Counting complex disordered states by efficient pattern matching: chromatic polynomials and Potts partition functions
Counting complex disordered states by efficient pattern matching: chromatic polynomials and Potts partition functions
复制标题
通过有效的模式匹配计算复杂的无序态：色多项式和 Potts 配分函数
DOI：
10.1088/1367-2630/11/2/023001
复制
发表时间：
2009
影响因子：
3.3
通讯作者：
Sebastian Stolzenberg
中科院分区：
文献类型：
--
作者：
M. Timme;F. V. Bussel;D. Fliegner;Sebastian Stolzenberg
计数问题，即在一定的约束条件下确定一个大系统的可能状态的数量，在许多科学领域起着重要的作用。在物理和化学、数学图论和计算机科学中，它们自然地出现在复杂的无序系统中。然而，计数问题是最难通过计算解决的问题之一。在这里，我们提出了一种新的方法来解决基准计数问题，即寻找图的色多项式。我们开发了一种面向顶点的符号模式匹配算法，该算法利用了波茨反铁磁体的色多项式和零温度配分函数在同一张图上的等价性。使用适当的计算机代数实现这种自下而上的算法，对于中等大小的图，新方法比标准的自上而下的方法要好几个数量级。作为第一个应用，我们计算了简单立方晶格样本的色多项式，这是第一次计算访问物理相关的三维晶格。该方法为其他几个计数问题提供了直接的推广。
Counting problems, determining the number of possible states of a large system under certain constraints, play an important role in many areas of science. They naturally arise for complex disordered systems in physics and chemistry, in mathematical graph theory, and in computer science. Counting problems, however, are among the hardest problems to access computationally. Here, we suggest a novel method to access a benchmark counting problem, finding chromatic polynomials of graphs. We develop a vertex-oriented symbolic pattern matching algorithm that exploits the equivalence between the chromatic polynomial and the zero-temperature partition function of the Potts antiferromagnet on the same graph. Implementing this bottom-up algorithm using appropriate computer algebra, the new method outperforms standard top-down methods by several orders of magnitude, already for moderately sized graphs. As a first application, we compute chromatic polynomials of samples of the simple cubic lattice, for the first time computationally accessing three-dimensional lattices of physical relevance. The method offers straightforward generalizations to several other counting problems.