#' Format a partition matrix
#'
#' @param P1 A partition matrix.
#' @return A formatted partition matrix.
#' @description Formats a partition matrix so that subsets in a partition will be ordered by the value of the smallest in each subset
# 1. subsets in a partition will be ordered by the value of the smallest in each subset
format_partition <- function(P1) {
P <- as.matrix(P1)
n_p <- ncol(as.matrix(P))
m_p <- nrow(as.matrix(P))
P.form <- matrix(0, ncol = n_p, nrow = m_p)
a <- rep(0, n_p)
for (i in 1:n_p) {
a[i] <- position_finder(P[, i])
}
num_emps <- which(a == 999999)
non_zero_cols <- which(a != 999999)
a_non_zero <- a[non_zero_cols]
for (i in 1:length(non_zero_cols)) {
col_ord_ind <- which(rank(a_non_zero) == i)
P.form[, i] <- P[, col_ord_ind]
}
return(P.form)
}
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