In a seminal paper on finding large matchings in sparse random graphs, Karp and Sipser proposed two algorithms for this task. The second algorithm has been intensely studied, but due to technical difficulties, the first algorithm has received less attention. Empirical results suggest that the first algorithm is superior. In this paper we analyze an adapted version of the first algorithm, the Reduce-Construct algorithm. We show that the Reduce-Construct algorithm finds a maximum matching in random $k=O(1)$-regular graphs in linear time in expectation, as opposed to $O(n^{3/2})$ time for the worst-case.