A Theoretical Study on Variable Ordering of Zero-Suppressed BDDs for Representing Frequent Itemsets

  • Shin-ichi Minato
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4755)

Abstract

Recently, an efficient method of database analysis using Zero-suppressed Binary Decision Diagrams (ZBDDs) has been proposed. BDDs are a graph-based representation of Boolean functions, now widely used in system design and verification. Here we focus on ZBDDs, a special type of BDDs, which are suitable for handling large-scale combinatorial itemsets in frequent itemset mining. In general, it is well-known that the size of ZBDDs greatly depends on variable ordering; however, in the specific cases of applying ZBDDs to data mining, the effect of variable ordering has not been studied well. In this paper, we present a theoretical study on ZBDD variable ordering for representing frequent itemsets. We show two instances of databases we composed, where the ZBDD sizes are exponentially sensitive to the variable ordering. We also show that there is a case where the ZBDD size must be exponential in any variable ordering. Our theoretical results are helpful for developing a good heuristic method of variable ordering.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Agrawal, R., Imielinski, T., Swami, A.N.: Mining association rules between sets of items in large databases. In: Buneman, P., Jajodia, S. (eds.) Proc. of the 1993 ACM SIGMOD International Conference on Management of Data. SIGMOD Record, 22(2), 207–216 (1993)Google Scholar
  2. 2.
    Bryant, R.E.: Graph-based algorithms for Boolean function manipulation. IEEE Transactions on Computers C-35(8), 677–691 (1986)CrossRefGoogle Scholar
  3. 3.
    Fujita, M., Fujisawa, H., Kawato, N.: Evaluation and implementation of boolean comparison method based on binary decision diagrams. In: ICCAD-88. Proc. of ACM/IEEE International Conf. on Computer-Aided Design, pp. 2–5 (1988)Google Scholar
  4. 4.
    Goethals, B.: Survey on frequent pattern mining (2003), http://www.cs.helsinki.fi/u/goethals/publications/survey.ps
  5. 5.
    Han, J., Pei, J., Yin, Y., Mao, R.: Mining frequent patterns without candidate generation: a frequent-pattern tree approach. Data Mining and Knowledge Discovery 8(1), 53–87 (2004)CrossRefMathSciNetGoogle Scholar
  6. 6.
    Iwasaki, H., Minato, S., Zeugmann, T.: A method of variable ordering for zero-suppressed binary decision diagrams in data mining applications. In: SWOD 2007. Proc. of The Third IEEE International Workshop on Databases for Next-Generation Researchers (to appear, 2007)Google Scholar
  7. 7.
    Minato, S.: Zero-suppressed BDDs for set manipulation in combinatorial problems. In: Proc. of 30th ACM/IEEE Design Automation Conference, pp. 272–277 (1993)Google Scholar
  8. 8.
    Minato, S.: Zero-suppressed BDDs and their applications. International Journal on Software Tools for Technology Transfer (STTT) 3(2), 156–170 (2001)MATHGoogle Scholar
  9. 9.
    Minato, S., Arimura, H.: Frequent pattern mining and knowledge indexing based on zero-suppressed BDDs. In: Bonchi, F., Boulicaut, J.-F. (eds.) KDID 2006. LNCS, vol. 3933, pp. 83–94. Springer, Heidelberg (2006)Google Scholar
  10. 10.
    Tani, S., Hamaguchi, K., Yajima, S.: The complexity of the optimal variable ordering problems of shared binary decision diagrams. In: Ng, K.W., Balasubramanian, N.V., Raghavan, P., Chin, F.Y.L. (eds.) ISAAC 1993. LNCS, vol. 762, pp. 389–398. Springer, Heidelberg (1993)Google Scholar
  11. 11.
    Zaki, M.J.: Scalable algorithms for association mining. IEEE Trans. Knowl. Data Eng. 12(2), 372–390 (2000)CrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2007

Authors and Affiliations

  • Shin-ichi Minato
    • 1
  1. 1.Graduate School of Information Science and Technology, Hokkaido University, Sapporo, 060-0814Japan

Personalised recommendations