We study two problems: (1) mining frequent sequences from a transactional database, and (2) incremental update of frequent sequences when the underlying database changes over time. We review existing sequence mining algorithms including GSP, PrefixSpan, SPADE, and ISM. We point out the large memory requirement of Pref ixSpan, SPADE, and ISM, and evaluate the performance of GSP. We discuss the high I/O cost of GSP, particularly when the database contains long frequent sequences. To reduce the I/O requirement, we propose an algorithm MFS, which could be considered as a generalization of GSP. The general strategy of MFS is to first find an approximate solution to the set of frequent sequences and then perform successive refinement until the ex...