The parallel complexity of growth models

This paper investigates the parallel complexity of several nonequilibrium growth models.Invasion percolation, Eden growth, ballistic deposition, andsolid-on-solid growth are all seemingly highly sequential processes that yield self-similar or self-affine random clusters. Nonetheless, we present fast parallel randomized algorithms for generating these clusters. The running times of the algorithms scale asO(log2N), whereN is the system size, and the number of processors required scales as a polynomial inN. The algorithms are based on fast parallel procedures for finding minimum-weight paths; they illuminate the close connection between growth models and self-avoiding paths in random environments. In addition to their potential practical value, our algorithms serve to classify these growth models as less complex than other growth models, such asdiffusion-limited aggregation, for which fast parallel algorithms probably do not exist.