Local computation algorithms for graphs of non-constant degrees

Abstract

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 quasi-polynomial dependence on d, while maintaining polylogarithmic dependence on n. In this thesis, we give randomized LCAs for computing maximal independent sets, maximal matchings, and approximate maximum matchings. Both time and space complexities of our LCAs on these problems are 2 0(log3 d)polylog(n), 2 0(log2 d)polylog(n) and 2 0(log3 d)polylog(n), respectively.