Subsequences and Supersequences of Strings

Stringology - the study of strings - is a branch of algorithmics which been the sub-ject of mounting interest in recent years. Very recently, two books [M. Crochemore and W. Rytter, Text Algorithms, Oxford University Press, 1995] and [G. Stephen, String Searching Algorithms, World Scientific, 1994] have been published on the subject and at least two others are known to be in preparation. Problems on strings arise in information retrieval, version control, automatic spelling correction, and many other domains. However the greatest motivation for recent work in stringology has come from the field of molecular biology. String problems occur, for example, in genetic sequence construction, genetic sequence comparison, and phylogenetic tree construction. In this thesis we study a variety of string problems from a theoretical perspective. In particular, we focus on problems involving subsequences and supersequences of strings

Parameterized Rural Postman Problem

The Directed Rural Postman Problem (DRPP) can be formulated as follows: given a strongly connected directed multigraph $D=(V,A)$ with nonnegative integral weights on the arcs, a subset $R$ of $A$ and a nonnegative integer $\ell$, decide whether $D$ has a closed directed walk containing every arc of $R$ and of total weight at most $\ell$. Let $k$ be the number of weakly connected components in the the subgraph of $D$ induced by $R$. Sorge et al. (2012) ask whether the DRPP is fixed-parameter tractable (FPT) when parameterized by $k$, i.e., whether there is an algorithm of running time $O^*(f(k))$ where $f$ is a function of $k$ only and the $O^*$ notation suppresses polynomial factors. Sorge et al. (2012) note that this question is of significant practical relevance and has been open for more than thirty years. Using an algebraic approach, we prove that DRPP has a randomized algorithm of running time $O^*(2^k)$ when $\ell$ is bounded by a polynomial in the number of vertices in $D$. We also show that the same result holds for the undirected version of DRPP, where $D$ is a connected undirected multigraph

Hardware-Assisted Secure Computation

The theory community has worked on Secure Multiparty Computation (SMC) for more than two decades, and has produced many protocols for many settings. One common thread in these works is that the protocols cannot use a Trusted Third Party (TTP), even though this is conceptually the simplest and most general solution. Thus, current protocols involve only the direct players---we call such protocols self-reliant. They often use blinded boolean circuits, which has several sources of overhead, some due to the circuit representation and some due to the blinding. However, secure coprocessors like the IBM 4758 have actual security properties similar to ideal TTPs. They also have little RAM and a slow CPU.We call such devices Tiny TTPs. The availability of real tiny TTPs opens the door for a different approach to SMC problems. One major challenge with this approach is how to execute large programs on large inputs using the small protected memory of a tiny TTP, while preserving the trust properties that an ideal TTP provides. In this thesis we have investigated the use of real TTPs to help with the solution of SMC problems. We start with the use of such TTPs to solve the Private Information Retrieval (PIR) problem, which is one important instance of SMC. Our implementation utilizes a 4758. The rest of the thesis is targeted at general SMC. Our SMC system, Faerieplay, moves some functionality into a tiny TTP, and thus avoids the blinded circuit overhead. Faerieplay consists of a compiler from high-level code to an arithmetic circuit with special gates for efficient indirect array access, and a virtual machine to execute this circuit on a tiny TTP while maintaining the typical SMC trust properties. We report on Faerieplay\u27s security properties, the specification of its components, and our implementation and experiments. These include comparisons with the Fairplay circuit-based two-party system, and an implementation of the Dijkstra graph shortest path algorithm. We also provide an implementation of an oblivious RAM which supports similar tiny TTP-based SMC functionality but using a standard RAM program. Performance comparisons show Faerieplay\u27s circuit approach to be considerably faster, at the expense of a more constrained programming environment when targeting a circuit

New Applications of the Nearest-Neighbor Chain Algorithm

The nearest-neighbor chain algorithm was proposed in the eighties as a way to speed up certain hierarchical clustering algorithms. In the first part of the dissertation, we show that its application is not limited to clustering. We apply it to a variety of geometric and combinatorial problems. In each case, we show that the nearest-neighbor chain algorithm finds the same solution as a preexistent greedy algorithm, but often with an improved runtime. We obtain speedups over greedy algorithms for Euclidean TSP, Steiner TSP in planar graphs, straight skeletons, a geometric coverage problem, and three stable matching models. In the second part, we study the stable-matching Voronoi diagram, a type of plane partition which combines properties of stable matchings and Voronoi diagrams. We propose political redistricting as an application. We also show that it is impossible to compute this diagram in an algebraic model of computation, and give three algorithmic approaches to overcome this obstacle. One of them is based on the nearest-neighbor chain algorithm, linking the two parts together

An image-space algorithm for hardware-based rendering of constructive solid geometry

A new approach to image-space hardware-based rendering of Constructive Solid Geometry (CSG) models is presented. The work is motivated by the evolving functionality and performance of computer graphics hardware. This work is also motivated by a specific industrial application --- interactive verification of five axis grinding machine tool programs. The goal is to minimise the amount of time required to render each frame in an animation or interactive application involving boolean combinations of three dimensional shapes. The Sequenced Convex Subtraction (SCS) algorithm utilises sequenced subtraction of convex objects for the purpose of interactive CSG rendering. Concave shapes must be decomposed into convex shapes for the purpose of rendering. The length of Permutation Embedding Sequences (PESs) used as subtraction sequences are shown to have a quadratic lower bound. In ma ny situations shorter sequences can be used, in the best case linear. Approaches to subtraction sequence encoding are presented including the use of object-space overlap information. The implementation of the algorithm is experimentally shown to perform better on modern commodity graphics hardware than previously reported methods. This work also examines performance aspects of the SCS algorithm itself. Overall performance depends on hardware characteristics, the number and spatial arrangement of primitives, and the structure and boolean operators of the CSG tree

Physical-Layer Security, Quantum Key Distribution and Post-quantum Cryptography

The growth of data-driven technologies, 5G, and the Internet place enormous pressure on underlying information infrastructure. There exist numerous proposals on how to deal with the possible capacity crunch. However, the security of both optical and wireless networks lags behind reliable and spectrally efficient transmission. Significant achievements have been made recently in the quantum computing arena. Because most conventional cryptography systems rely on computational security, which guarantees the security against an efficient eavesdropper for a limited time, with the advancement in quantum computing this security can be compromised. To solve these problems, various schemes providing perfect/unconditional security have been proposed including physical-layer security (PLS), quantum key distribution (QKD), and post-quantum cryptography. Unfortunately, it is still not clear how to integrate those different proposals with higher level cryptography schemes. So the purpose of the Special Issue entitled “Physical-Layer Security, Quantum Key Distribution and Post-quantum Cryptography” was to integrate these various approaches and enable the next generation of cryptography systems whose security cannot be broken by quantum computers. This book represents the reprint of the papers accepted for publication in the Special Issue

Computational methods in string theory and applications to the swampland conjectures

The goal of the swampland program is the classification of low energy effective theories which can be consistently coupled to quantum gravity. Due to the vastness of the string landscape most results of the swampland program are still conjectures, yet the web of conjectures is ever growing and many interdependencies between different conjectures are known. A better understanding or even proof of these conjectures would result in restrictions on the allowed effective theories. The aim of this thesis is to develop the necessary mathematical tools to explicitly test the conjectures in a string theory setup. To this end the periods of Calabi-Yau manifolds are computed numerically as well as analytically. Furthermore, tools applicable to general string phenomenological models are discussed, including the computation of target space Calabi-Yau metrics, line bundle cohomologies and Strebel differentials. These periods are used to test two conjectures, the refined swampland distance conjecture as well as the dS conjecture. The first states that an effective field theory is only valid up to a certain value of field excursions. If larger field values are included, the effective description breaks down due to an infinite tower of states becoming exponentially light. The conjecture is tested explicitly by computing the distances in the moduli space of CY manifolds. Challenging this conjecture requires the computation of the periods of different Calabi-Yau spaces. The dS conjecture on the other hand forbids vacua with positive cosmological constant. To test this conjecture, the KKLT construction is examined in detail and some steps of the construction are carried out explicitly. Moreover, the validity of the assumed effective theory in a warped throat is investigated. Besides these traditional approaches more exotic ones are followed, including the construction of dS theories using tachyons as well as modifying the signature of space time.Das Ziel des Swampland Programms ist die Klassifizierung effektiver, zu Quantengravitationstheorien vervollständigbarer Theorien. Aufgrund der enormen Anzahl an möglichen Stringvacua, zusammengefasst in der sogenannten Stringlandschaft, sind die meisten der bisherigen Resultate des Programms Vermutungen. Jedoch existiert ein beständig wachsendes dichtes Netz aus Abhängigkeiten zwischen diesen Vermutungen. Ein besseres Verständnis oder ein Beweis dieser Vermutungen würde die erlaubten Niederenergietheorien einschränken. Das Ziel dieser Arbeit ist deshalb die Entwicklung mathematischer Methoden, die explizite Tests der Swampland Vermutungen in stringtheoretischen Modellen ermöglichen. Insbesondere werden Perioden von Calabi-Yau Mannigfaltigkeiten auf numerischem und analytischem Weg berechnet. Darüber hinaus werden Methoden zur Berechnung von Calabi-Yau Metriken, Linienbündelkohomologien und Strebeldifferentialen behandelt. Diese werden zur Überprüfung zweier Vermutungen eingesetzt, zum Test der Swampland Distanzvermutung sowie zum Test der dS Vermutung. Erstere besagt, dass eine effektive Theorie nur bis zu bestimmten Feldwerten gültig sein kann. Werden diese überschritten werden unendlich viele nicht berücksichtigte Zustände exponentiell leicht und die verwendete effektive Beschreibung bricht zusammen. Diese Vermutung wird durch eine explizite Berechnung von Distanzen zwischen effektiven Theorien in Calabi-Yau Moduliräumen getestet. Die dS Vermutung verbietet hingegen stabile Vacua mit positiver kosmologischer Konstante. Um diese Vermutung zu überprüfen, wird ein Teil der KKLT-Konstruktion explizit durchgeführt. Darüber hinaus wird die Validität der zugrundeliegenden effektiven Theorie in einem warped throat analysiert. Neben diesen traditionellen Herangehensweisen werden exotischere Ansätze für die Konstruktion von dS Räumen untersucht. Dies umfasst Tachyonenkondensation sowie andere Raumzeitsignaturen