We present a technique for transforming classical approximation algorithms into constant-time algorithms that approximate the size of the optimal solution. Our technique is applicable to a certain subclass of algorithms that compute a solution in a constant number of phases. The technique is based on greedily considering local improvements in random order.The problems amenable to our technique include Vertex Cover, Maximum Matching, Maximum Weight Matching, Set Cover, and Minimum Dominating Set. For example, for Maximum Matching, we give the first constant-time algorithm that for the class of graphs of degree bounded by $d$, computes the maximum matching size to within $\eps n$, for any $\eps > 0$, where $n$ is the number of nodes in the graph. The running time of the algorithm is independent of $n$, and only depends on $d$ and $\eps$.