The Dispersion Time of Random Walks on Finite Graphs

Riveria, N., Sauerwald, T., Stauffer, A. and Sylvester, J. (2019) The Dispersion Time of Random Walks on Finite Graphs. In: 31st ACM Symposium on Parallelism in Algorithms and Architectures (SPAA '19), Phoenix, AZ, USA, 22-24 Jun 2019, pp. 103-113. ISBN 9781450361842 (doi: 10.1145/3323165.3323204)

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Abstract

We study two random processes on an n-vertex graph inspired by the internal diffusion limited aggregation (IDLA) model. These processes can also be regarded as protocols for allocating jobs in a distributed network of servers. In both processes n particles start from an arbitrary but fixed origin. Each particle performs a simple random walk until it first encounters an unoccupied vertex, at which point the vertex becomes occupied and the random walk terminates. In one of the processes, called Sequential-IDLA, a single particle moves until settling and only then does the next particle start whereas in the second process, called Parallel-IDLA, all unsettled particles move simultaneously. The second process is akin to running the first in parallel. Our main goal is to analyze the so-called dispersion time of these processes, which is the maximum number of steps performed by any of the n particles. In order to compare the two processes, we develop a coupling which shows the dispersion time of the Parallel-IDLA stochastically dominates that of the Sequential-IDLA; however, the total number of steps performed by all particles has the same distribution in both processes. This coupling also gives us that dispersion time of Parallel-IDLA is bounded in expectation by dispersion time of the Sequential-IDLA up to a multiplicative łog n factor. Moreover, we derive asymptotic upper and lower bound on the dispersion time for several graph classes, such as cliques, cycles, binary trees, d-dimensional grids, hypercubes and expanders. Most of our bounds are tight up to a multiplicative constant.

Item Type:Conference Proceedings
Status:Published
Refereed:Yes
Glasgow Author(s) Enlighten ID:Sylvester, Dr John
Authors: Riveria, N., Sauerwald, T., Stauffer, A., and Sylvester, J.
College/School:College of Science and Engineering > School of Computing Science
ISBN:9781450361842

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