An algorithm like PrefixSpan uses an horizontal database and do database projections. PrefixSpan is said to use a "pattern growth" approach.
For me, I categorize the algorithms as follows:
: Horizontal database, pattern-growth, database projection, do a depth-first search
: Vertical database encoded as bitvectors, generate candidates, do a depth-first search
: Vertical database, generate candidates, do a breadth first search
: patterngrowth, horizontal database I think... and probably also do database projection but I dont remember
and there are other algorithms.
If you did not read my survey of sequential pattern mining
, it gives some good overview.