1. Sequence Mining Automata: a New Technique for Mining Frequent Sequences Under Regular Expressions Roberto Trasarti, Francesco Bonchi, Bart Goethals

2. Problem Definition (1): Given a database of sequences D, the support of a sequence S ∈ Σ∗ is the number of sequences in D that are supersequences of S: sup(S) = | {T ∈ D | S ⊑ T} |. Given a Regular Expression R a sequence s is valid if can be generated by R. A B A C B A Sequence s: 1 Minimum support: 3 RE: A*BC* A A A B B C A B C C D A B A A B B C 2 C B A A B D A A A B 3 A A B Subsequence: Support: 3 Subsequence: Support: 2 … B C

3. Previous approaches and our contribution: Previous approaches [1,2,3] solve the problem focusing on its search space, exploiting in different ways the pruning power of the regular expression R over unpromising patterns. The idea behind our solution is to focus on the input dataset and the given regular expression: reading the input database we produce for each sequence in the database, all and only the valid patterns contained in the sequences. [1] H. Albert-Lorincz and J.-F. Boulicaut. Mining frequent sequential patterns under regular expressions: A highly adaptive strategy for pushing contraints. In Proc. of SDM’03. [2] M. N. Garofalakis, R. Rastogi, and K. Shim. Spirit: Sequential pattern mining with regular expression constraints. In Proceedings of VLDB’99. [3] J. Pei, J. Han, andW.Wang. Mining sequential patterns with constraints in large databases. In Proc. of CIKM’02. A B ... A C A B C A ... B ... A A ... ... C ... C A B ... A B A C B A A A A B B C ...

4. Sequence Mining Automata (1): Our subsequences mining automata SMA is a specialized kind of Petri Net, which can be constructed from a DFA by transforming each edge of the DFA in a transition with its two arcs from its input place and to its output place. Moreover it has the following peculiarities: • Transitions do not consume tokens• Parallel execution • External signal The initial marking consists of only the token representing the empty sequence ε in the starting places. External signal Example RE: A*B(B|C)D*E

5. Sequence Mining Automata (2): Each transition applies an process which is activated only if the external signal is equal to the label of the edge. This process produces a new set of tokens in the destination place. External signal Example RE: A*B(B|C)D*E

6. Sequence Mining Automata (3 Example): Given R ≡ A∗B(B|C)D∗E S ≡ ACDBFAEBCFDE

7. One-Pass Solution (SMA-1P) and Full-Cut (SMA-FC) Simply using the SMA on each transactions and at the end compute the support for each sequences extracted filtering using the support threshold. The support threshold is not used during the process of generation. We compute All the sequences in the dataset w.r.t the RE. A D B B E C Given a SMA a valid set of cuts is a partition p1, . . . , pn of the places of the SMA such as does not exist a path from a place in pj to a place in pi if j > i. For each cut we apply the SMA-1P on all the DB. At the end of the i-th scan we obtain an intermediate information about frequent patterns that can be used in subsequent scans by removing the infrequent tokens.

8. Experiments (Synthetic Data): (D=dataset size, N=number of items, C=average length)

9. Experiments (Mobility data): From San Jose to San Francisco and back – via CA-101 (west-bound of the bay), i.e., passing through San Mateo (cell H9 of our map); or via I-880 (east-bound of the bay), i.e., passing through Hayward (cell J8 of our map).

10. Conclusions: We have introduced “Sequence Mining Automata”, a new mechanism for mining frequent sequences under regular expressions. Around this basic mechanism we built a family of algorithms embedding different techniques. The efficiency of our proposal has been thoroughly proven empirically. The SMA is a very simple and fundamental mechanism opening the door to many possible extensions.