  In his Ph.D. thesis, Burak Ozbagci described an algorithm computing
signatures of Lefschetz fibrations where the input is a factorization of the
monodromy into a product of Dehn twists. In this note, we give a reformulation
of Ozbagci's algorithm which becomes much easier to implement. Our main tool is
Wall's non-additivity formula applied to what we call partial fiber sum
decomposition of a Lefschetz fibration over 2-disk. We show that our algorithm
works for bordered Lefschetz fibrations over disk and it yields a formula for
the signature of branched covers where the branched loci are regular fibers. As
an application, we give the explicit monodromy factorization of a Lefschetz
fibration over disk whose total space has arbitrarily large positive signature
for any positive fiber genus.
