平面图标准化的对数空间算法
A Log-space Algorithm for Canonization of Planar Graphs
摘要：
图同构是一个计算问题的主要例子，其复杂性的最佳已知下限和上限之间存在很大差异。我们弥合这一差距的一个自然的和重要的特殊情况下，平面图同构，提出了一个上限，匹配已知的对数空间硬度[Lindell'92]。事实上，我们证明了形式上更强的结果，平面图的标准化是在对数空间。这改进了先前已知的AC 1的上界[MillerReif'91]。我们的算法首先构造连通平面图的双连通分支树，然后将每个双连通分支细化为三连通分支树。下一步是将双连通平面图的同构和标准化问题减少到3-连通平面图的问题，这些问题在[DattaLimayeNimbhorkar'08]中已知是在对数空间中的。这是通过使用上述分解，并通过对用于树规范化的Lindell算法进行重大修改以及空间复杂度分析的变化来实现的。从连通的情况下减少到双连通的情况下，需要进一步的新思想，包括一个非平凡的情况下分析和群论引理绑定的有色3-连通平面图的自同构的数量。这个引理对于在对数空间中进行归约是至关重要的。
英文摘要：
Graph Isomorphism is the prime example of a computational problem with a wide difference between the best known lower and upper bounds on its complexity. We bridge this gap for a natural and important special case, planar graph isomorphism, by presenting an upper bound that matches the known logspace hardness [Lindell'92]. In fact, we show the formally stronger result that planar graph canonization is in logspace. This improves the previously known upper bound of AC1 [MillerReif'91]. Our algorithm first constructs the biconnected component tree of a connected planar graph and then refines each biconnected component into a triconnected component tree. The next step is to logspace reduce the biconnected planar graph isomorphism and canonization problems to those for 3-connected planar graphs, which are known to be in logspace by [DattaLimayeNimbhorkar'08]. This is achieved by using the above decomposition, and by making significant modifications to Lindell's algorithm for tree canonization, along with changes in the space complexity analysis. The reduction from the connected case to the biconnected case requires further new ideas, including a non-trivial case analysis and a group theoretic lemma to bound the number of automorphisms of a colored 3-connected planar graph. This lemma is crucial for the reduction to work in logspace.
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