Monte Carlo simulations are one of the most important numerical techniques for investigating statistical physical systems. Among these systems, spin models are a typical example which also play an essential role in constructing the abstract mechanism for various complex systems. Unfortunately, traditional Monte Carlo algorithms are afflicted with "critical slowing down" near continuous phase transitions and the efficiency of the Monte Carlo simulation goes to zero as the size of the lattice is increased. To combat critical slowing down, a very different type of collective-mode algorithm, in contrast to the traditional single-spin-flipmode, was proposed by Swendsen and Wang in 1987 for Potts spin models. Since then, there has been an explosion of work attempting to understand, improve, or generalize it. In these so-called "cluster" algorithms, clusters of spin are regarded as one template and are updated at each step of the Monte Carlo procedure. In implementing these algorithms the cluster labeling is a major time-consuming bottleneck and is also isomorphic to the problem of computing connected components of an undirected graph seen in other application areas, such as pattern recognition.
A number of cluster labeling algorithms for sequential computers have long existed. However, the dynamic irregular nature of clusters complicates the task of finding good parallel algorithms and this is particularly true on SIMD (single-instruction-multiple-data machines. Our design of the Hierarchical Cluster Labeling Algorithm aims at alleviating this problem by building a hierarchical structure on the problem domain and by incorporating local and nonlocal communication schemes. We present an estimate for the computational complexity of cluster labeling and prove the key features of this algorithm (such as lower computational complexity, data locality, and easy implementation) compared with the methods formerly known. In particular, this algorithm can be viewed as a generalized scan scheme applicable to problem domains of any high dimension and of arbitrary geometry (scan is an important primitive of parallel computing). In addition, from implementation results, the hierarchical cluster labeling algorithm has proved to work equally well on MIMD machines, though originally designed for SIMD machines.
Based on this success, we further study the hierarchical structure hidden in the algorithm. Hierarchical structure is a conceptual framework frequently used in building models for the study of a great variety of problems. This structure serves not only to describe the complexity of the system at different levels, but also to achieve some goals targeted by the problem, i.e., an algorithm to solve the problem. In this regard, we investigate the similarities and differences between this algorithm and others, including the FFT and the Barnes-Hut method, in terms of their hierarchical structures.
