Local Algorithms for Sparse Spanning Graphs
Abstract
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.
- Publication:
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arXiv e-prints
- Pub Date:
- February 2014
- DOI:
- 10.48550/arXiv.1402.3609
- arXiv:
- arXiv:1402.3609
- Bibcode:
- 2014arXiv1402.3609L
- Keywords:
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- Computer Science - Data Structures and Algorithms
- E-Print:
- Upper bounds for expanding graphs and minor free graphs