space-efficient population protocols for leader election and exact majority on graphs with good expansion properties. We believe our results will prove generally useful, as they allow efficient technology transfer between the well-mixed (clique) case, and the under-explored spatial setting.

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Section: Dynamics Of Interacting Particles and Token Shuffling Protocolsmentioning

confidence: 99%

Fast Graphical Population Protocols

Alistarh,

Gelashvili,

Rybicki

2021

Preprint

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“…Other superficially related processes include coalescing random walks [6,30], and coalescing branching walks [7,36]. See also [13] for a survey on multiple random walks.…”

Section: Related Workmentioning

confidence: 99%

How to Spread a Rumor: Call Your Neighbors or Take a Walk?

Giakkoupis,

Mallmann-Trenn,

Saribekyan

2020

Preprint

Self Cite

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We study the problem of randomized information dissemination in networks. We compare the now standard push-pull protocol, with agent-based alternatives where information is disseminated by a collection of agents performing independent random walks. In the visit-exchange protocol, both nodes and agents store information, and each time an agent visits a node, the two exchange all the information they have. In the meet-exchange protocol, only the agents store information, and exchange their information with each agent they meet.We consider the broadcast time of a single piece of information in an n-node graph for the above three protocols, assuming a linear number of agents that start from the stationary distribution. We observe that there are graphs on which the agent-based protocols are significantly faster than push-pull, and graphs where the converse is true. We attribute the good performance of agent-based algorithms to their inherently fair bandwidth utilization, and conclude that, in certain settings, agent-based information dissemination, separately or in combination with push-pull, can significantly improve the broadcast time.The graphs considered above are highly non-regular. Our main technical result is that on any regular graph of at least logarithmic degree, push-pull and visit-exchange have the same asymptotic broadcast time. The proof uses a novel coupling argument which relates the random choices of vertices in push-pull with the random walks in visit-exchange. Further, we show that the broadcast time of meet-exchange is asymptotically at least as large as the other two's on all regular graphs, and strictly larger on some regular graphs.As far as we know, this is the first systematic and thorough comparison of the running times of these very natural information dissemination protocols.

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“…Remark 2. The random voting-DAG H(v 0 ) can be viewed as the trajectory of a Coalescing and Branching random walk or, for short, COBRA walk (see [3], [6], [9] for recent research). A COBRA walk is a discrete process on a graph G where vertices are occupied by particles.…”

Section: Model and Proof Strategymentioning

confidence: 99%

Best-of-Three Voting on Dense Graphs

Kang

Rivera

2019

The 31st ACM Symposium on Parallelism in Algorithms and Architectures

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Given a graph G of n vertices, where each vertex is initially attached an opinion of either red or blue. We investigate a random process known as the Bestof-three voting. In this process, at each time step, every vertex chooses three neighbours at random and adopts the majority colour. We study this process for a class of graphs with minimum degree d = n α , where α = Ω (log log n) −1 . We prove that if initially each vertex is red with probability greater than 1/2 + δ, and blue otherwise, where δ ≥ (log d) −C for some C > 0, then with high probability this dynamic reaches a final state where all vertices are red within O (log log n) + O log δ −1 steps .

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Tight Bounds for Coalescing-Branching Random Walks on Regular Graphs

Cited by 6 publications

References 24 publications

Fast Graphical Population Protocols

Fast Graphical Population Protocols

How to Spread a Rumor: Call Your Neighbors or Take a Walk?

Best-of-Three Voting on Dense Graphs

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