# How to calculate the max possible earth mover distance between histograms, given the buckets?

I have two histograms and am able to calculate the EMD / Wasserstein metric between them using the algorithm described here. In order to better communicate the implication of this metric, I want to normalize the calculated EMD by its max possible value - essentially producing a result that is, "what percent of the worst case scenario has occurred."

I know that in general, EMD has no upper bound. That said, this is a case where I know the total masses of both distributions, and the buckets in which those masses may lie. Is there a way to calculate the largest possible EMD between two known histograms?

So far I can tell that given only the range of buckets, the max possible EMD between two distributions of equal mass is the total distance between the smallest and largest bucket, multiplied by the mass. For example, with four buckets and mass of X, the max EMD is X.

I want to extend this to cases where

1. Mass is arbitrarily distributed in the 'empirical' histogram
2. And optionally, where mass is not equal between the two distributions