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Self-Assembly with Geometric Tiles
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Transcript of Self-Assembly with Geometric Tiles
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Self-Assembly with Geometric TilesICALP 2012
Bin Fu University of Texas – Pan AmericanMatt Patitz University of ArkansasRobert Schweller (Speaker) University of Texas – Pan AmericanRobert Sheline University of Texas – Pan American
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Outline
• Basic Tile Assembly Model• Geometric Tile Assembly Model
– Basic Model– Planar Model– More efficient n x n squares
• Future Directions
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Tile Assembly Model(Rothemund, Winfree, Adleman)
T = G(y) = 2G(g) = 2G(r) = 2G(b) = 2G(p) = 1G(w) = 1
t = 2
Tile Set:
Glue Function:
Temperature:
x ed
cba
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T = G(y) = 2G(g) = 2G(r) = 2G(b) = 2G(p) = 1G(w) = 1
t = 2
d
e
x ed
cba
Tile Assembly Model(Rothemund, Winfree, Adleman)
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T = G(y) = 2G(g) = 2G(r) = 2G(b) = 2G(p) = 1G(w) = 1
t = 2d
e
x ed
cba
Tile Assembly Model(Rothemund, Winfree, Adleman)
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T = G(y) = 2G(g) = 2G(r) = 2G(b) = 2G(p) = 1G(w) = 1
t = 2d
e
x ed
cba
b c
Tile Assembly Model(Rothemund, Winfree, Adleman)
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T = G(y) = 2G(g) = 2G(r) = 2G(b) = 2G(p) = 1G(w) = 1
t = 2d
e
x ed
cba
b c
Tile Assembly Model(Rothemund, Winfree, Adleman)
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T = G(y) = 2G(g) = 2G(r) = 2G(b) = 2G(p) = 1G(w) = 1
t = 2d
e
x ed
cba
b c
Tile Assembly Model(Rothemund, Winfree, Adleman)
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T = G(y) = 2G(g) = 2G(r) = 2G(b) = 2G(p) = 1G(w) = 1
t = 2d
e
x ed
cba
b ca
Tile Assembly Model(Rothemund, Winfree, Adleman)
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T = G(y) = 2G(g) = 2G(r) = 2G(b) = 2G(p) = 1G(w) = 1
t = 2d
e
x ed
cba
b ca
Tile Assembly Model(Rothemund, Winfree, Adleman)
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T = G(y) = 2G(g) = 2G(r) = 2G(b) = 2G(p) = 1G(w) = 1
t = 2d
e
x ed
cba
b ca
Tile Assembly Model(Rothemund, Winfree, Adleman)
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T = G(y) = 2G(g) = 2G(r) = 2G(b) = 2G(p) = 1G(w) = 1
t = 2d
e
x ed
cba
b ca
Tile Assembly Model(Rothemund, Winfree, Adleman)
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T = G(y) = 2G(g) = 2G(r) = 2G(b) = 2G(p) = 1G(w) = 1
t = 2
x ed
cba
a b c
d
e
Tile Assembly Model(Rothemund, Winfree, Adleman)
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T = G(y) = 2G(g) = 2G(r) = 2G(b) = 2G(p) = 1G(w) = 1
t = 2
x ed
cba
x
a b c
d
e
Tile Assembly Model(Rothemund, Winfree, Adleman)
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T = G(y) = 2G(g) = 2G(r) = 2G(b) = 2G(p) = 1G(w) = 1
t = 2
a b c
d
e
x
x ed
cba
Tile Assembly Model(Rothemund, Winfree, Adleman)
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T = G(y) = 2G(g) = 2G(r) = 2G(b) = 2G(p) = 1G(w) = 1
t = 2
x ed
cba
a b c
d
e
x x
Tile Assembly Model(Rothemund, Winfree, Adleman)
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T = G(y) = 2G(g) = 2G(r) = 2G(b) = 2G(p) = 1G(w) = 1
t = 2
x ed
cba
a b c
d
e
x x
x
Tile Assembly Model(Rothemund, Winfree, Adleman)
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T = G(y) = 2G(g) = 2G(r) = 2G(b) = 2G(p) = 1G(w) = 1
t = 2
x ed
cba
a b c
d
e
x x
x x
Tile Assembly Model(Rothemund, Winfree, Adleman)
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Geometric Tile Model
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Geometric Tiles
Geometry Region
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Geometric Tiles
Geometry Region
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Geometric Tiles
Compatible Geometries
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Geometric Tiles
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Geometric Tiles
Incompatible Geometries
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Geometric Tiles
Incompatible Geometries
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n x n Results
Tile Complexity
)logloglog(
nnO
Geometric Tiles
Normal Tiles*
)log( nO
)logloglog(
nn
)log( n
Upper bound Lower bound
Planar Geometric Tiles
[*Winfree, Rothemund, Adleman, Cheng, Goel,Huang STOC 2000, 2001]
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n x n Squares, root(log n) tiles
log n0 1 0 1 1
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Assembly of n x n Squares
n
log n
0 1 1 0 0
1 1 1 1 11 1 1 1 0
0 1 0 1 1
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Assembly of n x n Squares
log n0 1 0 1 1
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2
log n
0 1 0 0 0 0 01 1 1 1 1 1 1 1 1
Assembly of n x n Squares-Build thicker 2 x log n seed row
)log()log(1
nOnkO k
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002
log n
0 1 0 0 0 0 01 1 1 1 1 1 1 1 1
Assembly of n x n Squares-Build thicker 2 x log n seed row
)log()log(1
nOnkO k
-But… can’t encode general binary strings:
0
-All the same
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log n
Assembly of n x n Squares
0
B3 B2 B1 B0
A3 A2 A1 A0
Key Idea:Geometry Decoding Tiles
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log n
Assembly of n x n Squares
0
0 1 0 0 0 0 01 1 1 1 1 1 1 1 1
B0
A0A1
B1
A2
B2
A3
B3
A0B1B2A3B0A1A2B3B0B1A2B3B0B1B2A3
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log n
Assembly of n x n Squares
0
0 1 0 0 0 0 01 1 1 1 1 1 1 1 1
B0
A0A1
B1
A2
B2
A3
B3
A0B1B2A3B0A1A2B3B0B1A2B3B0B1B2A3
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Assembly of n x n Squares
1
2
0
2
0
A2
B3
A3
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log n
Assembly of n x n Squares
0
0 0 0 01 1 1 1
B0
A0A1
B1
A2
B2
A3
B3
A0B1B2A3B0A1A2
1
2
0
2
0
A2
B3
A3
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log n
Assembly of n x n Squares
0
0 1 0 0 0 0 01 1 1 1 1 1 1 1 1
B0
A0A1
B1
A2
B2
A3
B3
A0B1B2A3B0A1A2B3B0B1A2B3B0B1B2A3
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log n
Assembly of n x n Squares
0
0 1 0 0 0 0 01 1 1 1 1 1 1 1 1
B0
A0A1
B1
A2
B2
A3
B3
A0B1B2A3B0A1A2B3B0B1A2B3B0B1B2A3
• build 2 x log n block:• Decode geometry into log n bit string
)log( n
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)logloglog(
nnO
)log( nO
)logloglog(
nn
)log( n
Upper bound Lower bound
n x n Results
Tile Complexity
Geometric Tiles
Normal Tiles*
[*Winfree, Rothemund, Adleman, Cheng, Goel,Huang STOC 2000, 2001]
Planar Geometric Tiles
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Planar Geometric Tile Assembly
Attachment requires a collision free path within the plane
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Planar Geometric Tile Assembly
Attachment requires a collision free path within the plane
Attachment not permitted in the planar model
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Planar Geometric Tile Assembly
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Planar Geometric Tile Assembly
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Planar Geometric Tile Assembly
Attachment not permitted in the planar model
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n x n Results
Tile Complexity
)logloglog(
nnO
Geometric Tiles
Normal Tiles*
)log( nO
)logloglog(
nn
)log( n
Upper bound Lower bound
Planar Geometric Tiles ?
[*Winfree, Rothemund, Adleman, Cheng, Goel,Huang STOC 2000, 2001]
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n x n Results
Tile Complexity
)logloglog(
nnO
Geometric Tiles
Normal Tiles*
)log( nO
)logloglog(
nn
)log( n
Upper bound Lower bound
Planar Geometric Tiles O( loglog n )
[*Winfree, Rothemund, Adleman, Cheng, Goel,Huang STOC 2000, 2001]
?
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1 0 1 0 0 1 1 0
log n
Planar Geometric Tile Assembly
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1
1
1
1
1
0101010
0
1
0
0
0
0
1
0
1
0 1
0 loglog n
• Build log n columns with loglog n tile typesPlanar Geometric Tile Assembly
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0101010
0
1
0
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0
0
1
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1
0 1
0 loglog n
• Build log n columns with loglog n tile typesPlanar Geometric Tile Assembly
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1
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0101010
0
1
0
0
0
0
1
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0 1
0 loglog n
• Build log n columns with loglog n tile types• Columns must assemble in proper order
Planar Geometric Tile Assembly
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1 0 1 0 0 1 1 0
log n
1
1
1
1
1
0101010
0
1
0
0
0
0
1
0
1
0 1
0 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
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• 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.
10
0
0 1
1
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• 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.
10
0
0 1
1
0
1
0
0
1
1
0
1
0 1
1
1
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• 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.
10
0
0 1
1
0
1
0
0
1
1
0
1
0 1
1
1
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1
0
0
0
1
0
Planar Geometric Tile Assembly
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1
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Planar Geometric Tile Assembly
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1
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1 0 1 0 0 1 1 0
log n
1
1
1
1
1
0101010
0
1
0
0
0
0
1
0
1
0 1
0 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.
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1 0 1 0 0 1 1 0
log n
1
1
1
1
1
0101010
0
1
0
0
0
0
1
0
1
0 1
0 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.
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1
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0
Planar Geometric Tile Assembly
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1
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Planar Geometric Tile Assembly
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Planar Geometric Tile Assembly
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1 0 1 0 0 1 1 0
log n
1
1
1
1
1
0101010
0
1
0
0
0
0
1
0
1
0 1
0 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
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n – log n
n – log n
log n
X
Y
)log(log n
Complexity:
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n x n Results
Tile Complexity
)logloglog(
nnO
Geometric Tiles
Normal Tiles*
)log( nO
)logloglog(
nn
)log( n
Upper bound Lower bound
Planar Geometric Tiles O( loglog n ) ?
[*Winfree, Rothemund, Adleman, Cheng, Goel,Huang STOC 2000, 2001]
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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
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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
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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.
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PeopleBin Fu
Matt Patitz
Robbie Schweller
Bobby Sheline
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79Barish, Shulman, Rothemund, Winfree, 2009
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DNA Origami Tiles
[Masayuki Endo, Tsutomu Sugita, Yousuke Katsuda, Kumi Hidaka, and Hiroshi Sugiyama, 2010]
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More DNA Origami Shapes
[Paul Rothemund, Nature 2006]
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Alphabet of Shapes, Built with DNA Tiles
[Bryan Wei, Mingjie Dai, Peng Yin, Nature 2012]
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83
n x n square’s with Geometric Tiles
Tile Complexity:
n - k
kk
n - k
)( /1 knk
x
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Assembly of n x n Squares
n - k
k
)( /1 knkO
Complexity:
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Assembly of n x n Squares
n – log n
log n)(log)(
2
log
/1
/1
nOnkO
n
nk
k
k
Complexity:
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Assembly of n x n Squares
n – log n
log n)(log)(
2
log
/1
/1
nOnkO
n
nk
k
k
Complexity:
seed row
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log n
0 1 0 0 0 0 01 1 1 1 1 1 1 1 1
Assembly of n x n Squares-Build thicker 2 x log n seed row
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n – log n
log n
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n – log n
n – log n
log n
X
Y
)log( N
Complexity: