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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\u201376, 1998) and M\u00e9rigot (Comput Graph Forum 30(5):1583\u20131592, 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.<\/jats:p>","DOI":"10.1007\/s00186-020-00703-z","type":"journal-article","created":{"date-parts":[[2020,2,12]],"date-time":"2020-02-12T08:03:22Z","timestamp":1581494602000},"page":"133-163","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":18,"title":["Semi-discrete optimal transport: a solution procedure for the unsquared Euclidean distance case"],"prefix":"10.1007","volume":"92","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-9856-0089","authenticated-orcid":false,"given":"Valentin","family":"Hartmann","sequence":"first","affiliation":[]},{"ORCID":"https:\/\/orcid.org\/0000-0001-7079-6313","authenticated-orcid":false,"given":"Dominic","family":"Schuhmacher","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2020,2,12]]},"reference":[{"key":"703_CR1","unstructured":"Altschuler J, Weed J, Rigollet P (2017) Near-linear time approximation algorithms for optimal transport via Sinkhorn iteration. 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