Internal Diffusion-Limited Aggregation: Parallel Algorithms and Complexity

TL;DR

This paper analyzes the computational complexity of internal DLA, showing it is CC-complete, and presents parallel algorithms that significantly reduce the time needed to grow and predict large clusters, especially in low dimensions.

Contribution

The paper establishes the CC-completeness of internal DLA prediction and introduces efficient parallel algorithms with proven speedups for cluster growth and prediction.

Findings

Predicting clusters is CC-complete, implying polynomial worst-case time.

Parallel relaxation algorithms can approximate cluster growth efficiently.

In one dimension, internal DLA prediction is in NC, achievable in logarithmic parallel time.

Abstract

The computational complexity of internal diffusion-limited aggregation (DLA) is examined from both a theoretical and a practical point of view. We show that for two or more dimensions, the problem of predicting the cluster from a given set of paths is complete for the complexity class CC, the subset of P characterized by circuits composed of comparator gates. CC-completeness is believed to imply that, in the worst case, growing a cluster of size n requires polynomial time in n even on a parallel computer. A parallel relaxation algorithm is presented that uses the fact that clusters are nearly spherical to guess the cluster from a given set of paths, and then corrects defects in the guessed cluster through a non-local annihilation process. The parallel running time of the relaxation algorithm for two-dimensional internal DLA is studied by simulating it on a serial computer. The numerical results are compatible with a running time that is either polylogarithmic in n or a small power of n. Thus the computational resources needed to grow large clusters are significantly less on average than the worst-case analysis would suggest. For a parallel machine with k processors, we show that random clusters in d dimensions can be generated in O((n/k + log k) n^{2/d}) steps. This is a significant speedup over explicit sequential simulation, which takes O(n^{1+2/d}) time on average. Finally, we show that in one dimension internal DLA can be predicted in O(log n) parallel time, and so is in the complexity class NC.