A Near-Optimal Parallel Algorithm for Finding Matroid Bases

We settle the classic question of the parallel complexity of computing a matroid basis, as first posed in the seminal work of Karp, Upfal, and Wigderson (FOCS 1985, JCSS 1988). Our algorithm runs in O(n^{1/3}\log^{1/3}n) rounds, matching the lower bound of KUW up to a \log^{2/3}(n) factor.