Let’s say you have 9 vials of liquid and the list of pairs to combine as follows:
1 & 2 (new batch)
5 & 8 (new batch)
9 & 4 (new batch)
9 & 6 (new 6 goes into existing 9)
3 & 5 (new 3 goes into existing 5)
7 & 1 (new 7 goes into existing 1)
2 & 5 (combine separate existing batches)
2 & 3 (same batch already)
How many separate batches do you have after each move? Are 2 & 4 in the same one?
Solution
This is solved with a Quick Union data structure; this structure has the following operations:
Ensures two IDs exist in the same set
This belongs to which set?
How many total sets?
Where 
and 
, the running time is 
.
My implementation is in git and below: