Apriori Itemset Generation

A frequent itemset is an itemset whose support is greater than some user-specified minimum support (denoted L_k, where k is the size of the itemset)
A candidate itemset is a potentially frequent itemset (denoted C_k, where k is the size of the itemset)

Apriori Algorithm

Apriori Algorithm: (by Agrawal et al at IBM Almaden Research Centre) can be used to generate all frequent itemset

Pass 1

Generate the candidate itemsets in C₁
Save the frequent itemsets in L₁

Pass k

Generate the candidate itemsets in Ck from the frequent

L_k

_-1

Join L_k_-1 p with L_k_-1q, as follows:

insert into

C_k

select

₁

_k-1

from

L_k

_-1

L_k

_-1

where

₁

_k-2

_k-1

Generate all (k-1)-subsets from the candidate itemsets in C_k
Prune all candidate itemsets from C_k where some (k-1)-subset of the candidate itemset is not in the frequent itemset L_k_-1

Scan the transaction database to determine the support for each candidate itemset in C_k
Save the frequent itemsets in L_k

Implementation: A working Apriori Itemset Generation program can be found on the Itemset Implementation page.

Example 1: Assume the user-specified minimum support is 50%

Given: The transaction database shown below

TID	A	B	C	D	E	F
T₁	1	0	1	1	0	0
T₂	0	1	0	1	0	0
T₃	1	1	1	0	1	0
T₄	0	1	0	1	0	1

The candidate itemsets in C₂ are shown below

Itemset X	*supp(X)*
{A,B}	25%
{A,C}	50%
{A,D}	25%
{B,C}	25%
{B,D}	50%
{C,D}	25%

The frequent itemsets in L₂ are shown below

Itemset X	*supp(X)*
{A,C}	50%
{B,D}	50%

Example 2: Assume the user-specified minimum support is 40%, then generate all frequent itemsets.

Given: The transaction database shown below

TID	A	B	C	D	E
T₁	1	1	1	0	0
T₂	1	1	1	1	1
T₃	1	0	1	1	0
T₄	1	0	1	1	1
T₅	1	1	1	1	0

Pass 1

C₁

Itemset X supp(X)

A ?

B ?

C ?

D ?

E ?

L₁

Itemset X supp(X)

A 100%

B 60%

C 100%

D 80%

E 40%

Pass 2

C₂

Itemset X	*supp(X)*
A,B	?
A,C	?
A,D	?
A,E	?
B,C	?
B,D	?
B,E	?
C,D	?
C,E	?
D,E	?

Nothing pruned since all subsets of these itemsets are infrequent

L₂

Itemset X	*supp(X)*
A,B	60%
A,C	100%
A,D	80%
A,E	40%
B,C	60%
B,D	40%
B,E	20%
C,D	80%
C,E	40%
D,E	40%

L₂ after saving only the frequent itemsets

Itemset X	*supp(X)*
A,B	60%
A,C	100%
A,D	80%
A,E	40%
B,C	60%
B,D	40%
C,D	80%
C,E	40%
D,E	40%

Pass 3

To create C3 only look at items that have the same first item (in pass k, the first k - 2 items must match)

C₃

	Itemset X	*supp(X)*
join AB with AC	A,B,C	?
join AB with AD	A,B,D	?
join AB with AE	A,B,E	?
join AC with AD	A,C,D	?
join AC with AE	A,C,E	?
join AD with AE	A,D,E	?
join BC with BD	B,C,D	?
join CD with CE	C,D,E	?

C₃ after pruning

Itemset X	*supp(X)*
A,B,C	?
A,B,D	?
A,C,D	?
A,C,E	?
A,D,E	?
B,C,D	?
C,D,E	?

Pruning eliminates ABE since BE is not frequent

Scan transactions in the database

L₃

Itemset X	*supp(X)*
A,B,C	60%
A,B,D	40%
A,C,D	80%
A,C,E	40%
A,D,E	40%
B,C,D	40%
C,D,E	40%

Pass 4

First k - 2 = 2 items must match in pass k = 4

C₄

	Itemset X	*supp(X)*
combine ABC with ABD	A,B,C,D	?
combine ACD with ACE	A,C,D,E	?

Pruning:

For ABCD we check whether ABC, ABD, ACD, BCD are frequent. They are in all cases, so we do not prune ABCD.

For ACDE we check whether ACD, ACE, ADE, CDE are frequent. Yes, in all cases, so we do not prune ACDE

L₄

Itemset X	*supp(X)*
A,B,C,D	40%
A,C,D,E	40%

Both are frequent

Pass 5: For pass 5 we can't form any candidates because there aren't two frequent 4-itemsets beginning with the same 3 items.