We study two random processes on an -vertex graph inspired by the internal diffusion limited aggregation (IDLA) model. In both processes particles start from an arbitrary but fixed origin. Each particle performs a simple random walk until first encountering an unoccupied vertex, and at which point the vertex becomes occupied and the random walk terminates. In one of the processes, called , only one particle moves until settling and only then does the next particle start whereas in the second process, called , all unsettled particles move simultaneously. 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 particles. In order to compare the two processes, we develop a coupling that 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 factor. Moreover, we derive asymptotic upper and lower bound on the dispersion time for several graph classes, such as cliques, cycles, binary trees, -dimensional grids, hypercubes and expanders. Most of our bounds are tight up to a multiplicative constant.
Description
Keywords
Random Walks on Graphs, Parallelization of random processes, Interacting particle systems
Journal Title
SPAA '19: Proceedings of the 31st ACM Symposium on Parallelism in Algorithms and Architectures
Conference Name
31st ACM Symposium on Parallelism in Algorithms and Architectures (SPAA 2019)