Brad Shutters

Self-Assembling Rulers for Approximating Generalized Sierpinski Carpets

Discrete self-similar fractals have been used as test cases for self-assembly, both in the laboratory and in mathematical models, ever since Winfree exhibited a tile assembly system in which the Sierpinski triangle self-assembles. For strict self-assembly, where tiles are not allowed to be placed outside the target structure, it is an open question whether any self-similar fractal can self-assemble. This has motivated the development of techniques to approximate fractals with strict self-assembly. Ideally, such an approximation would produce a structure with the same fractal dimension as the intended fractal, but with specially labeled tiles at positions corresponding to points in the fractal. We show that the Sierpinski carpet, along with an infinite class of related fractals, can approximately self-assemble in this manner. Our construction takes a set of parameters specifying a target fractal and creates a tile assembly system in which the fractal approximately self-assembles. This construction introduces rulers and readers to control the self-assembly of a fractal structure without distorting it. To verify the fractal dimension of the resulting assemblies, we prove a result on the dimension of sets embedded into discrete fractals. We also give a conjecture on the limitations of approximating self-similar fractals.

Incompatible quartets, triplets, and characters

We study a long standing conjecture on the necessary and sufficient conditions for the compatibility of multi-state characters: There exists a function f(r) such that, for any set C of r-state characters, C is compatible if and only if every subset of f(r) characters of C is compatible. We show that for every r≥2, there exists an incompatible set C of Ω(r2)r-state characters such that every proper subset of C is compatible. This improves the previous lower bound of f(r)≥r given by Meacham (1983), and f(4)≥5 given by Habib and To (2011). For the case when r=3, Lam, Gusfield and Sridhar (2011) recently showed that f(3)=3. We give an independent proof of this result and completely characterize the sets of pairwise compatible 3-state characters by a single forbidden intersection pattern.Our lower bound on f(r) is proven via a result on quartet compatibility that may be of independent interest: For every n≥4, there exists an incompatible set Q of Ω(n2) quartets over n labels such that every proper subset of Q is compatible. We show that such a set of quartets can have size at most 3 when n=5, and at most O(n3) for arbitrary n. We contrast our results on quartets with the case of rooted triplets: For every n≥3, if R is an incompatible set of more than n−1 triplets over n labels, then some proper subset of R is incompatible. We show this bound is tight by exhibiting, for every n≥3, a set of n−1 triplets over n taxa such that R is incompatible, but every proper subset of R is compatible.

A Simple Characterization of the Minimal Obstruction Sets for Three-State Perfect Phylogenies

Abstract We give a characterization of the minimal obstruction sets for the existence of a perfect phylogeny for a set of three-state characters that can be inferred by testing each pair of characters. This leads to a O ( m 2 n + p ) time algorithm for outputting all p minimal obstruction sets for a set of m three-state characters over a set of n taxa.

Fixed-Parameter Algorithms for Finding Agreement Supertrees

We study the agreement supertree approach for combining rooted phylogenetic trees when the input trees do not fully agree on the relative positions of the taxa. We consider two ways to deal with such conflict. The first is to contract a set of edges in the input trees so that the resulting trees have an agreement supertree. We show that this problem is NP-complete and give a fixed-parameter tractable (FPT) algorithm for the problem parameterized by the number of input trees and the number of edges contracted. The second approach is to remove a set of taxa from the input trees so that the resulting trees have an agreement supertree. Guillemot and Berry (2010) gave an FPT algorithm for this problem when the input trees are all binary. We give an FPT algorithm for the more general case where the input trees are allowed to have arbitrary degree.

Fixed-Parameter algorithms for finding agreement supertrees

We study the agreement supertree approach for combining rooted phylogenetic 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. We show that this problem is NP-complete and give an FPT algorithm for the problem parameterized by the number of input trees and the number of edges contracted. The second approach is to remove a set of taxa from the input trees so that the resulting trees have an agreement supertree. An FPT algorithm for this problem when the input trees are all binary was given by Guillemot and Berry (2010). We give an FPT algorithm for the more general case when the input trees have arbitrary degree.

Self-assembling rulers for approximating generalized sierpinski carpets

Discrete self-similar fractals have been studied as test cases for self-assembly ever since Winfree exhibited a tile assembly system in which the Sierpinski triangle self-assembles. For strict self-assembly, where tiles are not allowed to be placed outside the target structure, it is an open question whether any self-similar fractal can self-assemble. This has motivated the development of techniques to approximate fractals with strict self-assembly. Ideally, such an approximation would produce a structure with the same fractal dimension as the intended fractal and with specially labeled tiles at positions corresponding to points in the fractal. We show that the Sierpinski carpet, along with an infinite class of related fractals, can approximately self-assemble in this manner. Our construction takes a set of parameters specifying a target fractal and creates a tile assembly system in which the fractal approximately selfassembles. This construction introduces rulers and readers to control the self-assembly of a fractal structure without distorting it. To verify the fractal dimension of the resulting assemblies, we prove a result on the dimension of sets embedded into discrete fractals. We also give a conjecture on the limitations of approximating self-similar fractals.

Improved lower bounds on the compatibility of quartets, triplets, and multi-state characters

We study a long standing conjecture on the necessary and sufficient conditions for the compatibility of multi-state characters: There exists a function f(r) such that, for any set C of r-state characters, C is compatible if and only if every subset of f(r) characters of C is compatible. We show that for every r≥2, there exists an incompatible set C of

Approximate Self-Assembly of the Sierpinski Triangle

\lfloor\frac{r}{2}\rfloor\cdot\lceil\frac{r}{2}\rceil + 1

Approximate self-assembly of the Sierpinski triangle

r-state characters such that every proper subset of C is compatible. Thus, f(r) ≥

Approximate Self-Assembly of the Sierpinski Triangle.

\lfloor\frac{r}{2}\rfloor\cdot\lceil\frac{r}{2}\rceil + 1