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Description
Many forms of programmable matter have been proposed for various tasks. We use an abstract model of self-organizing particle systems for programmable matter which could be used for a variety of applications, including smart paint and coating materials for engineering or programmable cells for medical uses. Previous research using this model has focused on shape formation and other spatial configuration problems, including line formation, compression, and coating. In this work we study foundational computational tasks that exceed the capabilities of the individual constant memory particles described by the model. These tasks represent new ways to use these self-organizing systems, which, in conjunction with previous shape and configuration work, make the systems useful for a wider variety of tasks. We present an implementation of a counter using a line of particles, which makes it possible for the line of particles to count to and store values much larger than their individual capacities. We then present an algorithm that takes a matrix and a vector as input and then sets up and uses a rectangular block of particles to compute the matrix-vector multiplication. This setup also utilizes the counter implementation to store the resulting vector from the matrix-vector multiplication. Operations such as counting and matrix multiplication can leverage the distributed and dynamic nature of the self-organizing system to be more efficient and adaptable than on traditional linear computing hardware. Such computational tools also give the systems more power to make complex decisions when adapting to new situations or to analyze the data they collect, reducing reliance on a central controller for setup and output processing. Finally, we demonstrate an application of similar types of computations with self-organizing systems to image processing, with an implementation of an image edge detection algorithm.
ContributorsPorter, Alexandra Marie (Author) / Richa, Andrea (Thesis director) / Xue, Guoliang (Committee member) / School of Music (Contributor) / Computer Science and Engineering Program (Contributor) / School of Mathematical and Statistical Sciences (Contributor) / Barrett, The Honors College (Contributor)
Created2016-12
Description
Many programmable matter systems have been proposed and realized recently, each often tailored toward a particular task or physical setting. In our work on self-organizing particle systems, we abstract away from specific settings and instead describe programmable matter as a collection of simple computational elements (to be referred to as particles) with limited computational power that each perform fully distributed, local, asynchronous algorithms to solve system-wide problems of movement, configuration, and coordination. In this thesis, we focus on the compression problem, in which the particle system gathers as tightly together as possible, as in a sphere or its equivalent in the presence of some underlying geometry. While there are many ways to formalize what it means for a particle system to be compressed, we address three different notions of compression: (1) local compression, in which each individual particle utilizes local rules to create an overall convex structure containing no holes, (2) hole elimination, in which the particle system seeks to detect and eliminate any holes it contains, and (3) alpha-compression, in which the particle system seeks to shrink its perimeter to be within a constant factor of the minimum possible value. We analyze the behavior of each of these algorithms, examining correctness and convergence where appropriate. In the case of the Markov Chain Algorithm for Compression, we provide improvements to the original bounds for the bias parameter lambda which influences the system to either compress or expand. Lastly, we briefly discuss contributions to the problem of leader election--in which a particle system elects a single leader--since it acts as an important prerequisite for compression algorithms that use a predetermined seed particle.
ContributorsDaymude, Joshua Jungwoo (Author) / Richa, Andrea (Thesis director) / Kierstead, Henry (Committee member) / Computer Science and Engineering Program (Contributor) / School of Mathematical and Statistical Sciences (Contributor) / Barrett, The Honors College (Contributor)
Created2016-05
Description
A storage system requiring file redundancy and on-line repairability can be represented as a Steiner system, a combinatorial design with the property that every $t$-subset of its points occurs in exactly one of its blocks. Under this representation, files are the points and storage units are the blocks of the Steiner system, or vice-versa. Often, the popularities of the files of such storage systems run the gamut, with some files receiving hardly any attention, and others receiving most of it. For such systems, minimizing the difference in the collective popularity between any two storage units is nontrivial; this is the access balancing problem. With regard to the representative Steiner system, the access balancing problem in its simplest form amounts to constructing either a point or block labelling: an assignment of a set of integer labels (popularity ranks) to the Steiner system's point set or block set, respectively, requiring of the former assignment that the sums of the labelled points of any two blocks differ as little as possible and of the latter that the sums of the labels assigned to the containing blocks of any two distinct points differ as little as possible. The central aim of this dissertation is to supply point and block labellings for Steiner systems of block size greater than three, for which up to this point no attempt has been made. Four major results are given in this connection. First, motivated by the close connection between the size of the independent sets of a Steiner system and the quality of its labellings, a Steiner triple system of any admissible order is constructed with a pair of disjoint independent sets of maximum cardinality. Second, the spectrum of resolvable Bose triple systems is determined in order to label some Steiner 2-designs with block size four. Third, several kinds of independent sets are used to point-label Steiner 2-designs with block size four. Finally, optimal and close to optimal block labellings are given for an infinite class of 1-rotational resolvable Steiner 2-designs with arbitrarily large block size by exploiting their underlying group-theoretic properties.
ContributorsLusi, Dylan (Author) / Colbourn, Charles J (Thesis advisor) / Czygrinow, Andrzej (Committee member) / Fainekos, Georgios (Committee member) / Richa, Andrea (Committee member) / Arizona State University (Publisher)
Created2021
Description
The chromatic number $\chi(G)$ of a graph $G=(V,E)$ is the minimum
number of colors needed to color $V(G)$ such that no adjacent vertices
receive the same color. The coloring number $\col(G)$ of a graph
$G$ is the minimum number $k$ such that there exists a linear ordering
of $V(G)$ for which each vertex has at most $k-1$ backward neighbors.
It is well known that the coloring number is an upper bound for the
chromatic number. The weak $r$-coloring number $\wcol_{r}(G)$ is
a generalization of the coloring number, and it was first introduced
by Kierstead and Yang \cite{77}. The weak $r$-coloring number $\wcol_{r}(G)$
is the minimum integer $k$ such that for some linear ordering $L$
of $V(G)$ each vertex $v$ can reach at most $k-1$ other smaller
vertices $u$ (with respect to $L$) with a path of length at most
$r$ and $u$ is the smallest vertex in the path. This dissertation proves that $\wcol_{2}(G)\le23$ for every planar graph $G$.
The exact distance-$3$ graph $G^{[\natural3]}$ of a graph $G=(V,E)$
is a graph with $V$ as its set of vertices, and $xy\in E(G^{[\natural3]})$
if and only if the distance between $x$ and $y$ in $G$ is $3$.
This dissertation improves the best known upper bound of the
chromatic number of the exact distance-$3$ graphs $G^{[\natural3]}$
of planar graphs $G$, which is $105$, to $95$. It also improves
the best known lower bound, which is $7$, to $9$.
A class of graphs is nowhere dense if for every $r\ge 1$ there exists $t\ge 1$ such that no graph in the class contains a topological minor of the complete graph $K_t$ where every edge is subdivided at most $r$ times. This dissertation gives a new characterization of nowhere dense classes using generalized notions of the domination number.
number of colors needed to color $V(G)$ such that no adjacent vertices
receive the same color. The coloring number $\col(G)$ of a graph
$G$ is the minimum number $k$ such that there exists a linear ordering
of $V(G)$ for which each vertex has at most $k-1$ backward neighbors.
It is well known that the coloring number is an upper bound for the
chromatic number. The weak $r$-coloring number $\wcol_{r}(G)$ is
a generalization of the coloring number, and it was first introduced
by Kierstead and Yang \cite{77}. The weak $r$-coloring number $\wcol_{r}(G)$
is the minimum integer $k$ such that for some linear ordering $L$
of $V(G)$ each vertex $v$ can reach at most $k-1$ other smaller
vertices $u$ (with respect to $L$) with a path of length at most
$r$ and $u$ is the smallest vertex in the path. This dissertation proves that $\wcol_{2}(G)\le23$ for every planar graph $G$.
The exact distance-$3$ graph $G^{[\natural3]}$ of a graph $G=(V,E)$
is a graph with $V$ as its set of vertices, and $xy\in E(G^{[\natural3]})$
if and only if the distance between $x$ and $y$ in $G$ is $3$.
This dissertation improves the best known upper bound of the
chromatic number of the exact distance-$3$ graphs $G^{[\natural3]}$
of planar graphs $G$, which is $105$, to $95$. It also improves
the best known lower bound, which is $7$, to $9$.
A class of graphs is nowhere dense if for every $r\ge 1$ there exists $t\ge 1$ such that no graph in the class contains a topological minor of the complete graph $K_t$ where every edge is subdivided at most $r$ times. This dissertation gives a new characterization of nowhere dense classes using generalized notions of the domination number.
ContributorsAlmulhim, Ahlam (Author) / Kierstead, Henry (Thesis advisor) / Sen, Arunabha (Committee member) / Richa, Andrea (Committee member) / Czygrinow, Andrzej (Committee member) / Fishel, Susanna (Committee member) / Arizona State University (Publisher)
Created2020