In the model of local computation algorithms (LCAs), we aim to compute the queried part of the output by examining only a small (sublinear) portion of the input. This key aspect of LCAs generalizes various other models such as parallel algorithms, local filters and reconstructors. For graph problems, design techniques for LCAs and distributed algorithms are closely related and have been proven useful in each other's context. 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. We consider the case where the 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 quasipolynomial in d and polylogarithmic in n; • for constant ∈ > 0, a randomized LCA that provides a (1 - ∈)-approximation to maximum matching with high probability, whose time and space complexities are polynomial in d and polylogarithmic in n.