Constructing a spanning tree of a graph is one of the most basic tasks in graph theory. We consider a relaxed version of this problem in the setting of local algorithms. The relaxation is that the constructed subgraph is a sparse spanning subgraph containing at most $(1+\epsilon)n$ edges (where $n$ is the number of vertices and $\epsilon$ is a given approximation/sparsity parameter). In the local setting, the goal is to quickly determine whether a given edge $e$ belongs to such a subgraph, without constructing the whole subgraph, but rather by inspecting (querying) the local neighborhood of $e$. The challenge is to maintain consistency. That is, to provide answers concerning different edges according to the same spanning subgraph. We first show that for general bounded-degree graphs, the query complexity of any such algorithm must be $\Omega(\sqrt{n})$. This lower bound holds for constant-degree graphs that have high expansion. Next we design an algorithm for (bounded-degree) graphs with high expansion, obtaining a result that roughly matches the lower bound. We then turn to study graphs that exclude a fixed minor (and are hence non-expanding). We design an algorithm for such graphs, which may have an unbounded maximum degree. The query complexity of this algorithm is $poly(1/\epsilon, h)$ (independent of $n$ and the maximum degree), where $h$ is the number of vertices in the excluded minor. Though our two algorithms are designed for very different types of graphs (and have very different complexities), on a high-level there are several similarities, and we highlight both the similarities and the differences.
翻译:构造图树的图形是图形理论中最基本的任务之一。 我们考虑在设置本地算法时, 这个问题的宽松版本。 放松的是, 构建的子图是一个分散的子图, 最多包含$(1 ⁇ ⁇ epsilon) n 的边缘( 美元是脊椎数, $\\ epsilon$ 是给定的近似/ 差分参数 ) 。 在当地设置中, 目标是快速确定给定的边缘 $ / 美元是否属于这样的子图( $1 / 美元 / 美元 / 美元 / / 美元 / 美元) 。 目标在于快速确定给定的边端是否属于这样的子值, 而不是构建整个子线 。 我们设计了一个( 方向 ) ( 水平 ) 本地值 / 美元 的直径直值, 最高值 和 最高值 值 值 。 因此, 不同的 算算算出一个小数 。 。 。 我们设计了一个不同的 。