1. Reducing Tile Complexity for Self-Assembly Through Temperature Programming Midwest Theory Day, December 10, 2006 Based on paper to appear in SODA 2006 by Ming-Yang Kao Northwestern University Robert Schweller Northwestern University

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) = 2 G(g) = 2 G(r) = 2 G(b) = 2 G(p) = 1 G(w) = 1 t = 2

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

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

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

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

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

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

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

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

12. Each Shape Requires a Distinct Tile Set

13. Programmable, General Purpose Tile Set?

14. ...

15. Multiple Temperature Model - temperature may go up and down (Aggarwal, Cheng, Goldwasser, Kao, Espanes, Schweller, SICOMP 2005)

16. Multiple Temperature Model - temperature may go up and down t (Aggarwal, Cheng, Goldwasser, Kao, Espanes, Schweller, SICOMP 2005)

17. Multiple Temperature Model - temperature may go up and down t (Aggarwal, Cheng, Goldwasser, Kao, Espanes, Schweller, SICOMP 2005) Tile Complexity:Number of Tiles Temperature Complexity:Number of Temperatures

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. two temperatures 3 3 3 1 t = 4

21. t = 4 6 two temperatures

22. Programmable, General Purpose Tile Set?...

23. Given: n 1011001 log n High Level Approach

24. Given: n 1011001 log n temp High Level Approach 1

25. Given: n 1011001 log n temp High Level Approach 1 1

26. Given: n 1011001 log n temp High Level Approach 10 10

27. Given: n 1011001 log n temp High Level Approach 1011... 0 1011010

28. temp High Level Approach 01... 1011010 0

29. temp High Level Approach 01... 1011010 0

30. temp High Level Approach 01... 1011010 0

31. Assembly of N x N Squares N - k k

32. Assembly of N x N Squares N - k k

33. Assembly of N x N Squares N - k k

34. Assembly of N x N Squares N - k k Complexity:

35. Assembly of N x N Squares N – log N log N Complexity:

36. Assembly of N x N Squares N – log N log N Complexity: seed row

37. Encoding a Single Bit 0 a Z g z g g g g g g g g g 01 0’1’ zz 1 0 1

38. a Encoding a Single Bit 0 a Z g z g g g g g g g g g 01 0’1’ zz 1 0 1 t =

39. a Encoding a Single Bit 0 a Z g z g g g g g g g g g 01 0’1’ zz 1 0 1 t = 0

40. a Encoding a Single Bit 0 a Z g z g g g g g g g g g 01 0’1’ zz 1 0 1 t = 0

41. a Encoding a Single Bit 0 a Z g z g g g g g g g g g 01 0’1’ zz 1 0 1 t = 0 0’ z

42. a Encoding a Single Bit 0 a Z g z g g g g g g g g g 01 0’1’ zz 1 0 1 t = 0 0’ Z

43. a Encoding a Single Bit 0 a Z g z g g g g g g g g g 01 0’1’ zz 1 0 1 t = 0 0’ Z

44. a Encoding a Single Bit 0 a Z g z g g g g g g g g g 01 0’1’ zz 1 0 1 t = 0 0’ Z

45. a Encoding a Single Bit 0 a Z g z g g g g g g g g g 01 0’1’ zz 1 0 1 t = 0 0’ Z

46. a Encoding a Single Bit 0 a Z g z g g g g g g g g g 01 0’1’ zz 1 0 1 t = 0 0’ Z

47. a Encoding a Single Bit 0 a Z g z g g g g g g g g g 01 0’1’ zz 1 0 1 t = 0 0’ Z

48. a Encoding a Single Bit 0 a Z g z g g g g g g g g g 01 0’1’ zz 1 0 1 t = 0 0’ Z x

49. a Encoding a Single Bit 0 a Z g z g g g g g g g g g 01 0’1’ zz 1 0 1 t = 0 0’ Z 1

50. a Encoding a Single Bit 0 a Z g z g g g g g g g g g 01 0’1’ zz 1 0 1 t = Z 1 1’ z

51. a Encoding a Single Bit 0 a Z g z g g g g g g g g g 01 0’1’ zz 1 0 1 t = Z 1’

52. a Encoding a Single Bit 0 a Z g z g g g g g g g g g 01 0’1’ zz 1 0 1 t = Z 1’ 1

53. a Encoding a Single Bit 0 a Z g z g g g g g g g g g 01 0’1’ zz 1 0 1 t = Z 1’ 1 a Z t = 0 0’

54. Goal: 1 0 1 0 0

55. a 0 s b temp:

56. a 1 s b Goal: 1 0 1 0 0 temp:

57. a 1 s Goal: 1 0 1 0 0 b X temp:

58. a 1 s Goal: 1 0 1 0 0 b Y temp:

59. a 1 s Goal: 1 0 1 0 0 b Y temp: a b

60. a 1 s Goal: 1 0 1 0 0 b Y temp: a b 0

61. a 1 s Goal: 1 0 1 0 0 b Y temp: a b 0 X

62. a 1 s Goal: 1 0 1 0 0 b Y temp: a b 0 Y a b

63. a 1 s Goal: 1 0 1 0 0 b Y temp: a b 0 Y a b 0

64. a 1 s Goal: 1 0 1 0 0 b Y temp: a b 0 Y a b 1

65. a 1 s Goal: 1 0 1 0 0 b Y temp: a b 0 Y a b 1 X

66. a 1 s Goal: 1 0 1 0 0 b Y temp: a b 0 Y a b 1 Y a b

67. a 1 s Goal: 1 0 1 0 0 b Y temp: a b 0 Y a b 1 Y a b 0

68. a 1 s Goal: 1 0 1 0 0 b Y temp: a b 0 Y a b 1 Y a b 0 X

69. a 1 s Goal: 1 0 1 0 0 b Y temp: a b 0 Y a b 1 Y a b 0 Y a b

70. a 1 s Goal: 1 0 1 0 0 b Y temp: a b 0 Y a b 1 Y a b 0 Y a b 0

71. a 1 s Goal: 1 0 1 0 0 b Y temp: a b 0 Y a b 1 Y a b 0 Y a b 0 X

72. 1 1 0 0 1 0 0 0 1 1 1 0 1 1

73. 1 1 0 0 1 0 0 0 1 1 1 1 0 0 1 1 0 0 1 0 0 0 1 1 1 1 0 1 1 1 0 0 1 0 0 0 1 1 1 1 1 0 1 1 0 0 1 0 0 0 1 1 1 1 1 1 1 1 0 0 1 0 0 1 0 0 0 0 0 0

74. Assembly of N x N Squares N – log N log N

75. Assembly of N x N Squares O(log N)

76. Assembly of N x N Squares O(log n)

77. Single Tile set, can build ALL n x n squares –O(1) tile complexity –O(log n) temperature complexity For any n x n square –O(log n / loglog n) –O(1) temperature complexity Is there a smooth tradeoff? –NO: For almost all n, no tileset can achieve both o(log n/ loglog n) tile complexity and o(log n) temperature complexity simultaneously. Results ( Adleman, Cheng, Goel, Huang STOC 2001 )

78. Single Tile set, can build ALL n x n squares –O(1) tile complexity –O(log n) temperature complexity For any n x n square –O(log n / loglog n) –O(1) temperature complexity Is there a smooth tradeoff? –NO: For almost all n, no tileset can achieve both o(log n/ loglog n) tile complexity and o(log n) temperature complexity simultaneously. Results ( Adleman, Cheng, Goel, Huang STOC 2001 )

79. Single Tile set, can build ALL n x n squares –O(1) tile complexity –O(log n) temperature complexity For any n x n square –O(log n / loglog n) –O(1) temperature complexity Is there a smooth tradeoff? –NO: For almost all n, no tileset can achieve both o(log n/ loglog n) tile complexity and o(log n) temperature complexity simultaneously. Results ( Adleman, Cheng, Goel, Huang STOC 2001 )

80. Single Tile set, can build ALL n x n squares –O(1) tile complexity –O(log n) temperature complexity For any n x n square –O(log n / loglog n) –O(1) temperature complexity Is there a smooth tradeoff? –NO: For almost all n, no tileset can achieve both o(log n/ loglog n) tile complexity and o(log n) temperature complexity simultaneously. Results ( Adleman, Cheng, Goel, Huang STOC 2001 )

81. Thanks for Listening Questions? http://www.cs.northwestern.edu/~schwellerr/ 4 th year Graduate Student Electrical Engineering and Computer Science Department Northwestern University Advisor: Ming-Yang Kao Email: schwellerr@cs.northwestern.edu