1. Compact Encodings of Graphs Shin-ichi Nakano (Gunma Univ.) Gunma

2. Compact Encodings of Graphs Shin-ichi Nakano (Gunma Univ.) Gunma Nagano Tokyo Olympic 1998

3. Compact Encodings of Graphs Shin-ichi Nakano (Gunma Univ.) Gunma Nagano Tokyo Olympic 1998

4. Compact Encodings of Graphs Shin-ichi Nakano (Gunma Univ.) Gunma Nagano Tokyo Olympic 1998 なかの ながの

5. Compact Encodings of Graphs Shin-ichi Nakano (Gunma Univ.) Gunma

6. Why we need Compact Encodings for Graphs? Although the computation power increase The size of graphs increase more quickly Internet link graphFine 3D meshEtc.

7. reversible Irreversible reversible encoding one can reproduce the original Irreversible encoding one cannot reproduce the original graph spanner remove unimportant edges for Movie Music

8. reversible Irreversible reversible encoding one can reproduce the original graph Irreversible encoding one cannot reproduce the original graph

9. Compact encoding 0 0 0 0 1 1 1 0 0 1 1 0 1 1 1 0 0 1 1 0 0 1 1 1 0 0 1 0 1 0 1 1 0 1 1 0 0 1 1 0 0 1 Simple Data Structure ( Adjacency list or Adjacency matrix) encode Compact string save memory space save transmission costs decode

10. Compact encoding with Query Support 0 0 0 0 1 1 1 0 0 1 1 0 1 1 1 0 0 1 1 0 0 1 1 1 0 0 1 0 1 0 1 1 0 1 1 0 0 1 1 0 0 1 Simple Data Structure ( Adjacency list or Adjacency matrix) encode Compact string save processing time (use only main memory) (no page fault) What is the degree of Vertex v 4 ? 5 Query decode

11. In this talk I will show you compact encodings for (ordered) Trees Floor plans Triangulations Compact Encodings:

12. Compact Encodings: Trees Ordered trees with 4 vertices ordered = the left-to-right ordering matters # of ordered trees with n vertices = n-th Catalan number =

13. Number of Trees num of vertices num of trees 45 514 642 7132 8429 91430 104862

14. A lower bound of the length of code If the num of objects is 8 then at least 3 = log 8 bits are needed to encode them we need to assign a different code for each object 000 001 010 011 100 101 110 111

15. A lower bound of the length Since the num of trees is C n, at least log C n bits are needed to encode a tree with n vertices log C n = 2n - o(n)

16. Compact Encodings: Trees D Start DU U U D 2m bit = 2n-2 bit encoding Optimal ( simple & best!) D D U U D U [ [ ] ] [ ] Nested parenthsis structure query support o(n) bit table Depth First Traversal

17. Query Jacobson[1989] Munro and Raman[2001] …. Rank(x) return the position of node x in preorder Select(i) return the i-th node in preorder Child(x,i) return the i-th child of node x Deg(x) return the num of child nodex of x ….. (see Geary et.al. [2006])(search with keyword “succinct”) O(1) time

18. In this talk I will show you compact encodings for (ordered) Trees Floor plans [Yamanaka & Nakano 2006, 2007] Triangulations Compact Encodings:

19. Compact Encodings: Floor plans A floorplan is a plane drawing in which every face is a rectangle Applications VLSI, etc.

20. Compact Encodings: Floor plans break each SW corner vertex

21. Compact Encodings: Floor plans break each SW corner vertex

22. Compact Encodings: Floor plans break each SW corner vertex A tree is derived

23. Compact Encodings: Floor plans break each SW corner vertex A tree is derived

24. Compact Encodings: Floor plans break each SW corner vertex A tree is derived add two more vertices at the bottom

25. Compact Encodings: Floor plans Depth First Traversal from the NW corner next ? Start wall or fence

26. Compact Encodings: Floor plans Depth First Traversal from the NW corner next ? Start

27. Compact Encodings: Floor plans Always we have only two choices for the next edge Start

28. Compact Encodings: Floor plans Always we have only two choices for the next edge Start 2m = (3n -4) bit encoding 0 0 0 0 1 1 1 1 0 0 0 0 1 10

29. Compact Encodings: Floor plans Are there more compact encodings? YES !

30. More Compact Encodings: Floor plans

31. North South WestEast

32. More Compact Encodings: Floorplans

33. Rectangular Dual Floorplan

34. More Compact Encodings: Floorplans v parent of v The parent of v is the westmost one among the north neighbor Spanning Tree

35. More Compact Encodings: Floorplans [[ ] ][][ ]]] []] [] [[] ][ []

36. More Compact Encodings: Floorplans [[ ] ][][ ]]] []] [] [[] [ [] ] 1 [] [] 2 3 5 4 7 6 9 10 12 11 8 8 8E8W 6

37. More Compact Encodings: Floorplans [[ ] ][][ ]]] []] [] [[] [ [] ] 1 [] [] 2 3 5 4 7 6 9 10 12 11 8 8 8E8W 6

38. More Compact Encodings: Floorplans [] [] 8E8W 6 2W3W3E2E5W4E6W7W7E8W9W1W4W5E9E 10W10E [ [ ] ] [ [[ ] ] [[ ] ] [ ] [ 8E 1 2 3 4 6 7 9 10 12 11 8 5 8 Nested parenthsis structure

39. More Compact Encodings: Floorplans [] [] 8E8W 6 2W3W3E2E5W4E6W7W7E8W9W1W4W5E9E 10W10E [ [ ] ] [ [[ ] ] [[ ] ] [ ] [ 8E 1 2 3 4 6 7 9 10 12 11 8 5 8 Nested parenthsis structure

40. More Compact Encodings: Floorplans [] [] 8E8W 6 2W3W3E2E5W4E6W7W7E8W9W1W4W5E9E 10W10E [ [ ] ] [ [[ ] ] [[ ] ] [ ] [ 8E 1 2 3 4 6 7 9 10 12 11 8 5 8 Nested parenthsis structure

41. More Compact Encodings: Floorplans

43. D D U U Tree DDUUDDUUDDUD….. 2m = 2n -2 bit

44. More Compact Encodings: Floorplans D D U U Tree DDUUDDUUDDUD….. 2 bit for each edge Nested parenthsis [ [ ] ] [ [ [ ] ] [ [ ] ] [ ] ] ….. 0111101110111101 ….. 2 bit for each edge

45. More Compact Encodings: Floorplans D D U U Tree DDUUDDUUDDUD….. 2 bit for each edge Nested parenthsis [ [ ] ] [ [ [ ] ] [ [ ] ] [ ] ] ….. 0111101110111101 ….. 2 bit for each edge

46. More Compact Encodings: Floorplans D D U U Tree DDUUDDUUDDUD….. 2 bit for each edge Nested parenthsis [ [ ] ] [ [ [ ] ] [ [ ] ] [ ] ] ….. 0111101110111101 ….. 2 bit for each edge0 bit for each edge 5m/3 bit encoding

47. More Compact Encodings: Floorplans D D U U Tree DDUUDDUUDDUD….. 2 bit for each edge Nested parenthsis [ [ ] ] [ [ [ ] ] [ [ ] ] [ ] ] ….. 0111101110111101 ….. 2 bit for each edge0 bit for each edge 5m/3 bit encoding query support o(n) bit table

48. In this talk I will show you compact encodings for (ordered) Trees Floor plans Triangulations [Yamanaka & Nakano 2005] Compact Encodings:

49. Compact Encodings: Triangulations

50. Realizer: partition of inner edges into 3 trees Red RootBlue Root Green Root to Green Root to Red Rootto Blue Root

51. Compact Encodings: Triangulations Realizer Red RootBlue Root Green Root to Green Root to Red Rootto Blue Root

52. Compact Encodings: Triangulations Realizer Blue + Red = Purple Green Spanning Tree to Green Root to Red Rootto Blue Root

53. Compact Encodings: Triangulations Realizer Blue + Red = Purple Green Spanning Tree to Green Root to Red Rootto Blue Root

54. Compact Encodings: Triangulations 1W2W2E3W4W4E3E5W6W6E7W7E5E8W9W 10W10E 1 3 2 4 5 6 7 8 9 10 11 12 13

55. Compact Encodings: Triangulations Tree DUDDUUDDUDUUDD… 2 bit for each edge Nested parenthsis [ [ [ [ [ ] ] [ [ [ [ ] ] ] [ ] ] ] [.. 0000111001101110011. 2 bit for each edge 2m bit encoding query support o(n) bit table

56. Compact Encodings: Triangulations Are there more compact encodings? YES ! 4.35n +o(n) bits (with query support) a complete list of all possible triangulations having at most (log n) /4 nodes L. C. Aleardi, O. Devillers and G. Schaeffer[2005] 4m/3-1 bit (without query support) X. He, M. Y. Kao and H. I. Lu [1999]

57. Compact Encodings: Triangulations Are there more compact encodings? May be! Let’s try together!

58. Compact Encodings: Triangulations