data mining 5: tree & graph mining algorithmsarim/ikn-tokuron/arimuradm5...data mining 5: tree...

情報知識ネットワーク特論 Data Mining 5: Tree & Graph Mining algorithms

有村博紀 Hiroki Arimura 北海道大学大学院情報科学研究科情報理工学専攻 Division of CS, Grad. School of Inf. Sci. & Tech. Hokkaido University email: {arim,kida}@ist.hokudai.ac.jpSlide http://www-ikn.ist.hokudai.ac.jp/ikn-tokuron/

． Updated: 01 DEC 2015

今日の内容

5回:木とグラフのマイニング / Mining of Trees and Graphs l  半構造データ / Semi-structured data

l  頻出順序木マイニング /Frequent Ordered Tree Mining •  ラベルつき順序木(ordered trees) •  右拡張手法(Rightmost expansion) •  アルゴリズム：FREQT

l  頻出無順序木マイニング/ Unordered tree mining l  グラフマイニング / Graph mining ポイント

l  木の列挙 / Enumeration of Trees

半構造マイニングとは何か？ WHAT IS SEMI-STRUCTURED DATA MINING? (OR GRAPH & TREE MINING)

PAKDD2008, 21 May 2008, Hiroki Arimura, Hokkaido University

7

Semi-structured Data Mining

l  have emerged in 90’s with the rapid progress of computer and network technologies． l  Web pages, collections of XML documents. l  Biological databases (Entrez, Genbank, PubMED) l  Sematic Web / Networks of Ubiquitous Knowledge

l  Demands for efficient methods to discover knowledge from large semi-structured data

l  However, Semi-structured data are .... l  Huge, Heterogeneous collections of

Weakly-structured data

l  Traditional data mining technology cannot work!

l  Data Mining for Sequence, Trees, and Graphs

<people> <person age=“25” id=“608”> <name>John</name> <email>[email protected]</email> </person> <person id=“609”> <name>Mary</name> <tel>555-4567</tel> </person> </people>

email

people

@age @id

608

name

#text #text

person

name @id

609

tel

#text

#text

person

25

[email protected]

John 555-4567

Mary


8 DEWS2003 浅井達哉「半構造データマイニングに関する研究動向」

与えられたラベルつき木やグラフの集合から，特徴的な部分グラフ（パターン）を発見すること

半構造データマイニングとは？

n 特徴的なパターン l 頻出パターン

多くの文書に共通して出現するパターン l  適パターン

２種類の文書集合を，うまく区別するパターン

B

P

#text FONT #text #text

@color @face #text

blue Times Minsup = 5(%)



半構造データマイニングの応用

n スキーマ発見

n 情報抽出

n 文書分類

n 検索の効率化

n データ圧縮

n 複雑な構造をもつデータからの知識発見 l 化学物質 l 遺伝子ネットワーク l ネットワークログ l インターネットのリンク構造 l 電子回路

半構造マイニングの歴史 HISTORY OF SEMI-STRUCTURED DATA MINING



半構造データマイニングのさきがけ

n Holder, Cook, Djoko [KDD’94] l グラフマイニングシステムSUBDUE l すべての出現を取り除いて得られるグラフの大きさが

小となるような部分グラフを発見（MDL原理）

l グラフの圧縮に応用

n Nestrov, Abiteboul, Motwani [SIGMOD’98] l 論理プログラムを用いたグラフマイニング

l 入力グラフをできるだけ広く覆うパターンを計算

l 半構造データからのスキーマ発見に応用



n Wang & Liu [KDD’97; TKDE, 2000] , Cong, Yi, Liu, Wang [SDM’02] l 木を経路の集合とみなして，頻出パス列を発見

l 各経路をアイテムとみなしたApriori風のアルゴリズム

n Miyahara, et al. [PAKDD’01，PAKDD’02] l グラフ変数つき順序木・無順序木パターンを発見

l 高速なマッチング手法＋単純な生成テスト法

木構造データからの頻出パターン発見手法

X

Y


14 DEWS2003

グラフ構造データからの頻出パターン発見手法

n  Inokuchi, Washio, Motoda [PKDD’00] l  AGM: “An Apriori-based Algorithm for Mining Frequent

Substructures from Graph Data”

n Kuramochi & Karypis [ICDM’01, ICDM’02] l  FSG: “Frequent Subgraph Discovery”

n Vanetik, Gudes, Shimony [ICDM’02] l  “Computing frequent graph patterns from Semi-structured

data”

n Yan & Han [ICDM’02] l  gSpan: “gSpan: Graph-based substructure pattern mining”

H

H

H

X XC C



頻出パス列発見手法 [Wang and Liu (KDD’97)]

email

people

@age @id

608

name

#text

#text

person

name @id

609

tel

#text

#text

person

25

[email protected]

John 555-4567

Mary

n  順序木をパスの列に分解

n  各パスをアイテムとみなして，Apriori風に頻出パス列を発見




email

people

@age @id

608

name

#text

#text

person

name @id

609

tel

#text

#text

person

25

[email protected]

John 555-4567

Mary

person person person person person

people people people people people people

@id

person

name

#text

person 頻出パス列を発見

Apriori

@id name

#text

person

再構造化

n  順序木をパスの列に分解

n  各パスをアイテムとみなして，Apriori風に頻出パス列を発見

パスのワイルドカードを扱えるよう拡張 [Cong, Liu, Yi, Wang (SDM2002)]




この手法の問題点

l 兄弟が同一ラベルをもつ場合，木の枝分かれ構

造が失われる

l Apriori風に解候補の探索を行うため，効率が悪い

すべての頻出順序木を厳密に発見する計算効率のよいアルゴリズムが必要



n FREQT [Asai et al. (SDM’02, PKDD’02)] l  右拡張を用いた効率のよい順序木の枚挙

l  右葉出現の漸増的な更新

n Treeminer [Zaki (SIGKDD’02)] l 効率のよい順序木の枚挙

l 我々の研究とは独立

頻出順序木パターンを発見する効率のよいアルゴリズム

木構造データからの頻出パターン発見手法

•  Efficient Substructure Discovery from Large Semi-structured Data, Tatsuya Asai, Kenji Abe, Shinji Kawasoe, Hiroki Arimura, Hiroshi Sakamoto, Setsuo Arikawa, Proc. Second SIAM International Conference on Data Mining 2002 (SDM‘02), 158-174, SIAM, 2002.

•  Optimized Substructure Discovery for Semi-structured Data, Kenji Abe, Shinji Kawasoe, Tatsuya Asai, Hiroki Arimura, Setsuo Arikawa, Proc. 6th European Conference on Principles and Practice of Knowledge Discovery in Databases (PKDD-2002), LNAI 2431, Springer, 1-14, August 2002.

•  Mohammed J. Zaki, Efficiently mining frequent trees in a forest, KDD'02, ACM, 71-80, 2002. (Also appeared in: IEEE Trans. Knowl. Data Eng. 17(8), 1021-1035, 2005)


20

Efficient Algorithms for Mining Frequent Ordered Trees from Semi-structured Data

Hiroki Arimura (Graduate School of IST, Hokkaido University, Japan) Joint work with Takeaki Uno (National Institute of Informatics) Tatsuya Asai* (Fujitsu Labs. Co. LTD.), Shin-ichi Nakano (Gunma University)

Partly Supported by: Grant-in-aid for Scientific Researchs on Specially Promoted Research "Semi-structured Mining" and Priority Area “Information Explosion” (2007) * Presently, working for Fujitsu Lab. Co. LTD.


21

Our Research Goal n Knowledge discovery methods that

supports human’s discovery from large semi-structured data

n Key: Efficient semi-structured data mining algorithm in theory and practice l Having theoretical performance guarantee l Fast and Lightweight in practice

User

Semi-structured Data

Data Mining Engine


22

History of Semi-structured Data Mining

～1995

1996

1997

1998

1999

2000

2001

2002

2003

Algorithm for finding subgraphs by MDL principle Subdue [Holder et al. (KDD’94)] Finding frequent paths [Wang and Liu (KDD’97)] Finding Semi-structured Schema [Nestrov, Abiteboul et al. (SIGMOD’98)]

Finding frequent subgraphs AGM [Inokuchi, Wahio, Motoda (PKDD’00, MLJ. 2003)]

Finding frequent / optimal ordered trees FREQT [Ours (SDM’02)]，Treeminer [Zaki (KDD’02)] Finding frequent subgraphs gSpan [Yan and Han (ICDM’02)], VG [Venetik, et al.(ICDM’02)] Finding frequent un-ordered trees UNOT [Ours (SDM’03)]，NK [Nijssen, Kok (MGTS’03)]

FSG [Kuramochi et al. (ICDM’01)

Frequent Maximal/Closed Trees & Graphs mining [Yan&Han '03; Termier et al.'04] and many algorithms in 2000s


23

Tree Mining

Association Rule, Frequent Itemsets

Simple Efficient algorithms

Expressive Computationally Hard

General graphs (AGM, FSG, gSpan)

Trees

Development of Efficient tree mining algorithms

Expressive patterns & Efficient algorithms

PAKDD2008, invited talk, 21 May 2008, Hiroki Arimura, Hokkaido University

24

The first question when we strarted in 2001

How to define a Tree Mining problem?

25

PAKDD2008, 21 May 2008,

Hiroki Arimura, Hokkaido

University

Itemset Mining Revisited

l Frequent Itemset Mining l  Finding all "frequent" sets of elements (items) appearing

no more than σ times in a given transaction data base [Agrawal, Srikant, VLDB'94]

l  Most popular data mining problem and basis for others

1 2 3 4 5

t1 ○ ○

t2 ○ ○

t3 ○ ○ ○ ○

t4 ○ ○ ○

t5 ○ ○ ○

1 2 3 4 5

t1 ○ ○

t2 ○ ○

t3 ○ ○ ○ ○

t4 ○ ○ ○

t5 ○ ○ ○

database

minsupσ= 2

The itemset lattice (2Σ, ⊆)

Frequent setsFrequent sets ∅, 1, 2, 3, 4, 12, 13, 14, 23, 24, 124

X = {2, 4} appears three times, thus

frequent

people: [ person: { age: “25”, id: “608”, name: John, email: [email protected] }, person: { id: “609”, name: Mary, tel: 555-4567 } ]


26

半構造データの例 / Examples <people> <person age=“25” id=“608”> <name>John</name> <email>[email protected]</email> </person> <person id=“609”> <name>Mary</name> <tel>555-4567</tel> </person> </people>

email

people

@age @id

608

name

#text #text

person

name @id

609

tel

#text

#text

person

25

[email protected]

John 555-45

67 Mary

l  HTML & XML documents

l  JSON texts l  Feature structures in NLP l  Attribute graphs in Computer

Graphics

JSON text **

** JSON: Javascript Object Notation, www.json.org/

XML document *

* XML: Extended Markup Language, http://www.w3.org/TR/xml11/

people: [ person: { age: “25”, idj: “608”, name: John, email:[email protected] }, person: { id: “609”, name: Mary, tel: 555-4567 } ]


27 DEWS2003

半構造データのモデル n  ラベルつき順序木 / Labeled

Ordered Trees l Each node has a label, which

corresponds to: •  Markup tag •  Attribute & value •  Text string

l The children of a node are ordered from left to right by the sibling relation

l Each node can have unbound number of children (unranked)

n  Other class of semi-structured data l  ラベルつき無順序木 / Labeled ordered

trees

l  ラベルつきグラフ / Vertex-labeled and edge-labeled graphs

<people> <person age=“25” id=“608”> <name>John</name> <email>[email protected]</email> </person> <person id=“609”> <name>Mary</name> <tel>555-4567</tel> </person> </people>

email

people

@age @id

608

name

#text #text

person

name @id

609

tel

#text

#text

person

25

[email protected]

John 555-45

67 Mary

n  半構造データの例 / Examples l  HTML & XML documents

l  JSON texts l  Feature structures in NLP l  Attribute graphs in Computer

Graphics


29

n  φ is 1-to-1. n  φ preserves parent-child relation. n  φ preserves (indirect) sibling

relation. n  φ preserves labels.

Matching between trees

Pattern tree T matches a data tree D (T occurs in D )

There is a matching function φ from T into D.

r

C

B A

B

A

C B

data tree D pattern tree T 1

2

3 4 5

6

7

8

9 10

11

A

C B A

C B A

C B

A

C B A

C B

matching function φ


30

Frequency of a pattern tree

Root occurrence list OccD(T) = {2, 8}

•  A root occurrence of pattern T: •  The node to which the root of T maps by a matching function

•  The frequency fr(T) = #root occurrences of T in D

r

C

B A

C B

A

A C B

B

P1

P2

A

C B D

T 1

2

3 4 5

6

7

8

9 10

11


31

Frequent Tree Mining Problem

n  Given: a colection S of labeled ordered trees and a minimum frequency threshold s

n  Task: Discover all frequent ordered trees in S (with frequency no less than s) without duplicates

A minimum frequency threshold (min-sup) s = 50%

PAKDD2008, invited talk, 21 May 2008, Hiroki Arimura, Hokkaido University

32

The second question

How to efficiently solve the Tree Mining problem?

33

PAKDD2008, 21 May 2008,


University

Theoretical performance guanrantee

33

n Output-polynomial (OUT-POLY) l Total time = poly(N, M)

n  polynomial-time enumeration, or amortized polynomial-delay (POLY-ENUM) l Amotized delay is poly(Input), or l Total time = M·poly(N)

n  polynomial-delay (POLY-DELAY) l Maximum of delay is poly(Input)

How to measure the time complexity of mining algorithms? l Exponentially many answers! l As enumeration algorithms

+ polynomial-space (POLY-SPACE)

Output size M

Delay D

Input

Input size M

Total Time T

Algorithm Outputs

Time

34

PAKDD2008, 21 May 2008,


University

Itemset Mining Revisited

Backtrack algorithm [1997-1998]

• Depth-first search (DFS) •  Vertical data layout

1 2 3 4 5

t1 ○ ○

t2 ○ ○

t3 ○ ○ ○ ○

t4 ○ ○ ○

t5 ○ ○ ○

database

Apriori algorithm [1994]

• Breadth-first search (BFS) • Horizontal data layout

•  2nd generation •  In-core algorithm •  Space efficient

•  1st generation •  External memory

algorithm


35

Frequent Ordered Tree Mining

n How to enumerate all structured patterns without duplicates

n Rightmost expansion technique

n Efficient DFS Algorithm l Freqt [Asai, Arimura, '02]


36

Efficent Algorithms

n Depth-first mining algorithm for frequent ordered trees l Freqt [Asai, Arimura et al., SIAM DM'02] l TreeMiner [Zaki, KDD'02]

n Performance guarantee l Polynomial time enumeration per pattern l Small memory footprint

n Key technique: Rightmost expansion


37

Key: How to enumerate Ordered Trees without Duplicates?

n A naive algorithm l Starting from a one-node tree l Grow a pattern tree by adding a new

node one by one

n Drawback l Exponentially many different ways to

generate a same pattern tree l Explicit duplication test needed


38

演習

n  問題 1：　サイズがk=0,1,2,3,4の順序木（ラベルなし）をすべて書き出せ．

n  Problem 1: Print all (un-labeled) ordered trees of size k, k up to 4 (in the increasing order of their sizes)

n  問題2：入力kに対して，サイズがkの順序木（ラベルなし）をすべて書き出せをすべて出力するプログラムをかけ．

n  Problem 2 (Difficult!) Give a computer program that, given a number k >= 0, prints (enumerates) all (unlabeled) ordered trees without duplicates.


39

Itemset mining revisited: BFS vs. DFS

Apriori algorithm [1994]

• Breadth-first search (BFS) • Horizontal data layout

•  1st generation •  External memory

algorithm

•  2nd generation •  Space efficient in-

core algorithm

Backtrack algorithm [1997-2000]

• Depth-first search (DFS) •  Vertical data layout • MaxMiner, Eclat, FP-growth

etc.

1 2 3 4 5

t1 ○ ○

t2 ○ ○

t3 ○ ○ ○ ○

t4 ○ ○ ○

t5 ○ ○ ○

1 2 3 4 5

t1 ○ ○

t2 ○ ○

t3 ○ ○ ○ ○

t4 ○ ○ ○

t5 ○ ○ ○

database

How to implement DFS for ordered tree

patterns?


40

An idea: Encode a tree as a sequence

n Encode an ordered tree T l by its depth-label sequence

((d(v1), l(v1)) , … , (d(vk), l(vk)) in the preorder traversal of T

ordered tree T A

B B

B A

C

C

0

1

2

3

depth

id 1 2 3 4 5 6 7 dep,lab 0A 1B 2A 3C 2B 1B 2C

code for T


41

Rightmost Expansion Technique for Ordered Trees

n Starting from a one node tree

n Grow the tree by l  Attaching a new

node v l  to the rightmost

branch l  to be the

youngest sibling

1

k-1 T

l k p


42

A family tree for frequent ordered trees

⊥

A

B

A

A

A

B

B

A

B

B

B

B A

B

B B

B

B

B

B

B

A

B

A A

B

A B

infrequent infrequent

frequent

frequent


43

DFS for frequent ordered trees

n  Enumarate all frequent ordered trees by backtracking n  A family tree based on the rightmost expansion

⊥

A

B

A

A

A

B

B

A

B

B

B

B A

B

B B

B

B

B

B

B

A

B

A A

B

A B

infrequentinfrequent

frequent

frequent


44

How to incrementally compute the frequency of each pattern?

n  Use RMO (rightmost-leaf occurrence)

Data tree D

A

C T

RMO(T) ={5,12,14,19,22}

A

C T’

B

The rightmost expansion

RMO(T’)={6,17}

B

r

CB A

C B

C CB

B AC

AAC

B

AC C

A C

1

2

3 4

5 6

7

8

9

10

11

12

13

14

15

16 17

18

19

20 21

22

B

BRMO(T)

RMO(T’)

Pattern

浅井達哉「半構造データマイニングに関する研究動向」 45 DEWS2003 浅井達哉「半構造データマイニングに関する研究動向」

FREQTの性能評価

n  Scalability q  Dataset: citeseers q  Minimum support: σ=3.0(%) fixed q  Increasing the data size from 0.3MB to 5.6MB.

0

0.5

1

1.5

2

0 50000 100000 150000 200000Number of nodes in a data tree

Run

time

(sec

)

178,285ノードに対して 1.39秒

# of nodes

Runtime (sec)


適パターン発見アルゴリズムOPTT [Abe et al. (PKDD’02)]

n  適パターン発見では、以下をみたすパターンを見つける q  positiveに頻出，negativeに非頻出

n  positiveにもnegativeにも同程度で出現するパターンは見つけない

n  木やグラフの分類に応用

Positive data Negative data

Pattern P

matched

unmatched


小化 • 分類誤差 • 情報エントロピー • Gini 指標

データマイニング Optimized data mining (IBM DM group,1996 - 2000) 計算学習理論 Agnostic PAC learning (1990s) 統計学 Vapnik＆Chervonenkis theory (1970s)

f(p): 不均

衡関

数

p: パターンが出現する正例の割合

良いパターン 8 positives 2 negatives

悪いパターン 9 positives 9 negatives

50% 100% 0%

適パターン発見問題

応用例：FREQT ウェブページからの頻出部分構造発見

<a href=“_”> #text_1 </a>

 #text_2 #text_3  #text_4 #text_5 #text_6 #text_7 #text_8 #text_9 

実験１：CGI生成されたウェブページからの反復構造抽出

実験２：映画データベースからの構造抽出

データ：IMDB （アクション映画×50）データサイズ：1.05 MB/102,493 ノードパターンサイズ：1218ノード（サポート50%）深さ優先探索版FREQT

• スキーマ発見に有効 •  DataGuide [Widom, Garcia-Molina et al. (VLDB’97)]

応用例： OPTT XMLデータからの適パターン発見

正例：アクション映画１５タイトル

負例：ファミリー映画１５タイトル

見つかった適パターンの例

出現数：正例１０，負例０<movie> <certification> <certif> sweden:15 </certif> </certification> </movie>

出現数：正例１，負例１２<movie> <title /> <genre> animation <genre> </movie>

アルゴリズムOPTT

• 文書分類に有効


Treeminer [Zaki (SIGKDD’02)]

n  効率のよい頻出順序木パターン発見アルゴリズム

n  FREQTと類似

q  順序木の枚挙法は，FREQTと同じ右拡張

q  木のマッチングは， FREQTと異なる

n  Treeminerでは，親子関係にギャップを許したマッチングを採用

n  我々の研究とは独立

応用例：ウェブログからのアクセスパターン発見

あるユーザのウェブアクセスログ

<userSession name=”ppp0-69.ank2.isbank.net.tr” …> <uedge source=“5938” target=“16470” utime=“7:53:46”/> <uedge source=“16470” target=“24754” utime=“7:56:13”/> <uedge source=“16470” target=“24755” utime=“7:56:36”/> <uedge source=“24755” target=“47387” utime=“7:57:14”/> <uedge source=“24755” target=“47397” utime=“7:57:28”/> <uedge source=“16470” target=“24756” utime=“7:58:30”/>

5938

16470

24754 24755

47387 47397

24756

ウェブ閲覧木

実験：１００９件のウェブアクセスログから頻出パターンを発見

Path/sair_listesi.html

Path/poems/akgun_akova/index.html

Akova/picture.html Akova/contents.html Akova/biyografi.html

見つかったパターンの例


52


53

Frequent Unordered Tree Mining

n How to enumerate all structured patterns under isomorphism

n Canonical form technique

n Efficient DFS Algorithm l Unot [Asai, Arimura, DS'03] l NK [Nijssen, Kok, MGTS’03]


54

Previous Works: Frequent Pattern Discovery for Graphs

Algorithms for discovering all the frequent substructures from graphs

n  AGM [Inokuchi, Washio, Motoda (PKDD’00)] l  Computing adjacent matrices for frequent subgraphs.

n  FSG [Kuramochi, Karypis (ICDM’01)] l  Computing adjacent matrices for frequent subgraphs.

n  [Vanetik, Gudes, Shimony (ICDM’02)] l  Joining frequent paths.

n  gSpan [Yan, Han (ICDM’02)] l  Computing DFS trees for frequent subgraphs.


55

Efficient DFS mining algorithm for frequent unordered trees n  UNOT: an efficient algorithm that finds all the

frequent labeled unordered trees in a given database. l  Efficient enumeration of unordered trees in O(1)

time per tree without duplicate l  Incremental computation

AAT1 T3 T4BB BB

BBAA

CC

CC

AA

BB BB

BBAA

CC

CC

AA

BB BB

BB AA

CC

CC

AAT2BB BB

BB AA

CC

CC＞


56

Difficulties in Mining Unordered Trees

1.  How to define unique representation

2.  How to generate all the patterns without duplications

3.  How to compute the occurrences


57

How to define the canonical representation ? Leftmost-biased Tree

n  The code for an ordered tree T : the depth-label sequence in priorder traversal of T

n Code(T) = ((depth(v1), label(v1)) , … , (depth(vk), label(vk)) A T

B B

B A

C

C

Code(T) = (0A,1B,2A,3C,2B,1B,2C)

n The ordered tree T with lexicographically maximum code


58

Leftmost-biased Tree

A T1 T3 T4 B B

B A

C

C

A

B B

B A

C

C

A

B B

B A

C

C

n The ordered tree T with lexicographically maximum code

A T2 B B

B A

C

C

(0A,1B,2A,3C,2B,1B,2C) (0A,1B,2B,2A,3C,1B,2C) (0A,1B,2C,1B,2A,3C,2B) (0A,1B,2C,1B,2B,2A,3C)

＞


59

How to avoid duplications ? Rightmost Expansion

n We can prove that the rightmost expansion only generates leftmost-biased trees only

Rightmost expansion

SAME


60

Frequent Unordered Tree Mining

n How to enumerate all structured patterns under isomorphism

n Canonical form technique

n Efficient DFS Algorithm l Unot [Asai, Arimura, DS'03] l NK [Nijssen, Kok, MGTS’03]


61

グラフ構造データからの頻出パターン発見



AGM [Inokuchi, Washio, Motoda (PKDD’00)]

n AGM = Apriori-based Graph Mining n 隣接行列表現を用いて，Apriori風に頻出部分グラ

フを発見するアルゴリズム

n パターンの生成法

l 隣接行列の正準形を効率よく計算 Cl Cl

Cl C C

H

Cl C Cl C Cl H

Cl 0 1 0 0 0 0 C 1 0 1 2 0 0 Cl 0 1 0 0 0 0 C 0 2 0 0 1 1 Cl 0 0 0 1 0 0 H 0 0 0 1 0 0

Code = 101020000100010

隣接行列の正準形

行と列を入れ替えてCodeが小となるような隣接行列トリクロロエチレン



FSG [Kuramochi & Karypis (ICDM’01)]

n FSG = Frequent SubGraph discovery n 隣接行列表現を用いて，Apriori風に頻出部分グラ

フを発見するアルゴリズム v1 v2 v3 v4

v1 0 1 0 0 v2 1 0 1 2 v3 0 1 0 0 v4 0 2 0 0

A

A B

A v2

v1

v3 v4

v1 v4 v3 v2

v1 0 0 0 1 v4 0 0 0 2 v3 0 0 0 1 v2 1 2 1 0

各ノードを，ラベルとランクの組ごとに分割



[Vanetik, Gudes, Shimony (ICDM’02)] n 頻出パスにbijective sumとspliceの操作を繰り返し，

Apriori風に頻出部分グラフを発見するアルゴリズム

Bijective sum Splice



gSpan [Yan & Han (ICDM’02)]

n gSpan = graph-based Substructure pattern mining

n 頻出部分グラフ発見アルゴリズム

n グラフのDFS木を効率よく生成



応用例：化学物質データからの知識発見

化学物質の発ガン性を判定するルールを発見

データ３００種類の化学物質（うち１８５種類が発ガン性あり）

H X

C C

C C

C

C

H 発ガン性あり

H

H

H

X X C C

発ガン性あり見つかったルール

今日の内容

5回:木とグラフのマイニング / Mining of Trees and Graphs l  半構造データ / Semi-structured data

l  頻出順序木マイニング /Frequent Ordered Tree Mining •  ラベルつき順序木(ordered trees) •  右拡張手法(Rightmost expansion) •  アルゴリズム：FREQT

l  頻出無順序木マイニング/ Unordered tree mining l  グラフマイニング / Graph mining ポイント

l  木の列挙 / Enumeration of Trees

data mining 5: tree & graph mining algorithmsarim/ikn-tokuron/arimuradm5...data mining 5: tree...

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