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表インターフェースのための　　属性付きグラフとアルゴ

リズムAttribute Graphs and Their Algorithms

for Table Interface

○ 本橋友江 ( 早稲田大学 )土田賢省 ( 東洋大学 )夜久竹夫 ( 日本大学 )

WAAP 110 May.11th, 2002

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Position and ScopeData Processing

NumberCharacter StringsPicture : Tables, Charts, Graphs,

PicturesScope

Tables in Spread Sheet or WordEditing, Drawing, Calculation

1. Introduction

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BackgroundEffectiveness of Table Processing depends on Representation Methods of Table. Known Representation Methods

Quad-Tree Representation For Search Algorithm

Rectangular Dual Graph Representation

For Plant Layout

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ProblemsIn present table processing systems,

Editing operation often cause unexpected results.We do not know which Representation Methods is applied.

Quad-tree Rep. and Rectangular Dual Graph Rep. are not for table editing and drawing.

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Example（ unexpected results）

２を指定列挿入

１２３４

＊ １２ ＊＊ ３＊ ４

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Our HistoryTree structured diagrams

Attributed marked treesCombinatorial drawing algorithms

Modular forms Attributed marked trees

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Our Purpose1. To propose Graph

Representation Methods for tables in consideration of editing and drawing.

2. To investigate mathematical properties of the graphs in the representation methods.

3. To introduce typical algorithms on graphs and evaluate their complexity.

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Our Results1. A Graph representation method

is proposed.2. Properties of the graph is shown

|nodes|in the graph = |cells| in the table.

The degrees of nodes ≦ 8.3. Algorithms for Wall move and

Column Insertion are introduced,

and runs in linear time.

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Representation Method of tablesQuad-Tree Representation

(for search)

NENW SW SE

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Rectangular Dual Graph Representation

(for プラントレイアウト )

Horizontal edgeVertical edge

１２

３ ４

５６

W

N

E

S

１ ２

３ ４ ６

５

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2. Tabular Diagrams

Definitions. An (n, m)-table T is a set

{(i, j) | 1≦i ≦n, 1 ≦ j ≦ m}

(2,3)-table T

(1,1) (1,2)(2,2)

(1,3)(2,1) (2,3)

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Definitions. 　 For an (n,m)-table T, A partial table S is a subset of T

s.t. S = {(i, j) | k ≦ i ≦ l, s ≦ j ≦ t}, 　　　　　 1 ≦ k ≦ l ≦n, 1 ≦ s ≦ t ≦m

A partition P {S1,S2,…,SN} is the set of partial tables s.t. S1∪S2∪…∪SN=T, Si∩Sj=φ

partition P2 over T

cell {(1,1), (2,1)}

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

{(1,3)}

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Definitions. For an (n, m)-table T,Row grid gr ： {0, 1, 2, ..., n} →R 　増Column grid gc

Grid g is a pair g=(gr, gc)A Tabular Diagram D= (T, P, g).

Tabular diagram D2=(T,P2,g2)

0 2 3 60

21

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Definitions. For a cell c={(i, j) | k ≦ i ≦ l, s ≦ j ≦ t},

north wall nw(c)=gr(k-1)south wall, east wall, west wall

0 2 3 60

21 c

nw(c)=1 sw(c)=2

ew(c)=6 　 ww(c)=2

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Condition*.For a tabular diagram

D=(T,P,G),1. T is a (n,m)-table, where

n,m≧3.2. P has perimeter cells with width=0 or height=0

Perimeter cells

0 2 3 6

0

21

2

00

6

*

* ******

* *

****

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Examples.

D2=(T,P2,g2)

We consider tabular diagramssatisfying Condition*.

D2*=(T*,P2*,g2*) satisfying　Condition*

0 2 3 60

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0 2 3 6

0

21

2

00

6

*

* ******

* *

****

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3. Graph Representation of Tables

Operations on Editing Tables Wall moving Insertion/deletion/unification

of Row/column/cell Requirements

computational complexitySuitable correspondence

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Definitions.An attribute graph is a 6-tuple

G=(V,E,L,λ,A,α), where(V,E) : multi edge graph L : label set for edgesλ : E→L is the label functionA is the set of attributeα : V→A is the attribute map

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Graph representation of tablesA Tabular Diagram D=(T,P,g) is repres

ented as an attribute graph GD=(VD,ED,L,λD,A,αD)

VD is identified by a partition Pnode⇔cell. We denote a node vc for the cell c,

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L={enw,esw,eew,eww},Equal north wall, equal south wall, …

A = R4, αD : VD →R4 are defined for vc 　 by αD(vc)=

(nw(c),sw(c),ew(c),ww(c)).

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ED and λD : ED→L are defined by the Rules,

Rule N If cells c and d have the equal north wall, i.e. nw(c)=nw(d),

and there is not such a cell between them,

then an edge [vc, vd] is in ED,

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λD [vc,vd] = enw. (equal north wall)The edge [vc,vd] is called a north wall edge. Rule S Rule E Rule W も同様

North wall edge

South Wall Edge

East Wall Edge

West Wall Edge

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Examples.

An attribute graph is called a tessellation graph, if it represents a tabular diagram.

dc

vd

vc

a vaa

d c

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Properties

Let GD be tessellation graph for a tabular diagram D of an (n,m)-table.

Proposition 3.1Number of nodes in GD

= number of cells in D

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Proposition 3.22|ED| 　= 6(2n-4) + 6(2m-4) + 8k + 16.

Where k is the number of 　　 non perimeter cells.The degree of node in GD

　 is at most 8.

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4.AlgorithmsMOVEEASTWALL

Output

GE=(VE,EE,L,λE,A,αE)

InputGD=(VD,ED,L,λD,A,αD)

Vc: a node δ≧0 : movement value

c

δ

D E

c

Moved Wall

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Node vc 　からEast wall edge を上にたどる

East wall edge をたどりながらEast Wall 変更

隣の列をWest wall edgeをたどりながらWest　Wall 変更

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INSERTCOLUMN

OutputGE

Input GD, Vc : a node

c

D E

Inserted Column

**

**

**

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Node vc 　からWest wall edge を上にたどる

West wall edge をたどりながら、1node ずつ追加Edge の更新

新しい列以東Wall Move East を行う

******

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For a tabular diagram D of (n,m)-table,

Theorem4.1Algorithm MOVEEASTWALL

runs in O(n) time.

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Theorem 4.2Algorithm INSERTCOLUMN runs in O(nm) time.

RemarkAlgorithm INSERTCOLUMN visits only nodes which needs to change attributes.

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summary1. An attribute Graph rep. method

is proposed.2. Properties of the graph is

shown|nodes|in the graph

= |cells| in the table. The degrees of nodes ≦ 8.

3. Algorithms for Wall move and Column Insertion runs in linear time.

5. Conclusion

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Future WorksXML Viewer for tessellation graph. Other Algorithms for Editing CommandsPractical Processing SystemsGraph Grammars for generate tessellation graphsConditions of tessellation graph

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East wall and west wall coordinates are same of x and y

West wall coordinates are same of x and y

East wall coordinates are same of x and y

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Representation of table by tessellation graphs (continued) Peripheral cells and corresponding nodes Cell a is represented by nodeCeiling of the cell a is represented by node zVirtual cells x and y has no valid heightsCeiling of the cell z is covered by cells x and y

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x y

z

example

a

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Cells x and y are of same ceiling coordinates

Cells x and y are of same floor coordinates

Cells x and y are of same ceiling and floor coordinates

x y

x y

x y

General rules

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Tables and tessellation graphs(3)A large inner cell and corresponding nodesThe cell a is represented by the node x.

x

a

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North wall coordinates are same of x and y

south wall coordinates are same of x and y

North wall and south wall coordinates are same of x and y

x

y

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East wall and west wall coordinates are same of x and y

East wall coordinates are same of x and y

West wall coordinates are same of x and y

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 4. CONCLUSION

Future WorksOther MethodsRelation to Mathematical TheoryGraph Grammars

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2. Known Results

図表昔から重要な視覚的な情報媒介コンピュータの分野でも欠かせない情報の伝達手段、思考ツール

表：様々な分野で、数多く考案・開発相関表　生活行為相関表、リーグ戦表、九九表行列表　生命表、曜日表、履修科目一覧表、万年暦三角・多角表　要因分析表、 P 型相関表（五角形）配列　魔方陣、掛け算表（インド）、碁盤、ダイヤモンド・ゲーム

（ 「図の体系」　出原他、日科技連より）

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2. Known Results

表計算ソフト利用目的

データ処理としての計算表形式の分かり易い資料作成

多数開発データ処理として、 EXCEL や Lotus資料作成として、 Word の表作成機能Gridy 、 Noel 、 SpreadCE 、 GS-Calc v5.0 、・・・

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1. Introduction

対象：資料作成のための表作成機能主流： Word の表作成機能

問題点表計算ソフトでは、編集結果の予測がつかないことが起こる

対象と操作に対応するデータ構造とその変換結果が明示されない利用者が思考錯誤して推測している

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1. Introduction

本研究の目標表に必要なデータをまとめ、形式的に定義表データに、属性付きグラフの構造を持たせる手法の提案表の編集操作のアルゴリズムをいくつか作成し、計算時間を評価

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1. Introduction

本研究の目的編集・描画を考慮した、表のデータ構造を提案表を属性付きグラフを用いてあらわす手法の提案表の編集操作のアルゴリズムをいくつか作成し、計算時間を評価

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Example 1. The following figure denotes a support partition {{(1,1), (2,1)}, {(1, 2)}, {(1, 3)}, {(2, 2), (2, 3)}} over (2, 3)-support.

 

   

 

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Example 2. 

     

 

 

   

 

 

   

       

           

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Applications of Tables計算幾何学、プラントレイアウト、回路設計、結晶群と離散構造、ソフトウェア工学（仕様書、表エディタ）

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Application of Our ResultsOur concept may be applied to general table document processing in common spread sheet systems or common document processor.

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Expectation of Our ResultsViewerOther algorithmsPractical processing systemsGraph grammars for generate our graphs

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Example

１ ２ ３

４

１ ２ ３

４

２を指定列挿入

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Example.Tabular diagram

D2=(T,P2,g2)

nw(c)=1 sw(c)=2 ew(c)=6 ww(c)=2 height(c)=1 width(c)=4

0 2 3 60

21 c

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Condition*(続き ). 2. P has perimeter cells of the

form {(k,s)} satisfying one of k=1 s∈{1,2,…,m}.k=n s∈{1,2,…,m}.k∈{1,2,…,n} s=1.k∈{1,2,…,n} s=m.

For a perimeter cell c={(k,s)},

width(c)=0 if s∈{1,m}. height(c)=0 if k∈{1,n}.