1. Frequent Closed Pattern Search By Row and Feature Enumeration

2. Outline Problem Definition
Related Work: Feature Enumeration Algorithms
CARPENTER: Row Enumeration Algorithm
COBBLER: Combined Enumeration Algorithm

3. Problem Definition
Frequent Closed Pattern: 1) frequent pattern: has support value higher than the threshold 2) closed pattern: there exists no superset which has the same support value
Problem Definition: Given a dataset D which contains records consist of features, our problem is to discover all frequent closed patterns respect to a user defined support threshold.

4. Related Work Searching Strategy: breadth-first & depth-first search
Data Format: horizontal format & vertical format
Data Compression Method:
diffset, fp-tree, etc.

5. Typical Algorithms CLOSET APRIORI CHARM feature enumeration
horizontal format
depth-first search
fp-tree technique
APRIORI
feature enumeration
horizontal format
breadth-first search
CHARM
vertical format
depth-first search
deffset technique

6. CARPENTER Motivation Algorithm Prune Method Experiment
CARPENTER stands for Closed Pattern Discovery by Transposing Tables that are Extremely Long
Motivation
Algorithm
Prune Method
Experiment

7. Motivation
Bioinformatic datasets typically contain large number of features with small number of rows.
Running time of most of the previous algorithms will increase exponentially with the average length of the transactions.
CARPENTER’s search space is much smaller than that of the previous algorithms on these kind of datasets and therefore has a better performance.

8. Algorithm The main idea of CARPENTER is to mine the dataset row-wise.
2 steps:
First, transpose the dataset
Second , search in the row enumeration tree.

9. Transpose Table Feature a, b, c, d. Row r1, r2 , r3, r4. r1 a b c r2
project on (r2 r3)
original table
transposed table
projected table

10. Row Enumeration Tree
r1 r2 r3 {bc}
r1r2r3r4 { }
r1 r2 {bc}
According to the transposed table, we build the row enumeration tree which enumerates row ids with a pre-defined order.
We do a depth first search in the row enumeration tree with out any prune strategies.
r1 r2 r4 {}
r1 r3 {bc}
r1 r3 r4 { }
r1 {abc}
minsup=2
bc: r1r2r3
bcd: r2r3
d: r2r3r4
r1 r4 {}
r2 r3 {bcd}
r2 r3 r4 {d }
{ }
r2 {bcd} a r1 b
r1 r2 r3 c d
r2 r3 r4
r2 r4 {d}
r3 {bcd}
r3 r4 {d }
r4 {d}

11. Prune Method 1
In the enumeration tree, the depth of a node is the corresponding support value.
Prune a branch if there won’t be enough depth in that branch, which means the support of patterns found in the branch will not exceed the minimum support.
minsup 4
r2 r3 {bcd}
r2 {bcd}
r2 r4 {d}
depth= 1
sup =1
2 sub-nodes
Max support value in branch “r2” will be 3, therefore prune this branch.

12. Prune Method 2
If rj has 100% support in projects table of ri, prune the branch of rj.
r2 {bcd}
r2 r3 {bcd}
r2 r3 r4 {d}
r2 r4 {d} b r3 c d
r3 r4 b c d r4
r2 r3 {bcd}
r2 r3 r4 {d}
r3 has 100% support in the projected table of “r2”, therefore branch “r2 r3” will be pruned and whole branch is reconstructed.

13. Prune Method 3
At any node in the enumeration tree, if the corresponding itemset of the node has been found before, we prune the branch rooted at this node.
r2 {bcd}
r2 r3 {bcd}
r2 r4 {d}
r3 {bcd}
r3 r4 {d}
Since itemset {bcd} has been found before, the branch rooted at “r3” will be pruned.

14. Performance We compare 3 algorithms, CARPENTER, CHARM and CLOSET.
Dataset (Lung Cancer) has 181 rows with features.
We set 3 parameters, minsup, Length Ratio and Row Ratio.

15. minsup Lung Cancer, 181 rows, length ratio 0.6,row ratio 1.
Running time of CARNPENTER changes from 3 to 14 second

16. Length Ratio Lung Cancer, 181 rows, sup 7 (4%), row ratio 1
Running time of CARPENTER changes from 3 to 33 seconds

17. Row Ratio Lung Cancer, 181 rows, length ratio 0.6,sup 7 (4%)
Running time of CARPENTER changes from 9 to 178 seconds

18. Conclusion
We propose an algorithm call CARPENTER for finding closed pattern on long biological datasets.
CARPENTER perform row enumeration instead of column enumeration since the number of rows in such datasets are significantly smaller than the number of features.
Performance studies show that CARPENTER is much more efficient in finding closed patterns compared to existing feature enumeration algorithms.

19. COBBLER
Motivation
Algorithm
Performance

20. Motivation
With the development of CARPENTER, existing algorithms can be separated into two parts.
Feature enumeration: CHARM, CLOSET, etc.
Row enumeration: CARPENTER
We have two motivations to combine these two enumeration methods

21. Motivation
1. We can see that these two enumeration methods have their own advantages on different type of data set. Given a dataset, the characteristic of its sub-dataset may change.
sub-dataset
dataset
project
more rows than features
more features than rows
2. Given a dataset with both large number of rows and features, a single row enumeration algorithm or a single feature enumeration method can not handle the dataset.

22. Algorithm There are two main points in the COBBLER algorithm
How to build an enumeration tree for COBBLER.
How to decide when the algorithm should switch from one enumeration to another.
Therefore, we will introduce the idea of dynamic enumeration tree and switching condition

23. Dynamic Enumeration Tree
We call the new kind of enumeration tree used in COBBLER the dynamic enumeration tree.
In dynamic enumeration tree, different sub-tree may use different enumeration method.
original
transposed r1
a b c r2
a c d r3
b c r4 d a
r1 r2 b
r1 r3 c
r1 r2 r3 d
r2 r4
We use the table as an example in later discussion

24. Single Enumeration Tree
abcd { }
r1r2r3r4 { }
r1r2r3 {c}
abc {r1}
ab {r1}
r1r2 {ac}
abd { }
r1r2r4 { }
ac {r1r2}
acd { r2}
a {r1r2}
r1r3 {bc}
r1r3r4 { }
r1 {abc}
ad {r2}
r1r4 { }
bc {r1r3}
bcd { }
r2r3r4 { }
r2r3 {c}
{ }
b {r1r3}
{ }
r2 {acd} r1
a b c r2
a c d r3
b c r4 d
bd { }
r2r4 {d }
c {r1r2r3}
cd {r2 }
r3 {bc}
r3r4 { }
d {r2r4}
Feature enumeration
r4 {d}
Row enumeration

25. Dynamic Enumeration Tree
abcd { }
abc {r1}
r1r2 {c}
ab {r1}
r1 {bc} r1 bc r2 cd
abd { }
a {r1r2}
ac {r1r2}
acd { r2}
r2 {cd}
a {r1r2}
ad {r2}
r1 {c}
r1r3 { c}
{ }
b {r1r3}
r3 { c}
abc: {r1}
ac: {r1r2}
acd: {r2} b r1 c
r1 r2 d r2
c {r1r2r3}
r2 {d }
d {r2r4}
Feature enumeration to Row enumeration

26. Dynamic Enumeration Tree
r1r2r3r4 { }
ab {}
r1r2r3 {c}
a {r2}
r1r2 {ac}
ac { r2}
r1r2r4 { }
r1 {abc}
b {r3}
bc {r3 }
r1r3 {bc}
r1r3r4 { }
r1 {abc}
c {r2r3 }
r1r4 { }
ac {r1 }
acd { }
a {r1}
ad { }
{ }
r2 {acd}
cd { }
ac: {r1r2}
bc: {r1r3}
c: {r1r2r3}
c {r1r3}
d {r4 }
b {r1 }
bc {r1 }
r3 {bc}
c {r1r2 }
r4 {d}
Row enumeration to Feature Enumeration

27. Dynamic Enumeration Tree
When we use different condition to decide the switching, the structure of the dynamic enumeration tree will change.
No matter how it switches, the result set of closed pattern will be the same as the result of the single enumeration .

28. Switching Condition
The main idea of the switching condition is to estimate the processing time of the a enumeration sub-tree, i.e., row enumeration sub-tree or feature enumeration sub-tree.
Define some characters.

29. Switching Condition

30. Switching Condition
Suppose r=10, S(f1)=0.8, S(f2)=0.5, S(f3)=0.5, S(f4)=0.3 and minsup=2
Then the estimated deepest node under f1 is f1f2f3, since
S(f1)*S(f2)*S(f3)*r=2 >minsup
S(f1)*S(f2)*S(f3)*S(f4)*r=0.6 < minsup

31. Experiments We compare 3 algorithms, COBBLER, CHARM and CLOSET+.
One real-life dataset and one synthetic data.
We set 3 parameters, minsup, Length Ratio and Row Ratio.

32. minsup
Synthetic data
Real-life data (thrombin)

33. Length and Row ratio
Synthetic data

34. Discussion
The combination of row and feature enumeration also makes some disadvantage
The cost to calculate the switching condition and the cost of bad decision.
The increased cost in pruning, maintain two set of pruning system.

35. Discussion
We may use other more complicated data structure in our algorithm to improve the performance, e.g., the vertical data format and diffset technique.
And more efficient switching condition may improve the algorithm further.

36. Conclusion
The COBBLER algorithm gives better performance on dataset where the advantage of switching can be shown, e.g., complex dataset or dataset has both large number of rows and features.
For simple characteristic data, a single enumeration algorithm may be better.

37. Future Work Using other data structure and technique in the algorithm.
Extend COBBLER to handle dataset that can not be fitted into memory.

38. Thanks