Scalable Algorithms for Association Mining

1. Scalable Algorithms for Association Mining Mohammed J. Zaki IEEE Transactions on Knowledge and Data Engineering, 2000 報告人:吳建良

2. Abstract • Frequent itemset • Vertical tid-list database format • Lattice-theoretic approach • Prefix-based and Maximal-clique-based partition • Pattern search strategy • Bottom-up, top-down and hybrid search • Require a few databases scan

3. Symbol Definition

4. Example

5. Itemset Enumeration: Lattice-theoretic approach • Partial order • Reflexive, Antisymmetric, Transitive • Partial ordered set poset • Lattice • Poset • Any two element have unique join and meet • join= =least upper bound(a, b) • meet= = greatest lower bound(a, b) • Atom • Immediately succeed least element

6. Power set lattice P(I) Gray circle: frequent itemset Black circle: maximal frequent itemset

7. Lemma • Lemma1: • All subsets of a frequent itemset are frequent • All supersets of an infrequent itemset are infrequent • Lemma2: • The maximal frequent itemsets uniquely determine all frequent itemsets

8. Support Counting • L(X): each database item X its tid-list • Support of k-itemset • Intersect the tid-list of any two of its (k-1)- itemset • Example • L(CD)=L(C) ∩ L(D) • L(CDW)=L(CD) ∩ L(CW)

9. Example

10. Lattice Decomposition:Prefix-Based Classes • Equivalence relation • binary relation ≡ : reflexive, symmetric, transitive • partitions the set P into disjoint subsets called equivalence classes • An equivalence relation θk on the lattice P(I) where p(X, k)=X[1:k], the k length prefix of X • θk : prefix-based equivalence relation • Lemma: • Each equivalence class [X]θk induced by the equivalence relation θk is a sublattice of P(I)

11. Example of Equivalence Class P(I) induced by θ1 [A]θ1 induced by θ2

12. Search for Frequent Itemsets • Bottom-up Search Algorithm:

13. Search for Frequent Itemsets cont. • Example for [A]θ1

14. Search for Frequent Itemsets cont. • Top-down Search Algorithm:

15. Search for Frequent Itemsets cont. • Example for [A]θ1 Gray circle: infrequent itemset Black circle: maximal frequent itemset White circle: minimal infrequent itemset

16. Search for Frequent Itemsets cont. • Hybrid Search Algorithm:

17. Search for Frequent Itemsets cont. • Example for [A]θ1, assume that AD and ADW are frequent

18. Generating Smaller Classes: Maximal Clique Approach • Pseudoequivalence relation • binary relation ≡ : reflexive, symmetric • partitions the set P into possible overlapping subsets called pseudoequivalence classes • k-association graph Gk=(V, E) • Vertex set • Edge set

19. Maximal Clique Approach cont. • Clique • A complete subgraph of a graph • Mk: the set of maximal cliques in Gk • A pseudoequivalence relation φk on the lattice P(I) • φk : maximal-clique-based pseudoequivalence relation • Bottom-up search: reduce the number of intersections Top-down search: lead to smaller maximum element size

20. Example

21. Experiment • 比較的演算法

22. Experimental Result

23. Experimental Result cont.

24. Experimental Result cont.