Post on 11-Jan-2016
description
Mining Frequent Item Sets by Opportunistic Projection
Junqiang Liu1,4, Yunhe Pan1, Ke Wang2, Jiawei Han3
1 Institute of Artificial Intelligence, Zhejiang University, China2 School of Computing Science, Simon Fraser University,
Canada3 Department of Computer Science, UIUC, USA
4 Dept. of CS, Hangzhou University of Commerce, China
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Outline
How to discover frequent item sets
Previous works
Our approach: Mining Frequent Item
Sets by Opportunistic Projection
Performance evaluations
Conclusions
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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 c d f g i m p
02 a b c f l m o
03 b f h j o
04 b c k p s
05 a c e f 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
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How To Discover Frequent Item Sets
Frequent item sets can be represented by a tree, which is not necessarily materailized.
Mining process: a process of tree construction, accompanied
by a process of projecting transaction subsets
( , )
(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)
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Frequent Item Set Tree - FIST
FIST is an ordered tree each node: (item,weight) the following are imposed
items ordered on a path (top-down) items ordered at children (left to right)
Frequent item set 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
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Frequent Item Set Tree (example)
( , )
(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)
01 a c d f g i m p02 a b c f l m o03 b f h j o04 b c k p s05 a c e f l m n p
01 c f m p02 b c f m05 c f m p
02 c f m03 f04 c p
01 f m p02 f m04 p05 f m p
01 m p02 m05 m p
01 p05 p
01 f m02 f m05 f m
01 m02 m05 m
01 m02 m05 m
01 m p02 m05 m p
01 p05 p
(i ,w): a FIST node
: the PTS of the node
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Factors relate toMining Efficiency and Scalability
The FIST construction strategy breadth first v.s. depth first
The PTS representation Memory-based representation: array-based, tree-
based, vertical bitmap, horizontal bitstring, etc. Disk-based representation
PTS projecting method and item counting method
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Previous Works
Research
StrategyPTS
Representation
ProjectingMethod
Remarks
Apriori breadth first original DB on the fly Repetitive DB ScansHuge FIST for denseExp. pattern matching
TreeProjection
breadth first original DB on the fly
FPGrowth
depth first FP-tree
recursively materialize conditional DB/Fptree
#of conditional FPtree in same order of mag. as # of fre. item sets
H-Mine depth first H-struct partially materialize sub H-struct
Not most eff. for sparseCall FP-Growth for densePartition for large
DepthProject
depth firsthorizontal bitstring
selective projection
Maximal fre. item sets
Less efficient than array-based for sparse & large
Less efficient than tree-based for dense
MAFIA depth firstvertical bitmap
recursively materialize
compressions
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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
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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
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.
a 3b 3c 4f 4m 3p 3
acfmp
acfmp
abcfm
bf
bcp
01f04
02 0503
FI L
fi l tered TVLA of the ori gi nal DB i n the exampl e
LQ
array
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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.
a 3b 3c 4f 4m 3p 3
acfmp
acfmp
abcfm
bf
bcp
01f04
02 0503
unfi l tered chi l d TVLA
c 3f 3m 3
01 02 05
c 3f 3m 3 c
fm
cfm
cfm
01 02 05FI L(a)
FI L
FI L(a)
fi l tered chi l d TVLA
parent TVLA
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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)
a 3b 3c 4f 4m 3p 3
acfmp
acfmp
abcfm
bf
bcp
01f0402
0503 a 3
b 3c 4f 4m 3p 3
acfmp
acfmp
abcfm
bf
bcp
01 f0402 05
a 3b 3c 4f 4m 3p 3
acfmp
acfmp
abcfm
bf
bcp
01 02 05a 3b 3c 4f 4m 3p 3
acfmp
acfmp
abcfm
bf
bcp
01 05
TVLAi n sl i de 10
NULL
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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
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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.
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 3b 3c 4f 4m 3p 3
a, 3
b, 1
f , 1
m, 2
c, 2 c, 1
f , 2 f , 1
b, 2
c, 1
m, 1
p, 2 p, 1
fi l tered TTF of ori gi nal DB i n the exampl e
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Bottom up pseudo projection of TTF (example)
a 3b 3c 4f 4m 3p 3
a 3b 3c 4f 4m3p 2
a, 3
b, 1
f , 1
m, 2
c, 2 c, 1
f , 2 f , 1
b, 2
c, 1
m, 1
p, 2 p, 1
a 3b 3c 4f 4m3p 2
a 3b 3c 4f 3m3p 3
a 3b 1c 3f 3m3p 2
a 3b 3c 2f 2m1p 1
a, 3
b, 1
f , 1
m, 2
c, 2 c, 1
f , 2 f , 1
b, 2
c, 1
m, 1
p, 1p, 2
a, 3
b, 1
f , 1
m, 2
c, 2 c, 1
f , 2 f , 1
c, 1
m, 1
p, 1p, 2
a, 3
f , 1
m, 2
c, 2 c, 1
f , 2 f , 1
b, 2
c, 1
m, 1
p, 1p, 2
a, 3
b, 1
f , 1
m, 2
c, 2 c, 1
f , 2 f , 1
b, 2
c, 1
m, 1
p, 1p, 2
a, 3
b, 1
m, 2
c, 2 c, 1
f , 2 f , 1
b, 2
c, 1
m, 1
p, 2
b, 2
b, 2
p, 1
f , 1
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Top down pseudo projection of TTF (example)
a 3b 3c 4f 4m 3p 3
a 3b 2c 4f 4m 3p 3
a, 3
b, 1
f , 1
m, 2
c, 2 c, 1
f , 2 f , 1
b, 2
c, 1
m, 1
p, 2 p, 1
a 1b 3c 4f 4m 3p 3
a 3b 2c 3f 4m 3p 3
a 2b 1c 3f 2m2p 3
a 3b 1c 3f 3m3p 3
a, 1
b, 1
f , 1
m, 2
c, 2 c, 1
f , 2 f , 1
b, 2
c, 1
m, 1
p, 1p, 2
a, 3
b, 1
f , 1
m, 2
c, 2 c, 1
f , 2 f , 1
c, 1
m, 1
p, 1p, 2
a, 3
m, 2
c, 2 c, 1
f , 2 f , 1
b, 1
c, 1
m, 1
p, 1p, 2
a, 3
b, 1
f , 1
m, 2
c, 2 c, 1
f , 2 f , 1
b, 2
c, 1
m, 1
p, 1p, 2
a, 2
b, 1
m, 2
c, 2 c, 1
f , 2 f , 1
b, 1
c, 1
m, 1
p, 2
b, 1
b, 1
p, 1
f , 1 f , 1
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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.
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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.
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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.
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Algorithm OpportuneProject
OpportuneProject(Database: D) begin create a null root for frequent item set
tree T; D’= BreadthFirst(T, D); GuidedDepthFirst(root_of_T, D’); end
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Performance Evaluation: Efficiency on BMS-POS (sparse)
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Performance Evaluation: Efficiency on BMS-WebView1 (sparse)
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Performance Evaluation: Efficiency on BMS-WebView2 (sparse)
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Performance Evaluation: Efficiency on Connect4 (dense)
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Performance Evaluation: Efficiency on T25I20D100kN20kL5k
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Performance Evaluation: Scalability on T25I20D1mN20kL5k
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Performance Evaluation: Scalability on T25I20D10mN20kL5k
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Performance Evaluation: Scalability on T25I20D100k~15mN20kL5k
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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
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Acknowledgement
We would like to thank Blue Martini Software, Inc.
for providing us the BMS datasets!
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References
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Thank you !!!Thank you !!!