Algorithms for Mining Maximal Frequent Itemsets -- A Survey Chaojun Lu.

Algorithms for Mining Maximal Frequent Itemsets

-- A Survey

Chaojun Lu

Introduction Frequent Itemset Extension Tree Common Techniques Some MFI-Mining Algorithms Concluding Remarks

Introduction

• Terminology and Notations

• Problem

• Solution

Terminology and Notations

set of items: I = { i1, i2, …, in}set of transactions: DB = {T1,T2,…,Tm},Ti I(k-)itemset: N I ( |N| = k )support of itemset N: supp(N)frequent itemset (fi)maximal frequent itemset (mfi)set of all frequent (k-)itemsets: FI, FIk set of all mfi: MFI

ProblemDiscover all maximal frequent itemsets in a given transaction database

SolutionTraversing the search space -- subset lattice of I -- and count support for itemset in DB

{1} {2} {3} {1,2} {1,3} {2,3} {1,2,3}

Solution(cont.)Traversing the search space by --• Brute-force: 2|I|

• Clever use of the Basic Property of itemsets: A B supp(A) supp(B) BP1: All subsets of a known frequent itemset are also frequent. BP2: All supersets of a known infrequent itemset are also infrequent.

Frequent Itemset eXtension Tree

• Purpose

• Idea

• Description

• Problem Re-formulated

PurposeTo provide a general framework for analyzing and comparing existent MFI mining algorithms.

IdeaLarger frequent itemsets are generated by extending known smaller frequent itemsets with suitable items. FIXTree captures and illustrates this extension process.

Description of FIXTree• Root: • Nodes: frequent itemset Each node N is associated with its candidate extensions CX(N) and frequent extensions FX(N) defined as:CX(N) = {x | xI and N{x} may be frequent}FX(N) = {x | xCX(N) and N{x} is frequent}• Parent-Child PC: C is a frequent extension of P, i.e. C = P{x} for some xFX(P).

({1,2,3,4,5}/{1,2,3,4})

1 ({2,3,4}/{2,4}) 2 ({3,4}/{3,4})

12 ({4}/{4}) 14 (/)

124 (/)

23 ({4}/) 24 (/)

3… 4…

Example

Problem Re-formulatedGenerate as small a FIXTree containing MFI as possible while searching the subset lattice of I.

Common Techniques

• Search Strategies

• Pruning Strategies

• Dynamic Reordering

• Data Representation for Fast Support Counting

• Frequency Determination

Search StrategiesWe can generate the FIXTree via:• Breadth-first• Depth-first • Hybrid

For MFI-mining, it’s unnecessary to generate and count all nodes. Instead, we try to generate as fewer nodes of the FIXTree as possible, so long as MFI can be identified.

Pruning StrategiesBasicPS1: Prune node N’s infrequent extension subtree.

1 ({2,3,4}/{2,4})

12 ({4}/{4}) 14 (/)13

Note: This strategy greatly improves a PURE DFS algorithm for mining long patterns.

Pruning Strategies(cont.)BasicPS2: Node N’s CX(N) comes from its parent-node P’s FX(P). Let N=P{x}, xFX(P), then CX(N) = {y | yFX(P) and y > x}

1 ({2,3,4}/{2,4})

12 ({4}/…) 14 (/…)

Pruning Strategies (cont.)MaxPS1: At node N, if NCX(N) M (a known fi/mfi), then N-subtree may be pruned.MaxPS2: At node N, if NCX(N) is frequent by support counting, then all N’s children may be pruned ( and a possible new mfi is produced).

1 ({2,3,4}/…)

12 14 13

123 124

1234

Look-ahead

Pruning Strategies(cont.)MaxPS3: At node N, NCX(N) is frequent, then all N’s right-hand-side siblings may be pruned. (Those branches won’t produce new mfi.)

({1,2,3,4,5}/{1,2,3,4})

2 ({3,4}/…) 3… 4…1…

Pruning Strategies(cont.)DFMaxPS: In DFS, AFTER the recursive call DFS(Ni), check if the leftmost path N{i,…,n}is frequent. If yes, then Ni’s right-hand-side siblings may be pruned. (These won’t produce new mfi.)

N(…/{1,2,…n})

Ni ({i+1,…,n}) N(i+1) NnN1 …

Pruning Strategies(cont.)EquivPS: At node N, if for some xCX(N), supp(N{x}) = supp(N), then N can be replaced by N{x}, with CX(N{x}) = CX(N)-{x}

N ({x,y,z}/…)

Ny… Nz…Nx…

Nx ({y,z}/…)

Nxy… Nxz…

Nxy… Nxz…Itemsets containing N but not x cannot be mfi

Dynamic Reordering• The item order in which to extend itemsets greatly affects MFI mining algorithms• Two heuristics:DR1 At node N, reorder all xFX(N) in supp(Nx) increasing order.

1 {2,3,4}

12 {4,3}

123124{3}

1243

13{4}

134

14

Dynamic Reordering(cont.)

DR2 Reorder items of FX() (i.e. FI1) in decreasing order of IF(x) with xFI1, where

IF(x) = {y | yFI1 and xy is infrequent}.

Notes:1. |M(x)| |FI1|-|IF(x)| where M(x) is the size of

the longest mfi containing x2. DR2 + DR1 for FI1.3. Compute FI1 and FI2 before use of DR2.

Data Representation• Data representation transaction set of items bitstring tid-list for each item(set) FP-tree vertical bitmap for each item(set) diffset• Count support on the entire DB or sub-DB?• Counting techniques

Frequency DeterminationWe can determine a frequent itemset N via:• Direct counting supp(N) in DB• A known frequent superset of N• Lower Bound of supp(N) exceeding minsup

Lower Bound Technique• Obtain a lower-bound on supp(N) based on support information of N’s subsets.

supp(N{x}) = supp(N)-drop(N,x) supp(N)-drop(M,x) where MN.supp(NX) supp(N)-drop(M,x) where MN.

Lower Bound Technique(cont.)• LB-PSWe already have supp(N),supp(N1),supp(N2),supp(N3), so we can computeSupp(N123) = supp(N)-drop(N,1)-drop(N,2)-drop(N,3) and check if it is minsup?If yes, then prune N2 and N3 branches. (cf. MaxPS3)

N2 ({3}/…) N3N1 ({2,3}/…)

N (…/{1,2,3})

Some MFI-Mining Algorithms

• Apriori

• Pincer- Search

• FP-growth

• Max-Miner

• DepthProject

• MAFIA

• GenMax

Apriori

Breadth-first

Key steps:

Given FIk

Generate Ck+1

Join (Extending FIk using BasicPS2)

Prune (BP2)

Support Counting Ck+1 to obtain FIk+1

Apriori(cont.)Symmetry of FI-mining problem

FIk

Count Ck+1

FIk+1IFk+1

Count Ck

IFk

{1,2,…,n}

extension

reduction

Extension-based vs Reduction-basedFrequent vs Infrequent

Pincer-Search

Hybrid Search (Top-down + Bottom-up)

Key steps: initially CMFI={I}

Given FIk-1, Ck , CMFI and MFI

Count Ck CMFI to obtain FIk , IFIk and new MFI

Use MFI to prune FIk (BP1, MaxPS)

Use IFIk to update CMFI

Generate Ck+1

Join (Extending FIk using BasicPS2)

Recover missing candidates

Prune (BP2)

Pincer-Search(cont.)

21 3 4 5

12345

1234 pruned

12 13 14 23 24 34

pruned

bottomup

topdown

FP-Growth

FP-tree: a compact form of DB/sub-DB

Key steps: FP-growth(N,N-tree)

if N-tree is a single path N{x,y,z}

then a possible mfi is found Nx Ny Nz

else { extend N with xFX(N)

construct Nx-tree

FP-growth(N{x},Nx-tree)}

FP-Growth(cont.)

fcabmp

f:4

c:3

a:3

m:2

p:2

b:1

m:1

b:1

c:1

b:1

p:1

p(mbacf/c)

m(bacf/acf)

b a c f

cp

pruned

p’s subDB:fcam,fcam,cbp’s FP-tree: cm’s subDB: fca,fca,vcabm’s FP-tree: fca

FP-Growth(cont.)

Depth-first

MaxPS (if used for MFI-mining)

Dynamic Reordering

Projected subDB

Without Candidate Generation?

Construct subDB for N CX(N)

Single path MaxPS

Mining frequent 1-itemset in subDB FX(N)

MaxMiner

Breadth-first + Pruning

Key Steps: At node N with CX(N)

Count NCX(N), N{x} for xCX(N) to get FX(N)

If NCX(N) is frequent, prune using MaxPS2

Reorder FX(N) using DR1

Generate N’s children N{x} for xFX(N)

with CX(N{x})={y | yFX(N) and y > x}

MaxPS3 + LB-PS

DepthProject

Depth-first + Pruning

Key Steps: At node N with CX(N), call DP(N,DB)

Count N{x} in DB to obtain FX(N)

Prune using DFMaxPS, MaxPS1

Project DB to obtain subDB (if necessary)


For each xFX(N):

DP(N{x}, subDB)

Output: a superset of MFI

DepthProject(cont.)

Projected DB

DB Proj.DB for {a} a ({b,c})

abc FX(a) bc [101] ab ac

acd c abc

abe b [1010]

bd

DepthProject(cont.)

Project DB for some nodes on a path

Bitstring representation

Byte Counting

Bucket Counting

MAFIA


Key Steps: At node N, call MAFIA(N, MFI)

If NCX(N) MFI then prune using MaxPS1

Count N{x} obtain FX(N) using EquivPS


For each xFX(N)

MAFIA(N{x}, MFI)

If on leftmost path, prune using DFMaxPS

MAFIA(cont.)

Data Representation

Vertical bitmap and byte counting

Bitmap of item(set) N - bmp(N)

Tran. j 0/1

N

N {x}

t(N {x}) = t(N)t(x) bmp(N) AND bmp(x)

GenMax


Key Steps

Compute FI1 and FI2

Reorder FI1 using DR2 + DR1

MFI = used for MaxPS1

LMFI( , FI1, MFI) //use diffsets

Return MFI

GenMax(cont.)

MFI-subset check: progressive focusing

LMFI(N,FX(N),LMFI)

For each xFX(N)

Generate N{x}with CX(N)

If NxCX(Nx) LMFI // MaxPS1

then return

Count CX(Nx) to obtain FX(Nx)

update LMFI to obtain newLMFI

LMFI(Nx, FX(Nx), newLMFI)

GenMax(cont.)

MFI-subset check optimization: check for local MFI

DR2

Data Representation: diffsets

Concluding Remarks

• Independent components can fit together nicely• Search strategy: hybrid• Pruning strategy and dynamic reordering• Data projection, bitmap representation, fast

counting, compression• Different algorithms perform well under different

MFI distributions• MAFIA and GenMax: current state-of-the-art

References

R. C. Agarwal, et al. Depth first generation of long patterns.

R. J. Bayardo. Efficiently mining long patterns from databases.

D. Burdick, et al. MAFIA: a maximal frequent itemset algorithm for transactional databases.

K. Gouda, et al. Efficiently mining maximal frequent itemsets.

J. Han, et al. Mining frequent patterns without candidate generation.

D-I Lin, et al. Pincer-search: an efficient algorithm for discovering the maximum frequent set.

Thank You!

Algorithms for Mining Maximal Frequent Itemsets -- A Survey Chaojun Lu.

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