Thesis (Ph. D.)--University of Washington, 2005. / Vita. Includes bibliographical references (p. 94-95).

Graphs are mathematical abstractions that are useful for solving many types of problems in computer science. In this dissertation, when we talk of graphs we refer to directed graphs (digraphs), which consist of a set of nodes and a set of edges between the nodes, where each edge has a direction. Numerous implementations of graphs exist in computer science however, there is a need for more systematic and complete categorisation of implementations together with some proof of correctness. Completeness is an issue because other studies only tend to discuss the useful implementations and completely or partially ignore the rest. There is also a need for a treatment of graph representations using triples instead pairs as the base component. In this dissertation, a solution to each of these deficiencies is presented. This dissertation is a taxonomic approach towards a comprehensive treatment of digraph representations. The difficulty of comparing implementations with each other is overcome by a creating a taxonomy of digraph implementations. Taxonomising digraph representations requires a systematic analysis of the two main building blocks of digraphs implementations namely maps and sets. The analysis presented in the first part of the dissertation includes a definition of the abstract data types to represent maps and sets together with a comprehensive and systematic collection of algorithms and data-structures required for the implementations thereof. These algorithms are then written and re-written in a common notation and are examined for any essential com¬ponents, differences, variations and common features. Based on this analysis the maps and sets taxonomies are presented. After the completion of maps and sets implementation foundations the dissertation continues with the main contribution: a systematic collection and implementation of other operators used for the manipulation of the base triple components of digraphs and the derivation of the the final taxonomy of digraphs by integrating the maps and sets implementations with the operators on the sets of triples. With the digraph taxonomy we can finally see relationships between implementations and we also can easily establish their similarities and differences. Furthermore, the taxonomy is also useful for further discussions, analysis and visualisation of the complete implementation topography of digraph implementations. / Dissertation (MSc (Computer Science))--University of Pretoria, 2006. / Computer Science / unrestricted

This thesis concerns an active research area within fractal geometry. In the first part, in Chapters 2 and 3, for directed graph iterated function systems (IFSs) defined on ℝ, we prove that a class of 2-vertex directed graph IFSs have attractors that cannot be the attractors of standard (1-vertex directed graph) IFSs, with or without separation conditions. We also calculate their exact Hausdorff measure. Thus we are able to identify a new class of attractors for which the exact Hausdorff measure is known. We give a constructive algorithm for calculating the set of gap lengths of any attractor as a finite union of cosets of finitely generated semigroups of positive real numbers. The generators of these semigroups are contracting similarity ratios of simple cycles in the directed graph. The algorithm works for any IFS defined on ℝ with no limit on the number of vertices in the directed graph, provided a separation condition holds. The second part, in Chapter 4, applies to directed graph IFSs defined on ℝⁿ . We obtain an explicit calculable value for the power law behaviour as r → 0⁺ , of the qth packing moment of μ[subscript(u)], the self-similar measure at a vertex u, for the non-lattice case, with a corresponding limit for the lattice case. We do this (i) for any q ∈ ℝ if the strong separation condition holds, (ii) for q ≥ 0 if the weaker open set condition holds and a specified non-negative matrix associated with the system is irreducible. In the non-lattice case this enables the rate of convergence of the packing L[superscript(q)]-spectrum of μ[subscript(u)] to be determined. We also show, for (ii) but allowing q ∈ ℝ, that the upper multifractal q box-dimension with respect to μ[subscript(u)], of the set consisting of all the intersections of the components of F[subscript(u)], is strictly less than the multifractal q Hausdorff dimension with respect to μ[subscript(u)] of F[subscript(u)].

Directed graphs are an intuitive and versatile representation of natural language meaning because they can capture relationships between instances of events and entities, including cases where entities play multiple roles. Yet, there are few approaches in natural language processing that use graph manipulation techniques for semantic parsing. This dissertation studies graph-based representations of natural language meaning, discusses a formal-grammar based approach to the semantic construction of graph representations, and develops methods for open-domain semantic parsing into such representations. To perform string-to-graph translation I use synchronous hyperedge replacement grammars (SHRG). The thesis studies this grammar formalism from a formal, linguistic, and algorithmic perspective. It proposes a new lexicalized variant of this formalism (LSHRG), which is inspired by tree insertion grammar and provides a clean syntax/semantics interface. The thesis develops a new method for automatically extracting SHRG and LSHRG grammars from annotated “graph banks”, which uses existing syntactic derivations to structure the extracted grammar. It also discusses a new method for semantic parsing with large, automatically extracted grammars, that translates syntactic derivations into derivations of the synchronous grammar, as well as initial work on parse reranking and selection using a graph model. I evaluate this work on the Abstract Meaning Representation (AMR) dataset. The results show that the grammar-based approach to semantic analysis shows promise as a technique for semantic parsing and that string-to-graph grammars can be induced efficiently. Taken together, the thesis lays the foundation for future work on graph methods in natural language semantics.

Worst-case analysis is often meaningless in practice. Some problems never reach the anticipated worst-case complexity. Other solutions get bogged down with impractical constants during implementation, despite having favorable asymptotic running times. In this thesis, we investigate these contrasts in the context of finding maximum flows in planar digraphs. We suggest analytic techniques that adapt to the problem instance, and present a structural property that concludes equivalence between shortest paths and maximum st-flow in planar graphs. The best known algorithm for maximum st-flow in directed planar graphs is an augmenting- paths algorithm with O(n) iterations. Using dynamic trees, each iteration can be implemented in O(log n) time. Long before, Itai and Shiloach showed that when s and t are on the boundary of a common face, the O(n)-iteration augmenting-paths algorithm is equivalent to Dijkstra's algorithm in the graph���s dual: the max st-planar st-flow problem can be solved with one single-source shortest-path computation. In this thesis we show that (a) when s and t are separated by p faces, the max st-flow can be found with at most 2p single-source shortest-path computations, which, using the linear-time shortest-paths algorithm for planar graphs, results in an O(np)-time algorithm, and (b) that the equivalence between augmenting-paths and Dijkstra's extends to the most general non-st-planar digraphs, using their half-infinite universal cover graph. / Graduation date: 2013

The cycle space of a strongly connected graph has a basis consisting of directed circuits. The concept of relevant circuits is introduced as a generalization of the relevant cycles in undirected graphs. A polynomial time algorithm for the computation of a minimum weight directed circuit basis is outlined. (author's abstract) / Series: Preprint Series / Department of Applied Statistics and Data Processing

Graph augmentation concerns the addition of edges to a graph to satisfy some connectivity property of a graph. Previous research in this field has been preoccupied with edge augmentation; however the research in this document focuses on the addition of vertices to a graph to satisfy a specific connectivity property: ensuring that all the nodes of the graph are contained within cycles. A distinction is made between graph augmentation (edge addition), and graph enlargement (vertex addition). This document expands on previous research into a graph matching problem known as the “shoe matching problem” and the role of a graph enlargement algorithm in finding this solution. The aim of this research was to develop new and efficient algorithms to solve the graph enlargement problem as applied to the shoe matching problem and to improve on the naïve algorithm of Sanders. Three new algorithms focusing on graph enlargement and the shoe matching problem are presented, with positive results overall. The new enlargement algorithms: cost-optimised, matrix, and subgraph, succeeded in deriving the best result (least number of total nodes required) in 37%, 53%, and 57% of cases respectively (measured across 120 cases). In contrast, Sanders’s algorithm has a success rate of only 20%; thus the new algorithms have a varying success rate of approximately 2 to 3 times that of Sanders’s algorithm. / Computing / M. Sc. Computing

Finding a visually pleasing layout from a set of vertices and edges is the goal of automatic graph drawing. A requirement that has been barely explored however, is that users would like to specify portions of their layouts that are not altered by such algorithms. For example the user may have put a lot of manual effort into fixing a portion of a large layout and, while they would like an automatic layout applied to most of the layout, they do not want their work undone on the portion they manually fixed earlier. CluStic, the system developed and evaluated in this thesis, provides this capability. CluStic maintain the internal structure of a cluster by giving it priority over other elements in the graph. After high priority element has been positioned, non-priority vertices may be placed at the most appropriate remaining positions. Furthermore CluStic produces layouts which also maintain common aesthetic criteria: edge crossing minimization, layout height and edge straightening. Our method in this thesis is to first conduct an initial exploration study where we cross compare four industrial tools: Cytogate, GraphDraw, Diagram.Net and GraphNet. A set of layouts are generated with these tools and the user is timed on a task to identify the longest path. Through this exploration study we develop out intuition and determined that Cytogate is the best performing tool for longest path identification. Given this experience we fully develop CluStic and conduct our main study where we cross compare it with Cytogate and a baseline Breadth-first Search algorithm. Results show that CluStic produces drawings of good quality, Clustic achieves a visualization efficiency score of 1,4 which is an increase compared to the BFS layout (-3,8). CluStic is outperformed by Cytogate which achieves a visualization efficiency score of 1,9 and therefore produces less visually pleasing drawings. However Clustic, unlike Cytogate can preserve initial static structures, thus when a graph contains elements in which their position cannot be altered CluStic is a better choice. / Målet med automatiserad grafritning är att utifrån en uppsättning noder och kanter hitta en layout som är visuellt tillfredställande. Ett delområde som inte utforskats nog är möjligheten till att låsa vissa komponenter i grafen som sedan inte får alterneras av grafritningsalgoritmen. En användare som exempel, strukturerar vissa delar av grafen manuellt och applicerar sedan automatisk layout av resterande element utan att förstöra den struktur som manuellt skapats. CluStic, grafritningsverktyget som skapats och utvärderats i denna masters uppsats fyller denna funktion. CluStic bevarar den interna strukturen för ett kluster genom att tilldela en högre prioritet för noder i klustret med avseende på övriga element i grafen. Efter att högprioritets element placerats tilldelas resterande element sina bäst tillgängliga positioner. Utöver detta så uppfyller CluStic några av de vanligaste estetiska mål inom grafritning: minimera antalet kantkorsningar, minimera höjden, och räta ut kanter. Metoden som används i denna master uppsatts var att först gör en inledande studie där vi undersöker fyra populära grafritnings verktyg: Cytogate, GraphDraw, Diagram.Net och GraphNet. En uppsättning grafer genereras av dessa verktyg och vi mäter hur lång tid det tar för en användare att hitta den längsta vägen i grafen. Genom denna studie konstaterar vi att Cytogate presenterade grafer med best kvalitet. Från kunskap samlad i den inledande studien utvecklar vi CluStic och utför uppsatsens huvud studie där vi jämför CluStic med avseende på Cytogate och en bas layout Breddenförst algoritm. CluStic uppnår ett visualiserings effektivitetsvärde på 1,4 vilket är en ökning jämtemot Bredden-först algoritmen (-3,8). CluStic levererar inte layouter som är mer visuellt tillfredställande än de som skapats av Cytogate som får ett visualiserings effektivitetsvärde på 1,9. CluStic tillskillnad från Cytogate bevarar den internt fixa strukturen mellan element med hög prioritet vilket gör CluStic till det bättre verktyget för grafer med statiska element.

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