How to classify intervals according to their overlapping?

I am looking for an algorithm that classifies (with letters) a set of intervals according to their overlap.

• The output should associate each overlapping interval with the same letter, thus constituting a unique group.

• Intervals that overlap with multiple groups will be classified with several letters corresponding to each unique group.

• Non-overlapping intervals also constitute a unique group.

Graphically the problem is as follows:

I am working in R and my intervals are these:

structure(list(Interval = c(1, 2, 3, 4, 5, 6, 7, 8, 9, 10), Start = c(5.3, 
6.5, 7.6, 7.8, 8, 8.3, 8.5, 8.7, 8.8, 9.9), End = c(7.5, 8.7, 
9.8, 10, 10.2, 10.5, 10.7, 10.9, 11, 12.1)), row.names = c(NA, 
-10L), spec = structure(list(cols = list(Interval = structure(list(), class = c("collector_double", 
"collector")), Start = structure(list(), class = c("collector_double", 
"collector")), End = structure(list(), class = c("collector_double", 
"collector"))), default = structure(list(), class = c("collector_guess", 
"collector")), delim = "\t"), class = "col_spec"), class = c("spec_tbl_df","tbl_df", "tbl", "data.frame"))

I think the ivs package may have the solution but I don’t know the procedure.