On Semantic Word Cloud Representation

We study the problem of computing semantic-preserving word clouds in which semantically related words are close to each other. While several heuristic approaches have been described in the literature, we formalize the underlying geometric algorithm problem: Word Rectangle Adjacency Contact (WRAC). In this model each word is associated with rectangle with fixed dimensions, and the goal is to represent semantically related words by ensuring that the two corresponding rectangles touch. We design and analyze efficient polynomial-time algorithms for some variants of the WRAC problem, show that several general variants are NP-hard, and describe a number of approximation algorithms. Finally, we experimentally demonstrate that our theoretically-sound algorithms outperform the early heuristics

Zipping Segment Trees

Stabbing queries in sets of intervals are usually answered using segment trees. A dynamic variant of segment trees has been presented by van Kreveld and Overmars, which uses red-black trees to do rebalancing operations. This paper presents zipping segment trees - dynamic segment trees based on zip trees, which were recently introduced by Tarjan et al. To facilitate zipping segment trees, we show how to uphold certain segment tree properties during the operations of a zip tree. We present an in-depth experimental evaluation and comparison of dynamic segment trees based on red-black trees, weight-balanced trees and several variants of the novel zipping segment trees. Our results indicate that zipping segment trees perform better than rotation-based alternatives

Engineering Top-Down Weight-Balanced Trees

Weight-balanced trees are a popular form of self-balancing binary search trees. Their popularity is due to desirable guarantees, for example regarding the required work to balance annotated trees. While usual weight-balanced trees perform their balancing operations in a bottom-up fashion after a modification to the tree is completed, there exists a top-down variant which performs these balancing operations during descend. This variant has so far received only little attention. We provide an in-depth analysis and engineering of these top-down weight-balanced trees, demonstrating their superior performance. We also gaining insights into how the balancing parameters necessary for a weight-balanced tree should be chosen - with the surprising observation that it is often beneficial to choose parameters which are not feasible in the sense of the correctness proofs for the rebalancing algorithm.Comment: Accepted for publication at ALENEX 202

A mathematical framework to compare classical field theories

This article is a summary of the Master's thesis I wrote under the supervision of Prof. Ion Stamatescu and Prof. James Weatherall as a result of more than a year of research. The original work contained a bit more than 140 pages, while in the present summary all less relevant topics were shifted to the appendix such that the main part does not exceed 46 pages to ease the reading. However, the appendix was kept in order to show which parts were omitted. In the article, a mathematical framework to relate and compare any classical field theories is constructed. A classical field theory is here understood to be a theory that can be described by a (possibly non-linear) system of partial differential equations and thus the notion includes but is not limited to classical (Newtonian) mechanics, hydrodynamics, electrodynamics, the laws of thermodynamics, special and general relativity, classical Yang-Mills theory and so on. To construct the mathematical framework, a mathematical category (in the sense of category theory) in which a versatile comparison becomes possible is sought and the geometric theory of partial differential equations is used to define what can be understood by a correspondence between theories and by an intersection of two theories under such a correspondence. This is used to define in a precise sense when it is meaningful to say that two theories share structure and a procedure (based on formal integrability) is introduced that permits to decide whether such structure does in fact exist or not if a correspondence is given. It is described why this framework is useful both for conceptual and practical purposes and how to apply it. As an example, the theory is applied to electrodynamics and, among other things, magneto-statics is shown to share structure with a subtheory of hydrodynamics.Comment: Summary of Master Thesi

Towards a Topology-Shape-Metrics Framework for Ortho-Radial Drawings

Ortho-Radial drawings are a generalization of orthogonal drawings to grids that are formed by concentric circles and straight-line spokes emanating from the circles\u27 center. Such drawings have applications in schematic graph layouts, e.g., for metro maps and destination maps. A plane graph is a planar graph with a fixed planar embedding. We give a combinatorial characterization of the plane graphs that admit a planar ortho-radial drawing without bends. Previously, such a characterization was only known for paths, cycles, and theta graphs, and in the special case of rectangular drawings for cubic graphs, where the contour of each face is required to be a rectangle. The characterization is expressed in terms of an ortho-radial representation that, similar to Tamassia\u27s orthogonal representations for orthogonal drawings describes such a drawing combinatorially in terms of angles around vertices and bends on the edges. In this sense our characterization can be seen as a first step towards generalizing the Topology-Shape-Metrics framework of Tamassia to ortho-radial drawings