Download presentation

Presentation is loading. Please wait.

Published byStone Clagg Modified over 2 years ago

1
Self-Assembly with Geometric Tiles ICALP 2012 Bin FuUniversity of Texas – Pan American Matt PatitzUniversity of Arkansas Robert Schweller (Speaker)University of Texas – Pan American Robert ShelineUniversity of Texas – Pan American

2
Outline Basic Tile Assembly Model Geometric Tile Assembly Model – Basic Model – Planar Model – More efficient n x n squares Future Directions

3
3 Tile Assembly Model (Rothemund, Winfree, Adleman) T = G(y) = 2 G(g) = 2 G(r) = 2 G(b) = 2 G(p) = 1 G(w) = 1 t = 2 Tile Set: Glue Function: Temperature: x ed cba

4
4 T = G(y) = 2 G(g) = 2 G(r) = 2 G(b) = 2 G(p) = 1 G(w) = 1 t = 2 d e x ed cba Tile Assembly Model (Rothemund, Winfree, Adleman)

5
5 T = G(y) = 2 G(g) = 2 G(r) = 2 G(b) = 2 G(p) = 1 G(w) = 1 t = 2 d e x ed cba Tile Assembly Model (Rothemund, Winfree, Adleman)

6
6 T = G(y) = 2 G(g) = 2 G(r) = 2 G(b) = 2 G(p) = 1 G(w) = 1 t = 2 d e x ed cba bc Tile Assembly Model (Rothemund, Winfree, Adleman)

7
7 T = G(y) = 2 G(g) = 2 G(r) = 2 G(b) = 2 G(p) = 1 G(w) = 1 t = 2 d e x ed cba bc Tile Assembly Model (Rothemund, Winfree, Adleman)

8
8 T = G(y) = 2 G(g) = 2 G(r) = 2 G(b) = 2 G(p) = 1 G(w) = 1 t = 2 d e x ed cba bc Tile Assembly Model (Rothemund, Winfree, Adleman)

9
9 T = G(y) = 2 G(g) = 2 G(r) = 2 G(b) = 2 G(p) = 1 G(w) = 1 t = 2 d e x ed cba bca Tile Assembly Model (Rothemund, Winfree, Adleman)

10
10 T = G(y) = 2 G(g) = 2 G(r) = 2 G(b) = 2 G(p) = 1 G(w) = 1 t = 2 d e x ed cba bca Tile Assembly Model (Rothemund, Winfree, Adleman)

11
11 T = G(y) = 2 G(g) = 2 G(r) = 2 G(b) = 2 G(p) = 1 G(w) = 1 t = 2 d e x ed cba bca Tile Assembly Model (Rothemund, Winfree, Adleman)

12
12 T = G(y) = 2 G(g) = 2 G(r) = 2 G(b) = 2 G(p) = 1 G(w) = 1 t = 2 d e x ed cba bca Tile Assembly Model (Rothemund, Winfree, Adleman)

13
13 T = G(y) = 2 G(g) = 2 G(r) = 2 G(b) = 2 G(p) = 1 G(w) = 1 t = 2 x ed cba abc d e Tile Assembly Model (Rothemund, Winfree, Adleman)

14
14 T = G(y) = 2 G(g) = 2 G(r) = 2 G(b) = 2 G(p) = 1 G(w) = 1 t = 2 x ed cba x abc d e Tile Assembly Model (Rothemund, Winfree, Adleman)

15
15 T = G(y) = 2 G(g) = 2 G(r) = 2 G(b) = 2 G(p) = 1 G(w) = 1 t = 2 abc d e x x ed cba Tile Assembly Model (Rothemund, Winfree, Adleman)

16
16 T = G(y) = 2 G(g) = 2 G(r) = 2 G(b) = 2 G(p) = 1 G(w) = 1 t = 2 x ed cba abc d e xx Tile Assembly Model (Rothemund, Winfree, Adleman)

17
17 T = G(y) = 2 G(g) = 2 G(r) = 2 G(b) = 2 G(p) = 1 G(w) = 1 t = 2 x ed cba abc d e xx x Tile Assembly Model (Rothemund, Winfree, Adleman)

18
18 T = G(y) = 2 G(g) = 2 G(r) = 2 G(b) = 2 G(p) = 1 G(w) = 1 t = 2 x ed cba abc d e xx xx Tile Assembly Model (Rothemund, Winfree, Adleman)

19
Geometric Tile Model

20
Geometric Tiles Geometry Region

21
Geometric Tiles Geometry Region

22
Geometric Tiles Compatible Geometries

23
Geometric Tiles

24
Incompatible Geometries

25
Geometric Tiles Incompatible Geometries

26
n x n Results Tile Complexity Geometric Tiles Normal Tiles* Upper boundLower bound Planar Geometric Tiles [*Winfree, Rothemund, Adleman, Cheng, Goel,Huang STOC 2000, 2001]

27
n x n Squares, root(log n) tiles log n 01011

28
Assembly of n x n Squares n log n

29
Assembly of n x n Squares log n 01011

30
Assembly of n x n Squares -Build thicker 2 x log n seed row

31
log n Assembly of n x n Squares -Build thicker 2 x log n seed row -But… cant encode general binary strings: 0 -All the same

32
log n Assembly of n x n Squares 0 B3B2B1B0 A3A2A1A0 Key Idea: Geometry Decoding Tiles

33
log n Assembly of n x n Squares B0A0A1B1A2B2A3B3A0B1B2A3B0A1A2B3B0B1A2B3B0B1B2A3

34
log n Assembly of n x n Squares B0A0A1B1A2B2A3B3A0B1B2A3B0A1A2B3B0B1A2B3B0B1B2A3

35
Assembly of n x n Squares A2 B3 A3

36
log n Assembly of n x n Squares B0A0A1B1A2B2A3B3A0B1B2A3B0A1A B3A3

37
log n Assembly of n x n Squares B0A0A1B1A2B2A3B3A0B1B2A3B0A1A2B3B0B1A2B3B0B1B2A3

38
log n Assembly of n x n Squares B0A0A1B1A2B2A3B3A0B1B2A3B0A1A2B3B0B1A2B3B0B1B2A3 build 2 x log n block: Decode geometry into log n bit string

39
Upper boundLower bound n x n Results Tile Complexity Geometric Tiles Normal Tiles* [*Winfree, Rothemund, Adleman, Cheng, Goel,Huang STOC 2000, 2001] Planar Geometric Tiles

40
Planar Geometric Tile Assembly Attachment requires a collision free path within the plane

41
Planar Geometric Tile Assembly Attachment requires a collision free path within the plane Attachment not permitted in the planar model

42
Planar Geometric Tile Assembly

43

44
Attachment not permitted in the planar model

45
n x n Results Tile Complexity Geometric Tiles Normal Tiles* Upper boundLower bound Planar Geometric Tiles ? [*Winfree, Rothemund, Adleman, Cheng, Goel,Huang STOC 2000, 2001]

46
n x n Results Tile Complexity Geometric Tiles Normal Tiles* Upper boundLower bound Planar Geometric Tiles O( loglog n ) [*Winfree, Rothemund, Adleman, Cheng, Goel,Huang STOC 2000, 2001] ?

47
log n Planar Geometric Tile Assembly

48
loglog n Build log n columns with loglog n tile types Planar Geometric Tile Assembly

49
loglog n Build log n columns with loglog n tile types Planar Geometric Tile Assembly

50
loglog n Build log n columns with loglog n tile types Columns must assemble in proper order Planar Geometric Tile Assembly

51
log n loglog n Build log n columns with loglog n tile types Columns must assemble in proper order Somehow cap each column with specified 0 or 1 tile type. Planar Geometric Tile Assembly

52
Build log n columns with loglog n tile types Columns must assemble in proper order Somehow cap each column with specified 0 or 1 tile type

53
Build log n columns with loglog n tile types Columns must assemble in proper order Somehow cap each column with specified 0 or 1 tile type

54
Build log n columns with loglog n tile types Columns must assemble in proper order Somehow cap each column with specified 0 or 1 tile type

55
Planar Geometric Tile Assembly

56

57
log n loglog n Build log n columns with loglog n tile types Columns must assemble in proper order Somehow cap each column with specified 0 or 1 tile type.

58
log n loglog n Build log n columns with loglog n tile types Columns must assemble in proper order Somehow cap each column with specified 0 or 1 tile type.

59
Planar Geometric Tile Assembly

60

61

62

63

64
1 0 0

65
1 0 0

66
1 0 0

67
1 0 0

68

69

70

71

72
log n loglog n Build log n columns with loglog n tile types Columns must assemble in proper order Somehow cap each column with specified 0 or 1 tile type. O( loglog n ) tile types

73
n – log n log n X Y Complexity:

74
n x n Results Tile Complexity Geometric Tiles Normal Tiles* Upper boundLower bound Planar Geometric Tiles O( loglog n )? [*Winfree, Rothemund, Adleman, Cheng, Goel,Huang STOC 2000, 2001]

75
Outline Basic Tile Assembly Model – Rectangles – n x n squares Geometric Tile Assembly Model – More efficient n x n squares Planar Geometric Tile Assembly Model – Even MORE efficient n x n squares (A strange game.. planarity restriction helps you…) Future Directions and Other Results

76
Other Results Simulation of temperature-2 systems with temperature-1 geometric tile systems. Simulation of many glue systems with single glue geometric tile systems. Compact Geometry Design Problem – Algorithms, lower bounds

77
Future Directions Lower bound for the planar model? – Is O(1) tile complexity possible in the planar model? – If not, what about log*(n)? What can be done with just 1 tile type? – Stay tuned for: One Tile to Rule Them All: Simulating Any Turing Machine, Tile Assembly System, or Tiling System with One Rotatable Puzzle Piece by: Erik Demaine, Martin Demaine, Sandor Fekete, Matthew Patitz, Robert Schweller, Andrew Winslow, Damien Woods. What about no rotation, but relative translation placement: – Check out One Tile... -EXTENDED VERSION! SPOILER ALERT: There is totally 1 universal tile that can do anything that can be done.

78
People Bin Fu Matt Patitz Robbie Schweller Bobby Sheline

79
79 Barish, Shulman, Rothemund, Winfree, 2009

80
DNA Origami Tiles [Masayuki Endo, Tsutomu Sugita, Yousuke Katsuda, Kumi Hidaka, and Hiroshi Sugiyama, 2010]

81
More DNA Origami Shapes [Paul Rothemund, Nature 2006]

82
Alphabet of Shapes, Built with DNA Tiles [Bryan Wei, Mingjie Dai, Peng Yin, Nature 2012]

83
83 n x n squares with Geometric Tiles Tile Complexity: n - k k k x

84
Assembly of n x n Squares n - k k Complexity:

85
Assembly of n x n Squares n – log n log n Complexity:

86
Assembly of n x n Squares n – log n log n Complexity: seed row

87
log n Assembly of n x n Squares -Build thicker 2 x log n seed row

88
n – log n log n

89
n – log n log n X Y Complexity:

Similar presentations

© 2017 SlidePlayer.com Inc.

All rights reserved.

Ads by Google