Example)
First FPtree/FPGrowth
(Complete scan determine frequent 1itemsets, then build header
TID 1 2 3 4 5 6 7 8 9 10 Items {A,B} {B,C,D} {A,C,D,E} {A,D,E} {A,B,C} {A,B,C,D} {B,C} {A,B,C} {A,B,D} {B,C,E}
B A C D E 8 7 7 5 3
FP-tree construction
After reading TID=1: null B:1 A:1 After reading TID=2: B:2 A:1 C:1 D:1
TID 1 2 3 4 5 6 7 8 9 10
Items {A,B} {B,C,D} {A,C,D,E} {A,D,E} {A,B,C} {A,B,C,D} {B,C} {A,B,C} {A,B,D} {B,C,E}
null
FP-Tree Construction
TID 1 2 3 4 5 6 7 8 9 10 Items {A,B} {B,C,D} {A,C,D,E} {A,D,E} {A,B,C} {A,B,C,D} {B,C} {A,B,C} {A,B,D} {B,C,E}
Transaction Database B:8 A:5 C:3 D:1 D:1 C:3 D:1
null A:2 C:1 E:1 D:1 E:1 D:1 E:1
Header table
Item B A C D E Pointer 8 7 7 5 3
Chain pointers help in quickly finding all the paths of the tree containing some given item.
Paths containing node E
null B:8 A:5 C:3 D:1 D:1 C:3 D:1 E:1 A:2 C:1 D:1 E:1 B:1 C:1 E:1 D:1 E:1 null A:2 C:1 D:1 E:1 D:1 E:1
Conditional FP-Tree for E
FP-Growth builds a conditional FP-Tree for E, which is the tree of itemsets ending in E. It is not the tree obtained in previous slide as result of deleting nodes from the original tree. Why? Because the order of the items can change.
Now, C has a higher count than B.
null B:1 C:1 E:1 A:2 C:1 D:1 E:1
Suffix E
(New) Header table A D:1 E:1 C D 2 2 2 null
B doesn't C:1 survive because it has support 1, which is lower than min support of 2.
Conditional FP-Tree for suffix E
A:2 C:1 D:1 D:1
The set of paths ending in E. Insert each path (after truncating E) into a new tree.
We continue recursively. Base of recursion: When the tree has a single path only. FI: E
Steps of Building Conditional FPTrees
1. Find the paths containing on focus item. 1. Read the tree to determine the new counts of the items along those paths. Build a new header. 1. Read again the tree. Insert the paths in the conditional FP-Tree according to the new order.
Suffix DE
null A:2 C:1 D:1 D:1 (New) Header table
A
2
The conditional FP-Tree for suffix DE null A:2
The set of paths, from the Econditional FP-Tree, ending in D. Insert each path (after truncating D) into a new tree. We have reached the base of recursion. FI: DE, ADE
Base of Recursion
We continue recursively on the conditional FP-Tree. Base case of recursion: when the tree is just a single path.
Then, we just produce all the subsets of the items on this path merged with the corresponding suffix.
Suffix CE
null C:1 A:1 C:1 (New) Header table The conditional FP-Tree for suffix CE null
The set of paths, from the Econditional FP-Tree, ending in C. Insert each path (after truncating C) into a new tree. We have reached the base of recursion. FI: CE
Suffix AE
(New) Header table The conditional FP-Tree for suffix AE null
null A:2
The set of paths, from the Econditional FP-Tree, ending in A. Insert each path (after truncating A) into a new tree. We have reached the base of recursion. FI: AE
Suffix D
null B:3 A:2 C:1 D:1 B:2 The set of paths ending in D. Insert each path (after truncating D) into a new tree. C:1 We continue recursively. Base of recursion: When the tree has a single path only. FI: D C:1 C:1 D:1 C:1 D:1 C:1 A:2 D:1 D:1 A:4 B:1 (New) Header table A B C 4 3 3 null Conditional FP-Tree for suffix D
Suffix CD
null A:4 B:2 C:1 B:1 The set of paths, from the D-conditional FP-Tree, ending in C. Insert each path (after truncating C) into a new tree. C:1 B:1 C:1 A:2 (New) Header table A B 2 2 null B:1 Conditional FP-Tree for suffix CD
We continue recursively. Base of recursion: When the tree has a single path only. FI: CD
Suffix BCD
(New) Header table null A:2 B:1 B:1 null Conditional FP-Tree for suffix CDB
The set of paths from the CD-conditional FP-Tree, ending in B. Insert each path (after truncating B) into a new tree.
We have reached the base of recursion. FI: BCD
Suffix ACD
(New) Header table null Conditional FP-Tree for suffix ACD null
The set of paths from the CD-conditional FP-Tree, ending in A. Insert each path (after truncating B) into a new tree.
We have reached the base of recursion. FI: ACD
Suffix C
null B:6 A:3 C:3 A:3 The set of paths ending in C. Insert each path (after truncating C) into a new tree. We continue recursively. Base of recursion: When the tree has a single path only. FI: C C:3 A:1 C:1 B:6 (New) Header table B A 6 4 Conditional FP-Tree for suffix C null A:1
Suffix AC
null B:6 A:3 B:3 A:1 (New) Header table B 3 Conditional FP-Tree for suffix AC null
The set of paths from the C-conditional FP-Tree, ending in A. Insert each path (after truncating A) into a new tree.
We have reached the base of recursion. FI: AC, BAC
Suffix BC
null B:6 (New) Header table B 3 Conditional FP-Tree for suffix BC null
The set of paths from the C-conditional FP-Tree, ending in B. Insert each path (after truncating B) into a new tree.
We have reached the base of recursion. FI: BC
Suffix A
null B:5 A:5 B:5 A:2 (New) Header B table 5 Conditional FP-Tree for suffix A null
The set of paths ending in A. Insert each path (after truncating A) into a new tree.
We have reached the base of recursion. FI: A, BA
Suffix B
(New) Header table Conditional FP-Tree for suffix B null B:8 null
The set of paths ending in B. Insert each path (after truncating B) into a new tree.
We have reached the base of recursion. FI: B
Array Technique
FP-Tree Construction
TID 1 2 3 4 5 6 7 8 9 10 Items {A,B} {B,C,D} {A,C,D,E} {A,D,E} {A,B,C} {A,B,C,D} {B,C} {A,B,C} {A,B,D} {B,C,E}
Transaction Database
Header table B A C D E 8 7 7 5 3 First pass on DB: Determine the header. Then sort it. Second pass on DB: Build the FP-Tree. Also build an array of counts.
FP-Tree Construction Reading 1
TID 1 2 3 4 5 6 7 8 9 10 Items {A,B} {B,C,D} {A,C,D,E} {A,D,E} {A,B,C} {A,B,C,D} {B,C} {A,B,C} {A,B,D} {B,C,E}
Transaction Database B:1 A:1
null
Header table B A C D E 8 7 7 5 3
A C D E 1
FP-Tree Construction Reading 2
TID 1 2 3 4 5 6 7 8 9 10 Items {A,B} {B,C,D} {A,C,D,E} {A,D,E} {A,B,C} {A,B,C,D} {B,C} {A,B,C} {A,B,D} {B,C,E}
Transaction Database B:2 A:1 C:1 D:1
null
Header table B A C D E 8 7 7 5 3
A C D E 1 1 1 1
FP-Tree Construction Reading 3
TID 1 2 3 4 5 6 7 8 9 10 Items {A,B} {B,C,D} {A,C,D,E} {A,D,E} {A,B,C} {A,B,C,D} {B,C} {A,B,C} {A,B,D} {B,C,E}
Transaction Database B:2 A:1 C:1 D:1
null A:1 C:1 D:1 E:1
Header table B A C D E 8 7 7 5 3
A C D E 1 1 1 1 1 1 2 1 1
FP-Tree Construction Reading 4
TID 1 2 3 4 5 6 7 8 9 10 Items {A,B} {B,C,D} {A,C,D,E} {A,D,E} {A,B,C} {A,B,C,D} {B,C} {A,B,C} {A,B,D} {B,C,E}
Transaction Database B:2 A:1 C:1 D:1
null A:2 C:1 D:1 E:1
A C D E 1 1 1 1 2 2 2 1 2
D:1 E:1
Header table B A C D E 8 7 7 5 3
FP-Tree Construction Reading 5
TID 1 2 3 4 5 6 7 8 9 10 Items {A,B} {B,C,D} {A,C,D,E} {A,D,E} {A,B,C} {A,B,C,D} {B,C} {A,B,C} {A,B,D} {B,C,E}
Transaction Database B:3 A:2 C:1 C:1 D:1
null A:2 C:1 D:1 E:1
A C D E 2 2 1 2 2 2 2 1 2
D:1 E:1
Header table B A C D E 8 7 7 5 3
FP-Tree Construction Reading 6
TID 1 2 3 4 5 6 7 8 9 10 Items {A,B} {B,C,D} {A,C,D,E} {A,D,E} {A,B,C} {A,B,C,D} {B,C} {A,B,C} {A,B,D} {B,C,E}
Transaction Database B:4 A:3 C:2 D:1
A C D E 3 3 2
null A:2 C:1 D:1 E:1 D:1 E:1
C:1 D:1
Header table B A C D E 8 7 7 5 3
3 3 2 3 1 2
FP-Tree Construction Reading 7
TID 1 2 3 4 5 6 7 8 9 10 Items {A,B} {B,C,D} {A,C,D,E} {A,D,E} {A,B,C} {A,B,C,D} {B,C} {A,B,C} {A,B,D} {B,C,E}
Transaction Database B:5 A:3 C:2 D:1
A C D E 3 4 2
null A:2 C:1 D:1 E:1 D:1 E:1
C:2 D:1
Header table B A C D E 8 7 7 5 3
3 3 2 3 1 2
FP-Tree Construction Reading 8
TID 1 2 3 4 5 6 7 8 9 10 Items {A,B} {B,C,D} {A,C,D,E} {A,D,E} {A,B,C} {A,B,C,D} {B,C} {A,B,C} {A,B,D} {B,C,E}
Transaction Database B:6 A:4 C:3 D:1
A C D E 4 5 2
null A:2 C:1 D:1 E:1 D:1 E:1
C:2 D:1
Header table B A C D E 8 7 7 5 3
4 3 2 3 1 2
FP-Tree Construction Reading 9
TID 1 2 3 4 5 6 7 8 9 10 Items {A,B} {B,C,D} {A,C,D,E} {A,D...