  A spanner of a graph is a subgraph that preserves lengths of shortest paths
up to a multiplicative distortion. For every $k$, a spanner with size
$O(n^{1+1/k})$ and stretch $(2k+1)$ can be constructed by a simple centralized
greedy algorithm, and this is tight assuming Erd\H{o}s girth conjecture.
  In this paper we study the problem of constructing spanners in a local
manner, specifically in the Local Computation Model proposed by Rubinfeld et
al. (ICS 2011).
  We provide a randomized Local Computation Agorithm (LCA) for constructing
$(2r-1)$-spanners with $\tilde{O}(n^{1+1/r})$ edges and probe complexity of
$\tilde{O}(n^{1-1/r})$ for $r \in \{2,3\}$, where $n$ denotes the number of
vertices in the input graph. Up to polylogarithmic factors, in both cases, the
stretch factor is optimal (for the respective number of edges). In addition,
our probe complexity for $r=2$, i.e., for constructing a $3$-spanner, is
optimal up to polylogarithmic factors. Our result improves over the probe
complexity of Parter et al. (ITCS 2019) that is $\tilde{O}(n^{1-1/2r})$ for $r
\in \{2,3\}$. Both our algorithms and the algorithms of Parter et al. use a
combination of neighbor-probes and pair-probes in the above-mentioned LCAs.
  For general $k\geq 1$, we provide an LCA for constructing $O(k^2)$-spanners
with $\tilde{O}(n^{1+1/k})$ edges using $O(n^{2/3}\Delta^2)$ neighbor-probes,
improving over the $\tilde{O}(n^{2/3}\Delta^4)$ algorithm of Parter et al.
  By developing a new randomized LCA for graph decomposition, we further
improve the probe complexity of the latter task to be
$O(n^{2/3-(1.5-\alpha)/k}\Delta^2)$, for any constant $\alpha>0$. This latter
LCA may be of independent interest.
