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Based on paper to appear in SODA 2006 by Ming-Yang Kao Northwestern University

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. Tile Model of Self-Assembly

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Based on paper to appear in SODA 2006 by Ming-Yang Kao Northwestern University

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  1. Reducing Tile Complexity for Self-Assembly Through Temperature ProgrammingMidwest 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 G(y) = 2 G(g) = 2 G(r) = 2 G(b) = 2 G(p) = 1 G(w) = 1 t = 2 T =

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

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

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

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

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

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

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

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

  12. Each Shape Requires a Distinct Tile Set

  13. Programmable, General Purpose Tile Set?

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

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

  16. Multiple Temperature Model (Aggarwal, Cheng, Goldwasser, Kao, Espanes, Schweller, SICOMP 2005) Multiple Temperature Model - temperature may go up and down t < t1 , t2 , ... , tr-1 , tr >

  17. Multiple Temperature Model (Aggarwal, Cheng, Goldwasser, Kao, Espanes, Schweller, SICOMP 2005) Multiple Temperature Model - temperature may go up and down t < t1 , t2 , ... , tr-1 , tr > 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: k N Tile Complexity:

  20. two temperatures t= 4 3 1 3 3

  21. two temperatures t = 4 6

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

  23. High Level Approach Given: n 1011001 log n

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

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

  26. High Level Approach Given: n 1011001 log n 1 0 temp 1 0

  27. High Level Approach Given: n 1011001 log n 1 0 1 1 0 . . . . . . temp 1 0 1 1 0 0 1

  28. High Level Approach . . . 0 0 1 . . . temp 1 0 1 1 0 0 1

  29. High Level Approach . . . 0 0 1 . . . temp 1 0 1 1 0 0 1

  30. High Level Approach . . . 0 0 1 . . . temp 1 0 1 1 0 0 1

  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 Complexity: k

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

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

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

  38. Encoding a Single Bit t = < 2 > 0 0 1 0 1 0’ 1’ z z 1 z Z g g g g g g g g a g g a

  39. Encoding a Single Bit t = < 2 > 0 0 1 0 1 0 0’ 1’ z z 1 z Z g g g g g g g g a g g a

  40. Encoding a Single Bit t = < 2 > 0 0 1 0 1 0’ 1’ 0 z z 1 z Z g g g g g g g g a g g a

  41. Encoding a Single Bit t = < 2 > 0 0 1 0 1 0’ 1’ 0 z z 1 0’ z z Z g g g g g g g g a g g a

  42. Encoding a Single Bit t = < 2 > 0 0 1 0 1 0’ 1’ 0 z z 1 0’ z Z g g Z g g g g g g a g g a

  43. Encoding a Single Bit t = < 2 > 0 0 1 0 1 0’ 1’ 0 z z 1 0’ z Z g g Z g g g g g g a g g a

  44. Encoding a Single Bit t = < 2 > 0 0 1 0 1 0’ 1’ 0 z z 1 0’ z Z g g Z g g g g g g a g g a

  45. Encoding a Single Bit t = < 2 > 0 0 1 0 1 0’ 1’ 0 z z 1 0’ z Z g g Z g g g g g g a g g a

  46. Encoding a Single Bit t = < 2 > 0 0 1 0 1 0’ 1’ 0 z z 1 0’ z Z g g Z g g g g g g a g g a

  47. Encoding a Single Bit t = < 2, 5 > 0 0 1 0 1 0’ 1’ 0 z z 1 0’ z Z g g Z g g g g g g a g g a

  48. Encoding a Single Bit t = < 2, 5 > 0 0 1 0 0 1 0’ 1’ z z 1 x 0’ z Z g g Z g g g g g g a g g a

  49. Encoding a Single Bit t = < 2, 5 > 0 0 0 1 0 1 0’ 1’ 0’ z z 1 1 z Z g g Z g g g g g g a g g a

  50. Encoding a Single Bit t = < 2, 5 > 0 0 1 0 1 0’ 1’ 1’ z z 1 1 z z Z g g Z g g g g g g a g g a

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