One-Dimensional Staged Self-Assembly

Erik Demaine, Sarah Eisenstat, Mashhood Ishaque, Andrew Winslow

Funding in part by NSF grant CBET-0941538.

Setting

• Theoretic tile model of assembling labeled 1D structures.• The problem addressed is an optimization problem:

``Given structure C, find the most efficient system that builds C.’’

• The results will be algorithmic/complexity-theoretic:

``The assembly model is equivalent to context-free grammars’’

``A good algorithm for finding the smallest SAS exists’’

Staged Tile Assembly Model

Tiles are labeled squares with glues on their sides. BA

Tiles are mixed in bins and bond at common glues. AB

Mixing events are done in sequential stages.ABC

C

The composite objects created are assemblies/strings. ABC

A F

AF

A BF

AF

F A B

A B

BA

C

B C

E

BA E B C

FA

?

BA

?

C

Second part

Specifying a Self-Assembly System (SAS)

A directed graph of mixing steps.

A set of starting tiles. A EB C DF

The size of a SAS is the number of edges.

Problem and Approach

Problem: given an assembly S, find the smallest self-assembly system uniquely constructing S.

Approach: self-assembly systems as encodings of strings:

≈ A EB C DF ≈ “FABCED”

Context-Free Grammars (CFGs)

S1

D

S2

F A C EB

S4

S3

S5

S1 S2DS2 S3S5

S3 S4BS4 FAS5 CE CFG deriving the string FABCED

Size of CFG

Converting between SAS and CFG

A F B C E D A F B C E D

S4

S3

S5

S2

S1

CFG and SAS Equivalence

The problem is SAS minimization: given a labeled assembly, find the smallest SAS that produces it.

There are good algorithms for finding small CFGs and they cannot be improved much.

Using the conversion, algorithms for finding small CFGs are equivalent to algorithms for finding small SASs.

Theorem: there are good algorithms for the SAS minimization problem and they cannot be improved much.

A General Mixing Example

BA CAssemblies combine in all possible ways.

B C

BA C

Mixing starts with all input assemblies.

C A

B C AB C A The mixing products are the distinctterminal assemblies.

B C A

Why This Definition Matters

Previous result only applied to mixings with one product.

In the general model, mixings may have many products.

BA

BA

BA C

AC

AB

Using many-product mixings can result in much smaller systems for some assemblies.

Assemblies with Small SASs

A static string

An interleaved string Shift permutations of the interleaving

A composition of these interleavings

There is a SAS for this string using many-product mixings withsize ≈ √(n/log(n)), n the string length.

The string constructed has the structure:

ADBECF AFBDCE AEBFCD

Interleaving Shift Permutations Efficiently with SASs

Idea: Tiles with same label, different glues.

Implementation: add unlabeled tiles to “swap” glues.

Do all shift perm. glue swaps in one many-product mixing.

A B C

D D D

ADBECF AEBFCDAFBDCE

D

D

D

D

Why Can No CFG Encode This String Well?

Use Lempel-Ziv factorization from compression theory:

Rytter (2005): The size of the Lempel-Ziv factorization of a string is smaller than the size of any CFG generating it.

ADBECF AEBFCD AFBDCE

Thus any CFG for the string has size ≈ n/log(n).

ADBE

CF

AEBF

CD

AFBD

CE≈ n/log(n) factors

CFG and SAS Non-equivalence

Under the general model that allows many-product mixings,some assemblies have:

If many-product mixings are allowed, some strings require a quadratically larger CFG to encode than a SAS.

A SAS with size ≈ √(n/log(n)).

No CFG with size smaller than ≈ n/log(n).

Theorem: some strings admit quadradically-smaller SASs if many-product mixings are allowed.

Summary of Results

Theorem: if many-product mixings are allowed, some strings admit much smaller SASs than otherwise.

Theorem: if many-product mixings are not allowed, there are good algorithms for the SAS minimization problem and they cannot be improved much.

Thank you.