Distance between Univariate Distributions and Distribution Approximation

I've attached below the link with the paper containing the algo (a combinatorial optimization problem, refer to Kantorovich-Rubinstein algo in the paper). The problem is the following: given a discrete distribution F of m =10,000 points, find a subset of F, called distribution G with n = 1,000 data points, that has the same quantiles of the initial F distribution. The inputs of the problem are: 1) the initial discrete distribution of 10,000 points (imagine this can be the result of a MonteCarlo simulation of a fat-tailed distribution, or a static array representing a fat-tail distrib) 2) the number n of data points of the output distribution (a subset of the data points of the initial distribution in 1)), say n=1,000 3) last n daily returns of the s&p500 The outputs are: i) the "reduced" distribution of n points sampled from the distribution in 1) above ii) the VaR and CVaR at various confidence levels (84%, 95%, 99%) of the two distributions (the VaR 84% of the two distrib shall match or be very close due to the optimization; same for the CVaRs at 95% and 99%) iii) plots of the the empirical pdf and cdf of the G and F distributions, including Q-Q plot (to show how close they are) iv) plotting the points in the correlation-distance semi-plane (corr on y axis from -1 to +1, distance on x axis from 0 to 1) where the correlation is between the points in G and the S&P500, and distance coordinate is the results of the various iterations of the optimization problem so that in the end we choose the distribution G that solves the optimization problem but that also has the minimum correlation to the market (hence the point in the semi-plane with min distance from the (0,0)). The R code should be written in blocks doing small parts of the job: 1) load inputs (from excel file) 2) output reduced distribution (into excel file) 3) output VaRs and CVaRs of the two distributions 4) plot empirical pdf and CDFs of the two distributions, and QQ plot 5) plot the correlation-distance in the semiplane