1. Combinatorial Optimization Problems in Self-Assembly (Given a shape, output an “efficient” tile-system for assembling it)

2. Shapes vs. Assemblies B C A Seed Assembly Shape d 2/23/2019

3. Combinatorial Optimization Problems
Program Size: Minimum Tile Set Problem
Given a shape, find the smallest tile-system that uniquely assembles into this shape
Running Time: Tile Concentrations Problem
Given a shape, and a tile system that uniquely assembles into this shape, assign concentrations to each tile-type to minimize the expected assembly time
2/23/2019

4. Minimum Tile Set Problem: Definitions
Given a tile system G
|G| is the number of tile-types
Let SG denote the single-tile assembly which contains the seed tile for G
Let A1 be an assembly which contains the seed. If A2 can be obtained by adding a single tile from G to A1, then A1 !G A2
A finite assembly A is uniquely produced by G if
SG !G* B ) B !G* A for all B
A shape X is uniquely produced by G if G uniquely produces an assembly A with shape X
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5. Minimum Tile Set Problem [Contd.]
Problem: Given a shape X, find a tile system G which uniquely produces X and minimizes |G|
The size of the problem is the number of tiles in X
Results:
Polynomial time tractable for trees and squares
NP-Hard in general: Reduction from SAT; the polynomial time algorithm for trees is used as a subroutine in the reduction (also use ideas from [LaBean and Lagoudakis ] )
The corresponding decision problem in in NP. Given a shape and a tile system, we can determine in polynomial time whether the tile system uniquely produces the given shape (Program verification, for comparison, is not even a decidable problem!)
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6. Membership in NP
Let the input shape X have N tiles. Let G be the “guessed” tile system
First, simulate the self-assembly process for N-1 accretion steps, starting from the seed tile for G and using tiles from G
Let A be the resulting assembly
If A does not have the same shape as X, then answer “NO”
If A is not terminal, answer “NO”
We now need to verify that A is uniquely produced
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7. Membership in NP [Contd.]
For each position (x,y) in A
Remove the tile at position (x,y) from A to obtain Ax,y
Simulate the self-assembly process again starting from the seed tile and using Ax,y as a template till no more tiles can be added
All the tiles added should have the same type as the tile in the corresponding position in Ax,y
If more than one tile-type can now be attached at position (x,y), then answer “NO” and exit
Answer “YES”
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8. Minimum Tile Set Problem [Contd.]
Theorem: The above algorithm decides whether G uniquely produces X in time O(|X|2 + |X|¢|G|)
Several interesting generalizations studied by Rothemund, Winfree, Cook, and Kempe (personal communication)
Open Problems:
More general classes of shapes than squares and trees?
Convex (ie. rectangles)
recti-convex (convex under axis-parallel lines)
shapes without holes
Approximation algorithms?
2/23/2019

9. Tile Concentrations Problem
Given a shape, and a tile system that uniquely assembles into this shape, assign concentrations to different tile-types to minimize the expected assembly time
Non-trivial, even for simple shapes
NP-Hard? Open.
Conjecture: #P-Hard
Let xT be the concentration of tile type T
Let R(x) be an exponential random variable with mean 1/x
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10. Tile Concentrations Problem: ExampleD C 3
Constraint: xA + xB + xC +xD + xE = 1 A C 1 C 2 S B E
Expected time = E[Max{RB(xB) + RC1(xC) + RC2(xC),
RB(xB)+RE(xE) + RC2(xC),
RA(xA) +RC1(xC)+RC2(xC),
.…..}]
Our approach: Replace R(x) by 1/x to obtain a convex program
=> O(log n) approximation for large classes of shapes
2/23/2019

11. Tile Concentrations Problem: ExampleD C 3
Constraint: xA + xB + xC +xD + xE = 1 A C 1 C 2 S B E
Expected time = E[Max{RB(xB) + RC1(xC) + RC2(xC),
RB(xB)+RE(xE) + RC2(xC),
RA(xA) +RC1(xC)+RC2(xC),
.…..}]
Our approach: Replace R(x) by 1/x to obtain a convex program
=> O(log n) approximation for large classes of shapes
2/23/2019

12. Tile Concentrations Problem: ExampleD C 3
Constraint: xA + xB + xC +xD + xE = 1 A C 1 C 2 S B E
Expected time = E[Max{RB(xB) + RC1(xC) + RC2(xC),
RB(xB)+RE(xE) + RC2(xC),
RA(xA) +RC1(xC)+RC2(xC),
.…..}]
Our approach: Replace R(x) by 1/x to obtain a convex program
=> O(log n) approximation for large classes of shapes
2/23/2019

13. More Open Problems: Shape Languages
Motivation: characterize “assemblable” languages
Given a tile system G, let S(G) be the set of all terminal shapes that G can assemble into
S(G) is a shape language
Open Problem: Complexity of testing membership in a shape language?
Partial Progress: O(n) space, polynomial time suffices, where n is the number of tiles in the input shape
Open Problem: Natural automata that recognize shape languages?
Eg: DFAs recognize 1-D shape languages
2/23/2019

14. Shape Languages vs. 4-Way Automata
4-Way Finite Automaton: Can move Left, Right, Up, Down on an input shape, and must either accept or reject the shape
4-Way automata can not accept all shape languages
4-Way automata can’t traverse mazes; the set of all mazes is (roughly) a shape language
Not all 4-Way Automata languages are shape languages
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15. Shape Languages vs. 4-Way Automata
C-Language: Vertical Lines with matchings “knobs” at the end
Easily recognized using
4-Way automata; but not
a shape language
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16. Robustness
In practice, self-assembly is a thermodynamic process. When T=2, tiles with 0 or 1 matches also attach; tiles held by total strength 2 also fall off at a small rate.
Currently, there are 1-10% errors observed in experimental self-assembly. [Winfree, Bekbolatov, ’03]
0.3% now
Need error rates of 10-12, or better, for a postage stamp
A major bottleneck for practical self-assembly
For example, one bit flip can double the height of a counter
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17. The Source of Errors: Thermodynamics
GA = Activation energy
GB = Bond energy +
2GB GB GA GA
Correct Growth
Incorrect Growth
Rate of correct growth ¼ exp(-GA)
Probability of incorrect growth ¼ exp(-GA + GB)
Constraint: 2 GB > GA (system goes forward)
) Rate has quadratic dependence on error probability
) Time to reliably assemble an n £ n square ¼ n5
2/23/2019