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.

翻译：最优传输(OT)与非平衡最优传输(UOT)在众多机器学习、统计学及工程应用中占据核心地位。一维OT问题易于求解，时间复杂度为O(n log n)，但此前一维UOT问题尚无高效算法。本文提出一种新方法，利用相应网络流问题的连续最短路径算法。通过采用适当的表示方式，我们将不改变最短路径代价的多步操作合并处理。我们证明该算法可在O(n log n)时间内求解一维UOT问题，从而填补了该研究空白。