Notes on the optimal encoding scheme for self-assembly

Days 10, 11 and 12 Of Comp Sci 480

Question

How do you encode an n-bit binary string?

Answer

Have a unique tile type to encode each bit separately, so we represent 1 bit per tile.

We’ve seen this before.

Is this optimal?

NO!

A brief history

• Rothemund and Winfree (2000): 1 bit per tile• Adleman, et. al. (2002): > 1 bit per tile (an

optimal number of bits per tile)– Works at temperature 3

• Soloveichik and Winfree (2004): Modified optimal encoding scheme of Adleman, et. al. to work at temperature 2– Encodes n bits using O(n/log n) tile types

Basic idea

• Start with a binary string x = xn-1xn-2 x∙∙∙ 0

• Choose k such that it is the smallest number satisfying n/log n ≤ 2k

• More math later…

• Break w up into n/k k-bit substrings and store each substring in a unique tile type

• Extract the bits from all the substrings until you have 1 bit per tile

• Use the value to start a binary counter, or something else…

An example (some details missing)…

For the sake of example, let k = 4 for the binary string x = 01100101

0L

0L

0

$

+

0

0

0101

101$

R

R

101

R

R

101

01

R

R

01

1

101

01

R

R

0

0

0

0

1

1

1

1

0L

0L

11R

R

100

01

100

0

00

0

11

1

11

1

00L

0

11

1

Can use the bits encoded in the north glues of the topmost row as the start value for a binary counter (or an input to a Turing machine)!

0101

R

R

0101

+

0101

0101+

10

+

+

10

0L

0L

1

10

110

R

R

0101

R

R

0101

+

0101

0101+

110

+

+

110

0L

110

0110L

R

1

0101

12

+

23

0110L

3

1

1

1

0

10

R

R

The general construction…

The general construction

Unpacking tiles broken down into several logical groups1. Seed row2. Extract a bit3. Copy substring being extracted4. Initiate unpacking of a new substring5. Copy substring not being extracted6. Copy a bit7. The right side

The seed row

wm-1L

2m-1

+

2m-22m-1

wm-2

2m-32m-2

+

2m-42m-3

w0

01

R

0

Since x = xn-1xn-2∙∙∙x0, we let m = n/k, i.e., the number of k-bit substrings in x.

Create these tile types:

Write x = wm-1wm-2∙∙∙w0 as the concatenation of m k-bit binary substrings, with wm-1 padded to the left with leading 0’s if needed.

The general construction

Unpacking tiles broken down into several logical groups Seed row2. Extract a bit3. Copy substring being extracted4. Initiate unpacking of a new substring5. Copy substring not being extracted6. Copy a bit7. The right side

Extract bit

For all x ϵ {0,1}j where 0 < j ≤ k-1 and b ϵ {0,1}, create the following tile types:

Notation: {0,1}j means “the set of all length j binary strings”

0110L

3

+

23

0101

12

R

1

0L

110

0110L

0101

R

R

0101

R

R

0L

0L

0101

R

R

0101

R

R

110

+

+

110

10

+

+

10

+

0101

0101+

+

0101

0101+

1

10

110

0L

0L

1

1

0

$

+

0

1

0

10

0

0101

101$

R

R

101

R

R

101

01

R

R

01

1

101

01

R

R

0

0

0

0

1

1

1

1

0L

0L

11R

R

100

01

100

0

00

0

11

1

11

1

00L

0L

11

1

bL

x

bxL

b

x

bx

The general construction

Unpacking tiles broken down into several logical groups Seed row Extract a bit3. Copy substring being extracted4. Initiate unpacking of a new substring5. Copy substring not being extracted6. Copy a bit7. The right side

Copy substring being extracted

For all x ϵ {0,1}j where 1 < j ≤ k-1, create the following tile types:

0110L

3

+

23

0101

12

R

1

0L

110

0110L

0101

R

R

0101

R

R

0L

0L

0101

R

R

0101

R

R

110

+

+

110

10

+

+

10

+

0101

0101+

+

0101

0101+

1

10

110

0L

0L

1

1

0

$

+

0

1

0

10

0

0101

101$

R

R

101

R

R

101

01

R

R

01

1

101

01

R

R

0

0

0

0

1

1

1

1

0L

0L

11R

R

100

01

100

0

00

0

11

1

11

1

00L

0L

11

1

x

+

+

x

x

R

R

x

The general construction

Unpacking tiles broken down into several logical groups Seed row Extract a bit Copy substring being extracted4. Initiate unpacking of a new substring5. Copy substring not being extracted6. Copy a bit7. The right side

Initiate unpacking of a new substring

For all x ϵ {0,1}k-1 and b ϵ {0,1}, create the following tile types:

0110L

3

+

23

0101

12

R

1

0L

110

0110L

0101

R

R

0101

R

R

0L

0L

0101

R

R

0101

R

R

110

+

+

110

10

+

+

10

+

0101

0101+

+

0101

0101+

1

10

110

0L

0L

1

1

0

$

+

0

1

0

10

0

0101

101$

R

R

101

R

R

101

01

R

R

01

1

101

01

R

R

0

0

0

0

1

1

1

1

0L

0L

11R

R

100

01

100

0

00

0

11

1

11

1

00L

0L

11

1

b

x

bx

$

The general construction

Unpacking tiles broken down into several logical groups Seed row Extract a bit Copy substring being extracted Initiate unpacking of a new substring5. Copy substring not being extracted6. Copy a bit7. The right side

Copy substring not being extracted

For all x ϵ {0,1}k, create the following tile types:

0110L

3

+

23

0101

12

R

1

0L

110

0110L

0101

R

R

0101

R

R

0L

0L

0101

R

R

0101

R

R

110

+

+

110

10

+

+

10

+

0101

0101+

+

0101

0101+

1

10

110

0L

0L

1

1

0

$

+

0

1

0

10

0

0101

101$

R

R

101

R

R

101

01

R

R

01

1

101

01

R

R

0

0

0

0

1

1

1

1

0L

0L

11R

R

100

01

100

0

00

0

11

1

11

1

00L

0L

11

1

+

x

x

+

x

+

+

x

x

R

R

x

The general construction

Unpacking tiles broken down into several logical groups Seed row Extract a bit Copy substring being extracted Initiate unpacking of a new substring Copy substring not being extracted6. Copy a bit7. The right side

Copy bit

For all b ϵ {0,1}, create the following tile types:

0110L

3

+

23

0101

12

R

1

0L

110

0110L

0101

R

R

0101

R

R

0L

0L

0101

R

R

0101

R

R

110

+

+

110

10

+

+

10

+

0101

0101+

+

0101

0101+

1

10

110

0L

0L

1

1

0

$

+

0

1

0

10

0

0101

101$

R

R

101

R

R

101

01

R

R

01

1

101

01

R

R

0

0

0

0

1

1

1

1

0L

0L

11R

R

100

01

100

0

00

0

11

1

11

1

00L

0L

11

1

bL

bL

b

b

b

$

+

b

The general construction

Unpacking tiles broken down into several logical groups Seed row Extract a bit Copy substring being extracted Initiate unpacking of a new substring Copy substring not being extracted Copy a bit7. The right side

The right side

Create the following tile types:

0110L

3

+

23

0101

12

R

1

0L

110

0110L

0101

R

R

0101

R

R

0L

0L

0101

R

R

0101

R

R

110

+

+

110

10

+

+

10

+

0101

0101+

+

0101

0101+

1

10

110

0L

0L

1

1

0

$

+

0

1

0

10

0

0101

101$

R

R

101

R

R

101

01

R

R

01

1

101

01

R

R

0

0

0

0

1

1

1

1

0L

0L

11R

R

100

01

100

0

00

0

11

1

11

1

00L

0L

11

1

0R

R

0

R

R

1R

R

1

Tile complexity (math)

Setup

For an arbitrary length n bit string w, represented as n/k k-bit substrings, how many tile types do we need to extract w?

Reminder: n is the number of bits in the input binary string.

Reminder: k chosen such that it is the smallest number satisfying n/log n ≤ 2k

A key fact

• 2k < 2n/log n• Assume otherwise…• 2k ≥ 2n/log n 2k-1 ≥ n/log n

– Contradiction to the definition of k being the smallest such number with n/log n ≤ 2k

• Seed row has 2*n/k unique tile types• We know 2k ≥ n/log n by our choice of k• Therefore: k ≥ log(n/log n) ; take the log of both sides of the

previous inequality• Then we have, for “large” values of n:

2*n/k≤ 2*n/log(n/log n)= 2*n/(log n - log log n) ; log (a/b) = log a - log b< 2*n/(log n - 0) ; 0 < log log n for n ≥ 4≤ 2n/log n + 1 ; x ≤ x + 1 for any x≤ 2n/log n + n/log n ; n/log n ≥ 1 for n ≥ 2= 3n/log n= O(n/log n)

The seed row

Tile complexity of the general construction

Unpacking tiles broken down into several logical groups Seed row: O(n/log n)2. Extract a bit3. Copy substring being extracted4. Initiate unpacking of a new substring5. Copy substring not being extracted6. Copy a bit7. The right side

Extract a bit• We need to compute 2*(2(|{0,1}1| + |{0,1}2| + + |{0,1}∙∙∙ k-1|))• 2*(2(|{0,1}1| + |{0,1}2| + + |{0,1}∙∙∙ k-1|)) = 2*(2*(2+4+ +2∙∙∙ k-1))

= 4*(2k - 2)< 4*2k

< 4*2*n/log n= O(n/log n)

bL

x

For all x ϵ {0,1}j where 0 < j ≤ k-1 and b ϵ {0,1}, create the following tile types:

Notation: {0,1}j means “the set of all length j binary strings” bxL

b

x

bx

Tile complexity of the general construction

Unpacking tiles broken down into several logical groups Seed row: O(n/log n) Extract a bit O(n/log n)3. Copy substring being extracted4. Initiate unpacking of a new substring5. Copy substring not being extracted6. Copy a bit7. The right side

Copy substring being extracted

x

+For all x ϵ {0,1}j where 1 < j ≤ k-1 and b ϵ {0,1}, create the following tile types: +

x

x

R

R

x

• We need to compute 2*(2(|{0,1}2| + |{0,1}3| + + |{0,1}∙∙∙ k-1|))• 2*(2(|{0,1}2| + |{0,1}3| + + |{0,1}∙∙∙ k-1|)) = 2*(2*(4+8+ +2∙∙∙ k-1))

= 4*(2k - 4)< 4*2k

< 4*2*n/log n= O(n/log n)

Tile complexity of the general construction

Unpacking tiles broken down into several logical groups Seed row: O(n/log n) Extract a bit O(n/log n) Copy substring being extracted O(n/log n)4. Initiate unpacking of a new substring5. Copy substring not being extracted6. Copy a bit7. The right side

Initiate unpacking of a new substring

b

xFor all x ϵ {0,1}k-1, and b ϵ {0,1}, create the following tile types:bx

$

• The size of the set {0,1}k-1 is 2k-1, and b can take on two possible values, so the total number of tile types created in this step is:

2*2k-1 = 2k < 2n/log n = O(n/log n)

Tile complexity of the general construction

Unpacking tiles broken down into several logical groups Seed row: O(n/log n) Extract a bit O(n/log n) Copy substring being extracted O(n/log n) Initiate unpacking of a new substring O(n/log n)5. Copy substring not being extracted6. Copy a bit7. The right side

Copy substring not being extracted

+

xFor all x ϵ {0,1}k, create the following tile types:x

+

x

+

+

x

x

R

R

x

• The size of the set {0,1}k is 2k, so the total number of tile types created in this step is:

3*2k < 3*2n/log n = O(n/log n)

Tile complexity of the general construction

Unpacking tiles broken down into several logical groups Seed row: O(n/log n) Extract a bit O(n/log n) Copy substring being extracted O(n/log n) Initiate unpacking of a new substring O(n/log n) Copy substring not being extracted O(n/log n) Copy a bit O(1) The right side O(1)

TOTAL: O(n/log n)

Summary

• We can encode an n-bit binary string using O(n/log n) unique tile types

Application to squares

• How do we apply the optimal encoding scheme to the self-assembly of an NxN square?

• What is the resulting tile complexity?– n = log N– Can encode n bits using O(n/log n) tile types– Tile complexity of square: O(log N / log log N)

• Homework problem

Improvement

• I presented the construction using “spacer” tiles in between each k-bit substring

• Can we remove the spacer tiles?• This wouldn’t give us an asymptotical

improvement over O(n/log n), but it might make implementation easier

• Definitely worth thinking about…

Improved (?) construction

Need to modify seed row:wm-1L

2m-1

+

2m-22m-1

wm-2

2m-32m-2

+

2m-42m-3

w0

01

R

0

wm-1L

2m-1

wm-2

2m-22m-1

w0R

01

Still get O(n/log n) tile types after deleting the “spacer” tile types.Need to combine “Copy substring being extracted” with “Copy substring not

being extracted” groups…

x

y

y

x

Among others, we will at least need these tile types… For all x ϵ {0,1}j, where 1 < j ≤ k-1, and for all y ϵ {0,1}k create the following tile types…

BEFORE…x

+

+

x

x

R

R

x

+

x

x

+

x

+

+

x

x

R

R

x

Copy substring being extracted Copy substring NOT being extracted

What is the tile complexity of just this group of tile types?

Tile complexity of modification• Tile complexity of our previous modification:“For all x ϵ {0,1}j, where 1 < j ≤ k-1, and for all y ϵ {0,1}k create the following tile types…”

|{0,1}j | 1 < j ≤ k-1 }|*|{0,1}k| = (2k-2) 2k

= 2k2k-2*2k

> 22k - 2*(2*n/log n)

= (2k)2- 2*(2*n/log n)≥ (n/log n)2 - 1/2*(n/log n)2

= 1/2*(n/log n)2

= Ω(n2/log2 n)= No good!

• It seems like the spacer tiles are needed, unless drastic changes are made.