Variants of local algorithms on sparse graphs

Suppose we want to construct some structure on a bounded-degree graph, e.g., an almost maximum matching, and we want to decide about each edge depending only on its constant-radius neighborhood. We examine and compare the strengths of different extensions of these local algorithms. A common extension is to use preprocessing, which means that we can make some calculation about the whole graph, and each local decision can also depend on this calculation. In this paper, we show that preprocessing is needless: if a nearly optimal local algorithm uses preprocessing, then the same can be achieved by a local algorithm without preprocessing, but with a global randomization.