1. Scalable Algorithms for Association Mining
Mohammed J. Zaki
IEEE Transactions on Knowledge and Data Engineering, 2000
2018/12/10
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 I A set of items D Database of transactions tid
Identifier of transaction
itemset
k-itemset
An itemset with k items
σ(X)
The support of an itemset X
frequent itemset
Its support ≧ minimum support Fk
The set of frequent k-itemsets
A→B
Association rule
Support：σ(A∪B)
Confidence：σ(A∪B) / σ(A)

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: Lemma2: 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.
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 Eclat Prefix-based, bottom-up search MaxEclat
Prefix-based, hybrid search
Clique
Maximal-clique-based, bottom-up search
MaxClique
Maximal-clique-based, hybrid search
Topdown
Maximal-clique-based, top-down search
AprClique
Maximal-clique-based, horizontal data layout, hash tree
Partition
Decompose database into nonoverlapping partition
Use vertical tid-list to generate local frequent itemsets
Merge all local frequent itemsets and compute global counts

22. Experimental Result

23. Experimental Result cont.

24. Experimental Result cont.