局部检查少的近生成树的构造
Constructing Near Spanning Trees with Few Local Inspections
摘要：
构造图的生成树是图论中最基本的任务之一。最近的几个局部图算法的研究的动机，我们考虑这个问题的以下变体。设G是连通有界度图。给定$G$中的一条边$e$，我们想决定$e$是否属于由$(1+ε)n$边组成的连通子图$G'$（对于预先指定的常数$ε>0$），其中对不同边的决定应该与同一子图$G'$一致。仅检查$G$中的{ \em常量}条边是否可以执行此任务？我们的主要结果是：（1）证明了如果$G$的每个$t$ -顶点子图都有扩展$1/(\log t)^{1+o(1)}$，则可以（确定性地）用较少的检查构造出$G$的稀疏生成子图$G'$。为此，我们分析了一个著名的最小重量生成树算法的“本地”版本。(2)我们表明，上述扩展要求是尖锐的，即使允许随机化。为此，我们构造了一族高围长的$3$ -正则图，其中每个$t$ -顶点子图都有扩展$1/(\log t)^{1-o(1)}$ .
英文摘要：
Constructing a spanning tree of a graph is one of the most basic tasks in graph theory. Motivated by several recent studies of local graph algorithms, we consider the following variant of this problem. Let G be a connected bounded-degree graph. Given an edge $e$ in $G$ we would like to decide whether $e$ belongs to a connected subgraph $G'$ consisting of $(1+ε)n$ edges (for a prespecified constant $ε>0$), where the decision for different edges should be consistent with the same subgraph $G'$. Can this task be performed by inspecting only a {\em constant} number of edges in $G$? Our main results are: (1) We show that if every $t$-vertex subgraph of $G$ has expansion $1/(\log t)^{1+o(1)}$ then one can (deterministically) construct a sparse spanning subgraph $G'$ of $G$ using few inspections. To this end we analyze a "local" version of a famous minimum-weight spanning tree algorithm. (2) We show that the above expansion requirement is sharp even when allowing randomization. To this end we construct a family of $3$-regular graphs of high girth, in which every $t$-vertex subgraph has expansion $1/(\log t)^{1-o(1)}$.
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