We consider the problem of finding an optimal transport plan between an absolutely continuous measure and a finitely supported measure of the same total mass when the transport cost is the unsquared Euclidean distance. We may think of this problem as closest distance allocation of some resource continuously distributed over Euclidean space to a finite number of processing sites with capacity constraints. This article gives a detailed discussion of the problem, including a comparison with the much better studied case of squared Euclidean cost. We present an algorithm for computing the optimal transport plan, which is similar to the approach for the squared Euclidean cost by Aurenhammer et al. (Algorithmica 20(1):61–76, 1998) and Mérigot (Comput Graph Forum 30(5):1583–1592, 2011). We show the necessary results to make the approach work for the Euclidean cost, evaluate its performance on a set of test cases, and give a number of applications. The later include goodness-of-fit partitions, a novel visual tool for assessing whether a finite sample is consistent with a posited probability density.

我们考虑在运输成本为非平方欧几里德距离的情况下，在相同总质量的绝对连续度量和有限支持度量之间找到最佳运输计划的问题。我们可以将这个问题视为连续在欧几里得空间上连续分布的某些资源到具有容量限制的有限数量的处理站点的最近距离分配。本文对这个问题进行了详细的讨论，其中包括与研究得更好的平方欧几里得成本进行比较。我们提出一种用于计算最佳运输计划的算法，该算法类似于Aurenhammer等人的平方欧几里得成本的方法。（Algorithmica 20（1）：61–76，1998）和Mérigot（Comput Graph Forum 30（5）：1583–1592，2011）。我们展示了必要的结果，以使该方法适用于欧几里得成本，在一组测试用例上评估其性能，并提供了许多应用程序。后者包括拟合优度分区，这是一种新颖的可视化工具，用于评估有限样本是否与假定的概率密度一致。