1,736 research outputs found
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
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