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Complexities for Generalized Models of Self-Assembly Gagan Aggarwal Stanford University Michael H. Goldwasser St. Louis University Ming-Yang Kao Northwestern.

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Presentation on theme: "Complexities for Generalized Models of Self-Assembly Gagan Aggarwal Stanford University Michael H. Goldwasser St. Louis University Ming-Yang Kao Northwestern."— Presentation transcript:

1 Complexities for Generalized Models of Self-Assembly Gagan Aggarwal Stanford University Michael H. Goldwasser St. Louis University Ming-Yang Kao Northwestern University Robert T. Schweller Northwestern University Some results were obtained independantly by Cheng, Espanes 2003

2 Tile Model of Self-Assembly (Rothemund, Winfree STOC 2000) Tile System: t : temperature, positive integer G: glue function T: tileset s: seed tile

3 How a tile system self assembles T = G(y,y) = 2 G(g,g) = 2 G(r, r) = 2 G(b,b) = 2 G(p,p) = 1 G(w,w) = 1 t = 2

4 How a tile system self assembles T = G(y,y) = 2 G(g,g) = 2 G(r, r) = 2 G(b,b) = 2 G(p,p) = 1 G(w,w) = 1 t = 2

5 How a tile system self assembles T = G(y,y) = 2 G(g,g) = 2 G(r, r) = 2 G(b,b) = 2 G(p,p) = 1 G(w,w) = 1 t = 2

6 How a tile system self assembles T = G(y,y) = 2 G(g,g) = 2 G(r, r) = 2 G(b,b) = 2 G(p,p) = 1 G(w,w) = 1 t = 2

7 How a tile system self assembles T = G(y,y) = 2 G(g,g) = 2 G(r, r) = 2 G(b,b) = 2 G(p,p) = 1 G(w,w) = 1 t = 2

8 How a tile system self assembles T = G(y,y) = 2 G(g,g) = 2 G(r, r) = 2 G(b,b) = 2 G(p,p) = 1 G(w,w) = 1 t = 2

9 How a tile system self assembles T = G(y,y) = 2 G(g,g) = 2 G(r, r) = 2 G(b,b) = 2 G(p,p) = 1 G(w,w) = 1 t = 2

10 How a tile system self assembles T = G(y,y) = 2 G(g,g) = 2 G(r, r) = 2 G(b,b) = 2 G(p,p) = 1 G(w,w) = 1 t = 2

11 How a tile system self assembles T = G(y,y) = 2 G(g,g) = 2 G(r, r) = 2 G(b,b) = 2 G(p,p) = 1 G(w,w) = 1 t = 2

12 New Models Multiple Temperature Model –temperature may go up and down Flexible Glue Model –Remove the restriction that G(x, y) = 0 for x!=y Multiple Tile Model –tiles may cluster together before being added Unique Shape Model –unique shape vs. unique supertile

13 New Models Multiple Temperature Model –temperature may go up and down Flexible Glue Model –Remove the restriction that G(x, y) = 0 for x!=y Multiple Tile Model –tiles may cluster together before being added Unique Shape Model –unique shape vs. unique supertile

14 New Models Multiple Temperature Model –temperature may go up and down Flexible Glue Model –Remove the restriction that G(x, y) = 0 for x!=y Multiple Tile Model –tiles may cluster together before being added Unique Shape Model –unique shape vs. unique supertile

15 New Models Multiple Temperature Model –temperature may go up and down Flexible Glue Model –Remove the restriction that G(x, y) = 0 for x!=y Multiple Tile Model –tiles may cluster together before being added Unique Shape Model –unique shape vs. unique supertile

16 Focus of Talk Multiple Temperature Model –Adjust temperature during assembly Flexible Glue Model –Remove the restriction that G(x, y) = 0 for x!=y Goal: Reduce Tile Complexity

17 Our Tile Complexity Results Multiple temperature model: k x N rectangles: beats standard model: Flexible Glue: N x N squares: beats standard model: ( Adleman, Cheng, Goel, Huang STOC 2001 ) (our paper)

18 Building k x N Rectangles k-digit, base N (1/k) counter: k N

19 Building k x N Rectangles k-digit, base N (1/k) counter: Tile Complexity: N k

20 S C1C1 C2C2 C3C3 0 ggp Build a 4 x 256 rectangle: t = 2 C0C0 g S3S3 S2S2 0 0 S S1S1

21 S C1C1 C2C2 C3C3 0123 0 ggp Build a 4 x 256 rectangle: t = 2 C0C0 g S3S3 S2S2 0 0 00 g g SC1C1 C2C2 C3C3 S1S1 S2S2 S3S3 S1S1 0 0 0 0 ggp

22 S C1C1 C2C2 C3C3 0123 0 ggp Build a 4 x 256 rectangle: t = 2 C0C0 g gg S3S3 S2S2 0 0 1 00 g g pr 0 SC1C1 C2C2 C3C3 S1S1 00 S2S2 S3S3 00 00 01 S1S1 p

23 S C1C1 C2C2 C3C3 0123 0 ggp Build a 4 x 256 rectangle: t = 2 C0C0 g gg S3S3 S2S2 0 0 1 00 g g pr 0 SC1C1 C2C2 C3C3 S1S1 0001 S2S2 S3S3 00 00 01 S1S1 gg

24 S C1C1 C2C2 C3C3 0123 0 ggp Build a 4 x 256 rectangle: t = 2 C0C0 g gg S3S3 S2S2 0 0 1 00 g g pr 0 SC1C1 C2C2 C3C3 S1S1 0001 S2S2 S3S3 00 00 01 S1S1 C0C0 C1C1 C2C2 C3C3 00 00 p

25 S C1C1 C2C2 C3C3 0123 0 ggp Build a 4 x 256 rectangle: t = 2 C0C0 g gg S3S3 S2S2 0 0 1 00 g g pr 0 SC1C1 C2C2 C3C3 S1S1 0001 S2S2 S3S3 00 00 01 S1S1 C0C0 C1C1 C2C2 C3C3 00 00 p 11 00 00 12 23

26 S C1C1 C2C2 C3C3 0123 0 ggp 12 P p 3 Build a 4 x 256 rectangle: t = 2 C0C0 g gg R 0 p r r S3S3 S2S2 0 0 1 23 00 g g pr 0 SC0C0 C1C1 C2C2 C3C3 S1S1 0001 C1C1 C2C2 C3C3 1122331223 S2S2 S3S3 00 00 00 00 00 00 00 00 00 00 00 00 00 00 C0C0 C1C1 C2C2 C3C3 C0C0 C1C1 C2C2 C3C3 01 S1S1 p

27 S C1C1 C2C2 C3C3 0123 0 ggp 12 P p 3 Build a 4 x 256 rectangle: t = 2 C0C0 g gg R 0 p r r S3S3 S2S2 0 0 1 23 00 g g pr 0 SC0C0 C1C1 C2C2 C3C3 S1S1 0001 C1C1 C2C2 C3C3 1122331223 S2S2 S3S3 00 00 00 00 00 00 00 00 00 00 00 00 00 00 P C0C0 C1C1 C2C2 C3C3 C0C0 C1C1 C2C2 C3C3 01 S1S1

28 S C1C1 C2C2 C3C3 0123 0 ggp 12 P p 3 Build a 4 x 256 rectangle: t = 2 C0C0 g gg R 0 p r r S3S3 S2S2 0 0 1 23 00 g g pr 0 SC0C0 C1C1 C2C2 C3C3 S1S1 0001 C1C1 C2C2 C3C3 1122331223 S2S2 S3S3 00 00 00 00 00 00 00 00 00 00 00 00 00 00 P 01 C0C0 C1C1 C2C2 C3C3 C0C0 C1C1 C2C2 C3C3 01 S1S1

29 S C1C1 C2C2 C3C3 0123 0 ggp 12 P p 3 Build a 4 x 256 rectangle: t = 2 C0C0 g gg R 0 p r r S3S3 S2S2 0 0 1 23 00 g g pr 0 SC0C0 C1C1 C2C2 C3C3 S1S1 0001 C1C1 C2C2 C3C3 1122331223 S2S2 S3S3 00 00 00 00 00 00 00 00 00 00 00 00 00 00 P 01 C0C0 C1C1 C2C2 C3C3 C0C0 C1C1 C2C2 C3C3 01 S1S1 R

30 S C1C1 C2C2 C3C3 0123 0 ggp 12 P p 3 Build a 4 x 256 rectangle: t = 2 C0C0 g gg R 0 p r r S3S3 S2S2 0 0 1 23 00 g g pr 0 SC0C0 C1C1 C2C2 C3C3 S1S1 0001 C1C1 C2C2 C3C3 1122331223 S2S2 S3S3 00 00 00 00 00 00 00 00 00 00 00 00 00 00 P 01 C0C0 C1C1 C2C2 C3C3 C0C0 C1C1 C2C2 C3C3 01 S1S1 R C0C0 C1C1 C2C2 …

31 S C1C1 C2C2 C3C3 0123 0 ggp 12 P p 3 Build a 4 x 256 rectangle: t = 2 C0C0 g gg R 0 p r r S3S3 S2S2 0 0 1 23 00 g g pr 0 SC0C0 C1C1 C2C2 C3C3 S1S1 0001 C1C1 C2C2 C3C3 1122331223 S2S2 S3S3 00 00 00 00 00 00 00 00 00 00 00 00 00 00 P 01 R … 00 11 0000 C0C0 C1C1 C2C2 C3C3 C0C0 C1C1 C2C2 C3C3 C0C0 C1C1 C2C2 01 S1S1

32 S C1C1 C2C2 C3C3 0123 0 ggp 12 P p 3 Build a 4 x 256 rectangle: t = 2 C0C0 g gg R 0 p r r S3S3 S2S2 0 0 1 23 00 g g pr 0 01 223P P P 33 33 33 33 33 212 33 33 33 33 1101 33 33 33 33 00 33 33 32 RP 333 2 3 3 2 3 C1C1 C2C2 C3C3 C0C0 C1C1 C2C2 C3C3 C0C0 C1C1 C2C2 C3C3 C0C0 C1C1 C2C2 C3C3 C0C0 C1C1 C2C2 C3C3 S1S1

33 Building k x N Rectangles k-digit, base N (1/k) counter: Tile Complexity: N k

34 2-temperature model 3 3 3 1 t = 4

35 2-temperature model t = 4 6

36 2-temperature model Kolmogorov Complexity (Rothemund, Winfree STOC 2000) Beats Standard Model (our paper)

37 Assembly of N x N Squares

38 k N - k k

39 Assembly of N x N Squares k N - k X Y k Complexity: ( Adleman, Cheng, Goel, Huang STOC 2001 )

40 N x N Squares --- Flexible Glue Model a b c d e f a 1 0 2 0 0 1 b 0 0 1 0 1 0 c 0 0 3 0 1 1 d 2 2 2 2 0 1 e 0 0 0 1 2 1 f 1 1 2 2 1 1 a b c d e f a 1 - - - - - b - 0 - - - - c - - 3 - - - d - - - 2 - - e - - - - 2 - f - - - - - 1 Standard Glue FunctionFlexible Glue Function Kolmogorov lower bounds: Standard Flexible (Rothemund, Winfree STOC 2000)

41 N x N Square --- Flexible Glue Model log N N – log N seed row

42 N x N Square --- Flexible Glue Model log N N – log N seed row Complexity:

43 N x N Square --- Flexible Glue Model goal: - seed binary counter to a given value - 2 log N 010000001111111111

44 ... 3334444445555 345012345012345 5 N x N Square --- Flexible Glue Model

45 ... 3334444445555 345012345012345 0 0 1 1 0 1 1 0 0 1 1 1 0 | | | | | | | | | | | | | 5 5 N x N Square --- Flexible Glue Model key idea:

46 453 555 21 b4b4 5 5 w5w5 p5p5 G(b 4, p 5 ) = 1 G(b 4, w 5 ) = 0 N x N Square --- Flexible Glue Model

47 p0 p1 p2 p3 p4 p5 b0 0 1 1 0 1 1 b1 1 1 0 1 0 1 b2 0 1 0 1 1 1 b3 0 0 1 0 1 0 b4 0 0 0 0 0 1 b5 1 1 1 1 1 0 given B = 011011 110101 010111 … encode B into glue function B = 011011 110101 010111 … N x N Square --- Flexible Glue Model 4 b4b4 5 p5p5

48 Complexity: 0 1 0 1 1 0 0 0 1 1 0 0 1 0 0 0 1 1 0 1 1 1 0 0 0 1 0 1 build block N x N Square --- Flexible Glue Model

49 0 1 0 1 1 0 0 0 1 1 0 0 1 0 0 0 1 1 0 1 1 1 0 0 0 1 0 1 0 1 0 1 1 0 0 0 1 1 0 0 1 0 0 0 1 1 0 1 1 1 0 0 0 1 1 0 0 1 0 1 1 0 0 0 1 1 0 0 1 0 0 0 1 1 0 1 1 1 0 0 0 1 1 1 0 1 0 1 1 0 0 0 1 1 0 0 1 0 0 0 1 1 0 1 1 1 0 0 1 0 0 0 0 1 0 1 1 0 0 0 1 1 0 0 1 0 0 0 1 1 0 1 1 1 0 0 1 0 0 1 0 1 0 1 1 0 0 0 1 1 0 0 1 0 0 0 1 1 0 1 1 1 0 0 1 0 1 0 0 1 0 1 1 0 0 0 1 1 0 0 1 0 0 0 1 1 0 1 1 1 0 0 1 0 1 1 0 1 0 1 1 0 0 0 1 1 0 0 1 0 0 0 1 1 0 1 1 1 0 0 1 1 0 0 0 1 0 1 1 0 0 0 1 1 0 0 1 0 0 0 1 1 0 1 1 1 0 0 1 1 0 1 0 1 0 1 1 0 0 0 1 1 0 0 1 0 0 0 1 1 0 1 1 1 0 0 1 1 1 0

50 2 x log N block N – log N log N

51 N – log N log N

52 N – log N X Y log N Complexity:

53 Shape Verification Unique Shape Problem Input: T, a tile system S, a shape Question: Does T uniquely assemble S. Standard:P (Adleman, Cheng, Goel, Huang, Kempe, Flexible Glue:P Espanes, Rothemund, STOC 2002) Unique Shape:Co-NPC (our paper) Multiple Temperature:NP-hard (our paper) Multiple Tile:Co-NPC (our paper)

54 x1x1 ** x2x2 x3x3 *TTTT ok c2c2 c1c1 c2c2 c2c2 c3c3 * * * * c1c1 0 1 1 SAT x1x1 ** x2x2 x3x3 *TTFF ok c2c2 c2c2 c1c1 c2c2 c2c2 c3c3 * * * * c1c1 0 0 1 Satisfied Not Satisfied (LaBean and Lagoudakis, 1999) Unique-Shape Model

55 ** x1x1 x2x2 x3x3 c1c1 c2c2 c3c3 * * * * * ** x1x1 x2x2 x3x3 c1c1 c2c2 c3c3 * * * * * Satisfied Not Satisfied ** Multiple Temperature Model

56 ** x1x1 x2x2 x3x3 c1c1 c2c2 c3c3 * * 0 1 c1c1 ok c2c2 c2c2 1 T T T T * * * SAT * * * ** x1x1 x2x2 x3x3 c1c1 c2c2 c3c3 * * 0 1 c1c1 ok c2c2 c2c2 0 T T c2c2 F F * * * NO * * * Satisfied Not Satisfied ** Multiple Temperature Model

57 Satisfied Not Satisfied Multiple Temperature Model ** x1x1 x2x2 x3x3 c1c1 c2c2 c3c3 * * 0 1 c1c1 ok c2c2 c2c2 1 T T T T * * * SAT * * * * ** x1x1 x2x2 x3x3 c1c1 c2c2 c3c3 * * 0 1 c1c1 ok c2c2 c2c2 0 T T c2c2 F F * * * NO * * * *

58 Satisfied Not Satisfied Multiple Temperature Model * x1x1 x2x2 x3x3 * * * * * * x1x1 x2x2 x3x3 * * * * *

59 Our Tile Complexity Results Multiple temperature model: k x N rectangles: beats standard model: Flexible Glue: N x N squares: beats standard model: ( Adleman, Cheng, Goel, Huang STOC 2001 ) (our paper)

60 Unique Shape Problem Results StandardP Flexible GlueP Multiple TemperatureNP-hard Unique ShapeCo-NPC Multiple TileCo-NPC (Adleman, Cheng, Goel, Huang, Kempe, Espanes, Rothemund, STOC 2002) (our paper)

61 Further Research time complexity –multiple temperature –multiple tile more than 2 temperatures –raising/lowering temperature many times –monotonically increasing temperatures

62

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