10 research outputs found
Approximate Self-Assembly of the Sierpinski Triangle
The Tile Assembly Model is a Turing universal model that Winfree introduced
in order to study the nanoscale self-assembly of complex (typically aperiodic)
DNA crystals. Winfree exhibited a self-assembly that tiles the first quadrant
of the Cartesian plane with specially labeled tiles appearing at exactly the
positions of points in the Sierpinski triangle. More recently, Lathrop, Lutz,
and Summers proved that the Sierpinski triangle cannot self-assemble in the
"strict" sense in which tiles are not allowed to appear at positions outside
the target structure. Here we investigate the strict self-assembly of sets that
approximate the Sierpinski triangle. We show that every set that does strictly
self-assemble disagrees with the Sierpinski triangle on a set with fractal
dimension at least that of the Sierpinski triangle (roughly 1.585), and that no
subset of the Sierpinski triangle with fractal dimension greater than 1
strictly self-assembles. We show that our bounds are tight, even when
restricted to supersets of the Sierpinski triangle, by presenting a strict
self-assembly that adds communication fibers to the fractal structure without
disturbing it. To verify this strict self-assembly we develop a generalization
of the local determinism method of Soloveichik and Winfree
Self-Assembly of 4-sided Fractals in the Two-handed Tile Assembly Model
We consider the self-assembly of fractals in one of the most well-studied
models of tile based self-assembling systems known as the Two-handed Tile
Assembly Model (2HAM). In particular, we focus our attention on a class of
fractals called discrete self-similar fractals (a class of fractals that
includes the discrete Sierpi\'nski carpet). We present a 2HAM system that
finitely self-assembles the discrete Sierpi\'nski carpet with scale factor 1.
Moreover, the 2HAM system that we give lends itself to being generalized and we
describe how this system can be modified to obtain a 2HAM system that finitely
self-assembles one of any fractal from an infinite set of fractals which we
call 4-sided fractals. The 2HAM systems we give in this paper are the first
examples of systems that finitely self-assemble discrete self-similar fractals
at scale factor 1 in a purely growth model of self-assembly. Finally, we show
that there exists a 3-sided fractal (which is not a tree fractal) that cannot
be finitely self-assembled by any 2HAM system
Contributions to computational phylogenetics and algorithmic self-assembly
This dissertation addresses some of the algorithmic and combinatorial problems at the interface between biology and computation.
In particular, it focuses on problems in both computational phylogenetics, an area of study in which computation is used to better understand evolutionary relationships, and algorithmic self-assembly, an area of study in which biological processes are used to perform computation.
The first set of results investigate inferring phylogenetic trees from multi-state character data. We give a novel characterization of when a set of three-state characters has a perfect phylogeny and make progress on a long-standing conjecture regarding the compatibility of multi-state characters.
The next set of results investigate inferring phylogenetic supertrees from collections of smaller input trees when the input trees do not fully agree on the relative positions of the taxa. Two approaches to dealing with such conflicting input trees are considered. The first is to contract a set of edges in the input trees so that the resulting trees have an agreement supertree. The second is to remove a set of taxa from the input trees so that the resulting trees have an agreement supertree. We give fixed-parameter tractable algorithms for both approaches.
We then turn to the algorithmic self-assembly of fractal structures from DNA tiles and investigate approximating the Sierpinski triangle and the Sierpinski carpet with strict self-assembly. We prove tight bounds on approximating the Sierpinski triangle and exhibit a class of fractals that are generalizations of the Sierpinski carpet that can approximately self-assemble.
We conclude by discussing some ideas for further research
Improved Lower and Upper Bounds on the Tile Complexity of Uniquely Self-Assembling a Thin Rectangle Non-Cooperatively in 3D
We investigate a fundamental question regarding a benchmark class of shapes
in one of the simplest, yet most widely utilized abstract models of algorithmic
tile self-assembly. Specifically, we study the directed tile complexity of a thin rectangle in Winfree's abstract Tile Assembly Model, assuming
that cooperative binding cannot be enforced (temperature-1 self-assembly) and
that tiles are allowed to be placed at most one step into the third dimension
(just-barely 3D). While the directed tile complexities of a square and a
scaled-up version of any algorithmically specified shape at temperature 1 in
just-barely 3D are both asymptotically the same as they are (respectively) at
temperature 2 in 2D, the bounds on the directed tile complexity of a thin
rectangle at temperature 2 in 2D are not known to hold at temperature 1 in
just-barely 3D. Motivated by this discrepancy, we establish new lower and upper
bounds on the directed tile complexity of a thin rectangle at temperature 1 in
just-barely 3D. We develop a new, more powerful type of Window Movie Lemma that
lets us upper bound the number of "sufficiently similar" ways to assign glues
to a set of fixed locations. Consequently, our lower bound,
, is an asymptotic improvement over the
previous best lower bound and is more aesthetically pleasing since it
eliminates the that used to divide . The proof of our
upper bound is based on a just-barely 3D, temperature-1 counter, organized
according to "digit regions", which affords it roughly fifty percent more
digits for the same target rectangle compared to the previous best counter.
This increase in digit density results in an upper bound of
, that
is an asymptotic improvement over the previous best upper bound and roughly the
square of our lower bound
Fractals, Randomization, Optimal Constructions, and Replication in Algorithmic Self-Assembly
The problem of the strict self-assembly of infinite fractals within tile self-assembly is considered. In particular, tile assembly algorithms are provided for the assembly of the discrete Sierpinski triangle and the discrete Sierpinski carpet.
The robust random number generation problem in the abstract tile assembly model is introduced. First, it is shown this is possible for a robust fair coin flip within the aTAM, and that such systems guarantee a worst case O(1) space usage. This primary construction is accompanied with variants that show trade-offs in space complexity, initial seed size, temperature, tile complexity, bias, and extensibility.
This work analyzes the number of tile types t, bins b, and stages necessary and sufficient to assemble n × n squares and scaled shapes in the staged tile assembly model.
Further, this work shows how to design a universal shape replicator in a 2-HAM self-assembly system with both attractive and repulsive forces