In the model of \emph{local computation algorithms} (LCAs), we aim to compute the queried part of the output by examining only a small (sublinear) portion of the input. Many recently developed LCAs on graph problems achieve time and space complexities with very low dependence on $n$, the number of vertices. Nonetheless, these complexities are generally at least exponential in $d$, the upper bound on the degree of the input graph. Instead, we consider the case where parameter $d$ can be moderately dependent on $n$, and aim for complexities with subexponential dependence on $d$, while maintaining polylogarithmic dependence on $n$. We present: a randomized LCA for computing maximal independent sets whose time and space complexities are quasi-polynomial in $d$ and polylogarithmic in $n$; for constant $ε> 0$, a randomized LCA that provides a $(1-ε)$-approximation to maximum matching whose time and space complexities are polynomial in $d$ and polylogarithmic in $n$.