An Efficient Algorithm for Unbalanced 1D Transportation

Optimal transport (OT) and unbalanced optimal transport (UOT) are central in many machine learning, statistics and engineering applications. 1D OT is easily solved, with complexity O(n log n), but no efficient algorithm was known for 1D UOT. We present a new approach that leverages the successive shortest path algorithm for the corresponding network flow problem. By employing a suitable representation, we bundle together multiple steps that do not change the cost of the shortest path. We prove that our algorithm solves 1D UOT in O(n log n), closing the gap.