木を用いた構造化並列プログラミング

High-level abstractions for parallel programming are still immature. Computations on complicated data structures such as pointer structures are considered as irregular algorithms. General graph structures, which irregular algorithms generally deal with, are difficult to divide and conquer. Because the divide-and-conquer paradigm is essential for load balancing in parallel algorithms and a key to parallel programming, general graphs are reasonably difficult. However, trees lead to divide-and-conquer computations by definition and are sufficiently general and powerful as a tool of programming. We therefore deal with abstractions of tree-based computations. Our study has started from Matsuzaki’s work on tree skeletons. We have improved the usability of tree skeletons by enriching their implementation aspect. Specifically, we have dealt with two issues. We first have implemented the loose coupling between skeletons and data structures and developed a flexible tree skeleton library. We secondly have implemented a parallelizer that transforms sequential recursive functions in C into parallel programs that use tree skeletons implicitly. This parallelizer hides the complicated API of tree skeletons and makes programmers to use tree skeletons with no burden. Unfortunately, the practicality of tree skeletons, however, has not been improved. On the basis of the observations from the practice of tree skeletons, we deal with two application domains: program analysis and neighborhood computation. In the domain of program analysis, compilers treat input programs as control-flow graphs (CFGs) and perform analysis on CFGs. Program analysis is therefore difficult to divide and conquer. To resolve this problem, we have developed divide-and-conquer methods for program analysis in a syntax-directed manner on the basis of Rosen’s high-level approach. Specifically, we have dealt with data-flow analysis based on Tarjan’s formalization and value-graph construction based on a functional formalization. In the domain of neighborhood computations, a primary issue is locality. A naive parallel neighborhood computation without locality enhancement causes a lot of cache misses. The divide-and-conquer paradigm is known to be useful also for locality enhancement. We therefore have applied algebraic formalizations and a tree-segmenting technique derived from tree skeletons to the locality enhancement of neighborhood computations.電気通信大学201

Skeleton Structures and Origami Design

In this dissertation we study problems related to polygonal skeleton structures that have applications to computational origami. The two main structures studied are the straight skeleton of a simple polygon (and its generalizations to planar straight line graphs) and the universal molecule of a Lang polygon. This work builds on results completed jointly with my advisor Ileana Streinu. Skeleton structures are used in many computational geometry algorithms. Examples include the medial axis, which has applications including shape analysis, optical character recognition, and surface reconstruction; and the Voronoi diagram, which has a wide array of applications including geographic information systems (GIS), point location data structures, motion planning, etc. The straight skeleton, studied in this work, has applications in origami design, polygon interpolation, biomedical imaging, and terrain modeling, to name just a few. Though the straight skeleton has been well studied in the computational geometry literature for over 20 years, there still exists a significant gap between the fastest algorithms for constructing it and the known lower bounds. One contribution of this thesis is an efficient algorithm for computing the straight skeleton of a polygon, polygon with holes, or a planar straight-line graph given a secondary structure called the induced motorcycle graph. The universal molecule is a generalization of the straight skeleton to certain convex polygons that have a particular relationship to a metric tree. It is used in Robert Lang\u27s seminal TreeMaker method for origami design. Informally, the universal molecule is a subdivision of a polygon (or polygonal sheet of paper) that allows the polygon to be ``folded\u27\u27 into a particular 3D shape with certain tree-like properties. One open problem is whether the universal molecule can be rigidly folded: given the initial flat state and a particular desired final ``folded\u27\u27 state, is there a continuous motion between the two states that maintains the faces of the subdivision as rigid panels? A partial characterization is known: for a certain measure zero class of universal molecules there always exists such a folding motion. Another open problem is to remove the restriction of the universal molecule to convex polygons. This is of practical importance since the TreeMaker method sometimes fails to produce an output on valid input due the convexity restriction and extending the universal molecule to non-convex polygons would allow TreeMaker to work on all valid inputs. One further interesting problem is the development of faster algorithms for computing the universal molecule. In this thesis we make the following contributions to the study of the universal molecule. We first characterize the tree-like family of surfaces that are foldable from universal molecules. In order to do this we define a new family of surfaces we call Lang surfaces and prove that a restricted class of these surfaces are equivalent to the universal molecules. Next, we develop and compare efficient implementations for computing the universal molecule. Then, by investigating properties of broader classes of Lang surfaces, we arrive at a generalization of the universal molecule from convex polygons in the plane to non-convex polygons in arbitrary flat surfaces. This is of both practical and theoretical interest. The practical interest is that this work removes the case from Lang\u27s TreeMaker method that causes TreeMaker to fail to produce output in the presence of non-convex polygons. The theoretical interest comes from the fact that our generalization encompasses more than just those surfaces that can be cut out of a sheet of paper, and pertains to polygons that cannot be lied flat in the plane without self-intersections. Finally, we identify a large class of universal molecules that are not foldable by rigid folding motions. This makes progress towards a complete characterization of the foldability of the universal molecule

Constrained Planarity and Augmentation Problems

A clustered graph C=(G,T) consists of an undirected graph G and a rooted tree T in which the leaves of T correspond to the vertices of G=(V,E). Each vertex m in T corresponds to a subset of the vertices of the graph called ``cluster''. c-planarity is a natural extension of graph planarity for clustered graphs, and plays an important role in automatic graph drawing. The complexity status of c-planarity testing is unknown. It has been shown by Dahlhaus, Eades, Feng, Cohen that c-planarity can be tested in linear time for c-connected graphs, i.e., graphs in which the cluster induced subgraphs are connected. In the first part of the thesis, we provide a polynomial time algorithms for c-planarity testing of specific planar clustered graphs: Graphs for which - all nodes corresponding to the non-c-connected clusters lie on the same path in T starting at the root of T, or graphs in which for each non-connected cluster its super-cluster and all its siblings in T are connected, - for all clusters m G-G(m) is connected. The algorithms are based on the concepts for the subgraph induced planar connectivity augmentation problem, also presented in this thesis. Furthermore, we give some characterizations of c-planar clustered graphs using minors and dual graphs and introduce a c-planar augmentation method. Parts II deals with edge deletion and bimodal crossing minimization. We prove that the maximum planar subgraph problem remains NP-complete even for non-planar graphs without a minor isomorphic to either K(5) or K(3,3), respectively. Further, we investigate the problem of finding a minimum weighted set of edges whose removal results in a graph without minors that are contractible onto a prespecified set of vertices. Finally, we investigate the problem of drawing a directed graph in two dimensions with a minimal number of crossings such that for every node the incoming and outgoing edges are separated consecutively in the cyclic adjacency lists. It turns out that the planarization method can be adapted such that the number of crossings can be expected to grow only slightly for practical instances

Model checking finite paths and trees

This thesis presents efficient parallel algorithms for checking temporal logic formulas over finite paths and trees. We show that LTL path checking is in AC1(logDCFL) and CTL tree checking is in AC2(logDCFL). For LTL with pastime and bounded modalities, which is an exponentially more succinct logic, we show that the path checking problem remains in AC1(logDCFL). Our results provide a foundation for efficient algorithms of various applications in monitoring, testing, and verification as well as for query processing for tree-datastructures, e.g. XML documents. The presented path and tree checking algorithms are based on efficient parallel evaluation strategies for monotone Boolean circuits. We reduce the evaluation of product circuits to the problem of evaluating one-input-face monotone planar Boolean circuits: for a monotone Boolean circuit that is a product of a tree and a path, we provide an AC1-reduction; for a monotone Boolean circuit that is a product of two trees, we provide an AC2-reduction. We develop a classification of Kripke structures with respect to the complexity of LTL model checking: Kripke structures for which the problem is PSPACE- complete, Kripke structures for which the problem is coNP-complete, and Kripke structures for which the problem is in NC.Wir präsentieren effiziente parallele Algorithmen zum Überprüfen der Erfülltheit von temporal logischen Formeln auf Pfaden und Bäumen. Wir zeigen, dass für die Logik LTL das Überprüfen von Ausführungspfaden in der Komplexitätsklasse AC1(logDCFL) liegt. Für die Logik CTL ist das Überprüfen von Bäumen in AC2(logDCFL). Für Erweiterungen von LTL mit Vergangenheit und beschränkten zeitlichen Modalitäten beweisen wir, dass Pfade ebenfalls in AC1(logDCFL) überprüft werden können, obwohl die Logik exponentiell kompakter ist als einfaches LTL. Unsere Resultate bielden eine Grundlage für effiziente Algorithmen für verschiedene Anwendungen in den Bereichen der Systemüberwachung, des Testens und der Verfikation sowie für die Anfragebearbeitung für Baumdatenstrukturen, wie zum Beispiel XML Dokumente. Die präsentierten Algorithmen zum Überprüfen von Pfaden und Bäumen basieren auf effizient parallelen Strategien zur Evaluierung von monotonen Boolschen Schaltkreisen. Wir reduzieren die Evaluierung von Produkt-Schaltkreisen auf das Problem der Evaluierung von monoton planaren Boolschen Schaltkreisen, bei denen sich alle Eingaben auf dem äußeren Rand befinden. Für monotone Boolsche Schaltkreise, die das Produkt von einem Baum und einem Pfad sind, geben wir eine AC1-Reduktion an. Für monotone Boolsche Schaltkreise, die das Produkt von zwei Bäumen sind, geben wir eine AC2-Reduktion an. Wir entwickeln eine Klassifizierung von Kripkestrukturen im Hinblick auf die Komplexität des Erfülltheitsproblems für LTL: Kripkestrukturen, für die das Problem PSPACE-vollständig ist, Kripkestrukturen, für die das Problem coNP- vollständig ist, und Kripkestrukturen, für die das Problem in NC liegt

Transformation of functional programs for identification of parallel skeletons

Hardware is becoming increasingly parallel. Thus, it is essential to identify and exploit inherent parallelism in a given program to effectively utilise the computing power available. However, parallel programming is tedious and error-prone when done by hand, and is very difficult for a compiler to do automatically to the desired level. One possible approach to parallel programming is to use transformation techniques to automatically identify and explicitly specify parallel computations in a given program using parallelisable algorithmic skeletons. Current existing methods for systematic derivation of parallel programs or parallel skeleton identification allow automation. However, they place constraints on the programs to which they are applicable, require manual derivation of operators with specific properties for parallel execution, or allow the use of inefficient intermediate data structures in the parallel programs. In this thesis, we present a program transformation method that addresses these issues and has the following attributes: (1) Reduces the number of inefficient data structures used in the parallel program; (2) Transforms a program into a form that is more suited to identifying parallel skeletons; (3) Automatically identifies skeletons that can be efficiently executed using their parallel implementations. Our transformation method does not place restrictions on the program to be parallelised, and allows automatic verification of skeleton operator properties to allow parallel execution. To evaluate the performance of our transformation method, we use a set of benchmark programs. The parallel version of each program produced by our method is compared with other versions of the program, including parallel versions that are derived by hand. Consequently, we have been able to evaluate the strengths and weaknesses of the proposed transformation method. The results demonstrate improvements in the efficiency of parallel programs produced in some examples, and also highlight the role of some intermediate data structures required for parallelisation in other examples

New Approaches to Classic Graph-Embedding Problems - Orthogonal Drawings & Constrained Planarity

Drawings of graphs are often used to represent a given data set in a human-readable way. In this thesis, we consider different classic algorithmic problems that arise when automatically generating graph drawings. More specifically, we solve some open problems in the context of orthogonal drawings and advance the current state of research on the problems clustered planarity and simultaneous planarity