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However the original definition of HUM involves multiple integration and thus a medical investigation for multi\u2010class receiver operating characteristic (ROC) analysis could suffer from huge computational cost when the formula is implemented naively. We introduce a novel graph\u2010based approach to compute HUM efficiently in this article. The computational method avoids the time\u2010consuming multiple summation when sample size or the number of categories is large. We conduct extensive simulation studies to demonstrate the improvement of our method over existing R packages. 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