arXiv subjects

Tim Wylie

Publications and source records attributed to Tim Wylie.

23 records · Page 2Linked to original sources

arXiv2016

Universal Shape Replicators via Self-Assembly with Attractive and Repulsive Forces

We show how to design a universal shape replicator in a self-assembly system with both attractive and repulsive forces. More precisely, we show that there is a universal set of constant-size objects that, when added to any unknown hole-free polyomino shape, produces an unbounded number of copies of that shape (plus constant-size garbage objects). The constant-size objects can be easily constructed from a constant number of individual tile types using a constant number of preprocessing self-assembly steps. Our construction uses the well-studied 2-Handed Assembly Model (2HAM) of tile self-assembly, in the simple model where glues interact only with identical glues, allowing glue strengths that are either positive (attractive) or negative (repulsive), and constant temperature (required glue strength for parts to hold together). We also require that the given shape has specified glue types on its surface, and that the feature size (smallest distance between nonincident edges) is bounded below by a constant. Shape replication necessarily requires a self-assembly model where parts can both attach and detach, and this construction is the first to do so using the natural model of negative/repulsive glues (also studied before for other problems such as fuel-efficient computation); previous replication constructions require more powerful global operations such as an "enzyme" that destroys a subset of the tile types.

cs.CG↗

arXiv2016

Concentration Independent Random Number Generation in Tile Self-Assembly

In this paper we introduce the \emph{robust random number generation} problem where the goal is to design an abstract tile assembly system (aTAM system) whose terminal assemblies can be split into $n$ partitions such that a resulting assembly of the system lies within each partition with probability 1/$n$, regardless of the relative concentration assignment of the tile types in the system. First, we show this is possible for $n=2$ (a \emph{robust fair coin flip}) within the aTAM, and that such systems guarantee a worst case $\mathcal{O}(1)$ space usage. We accompany our primary construction with variants that show trade-offs in space complexity, initial seed size, temperature, tile complexity, bias, and extensibility, and also prove some negative results. As an application, we combine our coin-flip system with a result of Chandran, Gopalkrishnan, and Reif to show that for any positive integer $n$, there exists a $\mathcal{O}(\log n)$ tile system that assembles a constant-width linear assembly of expected length $n$ for any concentration assignment. We then extend our robust fair coin flip result to solve the problem of robust random number generation in the aTAM for all $n$. Two variants of robust random bit generation solutions are presented: an unbounded space solution and a bounded space solution which incurs a small bias. Further, we consider the harder scenario where tile concentrations change arbitrarily at each assembly step and show that while this is not possible in the aTAM, the problem can be solved by exotic tile assembly models from the literature.

cs.FL↗

arXiv2016

An Interesting Gadget for Chain Pair Simplification

In this paper we present an interesting gadget based on the chain pair simplification problem under the discrete Fréchet distance (CPS-3F), which allows the construction of arbitrarily long paths that must be chosen in the simplification of the two curves. A pseudopolynomial time reduction from set partition is given as an example. For clarification, CPS-3F was recently shown to be in \textbf{P}, and the reduction is merely to show how the gadget works.

cs.CG↗

arXiv2016

Intermittent Map Matching with the Discrete Fréchet Distance

In this paper we focus on the map matching problem where the goal is to find a path through a planar graph such that the path through the vertices closely matches a given polygonal curve. The map matching problem is usually approached with the Fréchet distance matching the edges of the path as well. Here, we formally define the discrete map matching problem based on the discrete Fréchet distance. We then look at the complexities of some variations of the problem which allow for vertices in the graph to be unique or reused, and whether there is a bound on the length of the path or the number of vertices from the graph used in the path. We prove several of these problems to be NP-complete, and then conclude the paper with some open questions.

cs.CG↗

arXiv2014

On the Chain Pair Simplification Problem

The problem of efficiently computing and visualizing the structural resemblance between a pair of protein backbones in 3D has led Bereg et al. to pose the Chain Pair Simplification problem (CPS). In this problem, given two polygonal chains $A$ and $B$ of lengths $m$ and $n$, respectively, one needs to simplify them simultaneously, such that each of the resulting simplified chains, $A'$ and $B'$, is of length at most $k$ and the discrete \frechet\ distance between $A'$ and $B'$ is at most $δ$, where $k$ and $δ$ are given parameters. In this paper we study the complexity of CPS under the discrete \frechet\ distance (CPS-3F), i.e., where the quality of the simplifications is also measured by the discrete \frechet\ distance. Since CPS-3F was posed in 2008, its complexity has remained open. However, it was believed to be \npc, since CPS under the Hausdorff distance (CPS-2H) was shown to be \npc. We first prove that the weighted version of CPS-3F is indeed weakly \npc\, even on the line, based on a reduction from the set partition problem. Then, we prove that CPS-3F is actually polynomially solvable, by presenting an $O(m^2n^2\min\{m,n\})$ time algorithm for the corresponding minimization problem. In fact, we prove a stronger statement, implying, for example, that if weights are assigned to the vertices of only one of the chains, then the problem remains polynomially solvable. We also study a few less rigid variants of CPS and present efficient solutions for them. Finally, we present some experimental results that suggest that (the minimization version of) CPS-3F is significantly better than previous algorithms for the motivating biological application.

cs.CG↗