may 3, 2006
DESCRIPTION
Complexities for the Design of Self-Assembly Systems. May 3, 2006. Robert Schweller Electrical Engineering and Computer Science Department Northwestern University Email: [email protected] http://www.cs.northwestern.edu/~schwellerr/. Outline. Background, Motivation Model - PowerPoint PPT PresentationTRANSCRIPT
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Robert Schweller Electrical Engineering and Computer Science Department Northwestern University Email: [email protected]
http://www.cs.northwestern.edu/~schwellerr/
May 3, 2006
Complexities for the Design of Self-Assembly Systems
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Outline
• Background, Motivation
• Model
• Temperature Programming
• Flexible Glue Self-Assembly
• Shape Verification
• Future Work
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A C
G C
T G C G
Molecular Building Blocks
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Molecular Building Blocks
A T A G CT A T C G
T G A T C G G AA C T A G C C T
A C T A G C C TA C T A G C C T
C T A G C C G TG A T C G G C A
G C T T G A C CC G A A C T G G
A G
A T
C G
A C
T C
T A G
C T
G
T A C
C G
C A
TA
T G
G C
G T A
T G
A A T
A G
CA
C T
T A T
C G
A C
T A G
C C
TA
C T A
G C
C T
A T A G CT A T C G
A T A G CT A T C G
G T A C AC A T G T
A T A
G C
T A T
C G
A T A
G C
T A T
C G
A T A
G C
T A T
C G
A T A
G C
T A T
C G
C G G T C
T T C C A
G A
C A G
T T A
G T
[John Reif’s Group,Duke University]
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DNA Scaffolding
[Sung Ha Park, Constantin Pistol, Sang Jung Ahn, John H. Reif, Alvin R. Lebeck, Chris Dwyer, and Thomas H. LaBean, 2006]
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Paul Rothemund, Nick Papadakis, Erik Winfree, PLoS Biology 2: e424 (2004)
340nm
Simulation of Cellular Automata
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Outline
• Background, Motivation
• Model
• Temperature Programming
• Shape Verification
• Flexible Glue Self-Assembly
• Future Work
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},...,1,0{: tG
},,,{ sTGt
Tile Model of Self-Assembly(Rothemund, Winfree STOC 2000)
Tile System:
t : temperature, positive integer
G: glue function
T: tileset , , ... { }r
r
w
g
p
y yb
r
b
r
b,
s: seed tile
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How a tile system self assembles
x dc
baST = G(y) = 2G(g) = 2G(r) = 2G(b) = 2G(p) = 1G(w) = 1
t = 2
S
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S a
How a tile system self assembles
x dc
baST = G(y) = 2G(g) = 2G(r) = 2G(b) = 2G(p) = 1G(w) = 1
t = 2
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S a
c
How a tile system self assembles
x dc
baST = G(y) = 2G(g) = 2G(r) = 2G(b) = 2G(p) = 1G(w) = 1
t = 2
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S a
c
d
How a tile system self assembles
x dc
baST = G(y) = 2G(g) = 2G(r) = 2G(b) = 2G(p) = 1G(w) = 1
t = 2
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S a b
c
d
How a tile system self assembles
x dc
baST = G(y) = 2G(g) = 2G(r) = 2G(b) = 2G(p) = 1G(w) = 1
t = 2
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S a b
c
d
x
How a tile system self assembles
x dc
baST = G(y) = 2G(g) = 2G(r) = 2G(b) = 2G(p) = 1G(w) = 1
t = 2
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S a b
c
d
x x
How a tile system self assembles
x dc
baST = G(y) = 2G(g) = 2G(r) = 2G(b) = 2G(p) = 1G(w) = 1
t = 2
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S a b
c
d
x x
x
How a tile system self assembles
x dc
baST = G(y) = 2G(g) = 2G(r) = 2G(b) = 2G(p) = 1G(w) = 1
t = 2
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S a b
c
d
x x
x x
How a tile system self assembles
x dc
baST = G(y) = 2G(g) = 2G(r) = 2G(b) = 2G(p) = 1G(w) = 1
t = 2
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Each Shape Requires a Distinct Tile Set
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Programmable, General Purpose Tile Set?
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Programmable, General Purpose Tile Set?
. . .
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Outline
• Background, Motivation
• Model
• Temperature Programming
• Flexible Glue Self-Assembly
• Shape Verification
• Future Work
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Multiple Temperature Model
Multiple Temperature Model
- temperature may go up and down
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Multiple Temperature Model
Multiple Temperature Model
- temperature may go up and down
},,,{ sTGtt
< t1 , t2 , ... , tr-1 , tr >
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Multiple Temperature Model
Multiple Temperature Model
- temperature may go up and down
},,,{ sTGtt
< t1 , t2 , ... , tr-1 , tr >
Tile Complexity: Number of Tiles
Temperature Complexity: Number of Temperatures
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Building k x n Rectangles
k-digit, base n(1/k) counter:0
0
0
S0
0
0
0
1 2
0
0
0
0
0
1
0
0
0
1
1
2
2
2
2
2
2
2
1
2
2
2
0
. . .k
N
k
n
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Building k x n Rectangles
k-digit, base n(1/k) counter:0
0
0
S0
0
0
0
1 2
0
0
0
0
0
1
0
0
0
1
1
2
2
2
2
2
2
2
1
2
2
2
0
. . .k
N
0
0
0
S0
0
0
0
1 2
0
0
0
0
0
1
0
0
0
1
1
2
2
2
2
2
2
2
1
2
2
2
0
. . .k
N
)( /1 knkO Tile Complexity:
n
k
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two temperatures
3
3
3
1
t = 4
k
N
j
k
N
j
n
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t = 4 6
k
N
j
two temperatures
n
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)nO(j 1/j
)n loglog
n logO(k
N
j
n
Tile Complexity:
two temperaturest = 4 6
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)nO(j 1/j
)n loglog
n logO(
)n loglog
n logΩ(Kolmogorov Complexity (Rothemund,
Winfree STOC 2000)
Beats Standard Model )k
nΩ(
1/k
k
N
j
n
Tile Complexity:
two temperaturest = 4 6
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Programmable, General Purpose Tile Set?
. . .
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Given: n 1011001
log n
High Level Approach
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Given: n 1011001
log nte
mp
High Level Approach
1
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Given: n 1011001
log nte
mp
High Level Approach
1
1
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Given: n 1011001
log nte
mp
High Level Approach
1 0
1 0
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Given: n 1011001
log nte
mp
High Level Approach
1 0 1 1 . . .
. . .
0
1 0 1 1 0 10
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tem
pHigh Level Approach
0 1. . .
. . .
1 0 1 1 0 10
0
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tem
pHigh Level Approach
0 1. . .
. . .
1 0 1 1 0 10
0
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tem
pHigh Level Approach
0 1. . .
. . .
1 0 1 1 0 10
0
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Assembly of n x n Squares
N - k
k
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Assembly of n x n Squares
n - k
k
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Assembly of n x n Squares
n - k
k
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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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Encoding a Single Bit
0
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Z
g
z
g
g
g
g
g
g
g
g
g
0 1
0’ 1’zz
10
1
t = < 2, 5 >
Z
1’
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a
Encoding a Single Bit
0
a
Z
g
z
g
g
g
g
g
g
g
g
g
0 1
0’ 1’zz
10
1
t = < 2, 5 >
Z
1’
1
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a
Encoding a Single Bit
0
a
Z
g
z
g
g
g
g
g
g
g
g
g
0 1
0’ 1’zz
10
1
t = < 2, 5 >
Z
1’
1
a
Z
t = < 2 >
0
0’
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Goal: 1 0 1 0 0
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a
0
s
Goal: 1 0 1 0 0
b
temp: < 4,9, 3,7, 4, 3,7, 4,9, 3,7, 4, 3,7, 4, 3 >
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a
1
s
b
Goal: 1 0 1 0 0
temp: < 4,9, 3,7, 4, 3,7, 4,9, 3,7, 4, 3,7, 4, 3 >
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a
1
s
Goal: 1 0 1 0 0
b
X
temp: < 4,9, 3,7, 4, 3,7, 4,9, 3,7, 4, 3,7, 4, 3 >
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a
1
s
Goal: 1 0 1 0 0
b
Y
temp: < 4,9, 3,7, 4, 3,7, 4,9, 3,7, 4, 3,7, 4, 3 >
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a
1
s
Goal: 1 0 1 0 0
b
Y
temp: < 4,9, 3,7, 4, 3,7, 4,9, 3,7, 4, 3,7, 4, 3 >
a
b
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a
1
s
Goal: 1 0 1 0 0
b
Y
temp: < 4,9, 3,7, 4, 3,7, 4,9, 3,7, 4, 3,7, 4, 3 >
a
b
0
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a
1
s
Goal: 1 0 1 0 0
b
Y
temp: < 4,9, 3,7, 4, 3,7, 4,9, 3,7, 4, 3,7, 4, 3 >
a
b
0
X
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a
1
s
Goal: 1 0 1 0 0
b
Y
temp: < 4,9, 3,7, 4, 3,7, 4,9, 3,7, 4, 3,7, 4, 3 >
a
b
0
Y
a
b
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a
1
s
Goal: 1 0 1 0 0
b
Y
temp: < 4,9, 3,7, 4, 3,7, 4,9, 3,7, 4, 3,7, 4, 3 >
a
b
0
Y
a
b
0
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a
1
s
Goal: 1 0 1 0 0
b
Y
temp: < 4,9, 3,7, 4, 3,7, 4,9, 3,7, 4, 3,7, 4, 3 >
a
b
0
Y
a
b
1
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a
1
s
Goal: 1 0 1 0 0
b
Y
temp: < 4,9, 3,7, 4, 3,7, 4,9, 3,7, 4, 3,7, 4, 3 >
a
b
0
Y
a
b
1
X
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a
1
s
Goal: 1 0 1 0 0
b
Y
temp: < 4,9, 3,7, 4, 3,7, 4,9, 3,7, 4, 3,7, 4, 3 >
a
b
0
Y
a
b
1
Y
a
b
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a
1
s
Goal: 1 0 1 0 0
b
Y
temp: < 4,9, 3,7, 4, 3,7, 4,9, 3,7, 4, 3,7, 4, 3 >
a
b
0
Y
a
b
1
Y
a
b
0
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a
1
s
Goal: 1 0 1 0 0
b
Y
temp: < 4,9, 3,7, 4, 3,7, 4,9, 3,7, 4, 3,7, 4, 3 >
a
b
0
Y
a
b
1
Y
a
b
0
X
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a
1
s
Goal: 1 0 1 0 0
b
Y
temp: < 4,9, 3,7, 4, 3,7, 4,9, 3,7, 4, 3,7, 4, 3 >
a
b
0
Y
a
b
1
Y
a
b
0
Y
a
b
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84
a
1
s
Goal: 1 0 1 0 0
b
Y
temp: < 4,9, 3,7, 4, 3,7, 4,9, 3,7, 4, 3,7, 4, 3 >
a
b
0
Y
a
b
1
Y
a
b
0
Y
a
b
0
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85
a
1
s
Goal: 1 0 1 0 0
b
Y
temp: < 4,9, 3,7, 4, 3,7, 4,9, 3,7, 4, 3,7, 4, 3 >
a
b
0
Y
a
b
1
Y
a
b
0
Y
a
b
0
X
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1 1 0 0 1 0 0 0 1 1 1 0 1 1
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1 1 0 0 1 0 0 0 1 1 1 0 1 1 1 1 0 0 1 0 0 0 1 1 1 1 0 0 1 1 0 0 1 0 0 0 1 1 1 1 0 1 1 1 0 0 1 0 0 0 1 1 1 1 1 0 1 1 0 0 1 0 0 0 1 1 1 1 1 1 1 1 0 0 1 0 0 1 0 0 0 0 0 0
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Assembly of n x n Squares
n – log n
log n
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Assembly of n x n Squares
O(log n)
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Assembly of n x n Squares
O(log n)
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91
Results
tile complexity temperature complexity
O(1) O(log n)
O(1))loglog
log(
n
nO
(Adleman, Cheng,Goel, Huang STOC 2001)
n x n squares
TemperatureProgramming
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92
Results
tile complexity temperature complexity
O(1) O(log n)
O(1))loglog
log(
n
nO
(Adleman, Cheng,Goel, Huang STOC 2001)
? < log nSmooth Trade off? n
n
loglog
log? <
n x n squares
TemperatureProgramming
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93
Results
tile complexity temperature complexity
O(1) O(log n)
O(1))loglog
log(
n
nO
(Adleman, Cheng,Goel, Huang STOC 2001)
? < log nSmooth Trade off? n
n
loglog
log? <
For almost all n, no tileset can achieve both o(log n/ loglog n) tile complexity and o(log n) temperature complexity simultaneously
n x n squares
TemperatureProgramming
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General Shapes?
• General Scaled Shapes [Soloveichik, Winfree 2004]
– O(Ks*) Tile complexity, single temperature
• Combined with Temperature Programming:– O(1) Tile Complexity– O(Ks) Temperature Complexity
• General, Constant Scaled Shapes?– O(1) Tile Complexity– O(|S|) Temperature Complexity
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Outline
• Background, Motivation
• Model and Basics
• Temperature Programming
• Flexible Glue Self-Assembly
• Shape Verification
• Future Work
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96
S a b
c
d
x x
x x
Only identical glues attract
x dc
baST = G(y,y) = 2G(g,g) = 2G(r,r) = 2G(b,b) = 2G(p,p) = 1G(w,w) = 1
t = 2
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S a b
c
d
x x
x x
Only identical glues attract
x dc
baST = G(y,y) = 2G(g,g) = 2G(r,r) = 2G(b,b) = 2G(p,p) = 1G(w,w) = 1
t = 2
We don’t have:G(p,b) = 1
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Flexible Glue Model
Remove the restriction that G(x, y) = 0 for x!=y
• Substantial Reduction in Tile Complexity
• Design of Flexible glues
yx
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99
n x n Squares --- Flexible Glue Model
a b c d e fa 1 0 2 0 0 1b 0 0 1 0 1 0c 0 0 3 0 1 1d 2 2 2 2 0 1e 0 0 0 1 2 1f 1 1 2 2 1 1
a b c d e fa 1 - - - - -b - 0 - - - -c - - 3 - - -d - - - 2 - -e - - - - 2 -f - - - - - 1
Standard Glue Function Flexible Glue Function
Kolmogorov lower bounds:
)loglog
log(
n
n
)log( n
Standard
Flexible
(Rothemund, Winfree STOC 2000)
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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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101
n x n Square --- Flexible Glue Model
goal: - seed binary counter to a given value
-
2
log n
0 1 0 0 0 0 0 01 1 1 1 1 1 1 1 1 1
)log( nO
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. . . 3 3 3 4 4 4 4 4 4 5 5 5 53 4 5 0 1 2 3 4 5 0 1 2 3 4 5
5
n x n Square --- Flexible Glue Model
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. . . 3 3 3 4 4 4 4 4 4 5 5 5 53 4 5 0 1 2 3 4 5 0 1 2 3 4 5
0 0 1 1 0 1 1 0 0 1 1 1 0 | | | | | | | | | | | | |
5
5
n x n Square --- Flexible Glue Model
key idea:
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4 53
555
21b4
5
5
w5
p5
G(b4, p5) = 1G(b4, w5) = 0
n x n Square --- Flexible Glue Model
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p0 p1 p2 p3 p4 p5b0 0 1 1 0 1 1b1 1 1 0 1 0 1b2 0 1 0 1 1 1b3 0 0 1 0 1 0b4 0 0 0 0 0 1b5 1 1 1 1 1 0
• given B = 011011 110101 010111 …
• encode B into glue function
B = 011011 110101 010111 …
n x n Square --- Flexible Glue Model
4b4
5p5
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• Complexity: )log( nO
0 1 0 1 1 0 0 0 1 1 0 0 1 0 0 0 1 1 0 1 1 1 0 0 0 1 0 1
nlog2• build block
n x n Square --- Flexible Glue Model
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0 1 0 1 1 0 0 0 1 1 0 0 1 0 0 0 1 1 0 1 1 1 0 0 0 1 0 10 1 0 1 1 0 0 0 1 1 0 0 1 0 0 0 1 1 0 1 1 1 0 0 0 1 1 00 1 0 1 1 0 0 0 1 1 0 0 1 0 0 0 1 1 0 1 1 1 0 0 0 1 1 10 1 0 1 1 0 0 0 1 1 0 0 1 0 0 0 1 1 0 1 1 1 0 0 1 0 0 00 1 0 1 1 0 0 0 1 1 0 0 1 0 0 0 1 1 0 1 1 1 0 0 1 0 0 10 1 0 1 1 0 0 0 1 1 0 0 1 0 0 0 1 1 0 1 1 1 0 0 1 0 1 00 1 0 1 1 0 0 0 1 1 0 0 1 0 0 0 1 1 0 1 1 1 0 0 1 0 1 10 1 0 1 1 0 0 0 1 1 0 0 1 0 0 0 1 1 0 1 1 1 0 0 1 1 0 00 1 0 1 1 0 0 0 1 1 0 0 1 0 0 0 1 1 0 1 1 1 0 0 1 1 0 10 1 0 1 1 0 0 0 1 1 0 0 1 0 0 0 1 1 0 1 1 1 0 0 1 1 1 0
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Assembly of n x n Squares
n – log n
log n
2 x log n block
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Assembly of n x n Squares
O(log n)
)log( N
Complexity:
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110
)n loglog
n log( (Adleman, Cheng,
Goel, Huang STOC 2001)
Tile Complexity for n x n squares
)n log(Flexible Glue:
Standard:
Assembly of n x n Squares
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Outline
• Background, Motivation• Model and Basics• Temperature Programming• Flexible Glue Self-Assembly
– Reduced Tile Complexity– Glue Design
• Shape Verification• Future Work
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Glue Design
ACGGT TGCCA
GGGAT
GTTGG
CGTAC
GACTC
CCCTA
CAACC
GCATG
CTGAG
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Glue Design- Standard
ACGGT TGCCA
GGGAT
GTTGG
CGTAC
GACTC
CCCTA
CAACC
GCATG
CTGAG
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Glue Design- Standard
ACGGT TGCCA
GGGAT
GTTGG
CGTAC
GACTC
CCCTA
CAACC
GCATG
CTGAG
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Glue Design- Standard
ACGGT
GGGAT
GTTGG
CGTAC
GACTC
Design n strings such that each pair of strings has Hamming distance
at least .
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Flexible Glue Design
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Flexible Glue Design
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Flexible Glue Design
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Flexible Glue Design
Input: - Graph G(V,E)
- separation parameter
Output:
A -labeling:
A labeling :V→{0,1}* such
that for some , , - , –if (u,v) G, HD((u), (v)) ≤
–if (u,v) G, HD((u), (v))
Minimize label length
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Graph Decomposition
Strategy:- Decompose graph- Label each Subgraph- Concatenate labels
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Graph Decomposition
Strategy:- Decompose graph- Label each Subgraph- Concatenate labels
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123
Graph Decomposition
Need Exact -labeling:Each pair of non-adjacent nodes haveexactly the same Hamming distance.
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Exact -labeling for Matchings
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Exact -labeling for Matchings
00000001111000001111011001100101011101001101101011001101
Hamming Distance:
Exactly
Word Length: O(n + )
Hadamard Code (n, )
Example: n=8, =4
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126
Exact -labeling for Matchings
Hadamard Code (n, )
Example: n=8, =4
00000001111000001111011001100101011101001101101011001101
Hamming Distance:
Exactly
Word Length: O(n + )
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Matching Decomposition
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Matching Decomposition
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Matching Decomposition
Length per Matching: O(n + )Number of Matchings: O(D)
Label Length: O(Dn + D)
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Tough Example
Length per Matching: O(n + )Number of Matchings: O(D)
Label Length: O(Dn + D)
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Star Graphs
Star Graph: All edges are adjacent to the same vertex
- Non-trivial application of the Hadamard Code yields:
length O(n) exact -labeling
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Star Destroyer
Algorithm Star Destroyer
1. Decompose graph into “large” stars:
- O(n) length per star
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133
Star Destroyer
Algorithm Star Destroyer
1. Decompose graph into “large” stars:
- O(n) length per star
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134
Star Destroyer
Algorithm Star Destroyer
1. Decompose graph into “large” stars:
- O(n) length per star
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135
Star Destroyer
Algorithm Star Destroyer
1. Decompose graph into “large” stars:
- O(n) length per star
2. Decompose remaining graph into matchings:
- O(n+) length per matching
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136
Star Destroyer
Algorithm Star Destroyer
1. Decompose graph into “large” stars:
- O(n) length per star
2. Decompose remaining graph into matchings:
- O(n+) length per matching
Star Destoyer yields length:
22 γmnnγm
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137
Star Destoyer )γmnnγmO( 22
Matching Algorithm
) DγDn O(
Word Length
Flexible Word Design Results
General Graphs
Other Work -Trees, Lines and Rings -Distance Labeling
-Application to applying DNA computing to digital signal processing
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Outline
• Background, Motivation
• Model and Basics
• Temperature Programming
• Flexible Glue Self-Assembly
• Shape Verification
• Future Work
![Page 139: May 3, 2006](https://reader038.vdocuments.us/reader038/viewer/2022102801/56815015550346895dbdfb1f/html5/thumbnails/139.jpg)
139
Shape Verification
Shape Verification Problem
Input: T, a tile system S, a shape
Question: Does T uniquely assemble S.
Standard: P (Adleman, Cheng, Goel, Huang, Kempe,
Flexible glue: P Espanes, Rothemund, STOC 2002)
Unique Shape: Co-NPC Multiple Temperature: NP-hard, Co-NP-hard Multiple Tile: NP-hard, Co-NP-hard
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*
Shape Verification: Unique-Shape Model
)(
)(
)(
321
321
321
xxx
xxx
xxx
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*
x1
x2
x3
*
)(
)(
)(
321
321
321
xxx
xxx
xxx
Shape Verification: Unique-Shape Model
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142
*
x1
x2
x3
*
* c2c1 c3 *
)(
)(
)(
321
321
321
xxx
xxx
xxx
Shape Verification: Unique-Shape Model
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143
*
x1
x2
x3
*
* c2c1 c3 *
x
x
x
1
0
x
x
)(
)(
)(
321
321
321
xxx
xxx
xxx
Shape Verification: Unique-Shape Model
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144
*
x1
x2
x3
*
* c2c1 c3 *
0
1
1
)(
)(
)(
321
321
321
xxx
xxx
xxx
Shape Verification: Unique-Shape Model
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145
*
x1
x2
x3
*
* c2c1 c3 *
c10
1
1
)(
)(
)(
321
321
321
xxx
xxx
xxx
Shape Verification: Unique-Shape Model
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146
*
x1
x2
x3
*
*
ok
c2c1 c3 *
c10
1
1
)(
)(
)(
321
321
321
xxx
xxx
xxx
Shape Verification: Unique-Shape Model
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147
*
x1
x2
x3
*
*
ok
ok
c2c1 c3 *
c10
1
1
)(
)(
)(
321
321
321
xxx
xxx
xxx
Shape Verification: Unique-Shape Model
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148
*
x1
x2
x3
*
*
ok
ok
c2c1
c2
c3 *
c10
1
1
)(
)(
)(
321
321
321
xxx
xxx
xxx
Shape Verification: Unique-Shape Model
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149
*
x1
x2
x3
*
*
ok
ok
c2c1
c2
c2
c3 *
c10
1
1
)(
)(
)(
321
321
321
xxx
xxx
xxx
Shape Verification: Unique-Shape Model
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150
*
x1
x2
x3
*
*
okok
ok
c2c1
c2
c2
c3 *
c10
1
1
)(
)(
)(
321
321
321
xxx
xxx
xxx
Shape Verification: Unique-Shape Model
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151
*
x1
x2
x3
*
*
ok
okok
ok
c2c1
c2
c2
c3 *
c10
1
1
)(
)(
)(
321
321
321
xxx
xxx
xxx
Shape Verification: Unique-Shape Model
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*
x1
x2
x3
*
*
ok
ok
ok
okok
ok
c2c1
c2
c2
c3 *
c10
1
1
)(
)(
)(
321
321
321
xxx
xxx
xxx
Shape Verification: Unique-Shape Model
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153
*
x1
x2
x3
*
*
ok
ok
ok
okok
ok
c2c1
c2
c2
c3 *
c10
1
1
)(
)(
)(
321
321
321
xxx
xxx
xxx
*
*
*
*
Shape Verification: Unique-Shape Model
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*
x1
x2
x3
*
*
ok
ok
ok
okok
ok
c2c1
c2
c2
c3 *
c10
1
1
)(
)(
)(
321
321
321
xxx
xxx
xxx
*
*
*
* T
Shape Verification: Unique-Shape Model
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155
*
x1
x2
x3
*
*
ok
ok
ok
okok
ok
c2c1
c2
c2
c3 *
c10
1
1
)(
)(
)(
321
321
321
xxx
xxx
xxx
*
*
*
* T T
Shape Verification: Unique-Shape Model
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156
*
x1
x2
x3
*
*
ok
ok
ok
okok
ok
c2c1
c2
c2
c3 *
c10
1
1
)(
)(
)(
321
321
321
xxx
xxx
xxx
*
*
*
* T T T
Shape Verification: Unique-Shape Model
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157
*
x1
x2
x3
*
*
ok
ok
ok
okok
ok
c2c1
c2
c2
c3 *
c10
1
1
)(
)(
)(
321
321
321
xxx
xxx
xxx
*
*
*
* T T T SAT
Satisfied
(LaBean and Lagoudakis, 1999)
Shape Verification: Unique-Shape Model
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158
x1
* *
x2
x3
* * T T T
ok
ok
ok
okok
ok
c2c1
c2
c2
c3 *
*
*
*
c10
1
1
SAT
x1
* *
x2
x3
* *
ok
ok
ok
c2ok
ok
c2c1
c2
c2
c3 *
*
*
*
c10
0
1
Satisfied
(LaBean and Lagoudakis, 1999)
Shape Verification: Unique-Shape Model
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159
x1
* *
x2
x3
* * T T T
ok
ok
ok
okok
ok
c2c1
c2
c2
c3 *
*
*
*
c10
1
1
SAT
x1
* *
x2
x3
* * T
ok
ok
ok
c2ok
ok
c2c1
c2
c2
c3 *
*
*
*
c10
0
1
Satisfied
(LaBean and Lagoudakis, 1999)
Shape Verification: Unique-Shape Model
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x1
* *
x2
x3
* * T T T
ok
ok
ok
okok
ok
c2c1
c2
c2
c3 *
*
*
*
c10
1
1
SAT
x1
* *
x2
x3
* * T F
ok
ok
ok
c2ok
ok
c2c1
c2
c2
c3 *
*
*
*
c10
0
1
Satisfied
(LaBean and Lagoudakis, 1999)
Shape Verification: Unique-Shape Model
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x1
* *
x2
x3
* * T T T
ok
ok
ok
okok
ok
c2c1
c2
c2
c3 *
*
*
*
c10
1
1
SAT
x1
* *
x2
x3
* * T F F
ok
ok
ok
c2ok
ok
c2c1
c2
c2
c3 *
*
*
*
c10
0
1
Satisfied Not Satisfied
(LaBean and Lagoudakis, 1999)
Shape Verification: Unique-Shape Model
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T T
ok ok ok
c2
SAT
*
*
T
ok
* T
x3
x2
x1
1
1
0
ok
c1
c1* *
c2 ok
*c2 c3
*
T F
ok c2 ok
c2
NO
*
*
T
ok
* T
x3
x2
x1
0
1
0
ok
c1
c1* *
c2 ok
*c2 c3
*
Satisfied Not Satisfied
Shape Verification: Multiple Temperature Model
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T T
ok ok ok
c2
SAT
*
*
T
ok
* T
x3
x2
x1
1
1
0
ok
c1
c1* *
c2 ok
*c2 c3
*
T T
ok ok ok
c2
NO
*
*
T
ok
* T
x3
x2
x1
1
1
0
ok
c1
c1* *
c2 ok
*c2 c3
*
Satisfied Not Satisfied
Shape Verification: Multiple Temperature Model
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T T
ok ok ok
c2
SAT
*
*
T
ok
* T
x3
x2
x1
1
1
0
ok
c1
c1* *
c2 ok
*c2 c3
*
T T
ok ok ok
c2
NO
*
*
T
ok
* T
x3
x2
x1
1
1
0
ok
c1
c1* *
c2 ok
*c2 c3
*
Satisfied Not Satisfied
Shape Verification: Multiple Temperature Model
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*
x3
x2
x1
*
*
x3
x2
x1
*
Satisfied Not Satisfied
Shape Verification: Multiple Temperature Model
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*
x1
x2
x3
*
0
0
0
c1
ok
ok
c2
c2
c2
ok
ok
ok *
*
*
* c1 c2 c3 *
FFT NO*
Shape Verification: Multiple Temperature Model
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167
*
x1
x2
x3
*
0
0
0
c1
ok
ok
c2
c2
c2
ok
ok
ok *
*
*
1
0
0
* c1 c2 c3 * *
FFT NO **
Shape Verification: Multiple Temperature Model
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168
*
x1
x2
x3
*
0
0
0
c1
ok
ok
c2
c2
c2
ok
ok
ok *
*
*
1
0
0
c1
c1
c1
c2
c2
c2
ok
ok
ok *
*
*
* c1 c1c2 c2c3 c3* * *
FFT NO ** FF F NO
Shape Verification: Multiple Temperature Model
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*
x1
x2
x3
*
0
0
0
c1
ok
ok
c2
c2
c2
ok
ok
ok *
*
*
1
0
0
c1
c1
c1
c2
c2
c2
ok
ok
ok *
*
*
0
1
0
* c1 c1c2 c2c3 c3* * * *
*
*
*
1
1
1
* *
c1
ok
ok
c2
c2
ok
*
*
*
c1 c2 c3 *
c3
c3
c3
FFT NO ** FF F NO * NO * FTT NO
...
Shape Verification: Multiple Temperature Model
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*
x1
x2
x3
*
0
0
0
c1
ok
ok
c2
c2
c2
ok
ok
ok *
*
*
1
0
0
c1
c1
c1
c2
c2
c2
ok
ok
ok *
*
*
0
1
0
* c1 c1c2 c2c3 c3* * * *
*
*
*
1
1
1
* *
c1
ok
ok
c2
c2
ok
*
*
*
c1 c2 c3 *
c3
c3
c3
FFT NO ** FF F NO * NO * FTT NO
...
Shape Verification: Multiple Temperature Model
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*
x1
x2
x3
*
0
0
0
c1
ok
ok
c2
c2
c2
ok
ok
ok *
*
*
1
0
0
c1
c1
c1
c2
c2
c2
ok
ok
ok *
*
*
0
1
0
* c1 c1c2 c2c3 c3* * * *
*
*
*
1
1
1
* *
c1
ok
ok
c2
c2
ok
*
*
*
c1 c2 c3 *
c3
c3
c3
FFT NO ** FF F NO * NO * FTT NO
...
Shape Verification: Multiple Temperature Model
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172
*
x1
x2
x3
*
0
0
0
c1
ok
ok
c2
c2
c2
ok
ok
ok *
*
*
1
0
0
c1
c1
c1
c2
c2
c2
ok
ok
ok *
*
*
0
1
0
* c1 c1c2 c2c3 c3* * * *
*
*
*
1
1
1
* *
c1
ok
ok
c2
c2
ok
*
*
*
c1 c2 c3 *
c3
c3
c3
FFT NO ** FF F NO * NO * FTT NO
...
*
x1
x2
x3
*
0
0
0
c1
ok
ok
c2
c2
ok
ok
ok
ok *
*
*
1
0
0
c1
c1
c1
c2
c2
c2
ok
ok
ok *
*
*
0
1
0
* c1 c1c2 c2c3 c3* * * *
*
*
*
1
1
1
* *
c1
ok
ok
c2
c2
ok
*
*
*
c1 c2 c3 *
c3
c3
c3
TTT SAT ** FF F NO * NO * FTT NO
...
Shape Verification: Multiple Temperature Model
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*
x1
x2
x3
*
*
x1
x2
x3
*
Satisfiable
Not Satisfiable
Shape Verification: Multiple Temperature Model
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174
Input Shape:
Shape Verification: Multiple Temperature Model
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175
Input Shape:
*
x1
x2
x3
*
0
0
0
c1
ok
ok
c2
c2
c2
ok
ok
ok *
*
*
1
0
0
c1
c1
c1
c2
c2
c2
ok
ok
ok *
*
*
0
1
0
* c1 c1c2 c2c3 c3* * * *
*
*
*
1
1
1
* *
c1
ok
ok
c2
c2
ok
*
*
*
c1 c2 c3 *
c3
c3
c3
FFT NO ** FF F NO * NO * FTT NO
...
Shape Verification: Multiple Temperature Model
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Input Shape:
*
x1
x2
x3
*
0
0
0
c1
ok
ok
c2
c2
c2
ok
ok
ok *
*
*
1
0
0
c1
c1
c1
c2
c2
c2
ok
ok
ok *
*
*
0
1
0
* c1 c1c2 c2c3 c3* * * *
*
*
*
1
1
1
* *
c1
ok
ok
c2
c2
ok
*
*
*
c1 c2 c3 *
c3
c3
c3
FFT NO ** FF F NO * NO * FTT NO
...
Equivalent to Co-SAT Co-NP-hard
Shape Verification: Multiple Temperature Model
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Input Shape:
Shape Verification: Multiple Temperature Model
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Input Shape:
*
x1
x2
x3
*
0
0
0
c1
ok
ok
c2
c2
ok
ok
ok
ok *
*
*
1
0
0
c1
c1
c1
c2
c2
c2
ok
ok
ok *
*
*
0
1
0
* c1 c1c2 c2c3 c3* * * *
*
*
*
1
1
1
* *
c1
ok
ok
c2
c2
ok
*
*
*
c1 c2 c3 *
c3
c3
c3
TTT SAT ** FF F NO * NO * FTT NO
...
Shape Verification: Multiple Temperature Model
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179
Input Shape:
Equivalent to SAT NP-hard
*
x1
x2
x3
*
0
0
0
c1
ok
ok
c2
c2
ok
ok
ok
ok *
*
*
1
0
0
c1
c1
c1
c2
c2
c2
ok
ok
ok *
*
*
0
1
0
* c1 c1c2 c2c3 c3* * * *
*
*
*
1
1
1
* *
c1
ok
ok
c2
c2
ok
*
*
*
c1 c2 c3 *
c3
c3
c3
TTT SAT ** FF F NO * NO * FTT NO
...
Shape Verification: Multiple Temperature Model
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Shape Verification Results
Standard P
Flexible Glue P Multiple Temperature NP-hard
Co-NP-hard
Unique Shape Co-NPC
Multiple Tile NP-hard Co-NP-hard
(Adleman, Cheng, Goel, Huang, Kempe,Espanes, Rothemund, STOC 2002)
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Outline
• Background, Motivation
• Model and Basics
• Temperature Programming
• Flexible Glue Self-Assembly
• Shape Verification
• Future Work
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Shape Replication
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Shape Replication
drop temperature
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Shape Replication
drop temperature
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Shape Replication
drop temperature
raise temperature
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Shape Replication
drop temperature
raise temperature
drop temperature
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Shape Replication
2
1 1
1 1
1 1
1 1
1 1
1 1
2 2
2 2
2 2
2 2
2 2
2 2
1 . . .
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Shape Replication
8 9765 4 3 2 1
10 11
12
13 14
1515
15
15
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Shape Replication
8 9765 4 3 2 1
10 11
12
13 14
1515
15
15
54 3 2 1
89
10
11
7
6
15
17
16
18
19
2021
22
raise temperature
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Shape Replication
8 9765 4 3 2 1
10 11
12
13 14
1515
15
15
54 3 2 1
89
10
11
7
6
15
17
16
18
19
2021
22
8 9765 4 3 2 1
10 11
12
13 14 15 16
17
18
19
20
21
8 9765 4 3 2 1
10 11
12
13 14
1515
15
15
raise temperature
drop temperature
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Staged Assembly
• Assembly takes place in stages, each stage with a different tile set– Large drops in Tile Complexity
• Filter-Free Staged Assembly– Can only add tiles, never remove– Still, large drops in Complexity
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• Lab Work– Experimental tests for Temperature
Programming– Flexible Glue Design
Future Work
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Thanks for Listening
Questions?
Robert Schweller Electrical Engineering and Computer Science Department Northwestern University Email: [email protected]
http://www.cs.northwestern.edu/~schwellerr/