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USC Computer Science Technical Reports, no. 802 (2003)

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Content Running Time and Program Size for Self-assembled Squares

Leonard Adleman
y z
University of Oklahoma
Ashish Goel
x
Stanford University
Ming-Deh Huang
{
University of Southern California
Pablo Moisset de Espan´ es
k
University of Southern California
University of Southern California
Abstract
Recently Rothemund and Winfree [8] have considered the program size complexity of constructing
squares by self-assembly. Here, we consider the time complexity of such constructions using a natural
generalization of the Tile Assembly Model defined in [8]. In the generalized model, the Rothemund-
Winfree construction of n n squares requires time (nlog n) and program size (log n). We present
a new construction for assembling n n squares which uses optimal time (n) and program size
(
logn
log logn
). This program size is also optimal since it matches the bound dictated by Kolmogorov
complexity. Our improved time is achieved by demonstrating a set of tiles for parallel self-assembly
of binary counters. Our improved program size is achieved by demonstrating that self-assembling sys-
tems can compute changes in the base representation of numbers. Self-assembly is emerging as a useful
paradigm for computation. In addition the development of a computational theory of self-assembly
promises to provide a new conduit by which results and methods of theoretical computer science might
be applied to problems of interest in biology and the physical sciences.
1 Introduction
Self-assembly is the ubiquitous process by which objects autonomously assemble into complexes. Na-
ture provides many examples: Atoms react to form molecules. Molecules react to form crystals and
supramolecules. Cells sometimes coalesce to form organisms. It has been suggested that self-assembly will

A preliminary version of this paper appeared in the 33rd ACM Symposium on Theory of Computing, 2001.
y
Professor of Computer Science and Molecular Biology, University of Southern California. Research supported by grants from
NASA/JPL, NSF, ONR, and DARPA. Email: adleman@usc.edu
z
Qi Cheng is at the Department of Computer Science, University of Oklahoma. Research supported by an NSF CAREER award.
Email: qcheng@ou.edu.
x
Ashish Goel is at the Department of Management Science and Engineering and (by courtesy) Computer Science, Stanford
University, Terman 311, Stanford CA 94305. Email: ashishg@stanford.edu. Research supported by NSF CAREER Award 0133968
and a grant from NASA.
{
Ming-Deh Huang is at the Department of Computer Science, University of Southern California. Email: huang@usc.edu.
Research supported in part by a grant from NASA and by NSF Grant CR-9820778.
k
Pablo Moisset de Espan´ es is at the Department of Computer Science, University of Southern California. Email: pmois-
set@usc.edu. Research supported in part by a grant from NASA/JPL.
1
ultimately become an important technology, enabling the fabrication of great quantities of intricate objects
such as computer circuits from inexpensive components such as DNA and inorganic nanocrystals. Despite
its importance, self-assembly is poorly understood. Recently, work related to DNA computation has led
to experimental systems for the investigation of self-assembly and its relation to computation [10, 6, 4, 3].
In addition certain theoretical aspects of self-assembly have been considered. Winfree [10, 11] proved that
self-assembling tile systems in a plane are capable of doing universal computation, and when restricted to
a line are exactly as powerful as discrete finite automata. Adleman [1] proposed a mathematical model
of self-assembly and analyzed the time complexity of linear polymerization. Rothemund and Winfree [8]
proposed the Tile Assembly Model of self-assembly and studied the program size complexity (the num-
ber of different tile-types used) of constructing n n squares. In this paper we extend the Tile Assembly
Model to include the time complexity of the assembly process. We then demonstrate a system of tiles
which assembles into an n n square while simultaneously achieving optimal ((n)) time and optimal
program size ((log n= log log n)). In contrast, the system proposed by Rothemund and Winfree takes time
(n log n) and uses (log n) program-size in the worst case. It is our hope that an understanding of simple
self-assembling systems will pave the way for a general theory of self-assembly.
To achieve the improved time complexity, we show how a binary counter that counts from 0 to n can
be assembled in expected time (n), as opposed to the (n log n) time for the same assembly in the work
of Rothemund and Winfree. Further, the assembly time has an exponentially decaying tail. The number
of increment steps required to count from 0 to n is exactly n. Thus our construction takes an amortized
time of (1) for an increment step, even though each increment step requires an addition of (log n) new
tiles. This is made possible by the parallelism inherent in our tile-assembly system. We also observe that no
system can result in a square being assembled in expected time less than
(n) in the model suggested by
Rothemund and Winfree and expanded by us in this paper.
In order to count to n, a log n-bit counter is required. Most of the
log n) tiles in the construction of
Rothemund and Winfree were used to produce the first row (the seed row) in the counter. The increment
steps in their counter assembly (and in ours) can be performed using a constant number of tiles, and the
fully assembled counter, which is roughly a log n n rectangle can be completed into a square using a con-
stant number of tiles. To eliminate the need to produce a seed-row that is log n bits long, we represent the
number n in base (log n= log log n) using only (log n= log log n) digits. Thus if the counter were to use
base (log n= log log n) rather than binary, only (log n= log log n) tiles would be required to assemble a
counter. This observation immediately allows us to reduce the number of tiles (the program size) required to
assemble a square to (log n= log log n), which matches the lower bound dictated by Kolmogorov complex-
ity. Unfortunately, this program size reduction comes at the cost of increased assembly time. To remedy this,
we introduce the notion of base conversion. We start out with a row of (log n= log log n) tiles representing
a number between 0 and n in base (log n= log log n). We then convert this number into binary using a
self-assembly process that simulates base conversion. Now the counting process can take place in binary,
allowing us to achieve the optimal time complexity and optimal program-size complexity simultaneously.
Section 2 presents a natural extension of the tile-assembly model of Rothemund and Winfree to include
the time-complexity of self-assembly. Section 3 shows how a log n n counter can be assembled in time
(n) and section 5 explains how base conversion can be simulated in the tile-assembly model. Section 6
sketches how an entire nn square can be constructed using the tools outlined in sections 3 and 5. Section 7
describes a more general running time analysis technique. Finally, we present some open problems and
conclusions in section 8.
2
2 Adding time complexity to the Tile Assembly Model
The Tile Assembly Model was originally proposed by Rothemund and Winfree [8]. It extends the theoretical
model of tiling by Wang [9] to include a mechanism for growth based on the physics of molecular self-
assembly. Since self-assembly is an emerging research field, there no widely accepted terminology. For this
reason the notation we use in this paper is somewhat ad-hoc and may not be appropriate to describe other
self-assembly problems such as the combinatorial optimizations presented in [2].
Informally each unit of an assembly is a square with glues of various types on each edge. The tile ”floats”
on a two dimensional plane and when two tiles collide they stick if their abutting sides have compatible glues.
Formally, a tile is an oriented unit square with the north, east, south and west edges labeled from some
alphabet of glues. We begin with a triple hT; g; i where T is a finite set of tiles, 2 Z
>0
is the
temperature, and g is the glue strength function from to N, where is the set of edge labels and N is
the set of natural numbers. It is assumed that null2 , g(x; y) = g(y; x) for x; y 2 , and g(null; x) = 0
for all x2 . For each tile i2 T , the labels of its four edges are denoted
N
(i),
E
(i),
S
(i), and
W
(i).
Given p = (x; y); p
0
= (x
0
; y
0
)2 Z
2
, we say p and p
0
are position adjacent iffjx x
0
j +jy y
0
j = 1.
A shape S is a finite subset of Z
2
, such that for all pairs of positions p, p
0
in S, either p = p
0
or there exists
a sequence of positions p
1
; p
2
;; p
n
such that p
1
= p, p
n
= p
0
and for all 1 k < n, p
k
and p
k+1
are
position adjacent. A configuration is a map from Z
2
to T[femptyg. For t2 T ,
(x;y)
t
is the configuration
such that
(x;y)
t
(i; j) = t iff (i; j) = (x; y) and empty otherwise. Let C and D be two configurations.
Suppose there exist some i 2 T and (x; y) 2 Z
2
such that C(x; y) = empty, D = C except at (x; y),
D(x; y) = i, and
g(
E
(i);
W
(D(x + 1; y)) + g(
W
(i);
E
(D(x 1; y)) +
g(
N
(i);
S
(D(x; y + 1)) + g(
S
(i);
N
(D(x; y 1)) :
Then we say that the position (x; y) in C is attachable, and we write C !
T
D to denote the transition from
C to D in attaching tile i to C at position (x; y). Informally, C !
T
D iff D can be obtained from C by
adding a tile to it such that the total strength of interaction in adding the tile to C is at least .
A seed configuration is a configuration of T such thatfpj(p)6= emptyg is a shape.
A tile system is a quadruple T =hT; S; g; i, where T; g; are as above and S is a set of seed configu-
rations called seed s-tiles.
We define the notion of a supertile (s-tile in brief) recursively as follows:
1. Every element in S is an s-tile.
2. For t2 T ,
(x;y)
t
is an s-tile.
3. if C !
T
D and C is an s-tile, then D is also an s-tile.
From this definition, it is clear that for every s-tile A,fpjA(p) 6= emptyg is a shape, and we will call
that set the shape of A and we will use the notation [A] to refer to it. Write A!
T
B for s-tiles A and B iff
there exist a2 A and b2 B such that a!
T
b.
Let !

T
denote the reflexive transitive closure of !
T
. A derived supertile of the tile system T is an
s-tile such that s !

T
A for some s 2 S. A terminal supertile of the tile system T is a derived supertile A
3
such that there is no s-tile B, different from A, such that A !

T
B. If there is a terminal supertile A such
that for any derived supertile B, B !

T
A, we say that the tile system uniquely produces A. We also say
that for a given shape S, a tile system uniquely produces S iff it uniquely produces some supertile A and
[A] = S.
Given a tile system T which uniquely produces A, we say that the program size complexity of the system
isjTj i.e. the number of tile types.
In this paper, we adopt the restriction, suggested by Rothemund and Winfree [8], that S contains a single
seed s, and that g(; ) = 0 for ; 2 with 6= . For a discussion of the lower bound on program-size
in the absence of the latter restriction, see open problem #2 in section 8. If the seed s consists of a single tile
we will say the system is a unit-seed tile system.
We now introduce the definition of the time complexity of self-assembly. A similar definition has also
been suggested by Winfree [12]. We associate with each tile i 2 T a positive probability P(i), such that
P
i2T
P(i) = 1. P is called a concentrations function for T . We assume that the tile system has an infinite
supply of each tile, and P(i) models the concentration of tile i in the system – the probability that tile i is
chosen when a tile is drawn at random. Now self-assembly of the tile system T corresponds to a continuous
time Markov process where the states are in a one-one correspondence with derived s-tiles, and the initial
state corresponds to the seed s. There is a transition of state B to C iff B !
T
C, and the rate of the
transition is P(i) if C is obtained from B by adding a tile i. Suppose the tile system uniquely produces an
s-tile A
T
. It would follow that A
T
is the unique sink state. Given the Markov process, the time for reaching
A
T
from s is a random variable. The time complexity for producing A
T
from s is defined as the expected
value of this random variable.
Informally our definition of time models a system wherein a seed ”floats” in solution encountering tiles
at random. The higher the concentration of a particular tile the higher the rate at which it is encountered.
When a tile is encountered which has sufficiently strong interaction with the seed, the tile is incorporated.
By this process of accretion the seed grows larger and larger.
We say a tile system produces an n n square iff it uniquely produces a terminal s-tile which is an
n n square of tiles. Then the time complexity of producing an n n square is the minimum of the time
complexity of all the tile systems which produce n n squares. The following theorem is immediate from
our formal model.
Theorem 2.1 The time complexity of producing an n n square is
(n).
3 Counting up ton in time (n)
The square construction of Rothemund and Winfree [8] occurs in two stages. They first show how to
assemble a log nn rectangle, and then extend it into an nn square. To assemble the log nn rectangle,
they simulate a counter that counts from 1 to n in binary. We are going to use the same general framework,
but will replace their counter assembly by a more efficient process. In their counter assembly, only one tile is
attachable to the assembly at any given time. In our assembly process, several tiles may be attachable at the
same time. This “parallelism” allows us to assemble a counter in time (n) as opposed to the (n log n)
assembly time for the counter described by Rothemund and Winfree. We call the process described below
the SA-counter.
4
3.1 The Tile System
We initially assume that n = 2
K
for some positive integer K – we will later show how this assumption can
be removed. While each tile is completely specified by its four glues, it is convenient for the purpose of
exposition to allow tiles to have labels. Each tile has one main label (either 0 or 1) which corresponds to
the bit represented by the tile. Further, the east-most zero in a binary number is any is labeled as 0

. The
fully assembled counter is going to be a rectangle, with K tiles in each row. Each row represents a number
between 0 and n – this number can be obtained by reading the main labels on the tiles in the row, with the
most significant bit being the westmost in the row. We describe the self-assembly process which results in
the counter being incremented.
The increment operation uses 8 different tiles as well as K special tiles that appear in the seed row only
(see Figure 1). The east-most zero in the i
th
row (auxiliary label 0

) is replaced by a 1-tile in the (i + 1)
st
row. Further, the assembly of the (i + 1)
th
row always starts as the attachment of a 1-tile on top of a 0

-tile.
All digits to the east of the east-most zero are replaced by 0-tiles. All digits to the west of the east-most
zero are replicated in the new row. If the east-most zero in the i
th
row is the least significant digit then the
position of the west-most zero in the (i + 1)
st
row must be determined by a sequential search running from
east to west.
We will assume that the temperature for this counter construction is 2. We now describe the tiles in each
of the rows and their glues. All tiles occur with the same probability, p
SA
which is a constant independent
of n.
0-tiles: There are the following 0-tiles: 0

E, 0

M, 0Z, 0C.
1-tiles: 1E, 1M, 1S, 1C.
The glues and their strengths are depicted pictorially in figure 1.
n
c s
a
s s
u b
s
u l
u
u
c c c c
n
n
c z
u
a
n
z
u
z
b
z
l
1 1
1
*
*
0
0 0
C
1
M
M S
0
C Z
E
E
Figure 1: The tiles for the SA-counter. The number of lines jutting from each edge of the tile represent the
strength of the bond (1 or 2) and the label on the edges represents the glue. Some edges do not have any
glues.
The label suffixes indicate the position in the row each row a tile may occupy and or the function in
the assembly.
E: The tile can be only in the east-most column of the counter.
5
M: The tile cannot be in the east-most column of the counter.
Z: Attach a 0-tile to the east.
C: Copy to the west the digit in the lower row.
S: Search to the west for the position of the first 0-tile in the lower row.
The Seed Row: The seed supertile S
K
is a row of K special tiles.
The tile system T
SA
(K) has a tile set consisting of all the tiles in figure 1. The glue strength function
is as indicated in figure 1. The temperature is 2, and there is a single seed s-tile S
K
. We will refer to the
Markov chain defined by the tile system as the SA-counter. We will assume that all tiles which are not in
S
K
have the same constant probability.
We reproduce a definition from [7] with minor modifications in the terminology to make it consistent
with the definitions in this paper.
Deterministic row-column (RC) property: Let T = hT; S; g; i be a unit-seed tile system and let be
a terminal s-tile of T and let S be the shape of . Let Let w
S
= minfxj9y s.t. (x; y) 2 Sg, i.e. the x
coordinate of the westmost tile. Define e
S
, n
S
and s
S
similarly.
We say has the C-property iff the shape of has no holes and for all w
S
x < e
S
there exists exactly
one y such that the strength of the glue between (x; y) and (x+1; y) is greater or equal than . Similarly,
we say has the R-property iff the shape of has no holes and for all s
S
y < n
S
there exists exactly one
x such that the strength of the glue between (x; y) and (x; y + 1) is greater or equal than . We say
has the RC-property iff it has the R-property and the C-property. We say an attachment of a tile t to a s-tile
X at a position p is deterministic iff t is the only attachable tile to X at p. We say has the deterministic
RC-property iff it has the RC-property and there exists a derivation of such that all attachments in that
derivation are deterministic.
This theorem, also from [7] will help us to prove correctness of the counter.
Theorem 3.1 If a unit-seed tile system T produces an s-tile that has the Strong RC-property, then T
uniquely produces .
Theorem 3.2 The tile system T
SA
(K) uniquely produces an s-tile which is a K 2
K
rectangle.
Proof: Consider a unit-seed tile system T
0
SA
(K) that results from modifying the K special tiles so T
0
can
assemble the seed-line starting from the unit seed at the eastmost position. This can be easily accomplished
by making all east-west bonds in the seed line of strength 2.
Clearly, there exists a derivation that assembles the rectangle row by row, i.e. no tile is attached in
row before the row below is totally assembled. Further, observe that in that derivation all attachments are
deterministic. Therefore, has the Strong RC property, and by Theorem 3.1, T
0
SA
(K) uniquely produces
a K n rectangle.
By contradiction, consider now the original tile system T
SA
(K) and assume T
SA
(K) does not produce
a K n rectangle. That would imply T
0
SA
(K) does not uniquely produce .
Minor modification of the seed row results in a tile system that uniquely produces an s-tile which is a
K n rectangle for any n 2
K
.
6
3.2 Analysis
Our counter construction is more involved than that proposed by Rothemund and Winfree [8] but exploits
“parallelism” to speed up the assembly process. The construction of Rothemund and Winfree takes time
O(n log n) in the model of running time described in Section 2.We show that the SA-counter gets assembled
in linear time in Section 4). Intuitively, the stronger (n) bound on the assembly time is due to the fact that
a row may start getting assembled even before the previous row is completely assembled.
In spite of having many possible derivations of the rectangle from the seed row, this particular tile system
has a property that makes the analysis of the running time easier. We reproduce the definition of partial order
systems introduced in [2], with minor adaptations.
Assume that we are given a tile system T and a shape S that is uniquely produced by T. Let A denote
the s-tile of T that has shape S. Let A(i; j) represent the tile-type at position (i; j). Consider a derivation
of A and let t
i;j
represent the time when the tile at position (i; j) attaches to the growing assembly. Define
a partial order on the tile positions in S such that (i; j) (p; q) iff t
i;j
t
p;q
for all possible derivations
of A using T.
Definition: A tile system T is said to be a partial order system iff it uniquely produces a shape S, and if
for all adjacent positions (i; j),(x; y) in S, either (i; j) (p; q), or (x; y) (i; j), or the strength of the
glues connecting tiles at positions (i; j) and (x; y) is zero.
Note that we can represent the partial order relation as a DAG G = (S; E), where (p; q)2 E iff p q.
Figure 2 depicts the assembly of a SA-counter counting from 0 to 7 in binary. The thin arrows indicate
the order relation in the process. For a arrow, the position at the tail must be filled before the position at the
head. Intuitively, a long path in the graph suggests long running time because it implies those positions will
be filled sequentially.
Lemma 3.3 The tile system T
SA
(K) is a partial order system.
Proof: For all rows (other than the seed row), the first tile to be added to that row is the one on top
of the 0

tile in the immediately lower row. This is because only tiles labeled as 0

have a strength 2 glue
on top. Since there is exactly one 0

tile per row, we observe that no tile can attach to the assembly if the
tile immediately to the south is not already in place. This gives the north-south order relation. Inside each
row, we note that for all positions p to the west of the 0

tile, an attachment is impossible if the position
immediately to the east of p is empty. Similarly, for all positions p to the east of the 0

tile, an attachment is
impossible if the position immediately to the east of p is empty. This completes the definition of the partial
order relation for all pairs of adjacent positions.
Let G
n
be the DAG representing the partial order relation in an SA-counter counting from 0 to n.
Lemma 3.4 The length of the longest directed path in G
n
is (n).
Proof: Let L be length of the longest path in G
n
. The longest path is the one containing the most
west-east edges. There is exactly one path containing all of them. It is the path containing all the vertices
corresponding to the 0

labeled tiles. Any path going from the bottom row to the top row will traverse
exactly n 1 north-south edges, hence L =
(n). The number of west-east and east-west edges traversed
is easily bounded by adding the distance from the east-most zero to the east-most column for all rows.
7
0 0
1
M
C
1
1
M
1
M
*
0
R
*
0
1
R
*
0
R
1
R
1
R
*
0
R
1
R C
1
C
1
C
1
C
1
0
M
*
0
C
0
C
0
M
*
0
Z
0
M
*
Seed row
Figure 2: Graph representing the assembly of a counter. The dashed line encloses the seed row. The thin
arrows indicate the order relation. The thick arrows represent the edges added to construct an Equivalent
Acyclic graph.
L n 1 +
P
k
j=1
(k j) 2
j
and
n 1 +
P
k
j=1
(k j) 2
j
n + k
P
k1
i=0
i2
ki
n(1 +
P
1
i=0
i2
i
) = O(n)
Theorem 3.5 The time complexity for building an SA-counter that counts from 0 to n is (n).
We will present the proof of Theorem 3.5 at the end of Section 4 after we develop some more machinery.
4 Running time Analysis.
For the purpose of our analysis, we transform the self-assembly process into another process which we call
the “sentinel” process. The sentinel process does not adhere to the model described in Section 2; in fact
there does not seem to be an easy implementation of the sentinel process. However, the sentinel process is
more amenable to analysis, and the time for this process to complete is an upper bound (in the stochastic
domination sense) on the completion time for the self-assembly process.
Given a directed graph G = (V; E) and vertices v; v
0
in V , we say v is a predecessor of v
0
iff (v; v
0
)2 E.
Given a vertex v, we define the predecessors of v in G, and we denote it as Pred
G
(v), to be the set of all
vertices v
0
such that v
0
is a predecessor of v.
8
Equivalent Acyclic Graph (EAG): Let T = hT; S; g; i be a tile system that uniquely produces some
s-tile , and let G = (V; E) be a DAG. G is an Equivalent Acyclic Graph (EAG) for T iff:
1. V = [].
2. G has exactly one source.
3. For all edges (p; p
0
)2 E, p is position-adjacent to p
0
.
4. Let be the seed s-tile. For all p 2 [] [], for all derived s-tiles
0
of T, if Pred
G
(p) [] and
p62 [
0
], then (p) is attachable to
0
at p.
Note that for a given T there may exist more than one EAG.
Lemma 4.1 For all tile systems T that uniquely produce an s-tile, there exists an EAG.
Proof: Let be the terminal s-tile produced by T, let be the seed s-tile and let n be the size of [] [].
Consider one arbitrary derivation of and the graph G = (V; E) constructed in the following way: For
1 i n call p
i
the position where the i
th
attachment occurs. Define V = [], E
1
=f(p
i
; p
j
)ji < jg and
E
b
=f(p; p
0
)jp2 [] and p
0
2 [][], and p and p
0
are position-adjacentg. Define a DAG G
0
= ([]; E
s
)
such that E
b
E
s
, G
0
is connected, for all edges (p; p
0
) in E
s
p and p
0
are position adjacent, and G
0
has a
single source. Define E = E
1
+ E
b
+ E
s
.
To complete the proof, we have to verify that G is an EAG. We start by proving G is a DAG. Because
of the way we constructed it, there cannot be a cycle in G
0
. We observe that all edges in E
1
connect vertices
in [] [], that all edges in E
b
go from a vertex in [] to a vertex in [] [] and that all edges in E
s
connect elements in []. Therefore, there cannot be a path from a vertex in [] [] to []. There are not
cycles in G
0
, so if there is a cycle in G it must contain exclusively elements in [] []. By the definition of
E
1
this is clearly impossible, hence G is a DAG. Now we prove that G has exactly one source, i.e. the one
in []. We claim that there cannot be a source in [] []: Assume there is one and let p
i
be that vertex. If
there is a vertex p2 [] such that p
i
and p are position-adjacent then, by definition of E
b
, p
i
has an incoming
edge and is not a source. Now consider the case when there is not such a p. Because of the definition of
E
1
we know that for all all outgoing edges (p
i
; p
j
), i < j. Therefore, in our derivation the attachment at i
happened before the attachment at j. If p
i
is a source, it means that the attachment at p
i
happened before
any of its position-adjacent locations were filled. No derivation can exhibit that behavior.
To verify the last condition of the definition of EAG, we simply observe that E was obtained from an
actual derivation.
In the previous proof we used a derivation of the s-tile to construct a suitable EAG. We say that the
EAG was induced by the derivation.
Sentinel process: LetT =hT; S; g; i be a tile system that uniquely produces an s-tile, let be that s-tile,
let P be a concentrations function for T, let G be a EAG for T and let g
s
be the source of G. Let S be the
set of all subgraphs G
0
= (V
0
; E
0
) of G such that g
s
2 V
0
and for all edges (p; p
0
) 2 V , (p; p
0
) 2 E
0
iff
p; p
0
2 V
0
.
We define S
T;G;P
to be the continuous time Markov process such that:
9
1. Each state in S
T;G;P
corresponds to a graph in S.
2. There is a transition from state s
0
to s
00
in S
T;G;P
iff the corresponding subgraphs G
0
= (V
0
; E
0
)
and G
00
= (V
00
; E
00
) are such that V
00
V
0
is a singleton and E
00
E
0
= f(u; v)jv 2 V
00
V and
u2 Pred
G
(v)g.
3. The probability associated with a transition from s to s
0
is P(v), where v is defined as before.
Note that S
T;G;P
has exactly one source state, corresponding to the source of G, and exactly one sink
state, corresponding to G. We define the completion time for the sentinel process as the time to go from
the source state to the sink state. Intuitively, the sentinel process is obtained by modifying the self-assembly
process by deleting some transitions in it, i.e. disallowing some bonds to form. This elimination of tran-
sitions will make the completion time of the sentinel process an upper bound for the running time of the
self-assembly process.
Completion time for the sentinel process: Consider a tile system T that uniquely produces an s-tile, and
a concentrations function P for T, and let c = min
i
(P(i)), i.e. the smallest concentration. Let G be an
EAG for T and let L be the length of the longest directed path in G. Let t be the completion time for
S
T;G;P
.
Lemma 4.2 E[t] = O(L=c). Further, t has an exponentially decaying tail.
Proof: Let P
1
; P
2
; : : :P
N
represent the N directed paths from the seed to any sink in the sentinel graph. At
each step in any path, the edge must go to a position-adjacent vertex and there are at most three candidates.
Therefore N is at most 3
L
e
2L
. Let S
l
denote the sum of all X
i;j
such that position (i; j) lies on path
P
l
. Then the completion time t = max
n
l=1
S
l
. S
l
is the sum of at most L mutually independent exponential
variables, each with mean less or equal to 1=c. Hence E [S
l
] L=c; let denote the value L=c. Clearly
= O(L=c). Using Chernoff bounds for exponential variables [5], it follows that Pr [S
l
> (1 + )]
((1 + )=e

)
L
. Hence Pr [t > (1 + )] N((1 + )=e

)
L
e
2L
((1 + )=e

)
L
= ((1 + )=e
2
)
L
. Let
us choose =
0
+ 4, where
0
> 0. Now
Pr

t > 5(1 +
0
)

Pr

t > (1 + 4 +
0
)

((1 +
0
)=e

0
)
L
:
This clearly gives an exponential tail bound. Now,
E [t] 5 (1 +
Z
1

0
=0
((1 +
0
)=e

0
)
L
d
0
);
which is O(L=c) as = O(L=c) and the integral is bounded by a constant for any value of L 1.
Stochastic dominance: Define t
i;j
to be the time at which the (i; j) position gets filled in the original
self-assembly process and t
0
i;j
to be the time at which it gets filled in the sentinel process. If (i; j) is in the
shape of the seed s-tile then assume t
i;j
= 0. Note that t
i;j
and t
0
i;j
are random variables. Let t and t
0
be the
10
random variables denoting the times at which the original self-assembly process and the sentinel assembly
complete.
A real valued random variable A is said to be stochastically dominated by another random variable B,
denoted A
sd
B, if for all x, Pr[A > x] Pr [B > x].
Lemma 4.3 For all (i; j) in the shape of the uniquely produced s-tile, t
i;j

sd
t
0
i;j
Proof: Let be the s-tile produced by T Let X
i;j
be an exponential random variable with mean
1=P(((i; j))), i.e. the reciprocal of the probability associated with the tile at position (i; j) in the final
s-tile. Let all the X
i;j
be independent. A tile attaches at position (i; j) in the self-assembly X
i;j
time after
this position becomes attachable
1
. We couple the sentinel process and the self-assembly process by setting
the values of X
i;j
to be the same for both processes. Define a
i;j
and a
0
i;j
to be the times at which tile position
(i; j) is attachable in the self-assembly process and the sentinel process, respectively. Let t be the earliest
time when a tile gets attached in the sentinel process but is still unattached in the self-assembly process. Let
(i; j) be this tile position. Clearly, a
0
i;j
< t. Therefore any tiles which had attached in the sentinel process
by time a
0
i;j
had also attached in the self-assembly process. Since the sentinel process was formed by disal-
lowing certain bonds in the self-assembly process, tile position (i; j) is also attachable in the self-assembly
process at time a
0
i;j
. Hence a
i;j
a
0
i;j
. But t
i;j
= a
i;j
+ X
i;j
and t
0
i;j
= a
0
i;j
+ X
i;j
. This implies that
t
i;j
t
0
i;j
, which is a contradiction. Since t
i;j
t
0
i;j
for each coupled experiment, t
i;j

sd
t
0
i;j
.
Lemma 4.3 in conjunction with the lemma 4.2 now allows us to conclude:
Theorem 4.4 E[t] = O(L=c). Further, t has an exponentially decaying tail.
Consider a unit-seed tile system T that uniquely produces a s-tile, and a constant valued concentrations
function C for T. Let c be the value of C.
Theorem 4.5 There exists an EAG G for T such that the time complexity of T is (L=c).
Proof: Let be the terminal s-tile of T, and let t be the time complexity of T. Consider an experiment,
i.e. the assembly of from the seed. Let
~
t be the completion time of the experiment.
Define p
1
= Pr

~
t 2t

. Clearly, E[
~
t] = t and by Markov’s inequality p
1
1=2.
Let
~
G be the EAG induced by the experiment, that can be constructed as we shown in the proof of
Theorem 4.1. Let
~
L be the length of the longest directed path in
~
G. Define p
2
= Pr
h
~
L=(4c)
~
t
i
. We will
now show that p
2
< 1=4.
Define = x
1
+ x
2
+ + x
~
L
where the x
0
i
s are independent exponential random variables with mean
1=c. Using the fact that exponential variables are the inter-arrival times of Poisson events we get:
Pr
h

~
L=(4c)
i
= Pr
h
(
~
L=4)
~
L
i
where (
~
L=4) is a Poisson variable with mean
~
L=4. Using Markov inequality again, we obtain Pr
h
(
~
L=4)
~
L
i

1=4p, hence Pr
h

~
L=(4c)
i
1=4. Since
sd
~
t we conclude that Pr
h
~
L=(4c)
~
t
i
1=4.
1
This fact follows from the fact that once a tile position becomes attachable it remains attachable till it actually attaches.
11
Since p
1
> p
2
, it must be the case that Pr
h
~
L=(4c)
~
t^
~
t 2t
i
6= 0. Therefore, there must exist a
derivation that induces an EAG G such that L=(4c) 2t and hence t =
(L=c), where L is the length of
the longest directed path in G. From Theorem 4.3 we know that t = O(L=c), completing the proof.
For completeness, we state the following lemma of trivial proof that gives a lower bound on the time-
complexity of a partial order system. Let T be a partial-order system and let G be the DAG describing the
partial order relation.
Lemma 4.6 For all concentrations functions P for T, the time complexity of T is
(L)
Theorem 3.5 can be obtained immediately from Lemma 3.4, Lemma 4.6, and Theorem 4.5. We present
the details below.
Proof of Theorem 3.5: Let G be the acyclic graph representing the partial order relation in the counter.
Lemma 3.4 states that the longest path in G has (n) length. The (n) running time follows from The-
orem 4.5. A suitable EAG is G, augmented with all edges of the form ((x + 1; y); (x; y)), where both
(x + 1; y) and (x; y) are in the seed row, and with edges of the form ((x; y); (x; y + 1)), where (x; y)’s are
in the seed row, (see Fig 2). It’s easy to see that the length of the longest directed path in the augmented
graph is still (n). To prove the time complexity is
(n), we simply invoke Lemma 4.6.
5 Simulating base conversion by self-assembly.
In Rothemund and Winfree’s construction (and in ours), the “seed row” represents a number written in
binary and consists of log n tiles, each encoding 0 or 1. The number is used as the starting point for a
counter. Since we are only allowed a seed s-tile consisting of a single tile, this seed row itself needs to
be assembled from a single tile. Each tile in the seed row needs to be distinct, and therefore the program
size complexity for assembling the seed row itself is
(log n). On the other hand, Kolmogorov complexity
dictates the program size complexity to be
(
log n
loglog n
). In order to achieve the Kolmogorov bound, we could
use a ”seed row” to represent a number written base b, where b is power of 2 such that
log n
log log n
b = 2
k
<
2 log n
log log n
;
for k a positive integer. Then the number of tiles to construct the seed row will be
h =
log n
log b
<
log n
log log n log log log n
= O(
log n
log log n
):
As in Rothemund and Winfree [8]’s counting scheme, we could use the “seed row” as a starting point for
a counter. If we use the counting strategy of Rothemund and Winfree, modified to count base b, we easily
achieve the optimal program size complexity or in the language of Rothemund and Winfree:
Theorem 5.1 K
2
SA
(N) = O(
log N
loglog N
).
Unfortunately if we simply follow Rothemund and Winfree [8]’s method, we pay a price in time com-
plexity. The Rothemund and Winfree construction begins with the seed row and grows through a succession
12
of s-tiles to produce a final rectangle. But for each s-tile produced in the process, there is exactly one tile
which can be incorporated. There are at least b = (
log n
loglog n
) distinct tiles which occur
(n) times in the
final rectangle, and at least one of these tiles must have probability O(
loglog n
log n
). It follows that such tiles
require
(
nlog n
loglog n
) time for assembling of the rectangle. In particular, the time is not linear. Even using the
parallel construction described in section 3, the time is not improved.
To overcome this time problem while preserving the O(
log n
loglog n
) program size, we adopt the strategy of
writing a ”seed column” base b and then converting it to base 2, forming a seed row for a counter. We then
employ the fast parallel counter described in section 3.
In the conversion part, we will employ some tiles, which have the capability to perform division-by-2.
These tiles have glues representing binary strings. The north glue represents last bit, the west glue represents
the prefix substring and the east glue represents the string itself. See Figure 5 (A) for the tile associated with
the string 0110, and Figure 5 (B) for cooperation of two tiles. We assume that the temperature is 3, but it is
not hard to see that the same idea also works for temperature 2.
0
x1
011 0110
0
0110
x1
0110
d
0110
d
01
1
011
d
x1
(A) Convert one bit.
(B) Convert two bits.
d
c*
c c d d c*
x
x x x x s
s
s
c
c
d
d
x d
0 0 0 1 1 1
0 00 00 00 00 00 00
0 0 1 11 11 11 11
01 01 0 1 1 0 0
(C) Convert 031 in base 4 to 001101 in base 2.
Figure 3: Converting base by self-assembly.
In the following tile descriptions, we identify a tile by the labels on its four sides: North(N), West(W),
East(E) and South(S), and a title symbol(TS). In our notations, a prefix “(2)” indicated a glue with strength
2, a prefix “(3)” indicated a glue with strength 3. The absence of prefix indicates glue with strength 1.
Suppose we want to convert b
1
b
h
in base b to binary base number, where b
i
if written in binary would
be b
i;1
b
i;2
b
i;k
. The set of tiles consists of
13
1. (Seed.) These tiles will form a seed column. For each b
i;1
b
i;2
b
i;k
, 1 i h,
TS: (b
i;1
b
i;2
b
i;k
)
s
N: if i < h, (3)S
i
; S: If i > 1, (3)S
i1
;
W: if i = h, (3)b
h;1
b
h;2
b
h;k
, else b
i;1
b
i;2
b
i;k
.
2. (division by 2.) For each binary string a
1
a
2
a
i
, 1 i k,
TS: (a
1
a
2
a
i
)
d
,
N: (2)a
i
, E: (3)a
1
a
2
a
i
,
W: if i 2, (3)a
1
a
2
a
i1
, otherwise x2,
S: if i 2, (2)x1, otherwise (2)x3.
3. (Base b copy.) For each binary string a
1
a
2
a
k
,
TS: (a
1
a
2
a
k
)
c
,
N: (2)x1, W: a
1
a
2
a
i
, E: a
1
a
2
a
i
, S: (2)x1.
4. (Last b copy tile.) For each binary string a
1
a
2
a
k
,
TS: (a
1
a
2
a
k
)
C
N: (2)x3, W: (3)a
1
a
2
a
i
, E: a
1
a
2
a
i
, S: (2)x1.
5. (Base 2 copy.) For a = 0 or 1,
TS: a
x
; N: (2)a; E: x2; S: (2)a; W: x2.
Note that the east side of the seed tiles, the south side of the first seed tile and the north side of the last
seed tile are not assigned with any glues. They are open for later use. Even if those glues are all different,
the number of distinct tiles is not increased.
Figure 5(C) is an example showing how to convert string 031 in base 4 to string 001101 in base 2.
In item 2, we use the most distinct tiles. which is
b +
b
2
+
b
4
+1 < 2b = O(
log n
log log n
):
Thus the number of distinct tiles in this conversion subroutine is O(
log n
loglog n
). Every tile appears at most
O(log n) times. We can arrange that each tile has probability greater than
(
log
2
n
nloglog n
), in which case, the
time to do the conversion is at most O(n).
6 Putting it all together
The self-assembly of an n n square begins with a seed tile which grows into a seed column consisting
of roughly b = (
log n
loglog n
) tiles representing a number m written base b. This seed row spawns a base
14

!!! !!! """ """ ### ### $$$ $$$ %%% %%% && && ’’’ ’’’ ’’’ (( (( (( ))) ))) ))) ** ** ** +++ +++ +++ +++ ,, ,, ,, ,, Base conversion
Initial Binary Counter Row
Binary Counter
Seed Tile
Base b Seed Line
Figure 4: Constructing a square by self-assembly.
15
conversion assembly as outlined in section 5 The result is a rectangle with top row representing m written
base 2. This top row becomes the starting point for a binary counter assembly as outlined in section 3.
The height of the resulting counter rectangle is a function H of m. H satisfies H(m + 1) = H(m) +
3. Hence by selecting an appropriate m, the base conversion assembly and the counter assembly have
dimensions such that height plus width equals n e, where e = 0; 1 or 2. We can grow e extra rows at
the base of the combined rectangle to bring the sum of the two dimensions exactly up to n. The resulting
rectangle acts as the basis for the completion of the nn square as described in [8] using diagonal elements
and filler tiles. Figure 4 describes this process pictorially.
Let T
n
be the variant of the tile system described above that assembles an n n square. Note that only
the seed row and base conversion tiles have to be adjusted to specify n. The tiles to build the square, the
diagonal and the filler tiles do not depend on n.
Theorem 6.1 For all positive integers n, there is a concentration functions P
n
for T
n
such that T
n
assem-
bles an nn square in O(n) time. Further, for all concentrations functions P
n
forT
n
, the square assembles
in
(n) time.
Proof:
T
n
is a partial-order system. We can use the DAG describing the partial-order relation as EAG. We
claim he length of the longest directed path in the DAG is (n), since by Lemma 3.4 the longest path
in the counter is of length (n), the total number of positions in the seed columns and base conversion is
O(log
2
(n)= log log(n)), and the longest path in the rest of the square is of length (n). By allocating a
combined probability of 1=5 to the tiles constructing the seed row, 1=5 to the tiles which perform the base
conversion, 1=5 to the tiles which perform the counter construction, 1=5 to the diagonal elements and 1=5
to the filler tiles, it follows from Theorem 4.4 that the time to construct the n n square is O(n). By
Lemma 4.6 the time complexity is
(n).
7 Bounding running time of a self assembly process
In section 3.2 we constructed the equivalent acyclic graph based on the partial order relation existing among
the times to fill every position of the counter. The technique is not always applicable. There are self-
assembly processes that do not exhibit the partial order relation. We present in this section a different
parallel counter that is not a partial order system and we will show how to bound the time to complete its
assembly.
The fully assembled counter is going to be a rectangle, with K tiles in each row. Each row represents a
number between 0 and n – this number can be obtained by reading the main labels on the tiles in the row,
with the most significant bit being the west-most in the row. We describe the self-assembly process which
results in the counter being incremented.
The increment operation happens in three stages and uses 15 different tiles (see figure 5). We start with
an “INERT” row (each tile carries the auxiliary label “I”). This row gets replicated into an “ACTIVE” row
(auxiliary label “A”) if and only if there is at least one 0-tile in the original INERT row. If all the tiles
in the INERT row are 1-tiles, the counter construction can not proceed any further and the self-assembly
process terminates. A “CARRY” row (auxiliary label “C”) then assembles on top of the ACTIVE row.
16
The bulk of the increment operation happens during this step. The CARRY row will represent the value
of the ACTIVE row incremented by one, except that the 0-tile (if any) which needs to change to a 1-tile
due to a carry propagation is still unchanged. Then an INERT row assembles on top of the CARRY row
which will convert such a tile (if any) to a 1-tile. The east-most tile in any row always carries the auxiliary
label “R” along with the auxiliary label corresponding to the row. Some tiles may also be marked special
(auxiliary label “S”) to aid in the carry propagation. Even though the above description seems serial, the
SA-counter need not assemble row-by-row; several different tiles (possibly belonging to different rows) may
be attachable at the same time.
We will assume that the temperature for this counter construction is 3. We now describe the tiles in each
of the rows and their glues. All tiles occur with the same probability, p
SA
which is a constant independent
of n.
INERT tiles: There are the following INERT tiles: 0L; 1L; 0LE; 1LE; 0LL; 1LL.
ACTIVE tiles: 0L; 1L; 0LE; 1LE.
CARRY tiles: 0L; 1L; 0LL; 0LE; 1LE. The glues and their strengths are depicted pictorially in figure 5.
IS
1 1
IR IR
0
I
0
a
k
L
IS
0
A
0
A
1 0
AR
1
AR
m
n n
a d
n
n
o
n
f
p
n
i
ACTIVE
TILES
q
0
C C
1
CS
0 0
CR CR
1
c
s
m
e
s
L
o
s
r
r
r
s
h
q p
J
CARRY
TILES
a
b
c
d
b
e
d
c
b k
i
b
J
f
k
h
INERT
TILES b
1
I
o
Figure 5: The tiles for the SA-counter. The number of lines jutting from each edge of the tile represent the
strength of the bond (1,2, or 3) and the label on the edges represents the glue. Some edges do not have any
glues.
Assembling ACTIVE rows: Notice that starting with an INERT row that consists of all 1-tiles, no ACTIVE
tiles can attach to the top of the row since all possible bonds are of strength 2 whereas the temperature
is 3. However, all 0-tiles in the INERT row can allow ACTIVE 0-tiles to attach on top through bonds
of strength 3. These newly attached ACTIVE 0-tiles provide strength 1 bonds that supplement the
strength 2 bonds between ACTIVE 1-tiles and INERT 1-tiles, allowing ACTIVE 1-tiles to attach in
positions adjacent to the already attached ACTIVE 0-tiles. The newly attached ACTIVE 1-tiles in
turn provide strength 1 bonds that allow adjacent ACTIVE 1-tiles to attach on top of INERT 1-tiles,
and so on. This allows the entire INERT row to be replicated into an ACTIVE row.
Assembling CARRY rows: Notice that any 0-tiles (except the east-most) in the ACTIVE row allow a
CARRY 0-tile to attach on top by means of bonds of strength 3. These tiles then allow CARRY
17
1-tiles to attach on the west through bonds of strength 1, which in turn allow more CARRY 1-tiles
to attach on the west. Notice that CARRY 0-tiles do not provide any glues on the east and therefore
can not facilitate attachment of CARRY 1-tiles to their east. This allows most of the ACTIVE row to
replicate into the CARRY row – we just need to bother about the east-most tile and those ACTIVE
1-tiles that do not have an ordinary ACTIVE 0-tile to their east. We will now consider two cases
depending on the east-most tile in the ACTIVE row. If the east-most tile in the ACTIVE row is a
0-tile then it will allow a east-most CARRY 1-tile to attach on its top through a bond of strength 3.
This then provides a bond of strength 1 to its west which will allow ACTIVE 1-tiles to assemble to its
west, thus completing the CARRY row. Notice that this CARRY row already represents the correct
result for the increment operation. If the east-most tile is an ACTIVE 1-tile, then it allows a east-most
CARRY 0-tile to attach on top. This tile then allows special CARRY 0-tiles (and not CARRY 1-tiles)
to bond to its west on top of the ACTIVE 1-tiles. Thus the entire sequence of contiguous ACTIVE
1-tiles on the east will have special CARRY 0-tiles attached on top. The CARRY row now represents
the correct result for the CARRY operation with one exception: the ACTIVE 0-tile which was im-
mediately to the west of the east-most sequence of ACTIVE 1-tiles has a CARRY 0-tile attached on
top, whereas the increment operation should convert it into a 1-tile due to carry propagation. This is
remedied in the next step.
Assembling INERT rows: The 0-tile which needs to be changed into a 1-tile can be “detected” as a
CARRY 0-tile which has a special CARRY 0-tile or a east-most CARRY 0-tile to its immediate east.
The description for the assembly of the INERT row is simple. All the tiles except ordinary CARRY
0-tiles can attach an INERT tile (with the other labels such as the bit-label, the east-most label, and
the special label remaining the same) on top. Now special INERT 0-tiles and east-most INERT 0-tiles
allow a special INERT 1-tile to attach (on top of a CARRY 0-tile) to their west, while all other INERT
tiles allow an INERT 0-tile to attach. This accomplishes the desired carry propagation and completes
the increment process.
The Seed Row: The seed s-tile S
K
is a row of K special tiles. The east-most tile in S
K
is identical to the
tile 0LE except that the bottom surface has no glue. The other K 1 tiles are identical to the tile 0L
except that there is no glue on the bottom surfaces
2
.
The tile system T
SA
(K) has a tile set consisting of all the tiles in figure 5 as well as the special tiles
needed in the s-tile S
K
. The glue strength function is as indicated in figure 5. The temperature is 3, and there
is a single seed s-tile S
K
. We will refer to the Markov chain defined by the tile system as the SA-counter.
We will assume that all tiles which are not in S
K
have the same constant probability.
The next theorem follows from our description of the tile system, and we omit the proof.
Theorem 7.1 The tile system T
SA
(K) uniquely produces an s-tile which is a K (3 2
K
2) rectangle.
Minor modification of the seed row results in a tile system that uniquely produces an s-tile which is a Kn
rectangle for any n 3 2
K
2.
2
In this section we are using s-tiles consisting of more than one tile as the seed, whereas our restricted model allows us to use
only s-tiles with a single tile as seed. We rectify this in section 6.
18
The equivalent acyclic graph: Look at a completed self assembly, and replace each bond by two directed
bonds, one in each direction. Each directed bond has the same strength as the original bidirected bond.
Then, remove all the directed bonds that satisfy any of the following criteria:
1. The bond goes from a higher to a lower row.
2. The bond goes from west to east in a non-ACTIVE row.
3. The bond goes from west to east in an ACTIVE row, and there is at least one zero-tile to the east of
the origin of the bond.
4. The bond goes from east to west in an ACTIVE row, and the destination tile of the bond is a zero-tile.
5. The bond goes from east to west in an ACTIVE row, the source and destination tiles are both one-tiles,
and there are no zero-tiles to the east of the source.
We can define an equivalent graph as the graph with vertices corresponding to the tiles, and directed arcs
corresponding to the directed bonds and labeled by the strength. The following lemma is immediate from
our construction.
Lemma 7.2 The sentinel graph is acyclic.
The sentinel process is a Markov chain obtained by modifying the Markov chain corresponding to the
SA-counter as follows. We consider each transition in the Markov chain for the SA-counter. Let this
transition correspond to adding a new tile X to an s-tile A. A tile Y in A is said to be a support tile if it
shares a side with X and there is an arc from Y to X in the sentinel graph; the strength of this arc is said to
be the support strength from Y . We retain this transition if the sum of the support strengths from the support
tiles is greater than the temperature and else we discard it. Any state in the Markov chain that is unreachable
from the source state is discarded.
Intuitively, the sentinel process is formed by taking the SA-counter and introducing a “sentinel” who
disallows transitions that require bond-formation from the west to the east, except when such transitions
are necessary for replication. It is not difficult to see that the sentinel process produces exactly the same
complete assembly as the SA-counter.
In the sentinel graph, there is exactly one bond that goes from the west-most column to the east, and this
happens just above the INERT row 011 : : :11 where the west-most 0 is the only tile that can “self-replicate”
into the active row and must then induce the 1-tiles to attach to its east. Further, this property is recursively
satisfied if we remove the west-most column and look at the sentinel graphs for the two sub-rectangles below
and above the 011 : : :11 row. This recursive structure makes the sentinel process more amenable to analysis.
The structure of the sentinel graph is illustrated in figure 6.
8 Conclusions and open problems
We have generalized the Tile Assembly Model of Rothemund and Winfree [8] to include time complexity
and have demonstrated a tile system for self-assembling n n squares that is optimal in both time and pro-
gram size. The long term goal of this research is the development of a mathematical-computational theory
19
C
1
A
1
1
I
C
1
A
1
1
I
C
1
A
1
IS
1
0
C CS
0 0
CR
A
0
A
1
AR
1
1
I
C
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CR
1
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1 0
AR
IS
1
IR
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0
C
0
CR
A
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AR
1
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0 1
IR
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C CR
1
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0 0
AR
IS
0
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IR
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A
0
I
0
0
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A
0
I
0
0
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A
0
I
0
I
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1 0
AR
IS
1
IR
0
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1
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0 0
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IR
I
0
G(4)
1
2
3
G (8)
G (8)
G (8)
G(8)
G(4)
Seed Row
Figure 6: The recursive structure of the sentinel graph.
20
of self-assembly that will combine classical theories such as thermodynamics and statistical mechanics with
modern theories such as combinatorics and computational complexity.
There are many open problems. For example:
1. In the model described here, s-tiles grow by accretion i.e. by the addition of single tiles to a growing
crystal. Modify the model to allow any two s-tiles to merge into a larger s-tile. In such a system the
lower bound of
(n) for the time to assemble nn squares no longer appears necessary. In this case,
what is the best possible time and can it be achieved simultaneously with optimal program size?
2. The Tile Assembly Model as presented here is ”irreversible” - once a tile sticks, it never ”unsticks”.
Generalize to a model that allows tiles to both attach to and detach from the assembly.
Acknowledgements
We would like to thank Paul Rothemund and Erik Winfree for several helpful discussions about their
work [8].
References
[1] L. M. Adleman. Towards a mathematical theory of self-assembly. Technical Report 00-722, Depart-
ment of Computer Science, University of Southern California, 2000.
[2] L. Adleman, Q. Cheng, A. Goel, M. Huang, D. Kempe, P. Moisset de Espan´ es, P. Rothemund: Combi-
natorial optimization problems in self-assembly. ACM Symposium on Theory of Computing (STOC)
2002. pages 23-32.
[3] H. Abelson, D. Allen, D. Coore, C. Hanson, G. Homsy, T. Knight, R. Nagpal, E. Rauch, G. Sussman
and R. Weiss. Amorphous computing. AI Memo 1665, August 1999.
[4] C. Mao, T. H. LaBean, J. H. Reif and N. C. Seeman. Logical computation using algorithmic self-
assembly of DNA triple-crossover molecules. Nature, 407:493–496, 2000.
[5] R. Motwani and P. Raghavan. Randomized Algorithms. Cambridge University Press, 1995.
[6] P. Rothemund. Using lateral capillary forces to compute by self-assembly. Proceedings of the National
Academy of Sciences, 97:984–989, 2000.
[7] P. Rothemund. Theory and Experiments in Algorithmic Self-Assembly University of Southern Cali-
fornia Ph.D. Thesis Copyright 2001.
[8] P. Rothemund and E. Winfree. The program-size complexity of self-assembled squares. STOC 2000.
[9] H. Wang. Proving theorems by pattern recognition. II. Bell Systems Technical Journal, 40:1-42, 1961.
[10] E. Winfree, F. Liu, L. A. Wenzler, and N. C. Seeman. Design and self-assembly of two-dimensional
DNA crystals. Nature, 394:539-544, 1998.
21
[11] E. Winfree. PhD thesis. Cal Tech, 1998.
[12] E. Winfree. Personal communication.
22

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Leonard Adleman, Qi Cheng, Ashish Goel, Ming-Deh Huang, Pablo Moisset de Espanes. "Running time and program size for self-assembled squares." Computer Science Technical Reports (Los Angeles, California, USA: University of Southern California. Department of Computer Science) no. 802 (2003).

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