Covering Arrays for Equivalence Classes of Words

Covering arrays for words of length t over a d letter alphabet are k × n arrays with entries from the alphabet so that for each choice of t columns, each of the dt t-letter words appears at least once among the rows of the selected columns. We study two schemes in which all words are not considered to be different. In the first case, words are equivalent if they induce the same partition of a t element set. In the second case, words of the same weighted sum are equivalent. In both cases we produce logarithmic upper bounds on the minimum size k = k(n) of a covering array. Most definitive results are for t = 2, 3, 4

Locating Arrays: Construction, Analysis, and Robustness

abstract: Modern computer systems are complex engineered systems involving a large collection of individual parts, each with many parameters, or factors, affecting system performance. One way to understand these complex systems and their performance is through experimentation. However, most modern computer systems involve such a large number of factors that thorough experimentation on all of them is impossible. An initial screening step is thus necessary to determine which factors are relevant to the system's performance and which factors can be eliminated from experimentation. Factors may impact system performance in different ways. A factor at a specific level may significantly affect performance as a main effect, or in combination with other main effects as an interaction. For screening, it is necessary both to identify the presence of these effects and to locate the factors responsible for them. A locating array is a relatively new experimental design that causes every main effect and interaction to occur and distinguishes all sets of d main effects and interactions from each other in the tests where they occur. This design is therefore helpful in screening complex systems. The process of screening using locating arrays involves multiple steps. First, a locating array is constructed for all possibly significant factors. Next, the system is executed for all tests indicated by the locating array and a response is observed. Finally, the response is analyzed to identify the significant system factors for future experimentation. However, simply constructing a reasonably sized locating array for a large system is no easy task and analyzing the response of the tests presents additional difficulties due to the large number of possible predictors and the inherent imbalance in the experimental design itself. Further complications can arise from noise in the system or errors in testing. This thesis has three contributions. First, it provides an algorithm to construct locating arrays using the Lovász Local Lemma with Moser-Tardos resampling. Second, it gives an algorithm to analyze the system response efficiently. Finally, it studies the robustness of the analysis to the heavy-hitters assumption underlying the approach as well as to varying amounts of system noise.Dissertation/ThesisMasters Thesis Computer Engineering 201

Algorithmic Methods for Covering Arrays of Higher Index

Covering arrays are combinatorial objects used in testing large-scale systems to increase confidence in their correctness. To do so, each interaction of at most a specified number t of factors is represented in at least one test; that is, the covering array has strength t and index 1. For certain systems, the outcome of running a test may be altered by variability of the interaction effect or by measurement error of the test result. To improve the efficacy of testing, one can ensure that each interaction of t or fewer factors is represented in at least λ tests. When λ \u3e 1, this leads to covering arrays of higher index. We explore two algorithmic methods for constructing covering arrays of higher index. One is based on the in-parameter-order algorithm, and the other employs a conditional expectation paradigm. We compare these two by performing experiments on real-world benchmarks and on uniform parameter sets

The Design and Analysis of Hash Families For Use in Broadcast Encryption

abstract: Broadcast Encryption is the task of cryptographically securing communication in a broadcast environment so that only a dynamically specified subset of subscribers, called the privileged subset, may decrypt the communication. In practical applications, it is desirable for a Broadcast Encryption Scheme (BES) to demonstrate resilience against attacks by colluding, unprivileged subscribers. Minimal Perfect Hash Families (PHFs) have been shown to provide a basis for the construction of memory-efficient t-resilient Key Pre-distribution Schemes (KPSs) from multiple instances of 1-resilient KPSs. Using this technique, the task of constructing a large t-resilient BES is reduced to finding a near-minimal PHF of appropriate parameters. While combinatorial and probabilistic constructions exist for minimal PHFs with certain parameters, the complexity of constructing them in general is currently unknown. This thesis introduces a new type of hash family, called a Scattering Hash Family (ScHF), which is designed to allow for the scalable and ingredient-independent design of memory-efficient BESs for large parameters, specifically resilience and total number of subscribers. A general BES construction using ScHFs is shown, which constructs t-resilient KPSs from other KPSs of any resilience ≤w≤t. In addition to demonstrating how ScHFs can be used to produce BESs , this thesis explores several ScHF construction techniques. The initial technique demonstrates a probabilistic, non-constructive proof of existence for ScHFs . This construction is then derandomized into a direct, polynomial time construction of near-minimal ScHFs using the method of conditional expectations. As an alternative approach to direct construction, representing ScHFs as a k-restriction problem allows for the indirect construction of ScHFs via randomized post-optimization. Using the methods defined, ScHFs are constructed and the parameters' effects on solution size are analyzed. For large strengths, constructive techniques lose significant performance, and as such, asymptotic analysis is performed using the non-constructive existential results. This work concludes with an analysis of the benefits and disadvantages of BESs based on the constructed ScHFs. Due to the novel nature of ScHFs, the results of this analysis are used as the foundation for an empirical comparison between ScHF-based and PHF-based BESs . The primary bases of comparison are construction efficiency, key material requirements, and message transmission overhead.Dissertation/ThesisM.S. Computer Science 201

On Approximability, Convergence, and Limits of CSP Problems

This thesis studies dense constraint satisfaction problems (CSPs), and other related optimization and decision problems that can be phrased as questions regarding parameters or properties of combinatorial objects such as uniform hypergraphs. We concentrate on the information that can be derived from a very small substructure that is selected uniformly at random. In this thesis, we present a unified framework on the limits of CSPs in the sense of the convergence notion of Lovasz-Szegedy that depends only on the remarkable connection between graph sequences and exchangeable arrays established by Diaconis-Janson. In particular, we formulate and prove a representation theorem for compact colored r-uniform directed hypergraphs and apply this to rCSPs. We investigate the sample complexity of testable r-graph parameters, and discuss a generalized version of ground state energies (GSE) and demonstrate that they are efficiently testable. The GSE is a term borrowed from statistical physics that stands for a generalized version of maximal multiway cut problems from complexity theory, and was studied in the dense graph setting by Borgs et al. A notion related to testing CSPs that are defined on graphs, the nondeterministic property testing, was introduced by Lovasz-Vesztergombi, which extends the graph property testing framework of Goldreich-Goldwasser-Ron in the dense graph model. In this thesis, we study the sample complexity of nondeterministically testable graph parameters and properties and improve existing bounds by several orders of magnitude. Further, we prove the equivalence of the notions of nondeterministic and deterministic parameter and property testing for uniform dense hypergraphs of arbitrary rank, and provide the first effective upper bound on the sample complexity in this general case

Among graphs, groups, and latin squares

A latin square of order n is an n × n array in which each row and each column contains each of the numbers {1, 2, . . . , n}. A k-plex in a latin square is a collection of entries which intersects each row and column k times and contains k copies of each symbol. This thesis studies the existence of k-plexes and approximations of k-plexes in latin squares, paying particular attention to latin squares which correspond to multiplication tables of groups. The most commonly studied class of k-plex is the 1-plex, better known as a transversal. Although many latin squares do not have transversals, Brualdi conjectured that every latin square has a near transversal—i.e. a collection of entries with distinct symbols which in- tersects all but one row and all but one column. Our first main result confirms Brualdi’s conjecture in the special case of group-based latin squares. Then, using a well-known equivalence between edge-colorings of complete bipartite graphs and latin squares, we introduce Hamilton 2-plexes. We conjecture that every latin square of order n ≥ 5 has a Hamilton 2-plex and provide a range of evidence for this conjecture. In particular, we confirm our conjecture computationally for n ≤ 8 and show that a suitable analogue of Hamilton 2-plexes always occur in n × n arrays with no symbol appearing more than n/√96 times. To study Hamilton 2-plexes in group-based latin squares, we generalize the notion of harmonious groups to what we call H2-harmonious groups. Our second main result classifies all H2-harmonious abelian groups. The last part of the thesis formalizes an idea which first appeared in a paper of Cameron and Wanless: a (k,l)-plex is a collection of entries which intersects each row and column k times and contains at most l copies of each symbol. We demonstrate the existence of (k, 4k)-plexes in all latin squares and (k, k + 1)-plexes in sufficiently large latin squares. We also find analogues of these theorems for Hamilton 2-plexes, including our third main result: every sufficiently large latin square has a Hamilton (2,3)-plex

Minimal Ramsey graphs, orthogonal Latin squares, and hyperplane coverings

This thesis consists of three independent parts. The first part of the thesis is concerned with Ramsey theory. Given an integer $q\geq 2$, a graph $G$ is said to be \emph{$q$-Ramsey} for another graph $H$ if in any $q$-edge-coloring of $G$ there exists a monochromatic copy of $H$. The central line of research in this area investigates the smallest number of vertices in a $q$-Ramsey graph for a given $H$. In this thesis, we explore two different directions. First, we will be interested in the smallest possible minimum degree of a minimal (with respect to subgraph inclusion) $q$-Ramsey graph for a given $H$. This line of research was initiated by Burr, Erdős, and Lovász in the 1970s. We study the minimum degree of a minimal Ramsey graph for a random graph and investigate how many vertices of small degree a minimal Ramsey graph for a given $H$ can contain. We also consider the minimum degree problem in a more general asymmetric setting. Second, it is interesting to ask how small modifications to the graph $H$ affect the corresponding collection of $q$-Ramsey graphs. Building upon the work of Fox, Grinshpun, Liebenau, Person, and Szabó and Rödl and Siggers, we prove that adding even a single pendent edge to the complete graph $K_t$ changes the collection of 2-Ramsey graphs significantly. The second part of the thesis deals with orthogonal Latin squares. A {\em Latin square of order $n$} is an $n\times n$ array with entries in $[n]$ such that each integer appears exactly once in every row and every column. Two Latin squares $L$ and $L'$ are said to be {\em orthogonal} if, for all $x,y\in [n]$, there is a unique pair $(i,j)\in [n]^2$ such that $L(i,j) = x$ and $L'(i,j) = y$; a system of {\em $k$ mutually orthogonal Latin squares}, or a {\em $k$-MOLS}, is a set of $k$ pairwise orthogonal Latin squares. Motivated by a well-known result determining the number of different Latin squares of order $n$ log-asymptotically, we study the number of $k$-MOLS of order $n$. Earlier results on this problem were obtained by Donovan and Grannell and Keevash and Luria. We establish new upper bounds for a wide range of values of $k = k(n)$. We also prove a new, log-asymptotically tight, bound on the maximum number of other squares a single Latin square can be orthogonal to. The third part of the thesis is concerned with grid coverings with multiplicities. In particular, we study the minimum number of hyperplanes necessary to cover all points but one of a given finite grid at least $k$ times, while covering the remaining point fewer times. We study this problem for the grid $\mathbb{F}_2^n$, determining the number exactly when one of the parameters $n$ and $k$ is much larger than the other and asymptotically in all other cases. This generalizes a classic result of Jamison for $k=1$. Additionally, motivated by the recent work of Clifton and Huang and Sauermann and Wigderson for the hypercube $\set{0,1}^n\subseteq\mathbb{R}^n$, we study hyperplane coverings for different grids over $\mathbb{R}$, under the stricter condition that the remaining point is omitted completely. We focus on two-dimensional real grids, showing a variety of results and demonstrating that already this setting offers a range of possible behaviors.Diese Dissertation besteht aus drei unabh\"angigen Teilen. Der erste Teil beschäftigt sich mit Ramseytheorie. Für eine ganze Zahl $q\geq 2$ nennt man einen Graphen \emph{$q$-Ramsey} f\"ur einen anderen Graphen $H$, wenn jede Kantenf\"arbung mit $q$ Farben einen einfarbigen Teilgraphen enthält, der isomorph zu $H$ ist. Das zentrale Problem in diesem Gebiet ist die minimale Anzahl von Knoten in einem solchen Graphen zu bestimmen. In dieser Dissertation betrachten wir zwei verschiedene Varianten. Als erstes, beschäftigen wir uns mit dem kleinstm\"oglichen Minimalgrad eines minimalen (bezüglich Teilgraphen) $q$-Ramsey-Graphen f\"ur einen gegebenen Graphen $H$. Diese Frage wurde zuerst von Burr, Erd\H{o}s und Lov\'asz in den 1970er-Jahren studiert. Wir betrachten dieses Problem f\"ur einen Zufallsgraphen und untersuchen, wie viele Knoten kleinen Grades ein Ramsey-Graph f\"ur gegebenes $H$ enthalten kann. Wir untersuchen auch eine asymmetrische Verallgemeinerung des Minimalgradproblems. Als zweites betrachten wir die Frage, wie sich die Menge aller $q$-Ramsey-Graphen f\"ur $H$ verändert, wenn wir den Graphen $H$ modifizieren. Aufbauend auf den Arbeiten von Fox, Grinshpun, Liebenau, Person und Szabó und Rödl und Siggers beweisen wir, dass bereits der Graph, der aus $K_t$ mit einer h\"angenden Kante besteht, eine sehr unterschiedliche Menge von 2-Ramsey-Graphen besitzt im Vergleich zu $K_t$. Im zweiten Teil geht es um orthogonale lateinische Quadrate. Ein \emph{lateinisches Quadrat der Ordnung $n$} ist eine $n\times n$-Matrix, gef\"ullt mit den Zahlen aus $[n]$, in der jede Zahl genau einmal pro Zeile und einmal pro Spalte auftritt. Zwei lateinische Quadrate sind \emph{orthogonal} zueinander, wenn f\"ur alle $x,y\in[n]$ genau ein Paar $(i,j)\in [n]^2$ existiert, sodass es $L(i,j) = x$ und $L'(i,j) = y$ gilt. Ein \emph{k-MOLS der Ordnung $n$} ist eine Menge von $k$ lateinischen Quadraten, die paarweise orthogonal sind. Motiviert von einem bekannten Resultat, welches die Anzahl von lateinischen Quadraten der Ordnung $n$ log-asymptotisch bestimmt, untersuchen wir die Frage, wie viele $k$-MOLS der Ordnung $n$ es gibt. Dies wurde bereits von Donovan und Grannell und Keevash und Luria studiert. Wir verbessern die beste obere Schranke f\"ur einen breiten Bereich von Parametern $k=k(n)$. Zusätzlich bestimmen wir log-asymptotisch zu wie viele anderen lateinischen Quadraten ein lateinisches Quadrat orthogonal sein kann. Im dritten Teil studieren wir, wie viele Hyperebenen notwendig sind, um die Punkte eines endlichen Gitters zu überdecken, sodass ein bestimmter Punkt maximal $(k-1)$-mal bedeckt ist und alle andere mindestens $k$-mal. Wir untersuchen diese Anzahl f\"ur das Gitter $\mathbb{F}_2^n$ asymptotisch und sogar genau, wenn eins von $n$ und $k$ viel größer als das andere ist. Dies verallgemeinert ein Ergebnis von Jamison für den Fall $k=1$. Au{\ss}erdem betrachten wir dieses Problem f\"ur Gitter im reellen Vektorraum, wenn der spezielle Punkt überhaupt nicht bedeckt ist. Dies ist durch die Arbeiten von Clifton und Huang und Sauermann und Wigderson motiviert, die den Hyperwürfel $\set{0,1}^n\subseteq \mathbb{R}^n$ untersucht haben. Wir konzentrieren uns auf zwei-dimensionale Gitter und zeigen, dass schon diese sich sehr unterschiedlich verhalten können

Extremal Problems on the Hypercube

PhDThe hypercube, Qd, is a natural and much studied combinatorial object, and we discuss various extremal problems related to it. A subgraph of the hypercube is said to be (Qd; F)-saturated if it contains no copies of F, but adding any edge forms a copy of F. We write sat(Qd; F) for the saturation number, that is, the least number of edges a (Qd; F)-saturated graph may have. We prove the upper bound sat(Qd;Q2) < 10 2d, which strongly disproves a conjecture of Santolupo that sat(Qd;Q2) = �� 1 4 + o(1) d2d��1. We also prove upper bounds on sat(Qd;Qm) for general m.Given a down-set A and an up-set B in the hypercube, Bollobás and Leader conjectured a lower bound on the number of edge-disjoint paths between A and B in the directed hypercube. Using an unusual form of the compression argument, we confirm the conjecture by reducing the problem to a the case of the undirected hypercube. We also prove an analogous conjecture for vertex-disjoint paths using the same techniques, and extend both results to the grid. Additionally, we deal with subcube intersection graphs, answering a question of Johnson and Markström of the least r = r(n) for which all graphs on n vertices may be represented as subcube intersection graph where each subcube has dimension exactly r. We also contribute to the related area of biclique covers and partitions, and study relationships between various parameters linked to such covers and partitions. Finally, we study topological properties of uniformly random simplicial complexes, employing a characterisation due to Korshunov of almost all down-sets in the hypercube as a key tool