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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
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
12
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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