  Constructing a sparse spanning subgraph is a fundamental primitive in graph
theory. In this paper, we study this problem in the Centralized Local model,
where the goal is to decide whether an edge is part of the spanning subgraph by
examining only a small part of the input; yet, answers must be globally
consistent and independent of prior queries.
  Unfortunately, maximally sparse spanning subgraphs, i.e., spanning trees,
cannot be constructed efficiently in this model. Therefore, we settle for a
spanning subgraph containing at most $(1+\varepsilon)n$ edges (where $n$ is the
number of vertices and $\varepsilon$ is a given approximation/sparsity
parameter). We achieve query complexity of
$\tilde{O}(poly(\Delta/\varepsilon)n^{2/3})$, ($\tilde{O}$-notation hides
polylogarithmic factors in $n$). where $\Delta$ is the maximum degree of the
input graph. Our algorithm is the first to do so on arbitrary bounded degree
graphs. Moreover, we achieve the additional property that our algorithm outputs
a spanner, i.e., distances are approximately preserved. With high probability,
for each deleted edge there is a path of $O(poly(\Delta/\varepsilon)\log^2 n)$
hops in the output that connects its endpoints.
