810 likes | 816 Views
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
E N D
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
Tile Model of Self-Assembly (Rothemund, Winfree STOC 2000) Tile System: t : temperature, positive integer G: glue function T: tileset s: seed tile
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 =
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 =
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 =
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 =
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 =
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 =
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 =
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 =
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 =
Multiple Temperature Model (Aggarwal, Cheng, Goldwasser, Kao, Espanes, Schweller, SICOMP 2005) Multiple Temperature Model - temperature may go up and down
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 >
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
Building k x N Rectangles k-digit, base N(1/k) counter: k N
Building k x N Rectangles k-digit, base N(1/k) counter: k N Tile Complexity:
two temperatures t= 4 3 1 3 3
two temperatures t = 4 6
High Level Approach Given: n 1011001 log n
High Level Approach Given: n 1011001 log n 1 temp
High Level Approach Given: n 1011001 log n 1 temp 1
High Level Approach Given: n 1011001 log n 1 0 temp 1 0
High Level Approach Given: n 1011001 log n 1 0 1 1 0 . . . . . . temp 1 0 1 1 0 0 1
High Level Approach . . . 0 0 1 . . . temp 1 0 1 1 0 0 1
High Level Approach . . . 0 0 1 . . . temp 1 0 1 1 0 0 1
High Level Approach . . . 0 0 1 . . . temp 1 0 1 1 0 0 1
Assembly of N x N Squares N - k k
Assembly of N x N Squares N - k k
Assembly of N x N Squares N - k k
Assembly of N x N Squares N - k Complexity: k
Assembly of N x N Squares N – log N Complexity: log N
Assembly of N x N Squares N – log N Complexity: seed row log N
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
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
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
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
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
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
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
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
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
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
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
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
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
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