A Combinatorial Algorithm for Approximating the Optimal Transport in the Parallel and MPC Settings

Optimal Transport is a popular distance metric for measuring similarity between distributions. Exact and approximate combinatorial algorithms for computing the optimal transport distance are hard to parallelize. This has motivated the development of numerical solvers (e.g. Sinkhorn method) that can exploit GPU parallelism and produce approximate solutions. We introduce the first parallel combinatorial algorithm to find an additive $\varepsilon$-approximation of the OT distance. The parallel complexity of our algorithm is $O(\log(n)/ \varepsilon^2)$ where $n$ is the total support size for the input distributions. In Massive Parallel Computation (MPC) frameworks such as Hadoop and MapReduce, our algorithm computes an $\varepsilon$-approximate transport plan in $O(\log (\log (n/\varepsilon))/\varepsilon^2)$ rounds with $O(n/\varepsilon)$ space per machine; all prior algorithms in the MPC framework take $\Omega(\log n)$ rounds. We also provide a GPU-friendly matrix-based interpretation of our algorithm where each step of the algorithm is row or column manipulation of the matrix. Experiments suggest that our combinatorial algorithm is faster than the state-of-the-art approximate solvers in the GPU, especially for higher values of $n$.