complexities for generalized models of self-assembly
DESCRIPTION
Complexities for Generalized Models of Self-Assembly. Gagan Aggarwal Stanford University Michael H. Goldwasser St. Louis University Ming-Yang Kao Northwestern University Robert T. Schweller Northwestern University. Some results were obtained independantly by Cheng, Espanes 2003. - PowerPoint PPT PresentationTRANSCRIPT
![Page 1: Complexities for Generalized Models of Self-Assembly](https://reader036.vdocuments.site/reader036/viewer/2022081515/56812a9f550346895d8e5f62/html5/thumbnails/1.jpg)
Complexities for Generalized Models of Self-Assembly
Gagan Aggarwal Stanford University
Michael H. Goldwasser St. Louis University
Ming-Yang Kao Northwestern University
Robert T. Schweller Northwestern University
Some results were obtained independantly by Cheng, Espanes 2003
![Page 2: Complexities for Generalized Models of Self-Assembly](https://reader036.vdocuments.site/reader036/viewer/2022081515/56812a9f550346895d8e5f62/html5/thumbnails/2.jpg)
},...,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
![Page 3: Complexities for Generalized Models of Self-Assembly](https://reader036.vdocuments.site/reader036/viewer/2022081515/56812a9f550346895d8e5f62/html5/thumbnails/3.jpg)
How a tile system self assembles
x dc
baST = G(y,y) = 2G(g,g) = 2G(r, r) = 2G(b,b) = 2G(p,p) = 1G(w,w) = 1
t = 2
S
![Page 4: Complexities for Generalized Models of Self-Assembly](https://reader036.vdocuments.site/reader036/viewer/2022081515/56812a9f550346895d8e5f62/html5/thumbnails/4.jpg)
S a
How a tile system self assembles
x dc
baST = G(y,y) = 2G(g,g) = 2G(r, r) = 2G(b,b) = 2G(p,p) = 1G(w,w) = 1
t = 2
![Page 5: Complexities for Generalized Models of Self-Assembly](https://reader036.vdocuments.site/reader036/viewer/2022081515/56812a9f550346895d8e5f62/html5/thumbnails/5.jpg)
S a
c
How a tile system self assembles
x dc
baST = G(y,y) = 2G(g,g) = 2G(r, r) = 2G(b,b) = 2G(p,p) = 1G(w,w) = 1
t = 2
![Page 6: Complexities for Generalized Models of Self-Assembly](https://reader036.vdocuments.site/reader036/viewer/2022081515/56812a9f550346895d8e5f62/html5/thumbnails/6.jpg)
S a
c
d
How a tile system self assembles
x dc
baST = G(y,y) = 2G(g,g) = 2G(r, r) = 2G(b,b) = 2G(p,p) = 1G(w,w) = 1
t = 2
![Page 7: Complexities for Generalized Models of Self-Assembly](https://reader036.vdocuments.site/reader036/viewer/2022081515/56812a9f550346895d8e5f62/html5/thumbnails/7.jpg)
S a b
c
d
How a tile system self assembles
x dc
baST = G(y,y) = 2G(g,g) = 2G(r, r) = 2G(b,b) = 2G(p,p) = 1G(w,w) = 1
t = 2
![Page 8: Complexities for Generalized Models of Self-Assembly](https://reader036.vdocuments.site/reader036/viewer/2022081515/56812a9f550346895d8e5f62/html5/thumbnails/8.jpg)
S a b
c
d
x
How a tile system self assembles
x dc
baST = G(y,y) = 2G(g,g) = 2G(r, r) = 2G(b,b) = 2G(p,p) = 1G(w,w) = 1
t = 2
![Page 9: Complexities for Generalized Models of Self-Assembly](https://reader036.vdocuments.site/reader036/viewer/2022081515/56812a9f550346895d8e5f62/html5/thumbnails/9.jpg)
S a b
c
d
x x
How a tile system self assembles
x dc
baST = G(y,y) = 2G(g,g) = 2G(r, r) = 2G(b,b) = 2G(p,p) = 1G(w,w) = 1
t = 2
![Page 10: Complexities for Generalized Models of Self-Assembly](https://reader036.vdocuments.site/reader036/viewer/2022081515/56812a9f550346895d8e5f62/html5/thumbnails/10.jpg)
S a b
c
d
x x
x
How a tile system self assembles
x dc
baST = G(y,y) = 2G(g,g) = 2G(r, r) = 2G(b,b) = 2G(p,p) = 1G(w,w) = 1
t = 2
![Page 11: Complexities for Generalized Models of Self-Assembly](https://reader036.vdocuments.site/reader036/viewer/2022081515/56812a9f550346895d8e5f62/html5/thumbnails/11.jpg)
S a b
c
d
x x
x x
How a tile system self assembles
x dc
baST = G(y,y) = 2G(g,g) = 2G(r, r) = 2G(b,b) = 2G(p,p) = 1G(w,w) = 1
t = 2
![Page 12: Complexities for Generalized Models of Self-Assembly](https://reader036.vdocuments.site/reader036/viewer/2022081515/56812a9f550346895d8e5f62/html5/thumbnails/12.jpg)
New Models• Multiple Temperature Model
– temperature may go up and down
• Flexible Glue Model– Remove the restriction that G(x, y) = 0 for x!=y
• Multiple Tile Model– tiles may cluster together before being added
• Unique Shape Model– unique shape vs. unique supertile
![Page 13: Complexities for Generalized Models of Self-Assembly](https://reader036.vdocuments.site/reader036/viewer/2022081515/56812a9f550346895d8e5f62/html5/thumbnails/13.jpg)
New Models• Multiple Temperature Model
– temperature may go up and down
• Flexible Glue Model– Remove the restriction that G(x, y) = 0 for x!=y
• Multiple Tile Model– tiles may cluster together before being added
• Unique Shape Model– unique shape vs. unique supertile
![Page 14: Complexities for Generalized Models of Self-Assembly](https://reader036.vdocuments.site/reader036/viewer/2022081515/56812a9f550346895d8e5f62/html5/thumbnails/14.jpg)
New Models• Multiple Temperature Model
– temperature may go up and down
• Flexible Glue Model– Remove the restriction that G(x, y) = 0 for x!=y
• Multiple Tile Model– tiles may cluster together before being added
• Unique Shape Model– unique shape vs. unique supertile
![Page 15: Complexities for Generalized Models of Self-Assembly](https://reader036.vdocuments.site/reader036/viewer/2022081515/56812a9f550346895d8e5f62/html5/thumbnails/15.jpg)
New Models• Multiple Temperature Model
– temperature may go up and down
• Flexible Glue Model– Remove the restriction that G(x, y) = 0 for x!=y
• Multiple Tile Model– tiles may cluster together before being added
• Unique Shape Model– unique shape vs. unique supertile
![Page 16: Complexities for Generalized Models of Self-Assembly](https://reader036.vdocuments.site/reader036/viewer/2022081515/56812a9f550346895d8e5f62/html5/thumbnails/16.jpg)
Focus of Talk• Multiple Temperature Model
– Adjust temperature during assembly
• Flexible Glue Model– Remove the restriction that G(x, y) = 0 for x!=y
Goal: Reduce Tile Complexity
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Our Tile Complexity Results
Multiple temperature model:
k x N rectangles: )loglog
log(
N
N
beats standard model: )(/1
k
N k
Flexible Glue:
N x N squares: )log( N
beats standard model: )loglog
log(
N
N (Adleman, Cheng,
Goel, Huang STOC 2001)
(our paper)
(our paper)
(our paper)
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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
)N( /1 kkO Tile Complexity:
N
k
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S
C1 C2 C3
0
g g p
Build a 4 x 256 rectangle: t = 2
C0
g
S3
S2 0
0
S
S1
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S
C1 C2 C3
0 1 2 30
g g p
Build a 4 x 256 rectangle: t = 2
C0
g
S3
S2 0
0
0 0
g
g
S C1 C2 C3
S1
S2
S3
S1
00
0
0g g p
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S
C1 C2 C3
0 1 2 30
g g p
Build a 4 x 256 rectangle: t = 2
C0
g
g g
S3
S2 0
0 1
0 0
g
g
p r0
S C1 C2 C3
S1 0 0
S2
S3
0 0
0 0
0 1
S1
p
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S
C1 C2 C3
0 1 2 30
g g p
Build a 4 x 256 rectangle: t = 2
C0
g
g g
S3
S2 0
0 1
0 0
g
g
p r0
S C1 C2 C3
S1 0 0 0 1
S2
S3
0 0
0 0
0 1
S1
g g
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S
C1 C2 C3
0 1 2 30
g g p
Build a 4 x 256 rectangle: t = 2
C0
g
g g
S3
S2 0
0 1
0 0
g
g
p r0
S C1 C2 C3
S1 0 0 0 1
S2
S3
0 0
0 0
0 1
S1
C0 C1 C2 C3
0 0
0 0
p
![Page 25: Complexities for Generalized Models of Self-Assembly](https://reader036.vdocuments.site/reader036/viewer/2022081515/56812a9f550346895d8e5f62/html5/thumbnails/25.jpg)
S
C1 C2 C3
0 1 2 30
g g p
Build a 4 x 256 rectangle: t = 2
C0
g
g g
S3
S2 0
0 1
0 0
g
g
p r0
S C1 C2 C3
S1 0 0 0 1
S2
S3
0 0
0 0
0 1
S1
C0 C1 C2 C3
0 0
0 0
p1 1
0 0
0 0
1 2
2 3
![Page 26: Complexities for Generalized Models of Self-Assembly](https://reader036.vdocuments.site/reader036/viewer/2022081515/56812a9f550346895d8e5f62/html5/thumbnails/26.jpg)
S
C1 C2 C3
0 1 2 30
g g p
1 2
Pp
3
Build a 4 x 256 rectangle: t = 2
C0
g
g g
R 0
p r
r
S3
S2 0
0 1
2 3
0 0
g
g
p r0
S C0C1 C2 C3
S1 0 0 0 1
C1 C2 C3
1 1 2 2 3 31 2 2 3
S2
S3
0 0
0 0
0 0
0 0
0 0
0 0
0 0
0 0
0 0
0 0
0 0
0 0
0 0
0 0
C0 C1 C2 C3 C0 C1 C2 C3
0 1
S1
p
![Page 27: Complexities for Generalized Models of Self-Assembly](https://reader036.vdocuments.site/reader036/viewer/2022081515/56812a9f550346895d8e5f62/html5/thumbnails/27.jpg)
S
C1 C2 C3
0 1 2 30
g g p
1 2
Pp
3
Build a 4 x 256 rectangle: t = 2
C0
g
g g
R 0
p r
r
S3
S2 0
0 1
2 3
0 0
g
g
p r0
S C0C1 C2 C3
S1 0 0 0 1
C1 C2 C3
1 1 2 2 3 31 2 2 3
S2
S3
0 0
0 0
0 0
0 0
0 0
0 0
0 0
0 0
0 0
0 0
0 0
0 0
0 0
0 0
P
C0 C1 C2 C3 C0 C1 C2 C3
0 1
S1
![Page 28: Complexities for Generalized Models of Self-Assembly](https://reader036.vdocuments.site/reader036/viewer/2022081515/56812a9f550346895d8e5f62/html5/thumbnails/28.jpg)
S
C1 C2 C3
0 1 2 30
g g p
1 2
Pp
3
Build a 4 x 256 rectangle: t = 2
C0
g
g g
R 0
p r
r
S3
S2 0
0 1
2 3
0 0
g
g
p r0
S C0C1 C2 C3
S1 0 0 0 1
C1 C2 C3
1 1 2 2 3 31 2 2 3
S2
S3
0 0
0 0
0 0
0 0
0 0
0 0
0 0
0 0
0 0
0 0
0 0
0 0
0 0
0 0
P
0 1
C0 C1 C2 C3 C0 C1 C2 C3
0 1
S1
![Page 29: Complexities for Generalized Models of Self-Assembly](https://reader036.vdocuments.site/reader036/viewer/2022081515/56812a9f550346895d8e5f62/html5/thumbnails/29.jpg)
S
C1 C2 C3
0 1 2 30
g g p
1 2
Pp
3
Build a 4 x 256 rectangle: t = 2
C0
g
g g
R 0
p r
r
S3
S2 0
0 1
2 3
0 0
g
g
p r0
S C0C1 C2 C3
S1 0 0 0 1
C1 C2 C3
1 1 2 2 3 31 2 2 3
S2
S3
0 0
0 0
0 0
0 0
0 0
0 0
0 0
0 0
0 0
0 0
0 0
0 0
0 0
0 0
P
0 1
C0 C1 C2 C3 C0 C1 C2 C3
0 1
S1
R
![Page 30: Complexities for Generalized Models of Self-Assembly](https://reader036.vdocuments.site/reader036/viewer/2022081515/56812a9f550346895d8e5f62/html5/thumbnails/30.jpg)
S
C1 C2 C3
0 1 2 30
g g p
1 2
Pp
3
Build a 4 x 256 rectangle: t = 2
C0
g
g g
R 0
p r
r
S3
S2 0
0 1
2 3
0 0
g
g
p r0
S C0C1 C2 C3
S1 0 0 0 1
C1 C2 C3
1 1 2 2 3 31 2 2 3
S2
S3
0 0
0 0
0 0
0 0
0 0
0 0
0 0
0 0
0 0
0 0
0 0
0 0
0 0
0 0
P
0 1
C0 C1 C2 C3 C0 C1 C2 C3
0 1
S1
R
C0 C1 C2…
![Page 31: Complexities for Generalized Models of Self-Assembly](https://reader036.vdocuments.site/reader036/viewer/2022081515/56812a9f550346895d8e5f62/html5/thumbnails/31.jpg)
S
C1 C2 C3
0 1 2 30
g g p
1 2
Pp
3
Build a 4 x 256 rectangle: t = 2
C0
g
g g
R 0
p r
r
S3
S2 0
0 1
2 3
0 0
g
g
p r0
S C0C1 C2 C3
S1 0 0 0 1
C1 C2 C3
1 1 2 2 3 31 2 2 3
S2
S3
0 0
0 0
0 0
0 0
0 0
0 0
0 0
0 0
0 0
0 0
0 0
0 0
0 0
0 0
P
0 1
R…
0 0
1 1
0 0 0 0
C0 C1 C2 C3 C0 C1 C2 C3 C0 C1 C2
0 1
S1
![Page 32: Complexities for Generalized Models of Self-Assembly](https://reader036.vdocuments.site/reader036/viewer/2022081515/56812a9f550346895d8e5f62/html5/thumbnails/32.jpg)
S
C1 C2 C3
0 1 2 30
g g p
1 2
Pp
3
Build a 4 x 256 rectangle: t = 2
C0
g
g g
R 0
p r
r
S3
S2 0
0 1
2 3
0 0
g
g
p r00 1
2 2 3 P
P
P
3 3
3 3
3 3
3 3
3 3
21 2
3 3
3 3
3 3
3 3
110 1
3 3
3 3
3 3
3 3
00
3 3
3 3
32
RP
33 3
2
3
3
2
3
C1 C2 C3 C0 C1 C2 C3 C0 C1 C2 C3 C0 C1 C2 C3C0C1 C2 C3
S1
![Page 33: Complexities for Generalized Models of Self-Assembly](https://reader036.vdocuments.site/reader036/viewer/2022081515/56812a9f550346895d8e5f62/html5/thumbnails/33.jpg)
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
)N( /1 kkO Tile Complexity:
N
k
![Page 34: Complexities for Generalized Models of Self-Assembly](https://reader036.vdocuments.site/reader036/viewer/2022081515/56812a9f550346895d8e5f62/html5/thumbnails/34.jpg)
2-temperature model
3
3
3
1
t = 4
k
N
j
k
N
j
![Page 35: Complexities for Generalized Models of Self-Assembly](https://reader036.vdocuments.site/reader036/viewer/2022081515/56812a9f550346895d8e5f62/html5/thumbnails/35.jpg)
2-temperature model
t = 4 6
k
N
j
![Page 36: Complexities for Generalized Models of Self-Assembly](https://reader036.vdocuments.site/reader036/viewer/2022081515/56812a9f550346895d8e5f62/html5/thumbnails/36.jpg)
2-temperature model
k
N
)( /1 jNjO )loglog
log(
N
NO
)loglog
log(
N
NKolmogorov Complexity (Rothemund,
Winfree STOC 2000)
Beats Standard Model )(/1
k
N k
k
N
(our paper)
(our paper)
![Page 37: Complexities for Generalized Models of Self-Assembly](https://reader036.vdocuments.site/reader036/viewer/2022081515/56812a9f550346895d8e5f62/html5/thumbnails/37.jpg)
Assembly of N x N Squares
![Page 38: Complexities for Generalized Models of Self-Assembly](https://reader036.vdocuments.site/reader036/viewer/2022081515/56812a9f550346895d8e5f62/html5/thumbnails/38.jpg)
Assembly of N x N Squares
kN - k
N - k
k
![Page 39: Complexities for Generalized Models of Self-Assembly](https://reader036.vdocuments.site/reader036/viewer/2022081515/56812a9f550346895d8e5f62/html5/thumbnails/39.jpg)
Assembly of N x N Squares
kN - k
N - k
X
Y
k
)loglog
log(
N
N
Complexity:
(Adleman, Cheng,Goel, Huang STOC 2001)
![Page 40: Complexities for Generalized Models of Self-Assembly](https://reader036.vdocuments.site/reader036/viewer/2022081515/56812a9f550346895d8e5f62/html5/thumbnails/40.jpg)
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)
![Page 41: Complexities for Generalized Models of Self-Assembly](https://reader036.vdocuments.site/reader036/viewer/2022081515/56812a9f550346895d8e5f62/html5/thumbnails/41.jpg)
N x N Square --- Flexible Glue Model
log N
N – log N
seed row
![Page 42: Complexities for Generalized Models of Self-Assembly](https://reader036.vdocuments.site/reader036/viewer/2022081515/56812a9f550346895d8e5f62/html5/thumbnails/42.jpg)
N x N Square --- Flexible Glue Model
log N
N – log N
seed row
)(log)(
2
log
/1
/1
NONkO
N
Nk
k
k
Complexity:
![Page 43: Complexities for Generalized Models of Self-Assembly](https://reader036.vdocuments.site/reader036/viewer/2022081515/56812a9f550346895d8e5f62/html5/thumbnails/43.jpg)
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
![Page 44: Complexities for Generalized Models of Self-Assembly](https://reader036.vdocuments.site/reader036/viewer/2022081515/56812a9f550346895d8e5f62/html5/thumbnails/44.jpg)
. . . 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
![Page 45: Complexities for Generalized Models of Self-Assembly](https://reader036.vdocuments.site/reader036/viewer/2022081515/56812a9f550346895d8e5f62/html5/thumbnails/45.jpg)
. . . 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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2 x log N block
N – log N
log N
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N – log N
N – log Nlog N
log N
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N – log N
N – log N
X
Y
log N
log N
)log( N
Complexity:
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Shape Verification
Unique Shape 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 (our paper)
Multiple Temperature: NP-hard (our paper)
Multiple Tile: Co-NPC (our paper)
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x1
* *
x2
x3
* T T T T
ok
ok
ok
okok
ok
c2c1
c2
c2
c3 *
*
*
*
c10
1
1
SAT
x1
* *
x2
x3
* T T F F
ok
ok
ok
c2ok
ok
c2c1
c2
c2
c3 *
*
*
*
c10
0
1
Satisfied Not Satisfied
(LaBean and Lagoudakis, 1999)
Unique-Shape Model
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* *
x1
x2
x3
c1 c2 c3
*
*
*
*
*
* *
x1
x2
x3
c1 c2 c3
*
*
*
*
*
Satisfied Not Satisfied
**
Multiple Temperature Model
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* *
x1
x2
x3
c1 c2 c3
*
*
0
1
c1
ok
c2
c2
1
T
ok
T
ok
ok
ok
T
ok
T
*
*
*
SAT
*
*
*
* *
x1
x2
x3
c1 c2 c3
*
*
0
1
c1
ok
c2
c2
0
T
ok
T
ok
ok
c2
F
ok
F
*
*
*
NO
*
*
*
Satisfied Not Satisfied
**
Multiple Temperature Model
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Satisfied Not Satisfied
Multiple Temperature Model
* *
x1
x2
x3
c1 c2 c3
*
*
0
1
c1
ok
c2
c2
1
T
ok
T
ok
ok
ok
T
ok
T
*
*
*
SAT
*
*
*
*
* *
x1
x2
x3
c1 c2 c3
*
*
0
1
c1
ok
c2
c2
0
T
ok
T
ok
ok
c2
F
ok
F
*
*
*
NO
*
*
*
*
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Satisfied Not Satisfied
Multiple Temperature Model
*
x1
x2
x3
*
*
*
*
*
*
x1
x2
x3
*
*
*
*
*
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Our Tile Complexity Results
Multiple temperature model:
k x N rectangles: )loglog
log(
N
N
beats standard model: )(/1
k
N k
Flexible Glue:
N x N squares: )log( N
beats standard model: )loglog
log(
N
N (Adleman, Cheng,
Goel, Huang STOC 2001)
(our paper)
(our paper)
(our paper)
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Unique Shape Problem Results
Standard P
Flexible Glue P
Multiple Temperature NP-hard
Unique Shape Co-NPC
Multiple Tile Co-NPC
(Adleman, Cheng, Goel, Huang, Kempe,Espanes, Rothemund, STOC 2002)
(our paper)
(our paper)
(our paper)
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Further Research
• time complexity– multiple temperature– multiple tile
• more than 2 temperatures– raising/lowering temperature many times– monotonically increasing temperatures
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