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：最优输运（Optimal Transport, OT）与非平衡最优输运（Unbalanced Optimal Transport, UOT）是许多机器学习、统计学与工程应用中的核心问题。一维OT易于求解，复杂度为O(n log n)，但此前尚无针对一维UOT的高效算法。我们提出了一种新方法，利用连续最短路径算法求解相应的网络流问题。通过采用合适的表示，我们将不改变最短路径代价的多个步骤捆绑在一起。我们证明，该算法能以O(n log n)的复杂度解决一维UOT问题，从而填补了这一空白。