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

1. 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

2. 1. Introduction Position and Scope • Data Processing • Number • Character Strings • Picture : Tables, Charts, Graphs, Pictures • Scope • Tables in Spread Sheet or Word • Editing, Drawing, Calculation

3. Background • Effectiveness of Table Processing depends on RepresentationMethods of Table. • Known RepresentationMethods • Quad-Tree Representation For Search Algorithm • Rectangular Dual Graph Representation For Plant Layout

4. Problems In 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.

5. Example（unexpected results） ２を指定 列挿入

6. Our History • Tree structured diagrams • Attributed marked trees • Combinatorial drawing algorithms • Modular forms • Attributed marked trees

7. Our Purpose • To propose Graph Representation Methods for tables in consideration of editing and drawing. • To investigate mathematical properties of the graphs in the representation methods. • To introduce typical algorithms on graphs and evaluate their complexity.

8. Our Results • A Graph representation method is proposed. • Properties of the graph is shown • |nodes|in the graph = |cells| in the table. • The degrees of nodes ≦ 8. • Algorithms for Wall move and Column Insertion are introduced, and runs in linear time.

9. Representation Method of tables • Quad-Tree Representation (for search) NE SE SW NW

10. Rectangular Dual Graph Representation (for プラントレイアウト) N ２ １ ４ E ３ W ６ ５ Horizontal edge Vertical edge S

11. 2. Tabular Diagrams Definitions. • An (n, m)-table T is a set {(i, j) | 1≦i ≦n, 1 ≦ j ≦ m} (2,3)-table T (1,2) (1,3) (1,1) (2,1) (2,2) (2,3)

12. Definitions.For an (n,m)-table T, • A partialtableS 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 {(1,2)} {(1,3)} {(1,1), (2,1)} cell {(2,2), (2,3)}

13. 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 6 0 1 2

14. 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 3 6 0 2 0 nw(c)=1 sw(c)=2 ew(c)=6ww(c)=2 1 c 2

15. Condition*. For a tabular diagram D=(T,P,G), • T is a (n,m)-table, where n,m≧3. • P has perimeter cells withwidth=0 or height=0 3 6 6 0 0 2 0 * * * * * 0 * * 1 * * 2 * * * * * 2 Perimeter cells

16. 3 6 0 2 0 1 2 D2=(T,P2,g2) Examples. 3 6 6 0 0 2 0 * * * * * 0 * * 1 * * 2 * * * * * 2 D2*=(T*,P2*,g2*) satisfyingCondition* We consider tabular diagrams satisfying Condition*.

17. 3. Graph Representation of Tables Operations on Editing Tables Wall moving Insertion/deletion/unification of Row/column/cell Requirements computational complexity Suitable correspondence

18. 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 edges • λ : E→L is the label function • A is the set of attribute • α : V→A is the attribute map

19. Graph representation of tables A Tabular Diagram D=(T,P,g) is represented as an attribute graph GD=(VD,ED,L,λD,A,αD) • VDis identified by a partition P • node⇔cell. • We denote a node vc for the cell c,

20. L={enw,esw,eew,eww}, • Equal north wall, equal south wall, … • A = R4, • αD : VD →R4 are defined for vcby αD(vc)= (nw(c),sw(c),ew(c),ww(c)).

21. EDand λD : ED→L are defined by the Rules, Rule NIf 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,

22. 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

23. a a va Examples. • An attribute graph is called a tessellation graph, if it represents a tabular diagram. vd d d c c vc

24. Properties Let GD be tessellation graph for a tabular diagram D of an (n,m)-table. Proposition 3.1 • Number of nodes in GD = number of cells in D

25. Proposition 3.2 • 2|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.

26. MOVEEASTWALL Output GE=(VE,EE,L,λE,A,αE) 4.Algorithms Moved Wall δ c c E D Input GD=(VD,ED,L,λD,A,αD) Vc: a node δ≧0 : movement value

27. 隣の列を West wall edge をたどりながら WestWall 変更 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

28. Inserted Column • INSERTCOLUMN * * c * * * * E D Input GD, Vc : a node Output GE

29. * * * * * * 新しい列以東 Wall Move East を行う West wall edge をたどりながら、 1node ずつ追加 Edgeの更新 Node vc　から West wall edge を上にたどる

30. For a tabular diagram D of (n,m)-table, Theorem4.1 • Algorithm MOVEEASTWALL runs in O(n) time.

31. Theorem 4.2 • Algorithm INSERTCOLUMN runs in O(nm) time. Remark • Algorithm INSERTCOLUMN visits only nodes which needs to change attributes.

32. 5. Conclusion summary • An attribute Graph rep. method is proposed. • Properties of the graph is shown • |nodes|in the graph = |cells| in the table. • The degrees of nodes ≦ 8. • Algorithms for Wall move and Column Insertion runs in linear time.

33. Future Works • XML Viewer for tessellation graph. • Other Algorithms for Editing Commands • Practical Processing Systems • Graph Grammars for generate tessellation graphs • Conditions of tessellation graph

35. 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

36. Representation of table by tessellation graphs (continued) • Peripheral cells and corresponding nodes • Cell a is represented by node • Ceiling of the cell a is represented by node z • Virtual cells x and y has no valid heights • Ceiling of the cell z is covered by cells x and y

37. a x y example z

38. General rules x y Cells x and y are of same ceiling coordinates x y Cells x and y are of same floor coordinates Cells x and y are of same ceiling and floor coordinates x y

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

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

41. 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

42. 4. CONCLUSION • Future Works • Other Methods • Relation to Mathematical Theory • Graph Grammars

43. 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）

44. 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、・・・

45. 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

46. 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

47. 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

48. 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.

49. Example 2.

50. 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）