1. Finding Frequent Itemsets by Transaction Mapping
Mingjun Song ,Sanguthevar Rajasekaan  
Proceedings of the 2005 ACM symposium on Applied computing
報告者：林靜怡
2006/01/13

2. Introduction Apriori algorithm needs many database scans
for each scan, frequent itemsets are searched by pattern matching
time-consuming for large frequent itemsets with long patterns.

3. TM Algorithm Vertical database representation Transaction mapping
Transaction ids of each itemset are mapped and compressed to continuous
transaction intervals in a different space
reducing the number of intersections

4. Lexicographic Prefix Tree

5. Lexicographic Prefix Tree (conti.)
generate candidate itemsets and test their frequency.
Each node in the tree stores a collection of frequent itemsets.

6. Lexicographic Prefix Tree (conti.)
Depth first--if the expansion of a node cannot possibly lead to the discovery of itemsets that have minimum support, then the node will
not be expanded and the search will backtrack.
When a frequent itemset that meets the minimum support requirement is found, it is output.

7. Transaction Mapping
Scan through the database once and identify all frequent 1-itemsets
sort them in descending order of frequency 1-itemsets

8. Transaction Mapping sup{1} = 5 sup{2} = 5 sup{3} = 4 sup{4} = 2
min_sup = 2
sup{1} = 5
sup{2} = 5
sup{3} = 4
sup{4} = 2
sup{5} = 1
sup{6} = 1 .
sup{20}=1
identify all frequent
1-itemsets
Frequent 1-itemsets：
1,2,3,4

9. Transaction Mapping(Conti.)
Scan through the database again
For each transaction, select items that are in frequent 1-itemsets
sort them according to the order of frequent 1-itemsets
insert them into the transaction tree

10. Transaction Tree At the beginning the root is the current node.
if the current node has a child node whose id is equal to this item, then just increment the count of this child by 1
otherwise create a new child node and
set its counter as 1.

11. Transaction Tree
root
1:1
2:1
2:1
3:1
3:1
4:1
3:1

12. Node Interval
a node u that has an associated interval of [s, e], where s is the relabeled start id, e is the relabeled end id.
If the node is the first child of it’s parent
s = start id of u’s parent
If not
s = the end id of its previous child+1
e = start id of u + counter - 1

13. Node Interval [1,5] [6,8] [1,2] [3,3] [6,6] [7,8] [1,2]
not first child
s=2+1=3
c=3+1-1=3
first child
s=1
c=1+2-1=2
first child
s=1
c=1+2-1=2
first child
s=1
c=1+5-1=5
[1,5]
[6,8]
[1,2]
[3,3]
[6,6]
[7,8]
[1,2]

14. output min_sup = 2 1 2 3 4 {1,2} {1,3} intersect [1,2] >2 {1,2,3,4}
<2
{1,2,4}
intersect
<2
{1,2}
intersect
[1,2] >=2
{1,2,3}
intersect
[1,2] >=2
2 3 4 1
3,4 2 3
{1,2,3} 4
{1,3} 2
3 4 4 3 4 3
{2,3}
{2,4} 4 3

15. Experiments OS：Windows 2000 CPU：DELL 2.4GHz Pentium PC RAM：1GB
Compiler：Visual C++

16. Experiments
synthetic data
real data

17. Experiments

18. Experiments

19. Experiments

20. Experiments