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An octal degree graph representation for the rectangular dissections

有田友和（桜美林大学）、本橋友江（関東学院大）

土田賢省（東洋大）、夜久竹夫（日大）

２００４年１２月１８日 ＷＡＡＰ １２５

日本大学文理学部 Title corrected

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資料

IASTED 　 SEA 　０２ (Cambridge,　2002)　

HJ2003　(Hangary-Japan　Sympos.　Discrete　Math.,　Tokyo)

IASTED　AI04　(Insbruck,　2004)

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対象 均一型矩形分割

不均一型矩形分割

操作（壁指向） 変形（壁移動、セル・行・列の追加・削除・移動、など）

特徴抽出（表の行合計計算、表構造の正誤など） 応用

図表（文書 ) 建物・フロアプラン（ OR) 地形図

1 1 11 1 11 1

1 2 22 2 2

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1.Introduction Motivation Aim Known Representation

Method Purpose

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Motivation Tables

Homogeneous Heterogeneous rectangular rectangulardissections dissection

Editing operations often cause unexpected results.

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Motivation (continues) Column Insertion at Right to Cell 1

Unexpected result Expected result

C3

B2

A1

B

C3

2

1 A

C3

B2

A1

Word

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Motivation

Excel does not allow this operation

300100

BA

70 200

300100

100

30

BA

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Aim Representation method for

rectangular dissection processing system.

Formalization of rectangular dissection editing operations.

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Known Representation Methods

Quad-Tree Representation [J.L.Bentley,1975] for Search Algorithm

Rectangular Dual Graph Representation [Kozminski,K and Kinnen,E,1984] for

Plant Layout

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Known Representation Methods

Quad-Tree Representation

NE NW

SWSE

NE NW SW SE

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Known Representation Methods Rectangular Dual Graph

Representation

Horizontal edgeVertical edge

１２

３ ４

５６

W

N

E

S

２

５

６４３

１

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Known Representation Methods

Example 1 Quad-tree Rep. has a weak power of expression.

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Known Representation Methods

Example 2 Rectangular Dual Graph Rep. may require higher complexity in editing operation.

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Known　results　(cont.)

Theorem. Decision problem of a graph

to be a (homogeneous) grid graph →O(n) (1990)

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Relate　Results

Data　Structures　:　Tessellation　graphs　(Motohashi　et.　al.　FOSE02-Matsuyama)

Viewer　(Kirishima　et.　al.,　LA02Summer,　IASTED　SE　02-Cambridge-USA)

Equivalent　condition　of　graphs　to　be　tessellation　graphs　(Kirishima　et.　al.,　　IASTED　AI　03-Insbruck)

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Purpose To propose a graph

representation method for tables in consideration of editing and drawing and to investigate mathematical properties.

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

To introduce a graph grammar

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2. Attribute Graphs for tables

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Example

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Def 2.1

Table T (2,3)-table Partition P {(1,1),(2,1)},{(1,2)}, . . . Grid g=(grow,gcolumn)

north wall of c nw(c)=1, sw(c)=2, east wall of c ew(c)=6, ww(c)=2.

0 2 4 60

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cell c

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Def 2.2 Tabular Diagram D=(T,P,g)

perimeter cellswith width=0 or height=0

We consider tabular diagramswith perimeter cells.

0 0 2 4 6 60

0

1

2

2

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Def 2.3 (Tessellation Graph)A tabular diagram D=(T,P,g) is

represented by an attribute graph G=(V,E,L,λ,A,α)

1st step put a node to a cell

D G

d c

vd

vc

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Graph representation 2nd step set edges and edge

labels nw(c) =nw(d)

sw(c) =sw(d)

ew(c) =ew(d)

ww(c) =ww(d)

D G

c d

c d

c

d

c

d

North wall edge South Wall Edge

East Wall Edge

West Wall Edge

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Graph representation

3rd step set an attribute

D G

vc(1,2,6,2)

nw(c)=1 sw(c)=2ew(c)=6 ww(c)=2

c

2 6

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Tessellation GraphExample

D GD

The tessellation Graphfor D □

d

vd

vc

d c

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Properties D : a tabular diagram of an (n,m)-table

GD : a tessellation graph for D. 1 node in GD corresponds to 1 cell

in D The degree of a node in GD ≦ 8.

Proposition 2.22|ED| = 6(2n-4) + 6(2m-4) + 8k + 16 (k : the number of non perimeter cells)

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　3D　:rectangularpiped　dissection:　24　degree　quasi　grid　graphs

Horizontal　4　links　(black)　x　nsew　４=　16　links

Vertical　4　links　(pink)　x　up-down　2　=　8　links　

Total　24　links

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Similar results to Proposition 2.2 hold for the 16 degree quasi grid graphs ; degree ≦ 16

for the 24 quasi grid graphs ; degree ≦ 24.

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Proposition　2.2　GD:a　rectangular　piped　dissection　graph　

for　D　of　an　(m,n,l)-cube.k　:　the　number　of　inner　cells　in　GD.For　#ED,　2x#ED　=　

20(n-2)(m-2)x2　+　20(m-2)(l-2)x2　+　20(l-2)(n-2)x2+　16(n-2)x4　+　16(m-2)x4　+　16(l-2)x4+　12　x　8+　24k

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Proposition (cf. Kundu)∃RD : A rectangular dissection with no wall crossing s.t. RD is not a T*-plan but is represented in a tessellation diagram

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Proof.A　spiral　graph　with　perimeter　cells　and　Its　corresponding　tessellation　graphs

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3. Algorithms

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RemarkAttributes for location of inner cells : not necessary

← Location information cited in perimeter cells

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3.1 ALGORITHM Unify-Cells(1) Input/Output Spec. Example

Input OutputGD=(VD,ED,L,λD,A,αD)

GE=(VE,EE,L,λE,A,αE)

vc, vd

D E

d

c c

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ccdcdc

(2) Illustration of Unify-Cells Mechanism

GD GE

(3) Theorem (Unify-Cells)Algorithm Unify-Cells runs in constant time.

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3.2 ALGORITHM Move-East-Wall

(1) INPUT/OUTPUT Spec. Exampleδ

D E

Moved Wall

c

Input OutputGD=(VD,ED,L,λD,A,αD)

GE=(VE,EE,L,λE,A,αE)

vcδ≧0 : movement value

cc cc

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cc cc

Traverse nodes upward through east wall edges .

Update east wall of each node, through east wall edges .

Update west wall of nodes in the right side column.

(2) Illustration of Move-East-Wall mechanisms

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3.3 ALGORITHM Insert-Column

D E

Inserted Column

Input OutputGD=(VD,ED,L,λD,A,αD)

GE=(VE,EE,L,λE,A,αE)

vc

cc cc

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Traverse nodes upward through west wall edges .

Add nodes and change edges.

Apply Move- East-Wall.

Insert-Column (cont)

cc cc

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complexity

e.w.movement

O(n)≦2n1cell-1node≦8Tessella

tionGraph

O(nm)≦nm1cell-1node

≦2n+2m

Rec.DualGraph

O(nm)≦nm1block-1node

≦ 5QuadTree

cell visit

cell-node rel.

node degree

model

3.4 2D Algorithms Comparison

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Algorithm Comparison (cont.)

O(1)O(n+m)Tessellation Graph

O(nm)O(nm)Rec. Dual Graph

O(nm)O(nm)Quad Tree

Unify cellsinsert column

model

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3.5 Theorem 3 [10, 2003]∃CS graph grammars to

generates the tessellation graphs,and the grid graphs, respectively.

Cf. There is no NLC graph grammar that generates the grid graphs (Janssen & Rozenberg).

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APPENDIX Multi Dim. Dissections

[2004, SCAI04]

Multi layer 2D: Multi layer rectangular dissections（ Stratum diagram）→16 degree quasi grid graphs

3D : Rectangular parallel piped dissections（Architect）→24 degree quasi grid graphs

Multi Dimension : General case→Wall linked quasi grid graphs

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A1.　Multi　layer　rectangular　dissection:　16　degree　quasi　grid　graph

Horizontal　2　links　x　nsew　=　8　links

Nsew　４ links　x　updown　2　=　8　links

Total　16　links

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A2.　3D　:rectangularpiped　dissection

:　24　degree　quasi　grid　graphs

Horizontal　4　links　(black)　x　nsew　４=　16　links

Vertical　4　links　(pink)　x　up-down　2　=　8　links　

Total　24　links

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Similar results to Proposition 2.2 hold for the 16 degree quasi grid graphs ; degree ≦ 16

for the 24 quasi grid graphs ; degree ≦ 24.

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Related　works• Browsers

– SVG and XML browsers [9, 2002]

• Quasi grid graphs SVG & XML

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4. Conclusion

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Summary Quasi grid graphs called tessellation

graphs for the rectangular dissections The node degrees ≦ 8.

Typical algorithms are shown, and their complexity approved.

A CS graph grammar that characterize the tessellation graphs is introduced.

Attribute graphs for 3D rectangular piped dissection; The node degrees ≦24

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Accession to the Aim Representation Method for Table Processing system is proposed.

Table editing operations are formalized.

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Future Works 1 Processing Systems on XML & SVGParsable graph grammars for tessellation graph

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Future　Works　2

～地形図への応用～多層の地形構造のためのデータ構造

地表プラス水面（２層）地層図

立体型の地形構造のためのデータ構造立体地層図

ブラウザ51