Local Computation Algorithms for Graphs of Non-constant Degrees

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. 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 with high probability, whose time and space complexities are polynomial in d and polylogarithmic in n .