Mining Frequent Item
Sets by Opportunistic
Projection
~From: Special Interest Group onKnowledge Discovery in
Data and Data Mining (SIGKDD 2002)
Junqiang Liu, Yunhe Pan, Ke Wang, Jiawei Han
碩專二 69121507 阮士峰
Outline






How to discover frequent item sets
Previous works
Our approach: Mining Frequent Item
Sets by Opportunistic Projection
Opportunistic Projection:
Observations and Heuristics
Performance evaluations
Conclusions
2
What Are Frequent Items Sets

What is a frequent item set?

set of items, X, that occurs together frequently in a
database, i.e., support(X) ≥ a given threshold

Example
tid
items
01
a
a
b
b
a
02
03
04
05
c
b
f
c
c
d
c
h
k
e
f
f
j
p
f
g i m p
l m o
o
s
l m n p
Given support threshold 3, frequent item
sets are as follows:
a:3, b:3, c:4, f :4, m:3, p:3,
ac:3, af :3, am:3, cf :3, cm:3, cp:3, fm:3,
acf :3, acm:3, afm:3, cfm:3,
acfm:3
3
How To Discover Frequent Item Sets

Frequent item sets can be represented by a tree,
which is not necessarily materailized.
( ,)
(a,3) (b,3) (c,4) (f,4) (m,3) (p,3)
(c,3) (f,3) (m,3) (f,3) (m,3) (p,3) (m,3)
(f,3) (m,3) (m,3)
(m,3)
(m,3)

Mining process:
 a process of tree construction, accompanied by
 a process of projecting transaction subsets
4
Frequent Item Set Tree - FIST

FIST is an ordered tree


each node: (item,weight)
the following are imposed



Frequent item set



items ordered on a path (top-down)
items ordered at children (left to right)
a path starting from the FIST root
its support is the ending node’s weight
PTS - projected transaction subset


Each FIST node has its own PTS, filtered or unfiltered
All transactions that support the frequent item set
represented by the node
5
Frequent Item Set Tree (example)
( ,)
01
02
03
04
05
(a,3)
(f,3)
01 f m
02 f m
05 f m
01 m
02 m
05 m
(f,3) (m,3)
01 m
02 m
05 m
(m,3)
c
b
f
c
c
(b,3)
01 c f m p
02 b c f m
05 c f m p
(c,3)
a
a
b
b
a
02 c f m
03 f
04 c p
(m,3)
(m,3)
d
c
h
k
e
f
f
j
p
f
g i m p
l m o
o
s
l m n p
(c,4)
01
02
04
05
f m p
f m
p
f m p
(f,3)
(m,3)
01 m p
02 m
05 m p
01 p
05 p
(f,4)
(m,3)
01 m p
02 m
05 m p
01 p
05 p
(p,3)
(p,3)
(m,3)
(m,3)
(i,w): a FIST node
: the PTS of the node
6
Factors relate to
Mining Efficiency and Scalability

The FIST construction strategy


The PTS representation



breadth first v.s. depth first
Memory-based representation: array-based, tree-based,
vertical bitmap, horizontal bitstring, etc.
Disk-based representation
PTS projecting method and item counting method
7
Our Approach: Mining Frequent Item Sets
by Opportunistic Projection

Philosophy:
The algorithm must adapt the construction strategy of FIST,
the representation of PTS, and the methods of item counting
in and projection of PTSs to the features of PTSs.

Main points:
Mining sparse data by projecting array-based PTS
 Intelligent projecting tree-based PTS for dense data
 Heuristics for opportunistic projection

8
Mining sparse data by
projecting array-based PTS
TVLA – threaded varied length array for sparse PTS






FIL– local frequent items list
LQ – linked queues
arrays
FIL a 3
Each local frequent item has
a FIL entry that consists of
an item, a count, & a pointer.
Each transaction is stored in
an array that is threaded to
FIL by LQ according to the
heading item in the imposing
order.
b
c
f
m
p
3
4
4
3
3
01
LQ
02
a
c
f
m
p
a
b
c
f
m
05
04
f
03
b
f
b
c
p
array
a
c
f
m
p
filtered TVLA of the original DB in the example
9
How to project TVLA for PTS



Arrays (transactions)
that support a node’s
first child are threaded
by the LQ attached to
the first entry of FIL.
(see previous figure)
TVLA for a child node’s
PTS has its own FIL
and LQ.
A child TVLA is
unfiltered if it shares
arrays with its parent,
filtered otherwise.
parent TVLA
FIL a 3
b 3
c 4
f 4
m 3
p 3
01
c 3
f 3
m 3
01
c 3
f 3
m 3
01
FIL(a)
FIL(a)
02
a
c
f
m
p
a
b
c
f
m
02
05
04
f
03
b
f
a
c
f
m
p
b
c
p
05
unfiltered child TVLA
02
c
f
m
05
c
c
f
f
m
m
filtered child TVLA
10
How to project TVLA for PTS (cont.)
Get next child’s PTS by shifting transactions threaded in the LQ
currently explored (current child’s PTS)

3
3
4
4
3
3
a
b
c
f
m
p
3
3
4
4
3
3
02
01
a
c
f
m
p
04
f
03
a
b
c
f
m
05
b
f
b
c
p
a
c
f
m
p

a
b
c
f
m
p
3
3
4
4
3
3
01
a
b
c
f
m
p
3
3
4
4
3
3
01
02
a
c
f
m
p
04
f
a
b
c
f
m
b
f
05
a
c
f
m
p
b
c
p

a
b
c
f
m
p

3
3
4
4
3
3
01
05
a
c
f
m
p
a
b
c
f
m
b
f
b
c
p
a
c
f
m
p

a
b
c
f
m
p
02
a
c
f
m
p
05
a
b
c
f
m
b
f
b
c
p
a
c
f
m
p
11
Intelligent projecting tree-based
PTS for dense data


Tree-based Representation of dense PTS,
inspired by FP-Growth
Novel projecting methods, totally differ
from FP-Growth


Bottom up pseudo projection
Top down pseudo projection
12
Tree-based Representation of
dense PTS

TTF - threaded transaction forest


IL - item list: each entry consists of an item, a count, and a pointer.
Forest: each node labeled by an item, associated with a weight.
a,3





Each local item in PTS has an
entry in the IL.
Each transaction in the PTS is
one path starting from a root
in the forest.
count is the number of
transactions represented by
the path.
All nodes of the same item
threaded by an IL entry.
TTF is filtered if only local
frequent items appear in TTF,
otherwise unfiltered.
a
b
c
f
m
p
3
3
4
4
3
3
b,1
c,2
c,1
f,2
f,1
m,2
m,1
b,2
c,1
f,1
p,2
p,1
filtered TTF of original DB in the example
13
Bottom up pseudo projection
of TTF (example)
a,3
a,3
a
b
c
f
m
p
3
3
4
4
3
3
b,1
c,2
c,1
f,2
f,1
m,2
m,1
b,2
c,1

f,1
p,2
a
b
c
f
m
p
3
3
4
4
3
2
b,1
c,2
c,1
f,2
f,1
m,2
m,1
a,3
b,2
c,1
p,2
p,1

f,1
a
b
c
f
m
p
3
3
4
4
3
2
p,1
b,1
c,2
c,1
f,2
f,1
m,2
m,1
b,2
c,1
f,1
p,2
p,1

a,3
3
1
3
3
3
2
b,1
c,2
c,1
f,2
f,1
m,2
m,1
p,2
b,2
c,1

a
b
c
f
m
p
f,1
p,1
a
b
c
f
m
p
3
3
2
2
1
1
b,1
c,2
c,1
f,2
f,1
m,2
m,1
p,2
a,3
b,2
c,1

a,3
f,1
p,1
a
b
c
f
m
p
3
3
4
3
3
3
b,2
c,2
c,1
f,2
f,1
m,2
m,1
p,2
b,2
c,1
f,1
p,1
14
a,3
a
b
c
f
m
p
3
3
4
3
3
3
b,2
c,2
c,1
f,2
f,1
m,2
m,1
p,2
b,2
c,1
f,1
p,1
15
Top down pseudo projection
of TTF (example)
a,3
a
b
c
f
m
p
3
3
4
4
3
3
b,1
c,2
c,1
f,2
f,1
m,2
m,1
a,1
b,2
c,1

f,1
p,2
a
b
c
f
m
p
1
3
4
4
3
3
p,1
b,1
c,2
c,1
f,2
f,1
m,2
m,1
a,3
b,2
c,1

f,1
a
b
c
f
m
p
3
2
4
4
3
3
p,1
p,2
b,1
c,2
c,1
f,2
f,1
m,2
m,1
b,1
c,1
f,1
p,1
p,2

a,3
2
1
3
2
2
3
b,1
c,2
c,1
f,2
f,1
m,2
m,1
p,2
b,1
c,1

a
b
c
f
m
p
f,1
p,1
a
b
c
f
m
p
3
1
3
3
3
3
b,1
c,2
c,1
f,2
f,1
m,2
m,1
p,2
a,3
b,2
c,1

a,2
f,1
p,1
a
b
c
f
m
p
3
2
3
4
3
3
b,1
c,2
c,1
f,2
f,1
m,2
m,1
p,2
b,1
c,1
f,1
p,1
16
Top down pseudo projection
of TTF (example)
( ,)
(a,3) (b,3) (c,4) (f,4) (m,3) (p,3)
(a,3) (a,3) (c,3) (a,3) (c,3) (f,3) (c,3)
(a,3)
(a,3)
(a,3)
(c,3)
(a,3)
Build FIST by top down pesudo projecting TTF
17
Opportunistic Projection:
Observations and Heuristics

Observation 1:



Upper portion of a FIST can fit in memory.
Transactions’ Number that support length k item sets decreases
sharply when k is greater than 2.
Heuristic 1:


Grow the upper portion of a FIST breadth first.
Grow the lower portion under level k depth first, whenever the
reduced transaction set can be represented by a memory based
structure, either TVLA or TTF.
18
Opportunistic Projection:
Observations and Heuristics(2)

Observation 2:



TTF compresses well at lower levels or denser branches, where
there are fewer local frequent items in PTSs and the relative
support is larger.
TTF is space expensive relative to TVLA if its compression ratio is
less than 6-t/n ( t: number of transactions, n: number of items in a
PTS).
Heuristic 2:

Represent PTSs by TVLA at high levels on FIST, unless the
estimated compression ratio of TTF is sufficiently high.
19
Opportunistic Projection:
Observations and Heuristics(3)

Observation 3:



PTSs shrink very quickly at high levels or sparse branches on FIST
where filtered PTSs are usually in form of TVLA.
PTSs at lower levels or dense branches shrink slowly where PTSs
are represented by TTF. The creation of filtered TTF involves
expensive pattern matching.
Heuristic 3:


Make a filtered copy for the child TVLA as long as there is free
memory when projecting a parent TVLA.
Delimitate the pseudo child TTF first and then make a filtered copy
if it shrinks substantially sharp when projecting a parent TTF.
20
Algorithm OpportuneProject
OpportuneProject(Database: D)
begin
create a null root for frequent item set tree T;
L=1
D’= BreadthFirst(T,L, D);
v = the null root of T;
GuidedDepthFirst(v, D’);
end

21
Performance Evaluation:
Efficiency on BMS-POS (sparse)
22
Performance Evaluation:
Efficiency on BMS-WebView1 (sparse)
23
Performance Evaluation:
Efficiency on BMS-WebView2 (sparse)
24
Performance Evaluation:
Efficiency on Connect4 (dense)
25
Performance Evaluation:
Efficiency on T25I20D100kN20kL5k
26
Performance Evaluation:
Scalability on T25I20D1mN20kL5k
27
Performance Evaluation:
Scalability on T25I20D10mN20kL5k
28
Performance Evaluation:
Scalability on T25I20D100k~15mN20kL5k
29
Conclusions

OpportuneProject

maximize efficiency and scalability for all data features
by combining

depth first with breadth first search strategies

array-based and tree-based representation for projected
transaction subsets

unfiltered, and filetered projections
30