[Paper Review] Internal Diffusion-Limited Aggregation: Parallel Algorithms and Complexity

Cristopher Moore, Jonathan Machta|ArXiv.org|Sep 15, 1999

Advanced Algebra and Logic4 citations

TL;DR

This paper investigates the computational complexity of internal diffusion-limited aggregation (internal DLA) in two and higher dimensions, proving it is CC-complete—implying inherent parallelization limits. It introduces a parallel relaxation algorithm that leverages the near-spherical shape of clusters to accelerate cluster prediction, achieving polylogarithmic or small-power parallel running time in simulations, with a theoretical O((n/k + log k)n^{2/d}) step complexity on k-processor machines, significantly outperforming sequential O(n^{1+2/d}) simulation.

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.

Motivation & Objective

• To analyze the computational complexity of internal DLA in two or more dimensions.
• To develop efficient parallel algorithms for simulating internal DLA cluster growth.
• To determine whether internal DLA can be predicted efficiently in parallel, especially in comparison to worst-case complexity bounds.
• To evaluate the practical performance of the proposed parallel relaxation algorithm through simulation.

Proposed method

• Prove that internal DLA prediction is CC-complete in d ≥ 2 dimensions, indicating inherent difficulty for parallel computation.
• Design a parallel relaxation algorithm that guesses the cluster shape based on path data and corrects errors via a non-local annihilation process.
• Use the near-spherical geometry of internal DLA clusters to guide the initial guess and reduce correction steps.
• Simulate the relaxation algorithm on a serial computer to estimate its parallel running time in two dimensions.
• Derive theoretical bounds on the number of steps required for cluster generation on a k-processor parallel machine.
• Show that in one dimension, internal DLA can be predicted in O(log n) parallel time, placing it in the NC complexity class.

Experimental results

Research questions

• RQ1 Is internal diffusion-limited aggregation computationally hard to predict in parallel, and what complexity class does it belong to in two or more dimensions?
• RQ2 Can the near-spherical structure of internal DLA clusters be exploited to design faster parallel algorithms?
• RQ3 What is the theoretical parallel time complexity of generating random internal DLA clusters on a k-processor machine?
• RQ4 Does the relaxation algorithm achieve better performance than worst-case analysis suggests in practice?
• RQ5 Can internal DLA in one dimension be efficiently parallelized, and does it lie in the NC complexity class?

Key findings

• Internal DLA in two or more dimensions is CC-complete, suggesting that even with parallel computation, the problem requires polynomial time in the worst case.
• The proposed parallel relaxation algorithm exhibits a running time that is either polylogarithmic or a small power of n in two dimensions, indicating better-than-worst-case performance.
• For a parallel machine with k processors, the algorithm runs in O((n/k + log k)n^{2/d}) steps in d dimensions, offering a significant speedup over sequential simulation.
• The theoretical speedup is substantial: from O(n^{1+2/d}) for sequential simulation to O((n/k + log k)n^{2/d}) for parallel execution.
• In one dimension, internal DLA can be predicted in O(log n) parallel time, placing it in the NC complexity class.
• Numerical simulations of the relaxation algorithm support the hypothesis that its running time grows slowly with n, contradicting worst-case complexity predictions.