35 th International Colloquium on Automata, Languages and Programming July 8, 2008
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Transcript of 35 th International Colloquium on Automata, Languages and Programming July 8, 2008
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35th International Colloquium on Automata, Languages and ProgrammingJuly 8, 2008
Randomized Self-Assembly for Approximate Shapes
Robert Schweller University of Texas – Pan American
In collaboration with
Ming-Yang Kao Northwestern University
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Outline
• Assembly Model
• Basic Constructions
• Probabilistic Assembly Model
• Main Result
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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:
S
Seed Tile:
x dc
baS
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x dc
baST = G(y) = 2G(g) = 2G(r) = 2G(b) = 2G(p) = 1G(w) = 1
t = 2
S
Tile Assembly Model(Rothemund, Winfree, Adleman)
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S a
x dc
baST = G(y) = 2G(g) = 2G(r) = 2G(b) = 2G(p) = 1G(w) = 1
t = 2
Tile Assembly Model(Rothemund, Winfree, Adleman)
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S a
c
x dc
baST = G(y) = 2G(g) = 2G(r) = 2G(b) = 2G(p) = 1G(w) = 1
t = 2
Tile Assembly Model(Rothemund, Winfree, Adleman)
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S a
c
d
x dc
baST = G(y) = 2G(g) = 2G(r) = 2G(b) = 2G(p) = 1G(w) = 1
t = 2
Tile Assembly Model(Rothemund, Winfree, Adleman)
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S a b
c
d
x dc
baST = G(y) = 2G(g) = 2G(r) = 2G(b) = 2G(p) = 1G(w) = 1
t = 2
Tile Assembly Model(Rothemund, Winfree, Adleman)
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S a b
c
d
x
x dc
baST = G(y) = 2G(g) = 2G(r) = 2G(b) = 2G(p) = 1G(w) = 1
t = 2
Tile Assembly Model(Rothemund, Winfree, Adleman)
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S a b
c
d
x x
x dc
baST = G(y) = 2G(g) = 2G(r) = 2G(b) = 2G(p) = 1G(w) = 1
t = 2
Tile Assembly Model(Rothemund, Winfree, Adleman)
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S a b
c
d
x x
x
x dc
baST = G(y) = 2G(g) = 2G(r) = 2G(b) = 2G(p) = 1G(w) = 1
t = 2
Tile Assembly Model(Rothemund, Winfree, Adleman)
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S a b
c
d
x x
x x
x dc
baST = G(y) = 2G(g) = 2G(r) = 2G(b) = 2G(p) = 1G(w) = 1
t = 2
Tile Assembly Model(Rothemund, Winfree, Adleman)
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How efficiently can you build an n x n square?
s
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How efficiently can you build an n x n square?
s
n
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How efficiently can you build an n x n square?
s
x
Tile Complexity:2n
n
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How efficiently can you build an n x n square?
s
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How efficiently can you build an n x n square?
s0 0 00
log n
-Use log n tile types to seedcounter:
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How efficiently can you build an n x n square?
s0 0 00
log n
-Use 8 additional tile types capable of binary counting:
-Use log n tile types capable ofBinary counting:
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How efficiently can you build an n x n square?
s0 0 00
log n
000
00
0 00
1 0 10
1 1 00
1 1 10
0 0 0
0 1
1 0
1
1
11
1
000 1
-Use 8 additional tile types capable of binary counting:
-Use log n tile types capable ofBinary counting:
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How efficiently can you build an n x n square?
s0 0 00
log n
000
00
0 00
1 0 10
1 1 00
1 1 10
0 0 01
0 0 11
0 1 01
0 1 11
1 0 01
1 0 11
1 1 11
1 1 01
1
11
1
000 1
-Use 8 additional tile types capable of binary counting:
-Use log n tile types capable ofBinary counting:
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How efficiently can you build an n x n square?
s0 0 00
000
000
00
0 00
1 0 10
1 1 00
1 1 10
0 0 01
0 0 11
0 1 01
0 1 11
1 0 01
1 0 11
1 1 11
1 1 01
1
1
11
1
0
0
0
0
1
0
0
0
0
1
0
0
1
1
0
0
0
0
1
0
1
0
1
0
0
1
1
0
1
1
1
0
0
0
0
1
1
0
0
1
0
1
0
1
1
1
0
1
0
0
1
1
1
0
1
1
0
1
1
1
1
1
1
1
-Use 8 additional tile types capable of binary counting:
-Use log n tile types capable ofBinary counting:
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How efficiently can you build an n x n square?
s0 0 00
000
000
00
0 00
1 0 10
1 1 00
1 1 10
0 0 01
0 0 11
0 1 01
0 1 11
1 0 01
1 0 11
1 1 11
1 1 01
1
1
11
1
0
0
0
0
1
0
0
0
0
1
0
0
1
1
0
0
0
0
1
0
1
0
1
0
0
1
1
0
1
1
1
0
0
0
0
1
1
0
0
1
0
1
0
1
1
1
0
1
0
0
1
1
1
0
1
1
0
1
1
1
1
1
1
1
n – log n
log n
x
y
Tile Complexity:O(log n)
With optimalcounter:Tile Complexity:O(log n / loglog n)
Meets lower bound:(log n / loglog n)
(Rothemund, Winfree 2000)
(Adleman, Cheng, Goel, Huang 2001)
(Rothemund, Winfree 2000)
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Outline
• Assembly Model
• Basic Constructions
• Probabilistic Assembly Model
• Main Result
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aS
Assign Relative Concentrations:
Probabilistic Assembly Model
b c
xd %5
%5
%5
%60
%20
%5
(Becker, Remila, Rapaport)
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Probabilistic Assembly Model
S
aS
Tileset =
b c
xd
G(y) = 2G(g) = 2G(r) = 2G(p) = 1G(w) = 1
t = 2
%5 %5
%5
%5
%60
%20
(Becker, Remila, Rapaport)
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Probabilistic Assembly Model
S
d
aS
Tileset =
b c
xd
%5 %5
%5
%5
%60
%20
G(y) = 2G(g) = 2G(r) = 2G(p) = 1G(w) = 1
t = 2
(Becker, Remila, Rapaport)
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Probabilistic Assembly Model
aS
d
aS
Tileset =
b c
xd
%5 %5
%5
%5
%60
%20
G(y) = 2G(g) = 2G(r) = 2G(p) = 1G(w) = 1
t = 2
(Becker, Remila, Rapaport)
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Probabilistic Assembly Model
aS
xd
aS
Tileset =
b c
xd
%5 %5
%5
%5
%60
%20
G(y) = 2G(g) = 2G(r) = 2G(p) = 1G(w) = 1
t = 2
(Becker, Remila, Rapaport)
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Probabilistic Assembly Model
aS
xd
S
d
aS
Tileset =
b c
xd
%5 %5
%5
%5
%60
%20
G(y) = 2G(g) = 2G(r) = 2G(p) = 1G(w) = 1
t = 2
(Becker, Remila, Rapaport)
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Probabilistic Assembly Model
aS b
xd
S
d
aS
Tileset =
b c
xd
%5 %5
%5
%5
%60
%20
G(y) = 2G(g) = 2G(r) = 2G(p) = 1G(w) = 1
t = 2
(Becker, Remila, Rapaport)
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Probabilistic Assembly Model
aS b c
xd
S
d
Two Terminal Shapes Produced
aS
Tileset =
b c
xd
%5 %5
%5
%5
%60
%20
G(y) = 2G(g) = 2G(r) = 2G(p) = 1G(w) = 1
t = 2
(Becker, Remila, Rapaport)
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aS
Tileset =
Probabilistic Assembly Model
b c
xd
%5 %5
%5
%5
%60
%20
S
S
d
S S ba
.20/.85 = %23.5.60/.85 = %70.6
.05/.85 = %5.9
G(y) = 2G(g) = 2G(r) = 2G(p) = 1G(w) = 1
t = 2
(Becker, Remila, Rapaport)
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aS
Tileset =
Probabilistic Assembly Model
b c
xd
%5 %5
%5
%5
%60
%20
S
S
d
S
S
d
a S
d
b
S ba
S
d
a S
d
b c
x
%23.5%70.6 %5.9
%75 %25
G(y) = 2G(g) = 2G(r) = 2G(p) = 1G(w) = 1
t = 2
(Becker, Remila, Rapaport)
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Generic Tileset for all Squares
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Generic Tileset for Approximate Squares
n(1-n (1+n
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Generic Tileset for Approximate Squares
n(1- n (1+n
() – Approximate Square Assembly
Given:
- - integer n
Design:A probabilistic tile system that will assemble an n’ x n’ square with:
(1-)n < n’ < (1+)n
With probability at least:
1 -
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High Level Idea
-Build random structure:-Dimensions are random-Internal pattern is random
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High Level Idea
-Build random structure:-Dimensions are random-Internal pattern is random
-Incorporate arithmetic tiles to extracta binary number from the randompattern
10110110
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Binary Counter
Bin
ary Co
un
ter
10110110
n
n
Finish off SquareOutput n approximation
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X
S
Line Estimation of n
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X
S
S
Line Estimation of n
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X
S
S
Line Estimation of n
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X
S
S
Line Estimation of n
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X
S
S
Line Estimation of n
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X
S
S
Line Estimation of n
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X
S
S
Line Estimation of n
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X
S
S
Line Estimation of n
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X
S
S
Line Estimation of n
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X
S
S
Line Estimation of n
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X
S
S
Line Estimation of n
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X
S
XS
Line Estimation of n
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X
S
XS
% c
% c/n
% c(n-1)/n
Line Estimation of n
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X
S
XS
% c
% c/n
% c(n-1)/n
Line Estimation of n
E[ Length ] = n
Length has Geometric distributionwith p = 1/n
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X
S
XS
% c
% c/n
% c(n-1)/n
Line Estimation of n[Becker, Rapaport Remila, 2006]
E[ Length ] = n
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X
S
XS
% c
% c/n
% c(n-1)/n
Line Estimation of n
- Assembles all n x n squares.
-Expected dimension specified by percentages.
-Geometric distribution:
Large variance
[Becker, Rapaport Remila, 2006]
E[ Length ] = n
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X
S
Improved Estimation of n: Binomial Distribution
S X
Key Idea
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X
S % c
% c/L
% c(L+1)(n-1)/Ln
Improved Estimation of n: Binomial Distribution
% c(L+1)/Ln
S X
Probability of placing a red tile given either a red or green tile is placed:
1/n
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X
S % c
% c/L
% c(L+1)(n-1)/Ln
Improved Estimation of n: Binomial Distribution
% c(L+1)/Ln
S X
Probability of placing a red tile given either a red or green tile is placed:
1/n
To compute estimation of n:Compute LENGTH / REDS
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X
S % c
% c/L
% c(L+1)(n-1)/Ln
Improved Estimation of n: Binomial Distribution
% c(L+1)/Ln
S X
1 0 0 1
1 0
1 1
1 0
0 1
0 0
1 0
1 1
0 0
1 0
0 0
1 1
1 0
1 0
1 1
1 0
0 1
0 1
0 0
1 1
0 1
0 0
1 0
0 1
1 1
0 0
1 0
1 1
1 1
0 0
1 0
0 1
0 1
1 0
1 0
1 0
0 0
1 1
1 0
0 0
1 0
1 1
1 0
Binary Counter Length: 10000Reds: 100
1 0
1 0
1 0
1 1
1 0
0 0
0 0
0 1
0 0
0 0
0 0
0 1
0 0
1 0 1 0
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X
S % c
% c/L
% c(L+1)(n-1)/Ln
Improved Estimation of n: Binomial Distribution
% c(L+1)/Ln
S X
1 0 0 1
1 0
1 1
1 0
0 1
0 0
1 0
1 1
0 0
1 0
0 0
1 1
1 0
1 0
1 1
1 0
0 1
0 1
0 0
1 1
0 1
0 0
1 0
0 1
1 1
0 0
1 0
1 1
1 1
0 0
1 0
0 1
0 1
1 0
1 0
1 0
0 0
1 1
1 0
0 0
1 0
1 1
1 01 0
1 0
1 0
1 1
1 0
0 0
0 0
0 1
0 0
0 0 1 0
0 0
0 0
0 1
0 0
1 0 1 0
0
Length: 10000Reds: 100
Compute Length / Reds: 100
Division tiles
Estimate for n
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Problem: Estimation Line too Long
S X
Length >> n :Too long for an n x n square…
Chernoff Bounds only yield high accuracy for Length >> n
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S
Multiple lines = HEIGHT: Determined by Geometric Distribution
WIDTH: Determined by Geometric Distribution
Solution: Estimation Frame
Phase 1: Build dimensions of frame.
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S
Multiple lines = HEIGHT: Determined by Geometric Distribution
WIDTH: Determined by Geometric Distribution
Solution: Estimation Frame
Phase 1: Build dimensions of frame.
Phase 2:Build Sampling Lines
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S
01
01
1
Sum Reds
Sum Reds
Sum Reds
01Sum Reds
Multiple lines = HEIGHT: Determined by Geometric Distribution
WIDTH: Determined by Geometric Distribution
Solution: Estimation Frame
Phase 1: Build dimensions of frame.
Phase 2:Build Sampling Lines
Phase 3: Sum Reds and Length for each Line
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S
01
01
1
Sum Reds
Sum Reds
Sum Reds
01Sum Reds
1 1 1
Su
m S
ub
tota
ls
Multiple lines = HEIGHT: Determined by Geometric Distribution
WIDTH: Determined by Geometric Distribution
Solution: Estimation Frame
Phase 1: Build dimensions of frame.
Phase 2:Build Sampling Lines
Phase 3: Sum Reds and Length for each Line
Phase 4: Sum subtotals
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S
01
01
1
Sum Reds
Sum Reds
Sum Reds
01Sum Reds
1 1 1
Su
m S
ub
tota
ls
Divide: Length / Reds
1001
Multiple lines = HEIGHT: Determined by Geometric Distribution
WIDTH: Determined by Geometric Distribution
Solution: Estimation Frame
Phase 1: Build dimensions of frame.
Phase 2:Build Sampling Lines
Phase 3: Sum Reds and Length for each Line
Phase 4: Sum subtotals
Phase 5: Compute Length to Reds ratio
OUTPUTEstimation
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HEIGHT
WIDTH
Solution: Estimation Frame
With high probability: HEIGHT < nWIDTH < n
Frame fits within n x n square
With high probability: HEIGHT x WIDTH >> n Chernoff Bounds imply:
Estimation is accurate with high probability:(1 - )n’ < n < (1 + ) n’
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10110110
Finish off SquareOutput n approximation
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Binary Counter
Bin
ary Co
un
ter
10110110
n
n
Finish off SquareOutput n approximation
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We have a fixed size O(1) tileset that:
-For any given - n > C( )
We can assign percentages such that:
With probability at least 1- , a size n’ x n’
square is assembled with
( 1- )n < n’ < ( 1+ )n
Binary Counter
Bin
ary Co
un
ter
10110110
n
n
Result:
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Probabilistic Results
• Generic size O(1) tileset that builds approximate squares.
• Approximation Frame has many potential applications– Encode arbitrary programs
• General shapes• Encode input of computational problems
• Simulation and experimental implementation• Extension to 3 dimensions
– Approximation accuracy increases for n x n x n cubes, possibly exact assembly of cubes with high probability
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Thanks for Listening
Questions?
Robert Schweller Computer Science Department University of Texas Pan American Email: [email protected]
http://www.cs.panam.edu/~schwellerr/