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## Self-Assembly with Geometric Tiles

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**ICALP 2012**Self-Assembly with Geometric Tiles • Bin Fu University of Texas – Pan American Matt Patitz University of Arkansas Robert Schweller (Speaker) University of Texas – Pan American Robert Sheline University of Texas – Pan American**Outline**• Basic Tile Assembly Model • Geometric Tile Assembly Model • Basic Model • Planar Model • More efficient n x n squares • Future Directions**Tile Assembly Model**(Rothemund, Winfree, Adleman) a b c x d e G(y) = 2 G(g) = 2 G(r) = 2 G(b) = 2 G(p) = 1 G(w) = 1 t = 2 T = Glue Function: Tile Set: Temperature:**Tile Assembly Model**(Rothemund, Winfree, Adleman) a b c x d e G(y) = 2 G(g) = 2 G(r) = 2 G(b) = 2 G(p) = 1 G(w) = 1 t = 2 T = e d**Tile Assembly Model**(Rothemund, Winfree, Adleman) a b c x d e G(y) = 2 G(g) = 2 G(r) = 2 G(b) = 2 G(p) = 1 G(w) = 1 t = 2 T = e d**Tile Assembly Model**(Rothemund, Winfree, Adleman) a b c x d e G(y) = 2 G(g) = 2 G(r) = 2 G(b) = 2 G(p) = 1 G(w) = 1 t = 2 T = e d b c**Tile Assembly Model**(Rothemund, Winfree, Adleman) a b c x d e G(y) = 2 G(g) = 2 G(r) = 2 G(b) = 2 G(p) = 1 G(w) = 1 t = 2 T = e d b c**Tile Assembly Model**(Rothemund, Winfree, Adleman) a b c x d e G(y) = 2 G(g) = 2 G(r) = 2 G(b) = 2 G(p) = 1 G(w) = 1 t = 2 T = e d b c**Tile Assembly Model**(Rothemund, Winfree, Adleman) a b c x d e G(y) = 2 G(g) = 2 G(r) = 2 G(b) = 2 G(p) = 1 G(w) = 1 t = 2 T = e d a b c**Tile Assembly Model**(Rothemund, Winfree, Adleman) a b c x d e G(y) = 2 G(g) = 2 G(r) = 2 G(b) = 2 G(p) = 1 G(w) = 1 t = 2 T = e d a b c**Tile Assembly Model**(Rothemund, Winfree, Adleman) a b c x d e G(y) = 2 G(g) = 2 G(r) = 2 G(b) = 2 G(p) = 1 G(w) = 1 t = 2 T = e d a b c**Tile Assembly Model**(Rothemund, Winfree, Adleman) a b c x d e G(y) = 2 G(g) = 2 G(r) = 2 G(b) = 2 G(p) = 1 G(w) = 1 t = 2 T = e d a b c**Tile Assembly Model**(Rothemund, Winfree, Adleman) a b c x d e G(y) = 2 G(g) = 2 G(r) = 2 G(b) = 2 G(p) = 1 G(w) = 1 t = 2 T = e d a b c**Tile Assembly Model**(Rothemund, Winfree, Adleman) a b c x d e G(y) = 2 G(g) = 2 G(r) = 2 G(b) = 2 G(p) = 1 G(w) = 1 t = 2 T = e x d a b c**Tile Assembly Model**(Rothemund, Winfree, Adleman) a b c x d e e x d a b c G(y) = 2 G(g) = 2 G(r) = 2 G(b) = 2 G(p) = 1 G(w) = 1 t = 2 T =**Tile Assembly Model**(Rothemund, Winfree, Adleman) a b c x d e e x x d a b c G(y) = 2 G(g) = 2 G(r) = 2 G(b) = 2 G(p) = 1 G(w) = 1 t = 2 T =**Tile Assembly Model**(Rothemund, Winfree, Adleman) a b c x d e x e x x d a b c G(y) = 2 G(g) = 2 G(r) = 2 G(b) = 2 G(p) = 1 G(w) = 1 t = 2 T =**Tile Assembly Model**(Rothemund, Winfree, Adleman) a b c x d e x x e x x d a b c G(y) = 2 G(g) = 2 G(r) = 2 G(b) = 2 G(p) = 1 G(w) = 1 t = 2 T =**Geometric Tiles**Geometry Region**Geometric Tiles**Geometry Region**Geometric Tiles**Compatible Geometries**Geometric Tiles**Incompatible Geometries**Geometric Tiles**Incompatible Geometries**n x n Results**Tile Complexity Upper bound Lower bound Normal Tiles* Geometric Tiles Planar Geometric Tiles [*Winfree, Rothemund, Adleman, Cheng, Goel,Huang STOC 2000, 2001]**n x n Squares, root(log n) tiles**0 1 0 1 1 log n**Assembly of n x n Squares**1 1 1 1 1 1 1 1 1 0 n 0 1 1 0 0 0 1 0 1 1 log n**Assembly of n x n Squares**log n 0 1 0 1 1**Assembly of n x n Squares**-Build thicker 2 x log n seed row 0 1 1 1 1 0 1 1 1 0 0 1 0 1 1 0 2 log n**Assembly of n x n Squares**-Build thicker 2 x log n seed row -But… can’t encode general binary strings: 0 1 1 1 1 0 1 1 1 0 0 1 0 1 1 0 -All the same 0 0 0 0 1 1 1 1 2 2 2 2 3 3 3 3 2 1 2 3 0 1 2 3 0 1 2 3 0 1 2 3 0 log n**Assembly of n x n Squares**Key Idea: Geometry Decoding Tiles A3 A2 A1 A0 B3 B2 B1 B0 0 0 0 0 1 1 1 1 2 2 2 2 3 3 3 3 2 1 2 3 0 1 2 3 0 1 2 3 0 1 2 3 0 log n**Assembly of n x n Squares**A3 A3 A3 A2 A2 A2 A1 A1 A0 A0 B3 B3 B3 B2 B2 B2 B1 B1 B1 B1 B0 B0 B0 B0 0 1 1 1 1 0 1 1 1 0 0 1 0 1 1 0 0 0 0 0 1 1 1 1 2 2 2 2 3 3 3 3 2 1 2 3 0 1 2 3 0 1 2 3 0 1 2 3 0 log n**Assembly of n x n Squares**A3 A3 A3 A2 A2 A2 A1 A1 A0 A0 B3 B3 B3 B2 B2 B2 B1 B1 B1 B1 B0 B0 B0 B0 0 1 1 1 1 0 1 1 1 0 0 1 0 1 1 0 0 0 0 0 1 1 1 1 2 2 2 2 3 3 3 3 2 1 2 3 0 1 2 3 0 1 2 3 0 1 2 3 0 log n**Assembly of n x n Squares**A3 A2 B3 0 2 2 0 1**Assembly of n x n Squares**A3 A3 A3 A2 A2 A2 A1 A1 A0 A0 B3 B3 B2 B2 B1 B1 B0 B0 1 0 0 0 1 0 1 1 0 0 0 0 0 1 1 1 1 2 2 2 2 2 2 3 3 3 3 2 1 2 3 0 1 2 3 0 0 1 1 2 3 0 1 2 3 0 log n**Assembly of n x n Squares**A3 A3 A3 A2 A2 A2 A1 A1 A0 A0 B3 B3 B3 B2 B2 B2 B1 B1 B1 B1 B0 B0 B0 B0 0 1 1 1 1 0 1 1 1 0 0 1 0 1 1 0 0 0 0 0 1 1 1 1 2 2 2 2 3 3 3 3 2 1 2 3 0 1 2 3 0 1 2 3 0 1 2 3 0 log n**Assembly of n x n Squares**• build 2 x log n block: • Decode geometry into log n bit string A3 A3 A3 A2 A2 A2 A1 A1 A0 A0 B3 B3 B3 B2 B2 B2 B1 B1 B1 B1 B0 B0 B0 B0 0 1 1 1 1 0 1 1 1 0 0 1 0 1 1 0 0 0 0 0 1 1 1 1 2 2 2 2 3 3 3 3 2 1 2 3 0 1 2 3 0 1 2 3 0 1 2 3 0 log n**n x n Results**Tile Complexity Upper bound Lower bound Normal Tiles* Geometric Tiles Planar Geometric Tiles [*Winfree, Rothemund, Adleman, Cheng, Goel,Huang STOC 2000, 2001]**Planar Geometric Tile Assembly**Attachment requires a collision free path within the plane**Planar Geometric Tile Assembly**Attachment requires a collision free path within the plane Attachment not permitted in the planar model**Planar Geometric Tile Assembly**Attachment not permitted in the planar model**n x n Results**Tile Complexity Upper bound Lower bound Normal Tiles* Geometric Tiles Planar Geometric Tiles ? [*Winfree, Rothemund, Adleman, Cheng, Goel,Huang STOC 2000, 2001]**n x n Results**Tile Complexity Upper bound Lower bound Normal Tiles* Geometric Tiles Planar Geometric Tiles O( loglog n ) ? [*Winfree, Rothemund, Adleman, Cheng, Goel,Huang STOC 2000, 2001]**Planar Geometric Tile Assembly**log n 1 0 1 0 0 1 1 0**Planar Geometric Tile Assembly**• Build log n columns with loglog n tile types 0 0 0 0 1 1 1 1 0 0 1 1 0 0 1 1 loglog n 0 1 0 1 0 1 0 1**Planar Geometric Tile Assembly**• Build log n columns with loglog n tile types 0 0 0 0 1 1 1 1 0 0 1 1 0 0 1 1 loglog n 0 1 0 1 0 1 0 1**Planar Geometric Tile Assembly**• Build log n columns with loglog n tile types • Columns must assemble in proper order 0 0 0 0 1 1 1 1 0 0 1 1 0 0 1 1 loglog n 0 1 0 1 0 1 0 1