  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.
