Let's say you have a list of similar sequences, such as
a a a a
a b a a a
x a a a a y
...
You want to detect a common aggregate of all these sequences, such as
x? a b? a a a y?
where operator ? specifies that element is optional.
What algorithm would you use?
Look at the sequence alignment algorithms used in bioinformatics.
More specifically, since you have a list, multiple sequence alignment. The Viterbi algorithm should do.
I think that if you convert your list to suffix tree then it will be very simple recursive solution but I am not sure about asymptotic complexity